Note: I found so many great probability activities as I was researching that I am going to put together another blog post dedicated to just that, so be sure to check back.

Vertical Alignment

Probability is an isolated skill. In CCSS, 6th grade and 8th grade do not have any probability concepts. However, multiplying fractions will be a key skill needed that does come from 5th and 6th grade and is covered again in 7th grade (7.NS.2). Be sure to review multiplying and simplifying fractions as well as vocabulary like numerator and denominator.

Teaching probability will reinforce so many important skills – solving proportions, converting fractions to decimals (for relative frequency), etc! This is a great unit to cover before reviewing for state tests to review these key concepts. 

Hook/Real-World

As I was putting this information together, I realized that pretty much every hook is an experiment or something that you should actually do with your students. Therefore, I will offer you this clip from the movie “21” where the Monty Hall problem is explained. This can cause some spirited debate once students have learned the basics of probability.

Make It Hands-On

Practically every probability problem can be acted out!  For introducing and explaining independent and dependent events, provide a baggie and colored tiles for each group.

For independent events, have students physically take one colored tile out of the bag and then replace it, so they could see the denominator stayed the same since the total number of tiles remained the same once replaced.

For dependent events, have students physically take one colored tile out of the bag and place it on their desk, so they could see that now the denominator, or the total number of tiles in the bag had changed. This helps them to see that they need to subtract one from the total. This also helps them see that they would need to subtract one from the numerator if there was more than one of that color. For example: what is the probability of drawing 2 yellow tiles without replacement in a bag with 2 yellow tiles and 3 pink tiles? P(y, y) = (2/5) (1/4) Assuming they got a yellow the first time, there is only 1 yellow left and 4 total tiles left.

Number Cubes/Dice

Your students may be tempted to make a silly mistake when solving simple probability with number cubes. Misconception: Let’s say that the problem was calculating the probability of rolling a 4; students would want to write 4/6 because they saw the 4. 

Solution:  Have students write out the numbers included on a number cube: 1 2 3 4 5 6 and circle the 4. Then they could see there was 1 possible outcome out of 6 total.

This is a great way for students to show their work. Probability of rolling an odd? Students write out 1 2 3 4 5 6 and circle all of the odd numbers to see that there are 3/6. You can also have students do this with the complement. Probability of not rolling a 3 (P’(3))? List out 1 2 3 4 5 6. Cross out the 3 since they want the probability of NOT getting a 3, and then circle the remaining numbers.

Theoretical vs. Experimental

Theoretical probability answers the question, “What should happen?” while experimental probability answers the question, “What actually happened?” Maneuvering the Middle has a fantastic Theoretical and Experimental Station Activity that will allow students to draw conclusions from multiple experiments. You can find this activity in our Probability Activity Bundle.

Other Tips + Useful Tools

• Usually I recommend anchor charts with math concepts or vocabulary, but for probability I recommend anchor charts on what is included in a deck of cards (suites, face cards, colors) and how many sides a dice has.

• This online probability generator includes editable spinners, bags of marbles, deck of cards, a random number generator, and a dice. You can use this to demonstrate concepts or for students to practice theoretical vs. experimental probability.
• This online Wheel of Names Spinner can be altered to include any word. 

Are you ready to teach probability? What tips do you have for teaching probability?

The post Teaching Probability in 7th Grade appeared first on Maneuvering the Middle.

I will be honest, I did not expect all of this goodness to exist out there for Systems of Equations! While I will definitely be linking to our fantastic resources, I am also linking some amazing task-like resources that will make sense of all of the typical system of equation word problems.

For Graphing

1. Systems of Two Linear Equations (Demos) This is a great way to introduce the concept that every point on a graph is a solution and the intersection is a solution to both equations. There is a whole class interaction aspect of this Desmos Activity that is pretty awesome! I would recommend this on Day 1 of graphing.
2. System of Equations Scavenger Hunt – This is a literal scavenger hunt complete with map. Students will practice graphing systems and solving for y. “This activity has a twist–figuring out where the friends are going to meet! So fun! This is one of my favorite resources!” -E.G. Perfect as a formative assessment on day 2 of graphing.
3. Algebra 1 Systems Performance Task – This Bowling Alley Performance task was designed specifically for Algebra 1 since it also includes a system of inequalities problem. The task asks students to compare the costs of food, shirts, and bowling ball weights at two competing bowling alleys.  It can be an alternative to assessment or a review before your unit test. It is available in our Algebra 1 Systems Activity Bundle.

All System of Equations Methods

4. The Custom Ink Project – This free activity (complete with handout and PowerPoint file) mixes a little art and a lot of math. Students design and price t-shirts, and then analyze their profits.
5. Dan Meyer Coin Problem – Gosh, I love this problem. Traditional coin problems can be dry and not engaging. This short video clip (30 second clip of a person putting coins into a coin counting machine) combined with a few question prompts will have all of your students trying to solve.
6. Desmos Card Sort This card sort activity would make a great station or warm up. It has a little bit of everything – sorting systems into the best method to solve and a few practice problems for each method of solving.
7. Dueling Discounts – Dan Meyer has another great way for students to approach systems. When provided with two coupons – 20% off and $20 off – what is the better deal on various items? At what dollar amount does it shift?

Other Helpful Things

8. These puzzles developed by Heather Sparks aren’t your typical math problems. They would make a great extension or bell ringer during your systems unit.
9. System of Equations Unit Review Error Analysis – Error Analysis develops both procedural and conceptual understanding. It allows students to think behind the HOW and into the WHY. Encourage mathematical discourse by assigning it as partner work at the end of your unit.
10. CCSS System of Equations Unit | CCSS Algebra 1 Systems Unit |  TEKS Algebra 1 Systems Unit |  – Not ready to jump into activities yet? Do you need a liiiiiiittle push in the right direction with instruction? Never fear. Our units are packed with planning resources, student handouts, independent practice, a unit review, and a unit assessment. 

What system of equation activities do you implement in your classroom?

The post 10 System of Equations Activities appeared first on Maneuvering the Middle.

Vertical Alignment

Before I jump into any topic, I like to see what students already know. Our units are complete with vertical planning, so I don’t have to search far to see that students have used equations to solve for unknown angles. Now we just need to make the jump to side lengths!

Plethora of Hook Opportunities

There are so many compelling ways to introduce Pythagorean Theorem to your students. In fact, the more context and real-world examples you can provide about the math they are learning, the more investment your students will have.

This Who Wants to Be a Millionaire video clip is kind of silly, but is a short and easy way to exclaim, “YOU MIGHT BE ASKED THIS ON A GAME SHOW FOR MONEY” or more likely, be asked on the street by someone making a video for TikTok. 

Day Mayer’s Three Act – The Taco Cart was recommended by many teachers in a Middle School Math Facebook group. I would show the first video and present it as, “By the end of this unit, you will be able to solve this problem.” This Desmos Adaptation is a great digital version.

Go Hands-On

Pythagorean Theorem is incredibly visual, but it is also incredibly kinetic. Let’s encourage students to manipulate and play to develop an understanding of how and why the Pythagorean Theorem works. You can do this in a variety of ways:

• Cheez Its or Starbursts

• Graph Paper – This PBS website gives you an outline on how to do this.
• Pythagorean Theorem Intro Activity – Another way students can “see” the theorem in action.

Know the Parts

One of the most common misconceptions students will face is identifying the legs and the hypotenuse on a triangle especially when the triangle is orientated in a different way.

Provide ample opportunities for students to label and annotate the right triangle before moving onto any calculations. Here is a non-exhaustive list of all the ways to help students labeling parts correctly:

• Start with the parts in the formula before using variables
  • hypotenuse^2 = leg ^2 + leg ^2

• Hypotenuse is the longest word and therefore the longest side
• Have students draw an arrow directly across the right angle (like I did in pink above) to the hypotenuse and label it first thing as hypotenuse = c first! This reinforces the idea that students should double check that it is a right angle before assuming it is a right angle.
• If a student feels hung up on the orientation of a right triangle (because it isn’t “right”), encourage students to turn their paper until the right triangle looks “right” to them.

Practice, Practice, Practice

There is a hierarchy of math practice. Students need to build up fluency, so they are able to tackle the more complex problem types. If they can’t square a number, take a square root of a number, or solve an equation for a variable, then they will struggle to solve application problems involving Pythagorean Theorem.

Pythagorean Theorem Maze Practice – This practice is scaffolded. One maze uses solely whole numbers, and then students can move on to problems involving rational numbers. 

Pythagorean Theorem He Said, She Said Activity – Error analysis is beneficial for procedural and conceptual understanding. It allows students to think behind the HOW and into the WHY. These types of problems are great for partner work and encourage mathematical discourse. 

Pythagorean Theorem Performance Task – If you are looking for an alternative to a unit test, may I recommend this math task? It provides multiple opportunities to show mastery of the Pythagorean Theorem, and it will encourage students to use multiple strategies of problem solving. What is the task? Students must design the layout of various attractions at a Harvest Festival. If you haven’t assigned a performance task in your classroom, be sure to check out these performance tasks tips and tricks.

Other Pythagorean Theorem Tips

• Does this side length make sense? Reasonableness is always something to be challenging students to consider. This develops their number sense (which is important when 8th graders are using calculators). When students are solving equations that are multi-step, I find that students can get so excited they leave off the last step. In the case of the Pythagorean Theorem, it is taking the final square root. I like to faux forget the last step and box my answer and wait for students to call me out! Then I would ask, “Does this answer make sense?” If I have side lengths of 12 and 9 inches, would the last side make sense at 225 inches?
• Posting an anchor chart of perfect squares can be a great way to reinforce students memorizing perfect squares. If students quickly know these math facts it will serve them well in Algebra 1, so this is a great time to work on that skill. Our 8th Grade Math and Algebra 1 Fast Pass have a list of perfect squares, so download the freebie and give it to your students to reference.

What tips do you have for teaching Pythagorean Theorem?

The post Read Before Teaching Pythagorean Theorem appeared first on Maneuvering the Middle.

This post is full, so be on the lookout for part 2 where we will discuss various systems activities for your classroom.

Vertical Alignment

Solving systems of equations starts in 8th grade and carries on being explicitly taught in Algebra 2. If you are an 8th grade teacher, you aren’t exactly starting from scratch.

Students will need to be proficient at:

• Solving for a single variable (in this case y) and graphing lines in slope-intercept form (for solving by graphing)
• Rational number operations (for elimination and substitution)
• Combining like terms and distributing (for substitution)

If your students aren’t up to speed with these skills, don’t fret! There is plenty of time to refresh and practice as you introduce systems of equations. I mention this to remind you that students will not be able to solve systems of equations problems without this foundation, so give your students time + practice!

System of Equations Overview

We can often jump in with the methods to solve so quickly that we don’t properly explain what we are solving for apart from saying x and y.

Remind students daily (or even before every problem) that solving systems means we are looking for a solution that makes both equations true! Then reinforce that this is where the lines will intersect on a graph.

Let’s dive into each method for solving systems of equations. 

Graphing

The method that I recommend starting with is solving systems of equations by graphing. Due to its visual nature, I find that it makes the abstract nature of solving 2 equations more concrete.

As you introduce this skill, start with the basics of a single line. Remind and reinforce that any point that exists on that single line is a solution to that equation! You can ask students to choose a point on the line, plug in the x and y values, and show them that the equation is a true statement! You can also do the opposite – have students choose and plug in a point that is not on the line to prove that it is not a solution to the equation.

Then you can move to the conclusion that the intersection of two lines is a solution to the system of equations! When students find the point of intersection, plug that point into both equations to find that it is a solution for both equations! (Wow! I am getting pretty excited writing about this!)

Other tips:

• If there was ever time to have your 8th grade or Algebra 1 students move their bodies around, it is this! Arms crossing for one solution, arms parallel for no solution, and arms overlapping for infinitely many solutions. 
• Students often get stuck with how to graph vertical and horizontal lines, such as y = 3 or x = 2. It is helpful to say it as a sentence, “The y-values will always be equal to 5.” Students can plot 2-3 points with a y-value of 3 and visually see that it will be a horizontal line that also intersects the y-axis at 3.

• Students will have to be proficient at solving for y! In fact, I wrote an entire blog post for this specific skill here.

Substitution

It is my recommendation that substitution is going to require a minimum of 2 days. For it to be done well, I recommend building a strong conceptual foundation of what substitution is before trying to actually solve anything by substituting.

I love these two ideas that I found while researching this topic. This first idea I can attribute to a teacher’s explanation in a Facebook Thread (I am going to summarize with bullet points to make it easier to follow):

• Write a whole number on the board.
• Ask students to write down as many different equivalent expressions of that number
• If the number is 6, then: 3(2), 4+2, 7-1 and so on. 
• Make the point that expressions can look different but still be equivalent.
• Next, add variables. x=6, then: x=3+3 and x=3(2).
• Now have students substitute their expressions for x: 3+3 = 3(2).

This blog post breaks down in painstaking detail how a visual aid (stars) makes substitution more concrete. While I used a similar technique with colored Post-it Notes, I think you will benefit from reading this teacher’s methodology. 

Other Tips:

• One solution, no solution, infinitely many solutions will still pop up in substitution, so be sure to review. If variables cancel out and the remaining statement is true, the system has infinitely many solutions. Example: 2=2. If variables cancel out and the remaining statement is false, the system has no solution. Example: 2=/=3.
• Distributing and managing negative signs is going to be a common error. Demonstrate and remind students that they can check their work by just plugging solutions in! This goes for elimination too!
• If you need a fun, no-prep, and free activity for substitution, I recommend playing this “Who Wants to Be a Millionaire” style game as a class.

Elimination

This is my personal favorite – maybe it is because I like when standard form actually comes in handy!  I am hesitant to admit that I did not know why the elimination method worked until I wrote this post, so I want to share what I learned.

The Additional Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. 

For any expressions, a, b, c, and d,

If a = b

And c = d

Then a+c = b+d

Whenever I present that type of information to my students, I prefer to give a few examples with numbers.

So let’s says a = 2, b = 2, c = 1, and d = 1

If 2=2

And 1=1

Then 2+1 = 2+1

If you didn’t follow that, perhaps this color coded image will make more sense?

Other Tips:

• Students seem to prefer elimination to substitution! I would give students the opportunity to practice changing y = mx+b into standard form. 
• Provide lots of practice multiplying to clear the fractions.
• Don’t forget to go back and solve for the other variable. This is why you should save elimination for after substitution. 

 What tips would you add to solving the system of equations?

The post 3 Steps to Mastering Systems of Equations appeared first on Maneuvering the Middle.

Fortunately, while that is the only way I ever learned division, there are now numerous other options for division, so students can feel successful. And though I believe the standard division algorithm is a necessity for math, it doesn’t have to be the only way students learn and internalize division.  

Before we jump into division, a quick reminder of the types of division we have. We have partitive division and measurement division.

In a Partitive division problem, the number of groups is known, and you are solving for the number in each group. Think of it like this – I have 12 cookies and want to divide them evenly by 4 children. How many cookies does each child receive?

In a measurement division problem, the number in each group is known, and you are solving for the number of groups. For example, I have 12 cookies and want to give each child 4 cookies. How many children can receive 4 cookies?

Now that we’ve reviewed the two types of division, here are 4 division strategies to try!

1. Partial Quotient Division

Partial quotient division does exactly what it sounds like – you work to find parts of the quotient (answer) and then add them together to find the overall quotient. 

Let’s take 432/18…

Here is a takeaway that will make teaching partial quotient division more accessible for your students; don’t try to have students do all of their multiples of 18 (which is normal when teaching the standard algorithm). It isn’t necessary. Instead, have students do friendly numbers. Depending on your dividend and divisor, this could be 10 or 100. In our case, 10 makes more sense, so 18×10=180.  Also, if I go ahead and find 18×2=36, it is painless to find 18×20=360. 

Another math tidbit I love is that instead of trying to multiply by 4, it is easier (to me) to multiply by 2 twice. 

You can see in the example above, that I was able to find that 18 went into 432 twenty times and then 18 went into 72 four times. I simply add the partial quotients of 20 and 4 to find the quotient of 24.

Lastly, partial quotients are a great tool because students can find the answer in a variety of ways. Take a look below for another way to solve 432/18.

2. Equivalent Ratios

This is a great strategy for division that reinforces ratios! This may not work for every division problem, but it is similar to partial quotients in that you can work in baby steps to get your quotient. Familiarize your students with these division rules:

• Numbers that end in an even number are divisible by 2.
• Numbers that end in a 0 and 5 are divisible by 5.
• Numbers that end in a 0 are divisible by 10.

Once your students have mastered those rules, you can introduce:

• Numbers with the last 2 digits divisible by 4 are divisible by 4 (ex: 1016, 3412, 1004)
• If the sum of the digits is divisible by 3, then the number is divisible by 3 (ex: 342. 3+4+2 = 9. 9 is divisible by 3, therefore, 342 is divisible by 3.) 

Even if your student cannot get all the way down to 1, they can still divide smaller and more manageable numbers by simplifying. In this example, you can think of it as partitive division, 25 represents the number in 1 group.

3. Multiplying Up

To use the method of Multiplying Up, rewrite the division problem as a multiplication problem. Let’s take the example of 1665/15 =111. Multiplying up asks “15 x ___ = 1655?”

We know that 15×100 = 1500 gets us close to 1665. We can work from there by adding 150 (10 groups of 15) and then again 15 (1 group of 15). 100 + 10 + 1 = 111. Therefore, 1500 + 150 +15 = 1665.

4. Repeated Subtraction

Repeated subtraction is similar to multiplying up (and partial quotient) but backwards. Obviously, these are all related! I think it works best when you are working with dividing numbers with unfriendly divisors (think 3 digit). 

Which division strategy will you try with your 5th grade or upper elementary students?

Related content: Check out this post on Dividing Fractions.

The post 4 Division Strategies for 5th Grade appeared first on Maneuvering the Middle.

According to the U.S. Department of Health and Human Services, “Movement increases the heart rate and stimulates brain function, which facilitates a child’s ability to learn,” so here are three “Get Up and Move” activities that I implemented in my classroom. 

[Note: Some of these activities are included in Maneuvering the Middle’s All Access subscription or can be purchased here, but I am going to explain how you might prepare for and execute these activities if you are not a middle school math teacher.]

Math Review Game: FLYSWATTER

The flyswatter game was around when I was a student – I remember playing it in my Spanish class!  Students love this game, and they ask for it daily.  Since it is a very low lift, I would use it as an incentive for finished work.

PLANNING

Project around 10 answers on the board.  (I don’t recommend writing the answers on the whiteboard with dry erase markers because they will get rubbed off.)  You need two flyswatters.  Tape a line on the floor parallel to the board, about five feet from the board.  This is where students will stop to hear their problem.  

The flyswatter game works with problems that are fluency-based.  We are going for speed here.  Here is a suggestion: fraction, decimal, and percent equivalents.  You could project these answers: 40%, 25%, 0.6, ½, etc…

EXECUTION

Divide students into two groups.  Boys versus girls is my go-to. They should be in 2 lines facing the board.  Set the expectation that you will not call out a problem until it is silent.  When the first two opponents are up, you could say, “Two fifths!”  The student who hit 40% first would win and stay in the game, and the other student would take a seat.  The fly swatters are passed on.  The winner would go to the back of the line, and the next two students would be up.  You keep playing until there is one player left.

Math Review Game:  SCAVENGER HUNT

This is by far my favorite activity.  You can read more details on how to execute a scavenger hunt in this post. It’s self-checking, and you could pull or work with a small group during this activity.

MATH REVIEW GAME: Cake Walk

PLANNING 

Similar to a cake walk, students will be up and walking in a circle, so consider that when arranging desks/chairs and/or giving directions for how you want them to walk. I had tables and would push them to the side of the room and arrange chairs in a large circle. Each chair needs to be numbered. 

Have your problems ready on a slideshow. Students will need scratch paper and a clipboard to write on.

EXECUTION

Play music while students walk around the chairs. Stop the music, students sit, and you project a problem. Students then work on the problem. Use a random number generator to pick the chair number. The student sitting in that chair gets a prize (could be bonus points on a test, free homework pass, sticker, piece of candy) if their work is correct. This activity is great for review before a unit test!

If you don’t have time to plan something special for students to get out of their seats, but you can feel the restlessness in the classroom, then you can still have students get up and move!  You can have students find a partner, work out a problem, and then move to find a new partner for the next problem. 

Math Concepts for Kinesthetic Learning

These concepts are perfect for students to use their body to act out. If you have more ideas, please comment below! 

1. Positive slope, negative slope, undefined, zero
2. Angles – acute, obtuse, 90 degrees, 180 degrees
3. Transformations – translate (slide), rotations, reflections, dilations (greater than 1? Act out e x p a n d i n g, less than 1? Act out shrinking)
4. Adding and subtracting decimals (act out lining up your buttons on a shirt with lining up decimals for adding and subtracting)
5. Coordinate plane (body is the y axis, arms are the x axis, negative space creates the quadrants)
6. Inequalities – arms for the greater than and less than signs
7. Types of Functions – linear, U for quadratic, and exponential
8. Parallel and perpendicular lines
9. Types of solutions – arms crossed for one solution, parallel arms for no solution, and arms on top of each other for infinite solutions)

How do you get students moving in your classroom?  

Check out this related post: Turn Any Worksheet Into a Math Activity

Maneuvering the Middle has been writing and publishing blog posts for almost a decade! This post was originally published in 2016 and has been updated for clarity and relevance.

The post 3 Math Review Games to Get Students Moving appeared first on Maneuvering the Middle.

1. Open Area Model

The area model demonstrates that when multiplying two numbers, you can find partial products and add them together to find the overall product. The open area model is my favorite alternative multiplication strategy and where I would start. After your students understand the basics of an array, the open area model is the next best model for students to fully see what is happening conceptually when multiplying (especially when multiplying 2 digits times 2+ digits). 

The steps are already built in. They know that each box needs to be filled in. This keeps students organized and (hopefully) more accurate. 

Additional benefits: this format will help students in high school Algebra multiplying and dividing polynomials.

2. Partial Product Multiplication

Partial product multiplication is the open area model without the boxes. Essentially, students are completing the exact same steps in the same order, but without the array format. While I wouldn’t start here, I think reinforcing the open area model with a partial product number sentence is the perfect progression. 

3. Distributive Property

Using the distributive property is a multiplication strategy that will later reinforce another important math skill. Have students break down one of the factors into its expanded form and then multiply. 

4. Chunking Using a Ratio Table

You can use a ratio table to split a factor into chunks (in this case 24 was broken down into 4 and 20) to find partial products that are then added together to find the final product. Have students take advantage of friendly math like doubling and multiplying by 10. 

5. Over and Under

In order to use friendly numbers, multiply by more (or less) groups than necessary. Then, subtract the extra groups, or add the missing groups. 

In this example, I traded 18 for 20. After finding the product of 31 and 20, I found the product of 31 and 2 (since 20 – 18 = 2). I subtracted this product of 62 from 620 to find the answer to our original problem 18 x 31.

In this case, I shot under by trading 31 for 30. I knew I would be missing one group of 18 so I added that to the product of 30 and 18.

What multiplication strategy is your favorite? Interested in more multiplication posts, check out our posts on Multiplication Facts and the Distributive Property.

The post 5 Multiplication Strategies appeared first on Maneuvering the Middle.

Today, let’s examine how proportional relationships go far beyond just solving for a missing number and how to set a foundation for success in Algebra.

THE STANDARDS

7.RP.2 Recognize and represent proportional relationships between quantities.

• 7.RP.2A  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane 
• 7.RP.2B  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportions.
• 7.RP.2C  Represent proportional relationships by equations. 
• 7.RP.2D  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r), where r is the unit rate.

VERTICAL ALIGNMENT

Visual Representations

These standards are heavily weighted with visual representations, including tables, graphs on the coordinate plane, and diagrams.  Some students will have been exposed to these multiple representations in 6th grade with standard 6.RP.3 and equivalent ratios.  The goal is for students to fluently connect the table, equation, graph, verbal description, and even diagram together.  When given one piece of information, students should be able to represent that same information in multiple ways. Students who can do this will have a firm foundation for 8th grade math and Algebra 1.

Ideas for Teaching Proportional Relationships

There are tons of great ideas and activities out there, but below are a few of my favorite “tried and true.”  They are all easy to incorporate and provide scaffolding for students who struggle in the math classroom.

1. Multiple Representations Graphic Organizer

Some students need to see everything together, and that is where this multiple representations graphic organizer comes in handy.  It is the perfect size for students to show their work. I really like how the components stay the same, but the given information changes.

These are perfect for introducing the various visual representations, but it also can be reused in a clear pocket to use in tutoring or in math intervention.  Once students are familiar with the graphic organizer, I would even give students butcher paper and various pieces of information. Then, they would use markers to represent the remaining information.  This is a quick informal assessment idea or can even function as partner work for a Friday.

2. Highlight the Unit Rate/Constant of Proportionality

A quick trick to help students see all the connections is to use a highlighter!  Model and have students highlight the unit rate/constant of proportionality within the different representations.  This is perfect for representing k in the equation and in the table or meeting 7.RP.2D by labeling (1, r) on the graph.

3. Emphasize Vocabulary

Hopefully, students are familiar with the term “unit rate” and how to find it.  Constant of proportionality sounds so cumbersome and challenging.  Unit rate is often emphasized when we are talking about unit price (cost per ounce, etc), so I think students have trouble seeing a graph or a table and also using the phrase “unit rate.”  Either way, students need to be familiar with the verbiage and understand what a question is asking. Here is a more detailed explanation for why they are the same.

Common Misconceptions

1. A relationship is not proportional unless (0,0) is visible in the table or graph
2. Dividing x/y to find k
3. General confusion about k
4. General disconnect between k and proportions
5. Proportional relationships only involve positive numbers
6. Mixing up x and y on the table, when it is not explicitly stated

Anchor Chart Ideas

Anchor charts are fabulous ways to showcase the content in a visual manner for students to reference.  They can easily be created before the lesson or as you are teaching, depending on the content.  The visual representations of proportional relationships are perfect for anchor charts.

Ideas for Struggling Students

1. Practice finding different points in the proportion that aren’t listed on a table.
2. Begin with the equation and use and input-output table to create a table.
3. Use a four corners graphic organizer.
4. Match multiple representations.
5. Practice graphing with four quadrants (from 6th grade).

Hopefully, this gives you some ideas for teaching proportions or even insight as to what knowledge your students are coming with.  I would love to hear other great activities or ideas you have used! Feel free to share in the comments.

Check out some of our other math concepts posts: percents  | ratios | integers

What concept would you like to hear more from us about? 

The post Teaching Proportional Relationships appeared first on Maneuvering the Middle.

I think that is why I love math so much; it can be broken down into smaller components.  The key is being able to practice the different steps and skills with student-centered math activities.

Note: I have utilized these activities in various levels of classes. At one point, I taught all three levels of 8th grade math – intervention, on-level, and advanced – within the same year. While the content of the activities may change, the activities are perfect for any level.

CUT AND PASTES

Students Practice Breaking Down the Process

Why I Love Them: While a little messy, cut and pastes keep students using their hands and doing math at the same time. They work well independently, in partners, or working with a small group. I like providing multiple incorrect answers as well. This keeps kids thinking and is a way to incorporate mathematical practices through error analysis.

When to Use Them: Cut and pastes are great for anything that requires a step-by-step process from solving equations to adding and subtracting integers. This is perfect for advanced kiddos who want to go straight to the answer or intervention students who need to focus on one step at a time.

Teacher Tip from Donna E: “Cut and Paste Activities. I actually laminate these so there is no cutting and pasting. I use these more as a sorting activity. It cuts down the time and still works out their misconceptions.” If you are going this route, you can also make it a group activity to reduce the number of individual activities you have to prepare.

SHOP CUT AND PASTES

CARD SORTS

Students Practice Differentiating Similarities and Differences

Why I Love Them: Card sorts are an excellent way to quickly assess a student’s understanding of the concept. They require higher order thinking skills, as students are required to analyze the given information and make a categorization. Although they take a bit of time upfront (cutting and laminating), they can be used over and over again. I have used card sorts for the real number system, proportional relationships, word problems, statistical and nonstatistical questions, properties of geometric figures, etc.

When to Use Them: They are fabulous as a classroom activity with pairs, an activity for early finishers, or an activity to keep skills fresh and improve fluency. In an advanced class, you could give the cards without the headers and ask students to sort them in any way possible. You would be surprised at what observations they are able to make. In an intervention setting, the headers will provide structure. 

Teacher Tip from Jordan:  “If you are at a 1-1 school or have access to technology, putting card sorts on Desmos is super easy and a great way to do activities like this digitally!” This blog post from Kate’s Math Lesson shows you how.

SHOP CARD SORTS

SOLVE AND COLORS

Students Practice Basic Math Skills 

Why I Love Them: Solve and colors are really perfect because kids get to color. Something about colored pencils in math makes the lesson more successful.

When to Use Them: My favorite use for solve and colors is for practice of basic math skills, whether that be adding and subtracting rational numbers or multiplying and dividing decimals. They are easy to leave for substitutes, to use after testing, or as fun homework assignments.

Teacher Tip from Tyne: “Use the coloring as the incentive to complete the math. Once the math was complete, I checked their work and then allowed students to start coloring.”

SHOP SOLVE AND COLORS

MATCHING CARDS

Students Practice Recognizing Multiple Representations

Why I Love Them: Matching cards require about the same amount of time upfront as a card sort, so get yourself some parent volunteers. I have found great success using matching cards to show multiple relationships. This can be depicted with ratio tables and graphs, proportional relationships, fraction, decimal, percents, and linear relationships.

When to Use Them: Again, these are perfect for sponge activities, review activities, or quick and easy lessons on Fridays. I personally loved using these over and over again with my intervention students. We would use matching cards to build number fluency with fractions, decimals, and percent representations, as well as many other necessary skills.

Teacher Tip from Tyne: Card Matches (and cut and pastes) can be great whole-class activities! When I taught 6th grade Ratio Unit, I used the Multiple Representation Cut and Paste for the entire class. Each student got a card and students had to make a group of 4 with the equivalent graph, table, equation, and verbal description. For more practice, 

SHOP MATCHING ACTIVITIES

TASK CARDS

Students Practice Individual Skills within a Small Group

Why I Love Them: Students can be working on the same concept with different types of problems. They are super flexible! In fact, we wrote an entire blog posts on task cards! Lots of teachers use them for scoot or various games, but my favorite is small groups. You can incorporate them into stations, use them for formative assessments, etc. Task cards are easy to prep and can be utilized multiple times throughout the year.

When to Use Them: When I pulled small groups, I loved using task cards, hands down. I would have a small group of students work on various problems that were all around a similar topic. I could easily scaffold the students within my small group based on the card, starting with the most basic problems and then moving on to a multi-step word problem.

Teacher Tip from Angelique: “I use task cards as an escape room activity. I have never had ALL group members engaged and truly holding each other accountable to complete the work.”

Find it, Fix It and She Said, He Said

Students practice analyzing work for errors

Why I Love Them: These error analysis type activities provide opportunities for students to not only solve the problem on their own, but evaluate how and why mistakes can be made on particular skills.

When to Use Them: I like to start with He Said, She Saids and build up to Find It, Fix It. He Said, She Said requires students to determine which answer out of 2 is correct. Find It, Fix It requires students to look at 4 different answers and determine which one is incorrect. I like using both of these activities for stations or for a whole group activity to practice skills learned that week.

Teacher Tip from Tyne: “I would post He Said, She Said and Find It, Fix It cards around the classroom, so student pairs could work together to solve. I would set a 3-5 minute timer and announce for pairs to rotate to keep students from gathering around the same problem and to also keep students on track to finish.”

If you haven’t used these student-centered math activities in class, I encourage you to try them out. All Access has all of these activities (and so so so much more), but you can shop individual activities on TpT.

The post Student-Centered Math Activities appeared first on Maneuvering the Middle.

Pros of a 100 Minute Class Period

• More time
• More instruction
• More support
• More practice
• At the end of the year a student has had double the amount of time in that class than a traditional schedule
• You should definitely be able to get through your scope and sequence with 100 minute class periods
• If you had 100 minute classes, you probably have less students over all

Cons of a 100 Minute Class Period

• 100 minutes is a loooooong time
• Students get distracted
• Classroom management is tough for that long of a time period

Things to Consider

• Students are with you for double the amount of time, but that does not mean that you simply extend a 50 minute lesson. How can you be efficient and productive with the time?
• Students need structure. How can you develop a routine that breaks up the 100 minutes but still provides structure?
• Students (and adults) have a short attention span. A good rule of thumb is that new learning should not take longer than 1 plus your students’ age, so if you teach 12 year olds, your notes should last no longer than 13 minutes (12+1). 

Below is just one way to structure your 100 minute class period. Sometimes things do not go according to plan, but it is always a good idea to have a structured routine for both yourself and your students, especially if you will be with them for so long.   

Do First/Bell Ringer/Warm Up 5-10 minutes

I used a very straightforward warm-up routine to get students working when they entered the classroom. The goal is that students can get started without needing assistance from me or their peers.

I used this time to:

• Greet students with a warm smile
• Check homework completion (if I assigned it)
• Take attendance

I would start a timer after the bell rang for 5 minutes and project it. When the timer went off, I spent the next 3-5 minutes either going over the warm up, going over last night’s homework, or a combination of both. 

Hook 2-5 minutes

These few minutes are a great time to introduce the objective and make real-world connections. It can also be utilized to review prior content that is connected or to have students review any new vocabulary. Anything that can create a bit of buy-in is beneficial.

Instruction 15-20 minutes 

The goal of instruction is to give students enough information to understand the concept, but not so much that you are doing all of the heavy lifting in class. It is a fine line to walk.

If you need more than that recommended amount of time for direct instruction, that is okay! Give students the opportunity to practice and engage in a meaningful way before returning to direct instruction. Another idea is to assign our student videos, since they adhere to this time recommendation. 

Remember that direct instruction isn’t your only option to teach a lesson. You could:

• Have students explore using manipulatives
• Think aloud or model
• Ask students to build upon previous knowledge
• Use the Building Thinking Classrooms approach

Lastly, I think it is important to note that if you are using our curriculum, you do not need to go over every single problem on a student handout. Work the problems ahead of time, decide which are the most important, and then save the rest for small group work time. 

Class Activity 20 minutes

This is the time period where students are engaging with the work in pairs or groups. In a 100 minute class, I recommend activities with movement, as well as collaboration. Sometimes we would do card sorts, but rather than sit at desks I would let students do the sort on the floor. Other times I would use stations or scavenger hunts to get kids up and moving or use math dates to have them work with various people. I would circulate and answer questions at this time. If you have a simple worksheet, make sure to read how to turn any worksheet into an activity.

Recap 5 minutes

As the activity wraps up, take a few minutes to recap what they have learned by asking students to summarize the lesson. Depending on the activity you could go over various responses or work a few of the difficult problems together.

Skill Practice 5 minutes

Most students have some need for remediation, gaps in their mathematical foundation, or need to expound upon their problem solving skills. Each day I would spend no more than five minutes addressing basic math skills. At the beginning of the year this was multiplication charts with various missing numbers or adding and subtracting decimals. I often spent several weeks on number sense by practicing converting between fractions, decimals, and percents.  

Station Work 30 minutes

I used this time to focus on small groups and remediation. I would work with small groups on their assignment, some students would work independently on a computer, while others would focus on concepts that they needed additional help with. This is also the time that my co-teacher would come into class, which was a life saver. You can read more about this on my math intervention schedule post.

You can read more about planning for and implementing stations here.

Clean Up/Close 5 minutes

By this time we are all wiped! It was time to wrap up, clean up, put away supplies, and get everything back in order.  

One Hundred minute classes never failed to wear me out, but the extra time was a gift! Especially when I think about the whirlwind of a 45 minute class. 

Who else has 100 minutes for math? How do you structure a 100 minute class? I would love to hear how you break it down!

Click to find out more about Maneuvering Math.

The post How To Structure a 100 Minute Class Period appeared first on Maneuvering the Middle.

1. Use a Scope and Sequence (grab ours!)

If you fail to plan, you plan to fail. If you aren’t provided a scope and sequence by your school, or if you haven’t made one yourself, the task of organizing 50+ math standards into 180 instructional days can be daunting.

Download our free pacing guides for 6th, 7th, 8th grade math, and Algebra 1!

You can see how our Maneuvering the Middle curriculum organizes the standards. From there, download a calendar on Google Sheets, fill in district holidays and professional development days, and work backwards from your state test.

2. Give Yourself Some Cushion

When planning for a unit, give a minimum of 3 days for cushion. No new material is introduced on these days. You can use these days for review, extra practice, or reteaching content that the majority of your students did not master prior. 

In a unit that is 15 days long, I would have one day for reviewing before the unit test, one day for an activity followed by a short quiz, and one additional day for an extra challenging skill that required more practice. That means that out of 5 days of the week, on average, I am only teaching new content 4 of those days. 

3. Be Familiar with Vertical Planning

The first year I taught 6th grade math, ordering rational numbers was in my first unit. My district provided a unit plan that gave me one day to teach this very complicated concept. When we got to ordering decimals and fractions, I modeled how to divide in order to convert fractions into decimals. My highest students raised their hands in protest – “You can’t divide a smaller number by a larger number!” Students lost their minds. That didn’t even include all of the students who forgot the standard algorithm for division. I was trying to cover too much in one lesson!

The lesson I learned that day was this – you need to know what your students already know and don’t know to successfully introduce new concepts. This will come in time, but take a moment to read vertical planning documents (provided in our All Access curriculum). This will prevent you from wasting entire days because you have to pivot on the fly.

4. Formative Assessments Keep You On Track

Knowing how your students are actually doing with the skills they are learning is paramount to staying on schedule each year. I recommend a daily formative assessment on the material covered that day while it is fresh. This can be a 2 question exit ticket or it can be as simple as collecting their classwork and checking problem #5. I like to keep these open-ended (why do students love to just circle multiple-choice answers?!). These are not always graded assignments; they are simply problems that tell me if I can move on to new material the next day or if we need to spend an extra day on the material.

What you are trying to avoid is students making it all the way to the unit test only to bomb spectacularly, and then you are bewildered and overwhelmed trying to plan next steps. 

5. Save the fun stuff for the alternative schedule days

I love when you can make math applicable to the real world! Projects or performance tasks are awesome! I like to save these types of assignments for days where the schedule is already a little crazy. This usually falls before holiday breaks, after pep rallies or on early release days. This is also when there are various field trips, students are more likely to be absent, and it doesn’t really make sense to cover new material. (Which leads me to a side rant that Fridays are also a bad day to cover new skills.)

Projects and performance tasks are a great way to spiral previous learned standards, so it provides a fun way to review.

6. Don’t Waste a Day for Absences

Don’t waste a single day! With an All Access membership, your students don’t have to lose an instructional day because you are sick or need to take a personal day. The student videos cover the student handouts. Instruction can continue even when you are absent! 

7. Consider a Self-Paced Classroom

If your students have a wide range of abilities, you may want to consider a self-paced classroom. We have two blog posts and podcast episodes that I highly recommend checking out if you are interested in letting students work at their own speed (What is the Grid Method? |  The Self-Paced Classroom). 

Students who need more time to master a skill can receive more of your attention since students who have already mastered a skill can move on without you. 

8. Sometimes You Need to Move On

As teachers, I believe that the ideal state is for 100% of students to master 100% of the concepts that we teach. This is not the reality we live in but we can work toward it. However, 100% of our students will master 0% of the concepts that we don’t teach.

Teachers have to cover all of the material in roughly 180 instructional days. I know it can feel impossible some days, but you can do it! If you find yourself stuck in a unit and the majority of your students are truly struggling, then maybe you need to revisit the material at a later date. Maybe your students are fatigued by the skill and moving onto something new will give them the mental clarity to try it again. 

How do you stay on track with your scope and sequence during the school year?

The post Staying on Track with Your Scope and Sequence appeared first on Maneuvering the Middle.

If you want to learn more, check out this book, Mathematize It!, that covers the topic of teaching how to solve word problems in much more detail.

Be sure to read: Part 1: Three Word Problem Types to Teach | Part 2: Three Steps to Solving Word Problems

Diagrams are a way to model what is happening in a word problem. Diagrams help provide students with the “operational sense” that they need in order to write an equation and solve a problem. We cannot rely on keywords to determine an operation.

Let’s dive into two examples. (You can see my example of open number line diagram here.)

Tape Diagrams in a Relationship Problem

Let’s start with an example problem:

After we have restated the problem, we use a diagram to represent the problem. Notice that this tape diagram is helpful because of the whole in relation to the part.

There are 2 parts to this marching band: those in percussion and those NOT in percussion. These 2 parts will make up the total number of members in the band (this is our whole).

We also know that there are 28 in percussion, so one part is 28. That leaves the unknown value as our other “part” – those not in percussion. So we will write the variable x in our model.

And now we can use the bar model to write an equation.  We can see we need to add the two parts together and set it equal to the whole, so we could write the equation x + 28 = 196.

The model helps a student develop the operational sense to write an equation with addition, but then perform subtraction to find the difference between the two values. They can see that their solution needs to be a value smaller than the total number of members.

Tape Diagrams in a Comparison Problem

Without problem solving strategies, a student may write an equation like this, p + d + c = 10.50. They have an equation with 3 different variables and then are stuck. They don’t know how to solve for d, the price of the drink. Let’s try a tape diagram.

We know the popcorn is three times as expensive as the candy and the drink is twice as expensive as the candy. Let’s create a tape diagram under each food item in our bar model. 

We know the popcorn is three times as expensive as the candy. This can be hard for students to grasp, so give them an example with simple numbers: “If candy is $1, how much is popcorn?” $3. We want students to see that popcorn is more expensive than candy and that the cost of 3 candies is equal to the cost of one popcorn. 

Similarly, the drink is twice as expensive as the candy so the drink is equivalent to 2c’s. 

Finally, we know the cost of the candy is represented with the variable c. Now we are ready to make a connection between our model and write an equation.

The model clearly shows that 6c = 10.50. We have taken a complex word problem and written a one-step equation to solve for the variable c. Once a student solves for the value of c, they can refer back to determine the cost of the drink and popcorn.

Grab our Problem Solving Poster freebie to display in your classroom!

How do you use tape diagrams in your classroom?

P.S. Check out these related posts: Math Problem Solving Strategies and How to Teach Word Problems and Problem Solving

The post Using Tape Diagrams to Solve Problems appeared first on Maneuvering the Middle.

(Noelle recently presented an amazing math training – Practical Problem Solving Strategies – this summer, and I was truly amazed at just how much I learned. I will be breaking down the training into 3 blog posts over the course of this month, so if you missed the training, be sure to check back here for more updates.)

If you want to learn more, check out this book, Mathematize It!, that covers the topic of teaching how to solve word problems in much more detail.

Be sure to read Part 1, Three Word Problem Types to Teach, and grab our freebie below!

Here are the 3 steps to solving word problems:

1. Restate the Problem Situation

Let’s use this word problem about Pedro as our example:

(Check back to last week’s blog post to understand why this scenario falls in the action category.)

By restating the problem, we want students to avoid seeing phrases like “leftover” and decide immediately that they must subtract. We want students to focus on the action taking place. Here is an example of restating the problem. You could have students do this verbally (think, pair, share style) or write down bullet points. Notice that there are no numbers present. 

• Pedro makes a pitcher of lemonade.
• He pours some for his friends and now he has some left over.
• How much lemonade did he start with?

2. Represent the Problem Situation

There are numerous ways to represent a problem: draw a picture, create a diagram or model, use manipulatives, or write an equation (don’t think numbers and variables; it can be something like “Pedro’s pitcher = leftovers + what he poured”).

The primary goal of this stage is that representations give students that “operational sense” they need in order to write an equation. We want them to know what to do next.

The 3 diagrams we use most in middle school math are open number lines, bar models, and ratio tables. For this problem, I recommend using an open number line.

You can also make the number line a vertical number line. Liquid in a pitcher as a vertical number line might make more sense visually to students. 

There was a starting amount of lemonade in the pitcher, but we actually don’t know what that is, so that is our unknown value that we will represent with the variable x on the number line.

Then Pedro pours 34 ounces of lemonade for his friends, here is our action or change, so we will represent that with a jump on the number line. Since we know our action describes removing lemonade from the pitcher, our jump will point down.

Now he looks at the pitcher and there are 20 ounces of lemonade left, this is the resulting value. The open number line allows us to identify what each value in the situation represents, and now we know that our unknown value was the starting amount. 

To finish out our lemonade problem, let’s use our representation to write an equation.

We have a starting amount, x, then we will subtract the amount he poured, 34, and that is equal to the amount remaining in the pitcher, 20.

Notice how the equation has subtraction in it, (which makes sense because the situation describes removing a quantity) but the operation we will perform to find the solution is addition. This is what we mean by developing operational sense for a situation.

The number line also helps students see that the solution needs to be a value greater than 20, since x is higher on the vertical number line. 

These representations can be so powerful as we help transition students from the concrete to the abstract!

Be sure to grab our free Problem Solving Posters that have examples of all of these representations.

3. Solve and Reflect

When we think about problem solving, we often think about the end goal being to find the solution, which of course is important.

But in order to grow our students as problem solvers, it’s important to also take time for individual and classroom reflection. Reflection is such an important part of making meaning. If students can construct mathematical arguments to justify their solutions, you know they fully understood the problem.

After students have worked through a word problem, encourage them to share the different models or equations that they came up with, and have them explain the operations they used to solve a problem. Students will learn from each other as they are exposed to different ways of thinking about the same problem.

P.S. Check out these related posts: Math Problem Solving Strategies and How to Teach Word Problems and Problem Solving

The post 3 Steps to Solving Word Problems appeared first on Maneuvering the Middle.

Noelle presented an amazing math training this summer on Practical Problem Solving Strategies. I was truly amazed at just how much I learned. I will be breaking down the training into 3 blog posts over the course of this month, so if you missed the training, be sure to check back here for more updates.

If you want to learn more, check out this book, Mathematize It!, that covers the topic of teaching how to solve word problems in much more detail.

Today we are going to talk about the 3 types or categories of word problems that you teach and your students may face: action, relationship, and comparison. The purpose of identifying word problem types is to force students to slow down and analyze what is happening in the word problem before jumping to computation.

You can also grab our problem solving posters freebie below!

Action Word Problems

Here is an example of an action word problem: 

How do we know that this is an action? Ask yourself:

• Did something occur?
• Was there some kind of change?

If yes, the word problem likely falls into the action category.

Relationship Word Problems

Here is an example of a relationship word problem: 

How do we know this is a relationship? Ask yourself:

• Are parts being described or referred to in relation to a whole? 
• Is a whole being described or referred to in relation to a part?

If yes, the word problem is a relationship. Here we can see the parts of the marching band relate to the total number of marching band members. 

Comparison Word Problems

And lastly, here is an example of a comparison word problem: 

How do we know this is a comparison?

Ask yourself:

• Is something in the word problem being described in comparison to something else?

In this word problem, we can see that the cost of popcorn is being described by the cost of the candy. 

Why is this helpful to know?

Why do students need to know this? Well, by observing and “making meaning” from the words and scenarios they are processing, students are less likely to rush to determine a path to the solution. 

Does this sound familiar? Students quickly perform some operations with the values given. In this first step of the problem-solving process, we want to take the focus off the values and direct students to notice what is being described in the problem.

The goal is not for them to be able to identify and put the word problem into the correct category. We simply want students to notice what is happening, and over time they will start to recognize patterns in word problems. 

Next week, I will dive deeper into how we take these word problem types and use them to help students with the first part of the problem solving process: restating the problem.

Grab our problem solving posters freebie!

In the meantime, can you identify the category these sample problems belong in?

1. Ricky buys a package of chicken to use throughout the week. On Monday, he uses 28 ounces to make chicken salad for lunch. On Thursday, he grills 53 ounces of chicken for dinner. If Ricky determines he has 37 ounces of chicken remaining to cook, how many ounces of chicken did he buy at the beginning of the week?
2. Gavin has two pet turtles, a red-eared slider and a map turtle. His red-eared slider weighs 2,680 grams and his map turtle weighs 670 grams. How many times bigger is the red-eared slider than the map turtle?
3. Ivory created a paper chain of her school colors, blue, green, and white, as a decoration for a pep rally. The blue section measured 5.5  feet long, the green section measured 4.25  feet long, and the white section measured 3.75  feet long. What is the total length of the paper chain?
4. A king-sized chocolate bar has a mass of 2.6 ounces. A regular-sized chocolate bar has a mass of 1.55 ounces. How many more ounces is the king-sized chocolate bar than the regular-sized chocolate bar?
5. A nature center has a stocked pond with an automatic fish feeder. The fish feeder has 70.5 pounds of fish food and releases food into the pond twice a day. If the feeder releases 2.6 pounds of food in the morning and 1.2 pounds of food in the evening, how many pounds of food are remaining in the feeder at the end of the day? 
6. The San Francisco Bay Area is hosting a triathlon, a race consisting of swimming, biking, and running. The athletes will swim for 0.75  miles and bike for 15.5  miles. If the total distance of the triathlon is 20.5  miles, how many miles is the running portion of the race?

The post 3 Types of Word Problems to Teach appeared first on Maneuvering the Middle.

However, there are so many interruptions that you can minimize to maximize the effectiveness of your classroom time. Every minute counts!

Let’s chat today about common classroom interruptions and some ways to combat these time wasters. Plus, I have included a freebie to help get your time back.

Locker and Bathroom Requests

Problem: You see a raised hand and anticipate a math question or observation only to be met with, “Can I go to the bathroom?” 

Or perhaps, a student left their math folder in their locker. Whatever the reason, a student needs to leave the learning environment and we all know class time is precious. 

Solution 1: Know what a student needs before you call on them. Our classroom poster pack has hand signs for RR, locker, answer, question, and more. You don’t need our classroom poster pack to implement this procedure, but having the visual will serve as a reminder for students.

Solution 2: Use our MATH FAST PASS to hold students accountable to the number of restroom breaks or locker requests you deem necessary based on the length of your class period. I am not a fan of limiting restroom breaks for students who genuinely need it (and passing periods do exist for this very reason), but this tool can serve as a way to manage that interruption.

Personally, 3 emergency passes per grading period usually did the trick!

This freebie also has some helpful math tables and concepts to keep in students’ folders or binders, so it is a win-win for math teachers everywhere. Before a student could ask to use the restroom, they have to have their MATH FAST PASS out for you to sign and date.

GRAB OUR MATH FAST PASS FREEBIE

First Aid Needs

Maybe it is a math teacher thing, but if you aren’t actively vomiting, then I think you are healthy enough to learn math.

Keep band-aids, mints, and paper towels in your classroom for these small ailments. Your nurse or front office staff will thank you.

A wet paper towel across the forehead will satisfy the needs of a student with a headache and the student will feel cared for. Another win!

Keep in mind that in most cases, the nurse won’t be able to offer anything stronger and they will be sent back to class anyway.

Pencils Needing to be Sharpened

Pencils. Pencils. Pencils. These writing utensils are a necessary evil in your classroom.

Start by establishing a pencil routine that you feel confident you have the stamina to maintain by the end of the year. Here are 2 ideas that I have personally used:

• Pencil Library. This is a trading system. A student who needs to sharpen their pencil can get up, drop off their old pencil, and grab a sharpened pencil. No need to run the sharpener. A student in my homeroom would sharpen the allotted amount of pencils for the day (around 10-20). The old traded pencils would get sharpened to use the next day.
• Pencil Parking Lot. This idea comes from To The Square Inch. I love this system! I like it because I can SEE the pencils. 5 pencils are clipped to the whiteboard. Students sign out a pencil by writing their name on the whiteboard with a dry erase marker. At the end of class, I would remind students to bring my pencils back and I would erase their names as I clipped the pencils back up. 

After my pencil sharpener broke, my pencil sharpener stopped being for public use. If your pencil sharpener is for students to use, I recommend sharing your expectations for use:

• How to ask to use it (use hand signs)
• When they can use it (never during instruction)
• Proper use (no colored pencils)

Cell Phones

Cell phones are tricky! Personally, I only ever worked in a school that had a school wide policy that they were off and stowed away. This made my job easier, but I know that is not always the case.

Phones aren’t going away anytime soon, so how do you manage the distraction? As I was doing research about this topic, I found that the teachers most successful with cell phones had 3 things in common:

• A why
• A clear and consistent routine and procedure
• Allowed the occasional use with boundaries around usage, complete with consequences

A Why

Students need buy-in before they detach from their device. You can show them studies (like this one from The University of Chicago) regarding how the mere presence of phones reduces available cognitive capacity.

Ask students to reflect on their own relationship with their phone. Does it distract you? Do you find yourself stopping what you are doing to check your phone? 

A Clear and Consistent Routine and Procedure

When developing a procedure for cell phones, it is important to be super clear about everything! Power struggles usually occur over ambiguity.  

Let’s say that the procedure is for students to put their phones in an over-the-door shoe rack during class. Here are all of the details to go over with your students:

• Phones go into the pocket before you sit down for class. (I would stand at the door for the first few days and send students over to the phone storage upon entry)
• Phones are turned off or on silent. I would explain to students that I don’t want any phones making sounds during instruction.
• I would have assigned pockets. Students need to place their phones in their designated spot.
• Phones will be picked up when class is dismissed. Exceptions will be made only with permission from me. (Example: they need to call their parents)
• Lastly, I would remind, remind, remind students at the beginning of class. It would be posted with their start-of-class instructions, it would be posted permanently on a wall, and I would also verbally tell students that, “If your phone isn’t put in the pocket in the next 10 seconds, I will have zero grace if you are caught with it or if it goes off.”*

(This procedure is just a suggestion to model the level of detail required.)

I recently saw a teacher, Mrs. O, suggest giving students brown paper bags to place their phones in, stapling the bag, and leaving it on their desk if they are struggling with looking at their phone during class. 

Boundaries + Consequences

If you have done everything from above, then I suspect you will have fewer problems (not saying zero!) with cell phones. When a student chooses to use their phone during class, they will know that a consequence is coming. Be sure to check with your administration, grade level team, or student handbook regarding appropriate consequences.

I wouldn’t recommend consequences that include first offense, second offense, or third offense since that is hard to track. 

Classroom interruptions are inevitable. How do you manage these common classroom interruptions?

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Problem solving can be a challenge to teach. Perhaps, it is because we are trying to teach students to manage and implement a myriad of skills: thinking, observing, investigating, reasoning through situations, and accessing prior knowledge. Check out these strategies to get students thinking!

LISTEN ON: APPLE PODCAST | SPOTIFY

Problem Solving with Word Problems

In the book The Young Child and Mathematics, Juanita Copley states that problem solving happens through “doing, talking, reflecting, discussing, observing, investigating, listening, and reasoning.”

We can often mistake different methods for decoding word problems as problem solving skills. R.U.B.I.E.S. and C.U.B.E.S. and all of the acronyms where students are supposed to underline, box, or circle may feel productive, but they cannot produce the skills needed to solve real-world problems. 

Why Are Problem Solving is so Difficult

There are a lot of different factors that play into this challenge and it could be a single isolated one or it could be a combination of several factors.

• Need for perseverance – some students can routinely go through all of the procedures involved in solving a math problem, but they haven’t yet acquired the perseverance needed to reason through a real-life situation 
• Reading skills — tasks and real-world applications require reading comprehension and decoding skills to fully understand the situation and apply the mathematical reasoning skills necessary to solve
• Multiple steps — requires students to create a plan with several steps and work through the plan
• Incorrect vocabulary usage — when solving word problems, sometimes students have been instructed to primarily look for specific words like “per” “of” “each” “sum”…and then they attempt to apply the operation — but we know that doesn’t always work and can also over simplify the process.

Shift the Focus from the Answer

This post and video from Phil Daro (co-author of the CCSS) says that based on their studies, teachers need to shift the focus from answer-getting to sense-making. “When the answer is the only goal, genuine learning is undermined.”

He says that teachers put too much emphasis on the answers; answers are part of the process, but they are not the only learning outcome. That wrong answers are part of the learning outcome. 

• Consider framing wrong “answers” as “discoveries.” If students reached a wrong answer, you talked more about why that approach doesn’t work instead of how to get the right answer.
• Consider giving students the answers before solving, removing some of the power, and instead spent class time figuring out different ways to get to the answer.

The goal of solving math problems is the critical thinking that happens along the way. Not the answer.

How do we teach problem solving?

Learning is not necessarily the solution to a specific question. We want students to apply sense-making to the rest of their lives.

I really want you to watch the video, but one of Daro’s suggestions is to provide students the correct answer before having students solve. It removes some of the power of finding the right answer, and puts an emphasis on figuring out different ways to get the answer. There are so many different ways to solve problems and it encourages students to think in this way.

Tip #1 – Allow flexibility in how students solve problems.

Showing your work is said so often that it can seem meaningless to a student. Give students options for ways they can show their work.

Grab a free REPRESENT IT bulletin board that will provide a visual aid for different ways to represent their work!

Here are a few examples:

• Write an equation
• Solve a simpler problem
• Draw a bar model
• Draw a picture
• Draw a graph
• Use an open number line
• Use manipulatives
• Guess and check
• Use logical reasoning
• Make a ratio table

Tip #2- Focus on quality over quantity.

Quality over quantity is going to mean different things for different teachers, depending on the number of students, the length of your class period and even the different concepts being covered.  Here is one example for helping students focus on “sense-making” in a problem. 

Let’s take this pretty basic word problem >> Mr. Roy overdrafts his account by $25.50 and then is charged a $10.35 fee by the bank. What is the change in Mr. Roy’s bank account?

Before modeling a think-aloud to discuss how to solve this problem (and move on quickly to the next), ask so many questions that students will have no choice but to THINK about what the problem is asking for. Shift the workload to the students. Here are a few I brainstormed:

• Who is Mr. Roy?
• What is an overdraft?
• Is a fee a good thing or bad thing? For who?
• Why is Mr. Roy charged a fee?
• What is the change measuring? 
• Does he have more money or less money?

I just asked 6 questions and haven’t even gotten to the math yet! Model for students the type of critical thinking and questioning you want students to use when they are solving solo.

Then have students just try out the problem. Circulate to see the different methods. Come together to discuss the different ways you saw the problem solved. 

Bring this problem to life by providing Monopoly money to students to “act” it out. 

I love seeing students be problem solvers and I think it is such a lifelong skill that they will always carry with them. In fact, they may not frequently need to calculate the equation of a line or the probability of a compound event, but each and everyday they will use their “sense-making skills” to solve problems and make decisions. 

If you want to focus on teaching problem solving, then All Access may be right for you!

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Let’s talk about 5 ways to use open number lines in middle school math!

1. Fluency

Adding friendly numbers – This method leaves one addend whole and adds the other addend in friendly amounts. 

In this example, 275 was added to 173 in friendly chunks of 200, 50, 20, and 5 to get the final sum of 448.

Adding to friendly numbers – This method adds a small amount first to get a partial sum that is friendly, then adds the remaining amount in friendly chunks.

In this example, 2 was added to 398 to get the friendly number of 400. Then, the remaining 112 was added in chunks of 100 and 12 to get the final sum of 512.

Distance (Subtraction) – Because subtraction is the distance between the two values, students can use an open number line to count up to the total distance between the two numbers.  Values closer together are better suited for this method.

Removal (Subtraction) – Removal takes away to find the difference between two values. Values further apart may be easier to subtract using removal.

Constant Difference – Add or subtract the same number from both values to make the numbers easier while keeping a constant difference. 

In this example, 620-370 was adjusted by subtracting 20 from each value (620-20=600 and 370-20=350)  and finding the difference between 600-350. (I could have added 30 to 370 and 620 too!)

2. Ratios and Conversions

Double Number Lines – Double number lines are useful for ratios and conversions. If you typically use a table for solving ratio problems, then an open number line is not too far off.

3. Ordering Numbers

You can read more about this in our Ordering Rational Numbers post, but ultimately, no problem that involves ordering numbers is complete without a number line. It provides context and is a way to actually show student thinking. 

While starting with a completely blank number line can work for some students, I recommend writing in whole numbers first.

Example: -½, -2.2, -1.5

Before placing a single number from above on an open number line, I would ask students to provide the integer numbers (for simplicity sake, I am referring to -3, -2, -1, and 0) that make sense with the numbers provided in the example.

Once determined, students will have a much easier job placing -1.5 between -2 and -1 than on a blank number line with -½ and -2.2.

Hint: Use a vertical number line when involving negative numbers.

4. Problem Solving

These problem solving strategies are best explained in word problem format. Please read the word problems to fully understand the open number line diagrams.

Active Situations – This method describes a situation in which quantities are joined or removed. There is a start, a change, and a result.

Part Part Whole – This describes a relationship that includes two or more parts that relate to a whole.

Additive Comparisons – These are situations that describe how two quantities compare to each other. They include a smaller quantity, a larger quantity, and the difference between the two quantities.

I had no idea there are so many ways to use number lines to solve so many different types of middle school math problems. Do you use open number lines?

If you are interested in learning more about open number lines, our Maneuvering Math Intervention Program will be adding a new module called Jump Start in time for Back to School Season! Jump Start is an introductory component to equip students with tools and strategies for number operations and problem solving.

It will help students reason with numbers, recognize and effectively represent problem-solving situations and approach math with increased confidence and tools to use throughout challenging concepts.

The post 5 Ideas for Open Number Lines appeared first on Maneuvering the Middle.

Before I begin, let’s talk about what a performance task is:

“A performance task is any learning activity or assessment that asks students to perform to demonstrate their knowledge, understanding and proficiency. Performance tasks yield a tangible product and/or performance that serves as evidence of learning.” (From Defined Learning)

Keep reading to get a free performance task for middle school math + algebra 1.

1. Prepare Your Students

My first mistake was assuming that I could successfully replace a multiple-choice test with a performance task and act like they are the same. Performance tasks require students to not just recall what they have learned but apply what they have learned to a problem. The effort is more sustained. A student cannot circle an answer and move on to the next problem like they can on a traditional assessment. 

If you plan to use a performance task as an assessment to be completed independently, I recommend you model and complete a performance task together with your class at some point during the unit. Consider it like a test review. 

You will most likely receive higher quality work because students know what the criteria for success is. You can show how you want diagrams labeled, or how thorough their explanations need to be, or whatever your heart desires. 

Performance tasks do not have to be an independent assignment or assessment! Our Maneuvering the Middle performance tasks are designed to be collaborative. Here are some ways you could assign performance tasks:

• Group Activity  – Provide roles so all group members contribute.
• Partners – Assign partners. You can have a partner responsible for turning in one performance task that they both contributed to, or each student is still responsible for turning in their own task, but are still able to work collaboratively. 

2. Provide Checkpoints and Opportunities for Feedback

A performance task typically encompasses everything learned in a unit, so the material is still very fresh to students. We want students to do well, right!? That is why I recommend checkpoints. Depending on the performance task, you may want to check students’ work before they proceed. You can do this a variety of ways:

• Draw a star on the performance tasks or circle a question number – when students reach this point, they raise their hand so you can check before they move on. (Bonus: this is also a great way to grade as you go! Instead of having a stack of paper to grade at the end, you can come up with a system on your clipboard that allows you to add or deduct points.)
• Circulate and keep circulating – You are probably already doing this, so keep on keeping on! I like to walk and add checkmarks to correct answers to give students confidence and prevent students from raising their hands to ask if the problem is correct. I want raised hands to be for questions, so I can prioritize appropriately.
• Check based on time elapsed – I also refer to this as chunking. Instead of telling students they have 50 minutes to complete the entire task, tell students they have 10 minutes for Section A, 15 minutes for Section B, and so on. Use these time restraints to circulate and check where students are and check for accuracy. “You should be wrapping up Section A. I am coming around to check while you work on Section B. You have 15 minutes to finish Section B.” *sets timer*

3. Complete the Performance Tasks Yourself Sans Answer Key

I recommend doing this for any assessment, assignment, or activity! Complete the performance tasks on your own without any assistance from a calculator (if students don’t get one) or without the answer key to guide you.

This will do 3 important things: 

1. This should give you an estimate for how long it will take a student to complete it. It is safe to assume that a student will take around 3 times longer to complete than the teacher.  If this exceeds a class period, you can make plans to spend two class periods on the performance task or shorten the task to accommodate. 
2. Most importantly, it will help you identify areas where students may feel confused or need extra support. If you hesitate or have to reference something to proceed, your students most likely will too! 
3. If you created the performance task from scratch, consider asking another teacher to complete it to receive feedback. This might save you a headache later. 

4. Students Who Need Motivation

Performance tasks require sustained effort which can be a challenge for some students. You still want to hold these students accountable for their work, but you can do a variety of tricks to keep them motivated. Students are generally motivated by choice and by learning about things that they care about. Can you alter the performance task to be about something that they love? Even if that means you take a pen and mark up their task on the fly. 

Is the task accessible to them? If not, find an entry point. Perhaps, they need a formula or a filled-in-example . You may have to alter the task a bit for this to happen, but students have different needs and that is ok. 

Last but not least, pulling a small group is a great place for students to feel supported and successful. You don’t have to do the task for them, but you can prompt them and keep a closer eye on them. 

5. When to Implement a Performance Task

I love how Marissa implements a performance task! She assigns performance tasks the day after a test to her students who mastered their unit test. Performance tasks act as a perfect extension for these students who are ready to move on. For the students who did not master concepts from the unit test and need reteaching, she pulls a small group. 

Maneuvering the Middle’s performance tasks are low-prep and also come in Google Slides format.

What tips do you have for implementing performance tasks?

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Let’s chat about why every math classroom can benefit from a word wall. You can also read more tips about implementing a word wall here.

What are the benefits of a math word wall?

1. BUILD A COMMON VOCABULARY BANK

Word walls are a necessity for math. In middle school, each unit introduces (on average) around 5 new vocabulary terms. If new terms are necessary for success in a unit, then your students better know them!

What better way for students to internalize these words than to see them, reference them, and then use them? A word wall makes this process accessible.

When you have trained your students to find new or unfamiliar words on the word wall, then you are setting them up to use their resources and promote independent thinking. 

For the most useful word wall, include the term, the definition, and a visual. It is inevitable that your students will need the occasional brain break and may need to stare off; if that is the case, I find it a good use of time if they can be staring at math words. You can read more about math vocabulary here.

2. SUPPORT BILINGUAL STUDENTS

All students benefit from easy access to frequently used words, their definition, and an image, but for bilingual students, it can be especially helpful! Word walls provide a visual aid for students who are learning English. 

For your Spanish speaking students, our new Middle School Math Word Wall Resource includes the Spanish term and Spanish definition to support your bilingual students even more!

3. INTRODUCE NEW TERMS AND REVIEW PREVIOUSLY LEARNED WORDS

Update 7/28/2023: Maneuvering the Middle now has a Middle School Math + Algebra 1 Word Wall.

As you can see in the video below, our Word Wall includes 190 essential math terms, their clear-cut definitions, and their visual representations.

We’ve included Spanish translations for all terms and definitions, ensuring a supportive and accessible learning experience for English Language Learners.

They were designed to be minimal prep and flexible to customize the formatting to suit your students’ unique needs.

That is a copious amount of words for students to know and use, which makes a word wall even more necessary. While they may only learn around 50 new words each year in math, they also will need to access words from previous grade levels. 

While students may be introduced to the term “product” in elementary school, that term will continue to appear in later grades. This is also true of words like equivalent, volume, area, difference, expression, and well, so on and so on. Math vocabulary terms continue from elementary to high school because math builds with each subsequent year. 

TIP: Don’t start the year with a complete word wall. Either add to your word wall as the vocabulary term is introduced or display terms unit by unit. 

The phrase “out of sight, out of mind” rings true. When students don’t see something every day, they tend to forget it. How does your classroom benefit from a word wall?

The post 3 Benefits of Math Word Walls appeared first on Maneuvering the Middle.

Here are a few tips I have for creating, maintaining, and maximizing the use of a middle school math word wall.

Keep reading to check out Maneuvering the Middle’s new Middle School Math Word Wall.

1. Structuring Your Word Wall

A word wall that includes the math term, the definition, and a visual representation will make the most impact.  The text should be visible from across the room, and should be in a place where a majority of students can see it.  I have many English Language Learners in my classes, so word walls are especially helpful for them.  Some of my fellow teachers even use the Spanish translation as additional leverage for students. (Keep reading to learn about a new resource from Maneuvering the Middle!)

I like my words to stay up all year as we learn and review the words.  This helps students use academic vocabulary and make connections between the different units.  I love when a student can see how a proportional relationship and a linear equation are connected!

At the beginning of the year, I only have the categories posted at the top of the wall.  I then include the words as they are introduced, then move on to the next category of words included in the next unit. 

2. WAYS TO USE

Incentivize Academic Vocabulary

While word walls make a classroom beautiful, their ultimate function is to be used.  This means that you need to make students aware of its existence and its usefulness.  

To do this, incentive using vocabulary terms from the word wall by rewarding students who use the vocabulary correctly when answering a question or explaining an answer.  When students are about to begin practicing a new skill, I ask students to point to the word that will help them if I am unavailable. 

During mad minute exercises, I include vocabulary that they can provide a definition with one word (example: quotient=division).  If they don’t know it, they can take an extra second to look up at the word wall.  

Flyswatter Game

If you want students to get familiar with your word wall, use the Fly Swatter Game.  This is a very engaging review game. If you are like me and don’t bother to cover up anything in your room before a test, this will help remind students where to look when they are stuck.  Two students face off with fly swatters in hand.  You give them a prompt such as “2, 4, 6, 8” are examples of ______”  And the first student to swat the word ‘multiples’ earns their team a point.

Non-Content Words

Word walls do not have to be specific to your content area.  I have character terms like tenacity, curiosity, courage, and community on my wall too.  These are our school values, and it is important that these words are referenced by students and me daily.

Flashlight Game

This game is great for those last few minutes of class as a sponge activity.  Turn off the lights and use a flashlight to point to a word on the wall.  Students can then shout out an example, the definition, or even a counterexample. 

3. Making a Word Wall

Knowing that teachers have more to do than hours in the day, creating the word posters is a task easily assigned to students who finish early or those students in your homeroom who are always asking how they can help. 

After a unit test, I would have early finishers complete this as an activity for the next unit.  I would give the word + definition + example + picture that I wanted them to use, and choose the best ones to go on the wall. I did this for several years before I wanted a more uniform look. 

There are also lots of resources on Pinterest or TPT to choose from. My advice is to batch this task during the summer. Print everything you need, laminate, and decide how you will store them. Then you can post the new words for each unit to your growing word wall. 

Update 7/28/2023: Maneuvering the Middle now has a Middle School Math + Algebra 1 Word Wall.

As you can see in the video below, our Word Wall includes 190 essential math terms, their clear-cut definitions, and their visual representations.

We’ve included Spanish translations for all terms and definitions, ensuring a supportive and accessible learning experience for English Language Learners.

They were designed to be minimal prep and flexible to customize the formatting to suit your students’ unique needs.

At first a word wall might seem excessive or more like an “extra” thing to do, but now that I have seen how my students use it, I am sold!  How do you use your word wall in your middle school classroom?  

Maneuvering the Middle has been publishing blog posts for 8 years. This post was originally published in November of 2016. It has been updated for relevance and clarity.

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Today, I am sharing ideas that support the personal financial literacy standards for middle school.

Let’s take a moment to at what is included in the Texas Personal Financial Literacy standards. *The standards below have been edited for conciseness.*

Comprehensive curriculum that is aligned to the TEKS can be a challenge to find. Fortunately, Maneuvering the Middle has already created the student handouts, homework, study guides, and assessments for this very subject.

In addition, Maneuvering the Middle has already created a variety of activities – scavenger hunts, card sorts, solve and color, online exploration activities, stations, and more to provide practice and keep Personal Financial Literacy engaging.

Think Long-Term

Remember that while we do teach the standards, we can emphasize the most long-term concepts.  While sixth grade calls for students to be able to explain the items on a credit report, we really want students to have a great understanding for what a credit report is and how short-term poor decisions can impact us for a long time.  In 8th grade, this concept is driven home with the emphasis on ways to save money with interest over time. To me this is a great takeaway for middle schoolers! 

Make It Relevant

I appreciate the vertical alignment of specific strands, specifically G and H. Sixth graders look at the larger scope of how career fields impact your lifetime income, 7th graders explore just how expensive it is to maintain a household, and lastly 8th graders are asked to devise a savings plan for college. This strand shows how personal this concept is to students’ futures. This is something that you will use in the real world (*gestures wildly*).

Activities to Try

Dollars and Sense – This free activity is similar to the game of Life. While it isn’t ink friendly, you could laminate and have a class set. Students choose a career, and have various opportunities to flip a coin to either incur a costly expense or financial favor. They decide how to spend and save their money on transportation, clothes, and housing. At the end of the game, they have to fill out a budget based on their choices. This is aligned to 6th, 7th, and 8th grade standards.

Financial Football – This computer-based activity requires students to answer financial multiple-choice questions between plays. The football aspect will engage some students, and it isn’t 100% aligned to one grade level’s standards, but overall, I think this would be a great extension for students over the course of the unit.

Comparing Salaries in Various Fields Project – In this project, students will research and compare annual salaries of various careers requiring different levels of education and calculate the effects of different salaries on lifetime income. The project even allows for a career fair! This project is directly aligned to TEKS 6.14G and 6.14H.

Real-Life Bills – Similar to Price of Right games, give students a category – water, electricity, cable, phone, rent, groceries, etc – and have groups guess what the average or median cost of those types of bills would be. You can use your own bills as a reference or Google averages in your area. 

Household Budgets and Percent Practice – Students will take on the role of an employee working for “Remote Possibilities”,  a company that helps clients who work remotely to determine the best location to live based on the client’s income, financial goals, and lifestyle desires. Students will understand and apply concepts of personal budgets and minimum household budgets. This project is directly aligned to TEKS 7.13B and 7.13D.

Planning and Saving for College Project – Students will take on the role of a financial advisor working for Scholarly Savers, a company that counsels families through various financial situations. Students will research the cost of colleges and create a savings plan for a fictional client. This project is directly aligned to TEKS 8.12G and 8.12C.

EMPHASIZE VOCABULARY

When writing the units and really digging into the standards, I was blown away by the level of new concepts and terms that are introduced.  Ask students to use the academic vocabulary, and spend a few minutes at the beginning of each class reviewing. To spice it up, you could do a quick fly-swatter game, Quizizz, or Kahoot.

The 6th grade standards have 10 new vocabulary words introduced! For comparison, most units average around 5. 

Common Misconceptions

Overall, I think the biggest challenge is that while these terms are familiar to us as adults, they are foreign to students.  Credit reports? Grants? Work-study? Like I said before, this unit is probably the most vocabulary dense unit in middle school math. 

• Grants are needs-based, while scholarships are needs- and merit-based
• Confusing total value with interest in the compound interest formula
• Confusing the terms “assets” and “liabilities” on a net worth statement
• A debit card is different that than the verb, “debit”

Anchor Chart

Anchor charts are fabulous ways to showcase the content in a visual manner for students to reference.  They can easily be created before the lesson or as you are teaching, depending on the content.

Do you have any other great ideas for teaching the personal financial literacy standards?  

Maneuvering the Middle has been publishing blog posts since 2014. This post was originally published in March 2018. It has been updated for relevance and clarity.

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Standards

Let’s start with the standards. While surface area is covered in 8th CCSS, it spans both 7th and 8th grade in the TEKS.

One thing to note is that in 7th grade, students will use a net to determine the surface area of 3D shapes. In 8th grade, students will have access to formulas to solve.

Lateral vs. Total Surface Area

In the standards above, lateral and total surface area are distinct. Students must be able to carefully read a problem and determine if the area of the bases should be included. I have students who would absolutely see the net below and start finding the surface area without even reading whether the question was asking for the lateral or total surface area. A tip here would be to ask students to write “no bases” when they see lateral surface area and draw Xs on the bases.

Problems in which lateral surface area is often seen:

• Surface area of a soup can label
• Surface area of an oatmeal canister
• The amount of square feet covered when rolling a paint roller or rolling pin 
• The amount of paint needed to paint the walls of a room

Tips for Teaching

Because 7th grade requires use of nets to solve for total and lateral surface area, students are going to need a method for organizing their work. Sara, a member of our curriculum team, recommends, “Find the area of each face and write the area on the net, then students add up each face to find the total or lateral surface area.”

This is also helpful when it comes to grading because you can give credit to students who may have made a computation error on one of the faces, impacting their final answer. This may be enough motivation for students to show their work. 

In the image above, the pen colors for the formulas and calculations match the shading for the corresponding faces. These are small things that can really help students follow along. (This image is a screen grab from our All Access Student Videos.)

You can also ask students to create a table to organize their work, similar to our Surface Area Student Handout 2 shown below. 

8th Grade Formula Chart

For 8th grade, the STAAR reference sheet for surface area looks like this (outlined in green).

You will need to repeat the phrases, “perimeter of the base,” “area of the base” and “height of the prism” until you lose your voice. This is a great time to rapidly cold call students to ask what each variable represents. 

Triangular and Rectangular Pyramids

Pyramids are included in 7th grade TEKS and 7th grade CCSS.  

A pyramid is composed of a base and triangular faces. The pyramid is named by the shape of the base – so triangular prisms have a triangle base while rectangular pyramids have a square or rectangle as a base.  Pyramids have one base and the lateral faces all come to one point, the vertex. 

It is important to differentiate between the lateral height (which is the height of the triangular face) and the height of the pyramid. This will be an important distinction when they transition to calculating volume in later units. 

Students sometimes struggled with differentiating between pyramids and triangular prisms.  To help students, ask,  “Are there more triangles or quadrilaterals?” If there are more triangles, then you have a pyramid. If there are more quadrilaterals, then you have a triangular prism. 

Activity Ideas

1. Make it hands on

In 7th grade, surface area is a brand new concept, so students will need time to understand what surface area is. Reusing amazon boxes, cereal boxes, or cracker boxes can be a great exercise to make the connection between 3D figures and their nets. You can provide student groups with a 3D box, they measure the dimensions (length, width, height), and then carefully cut to form the rectangular prism’s net. They will see those same dimensions that form the box’s net. 

In addition, bring in soup cans (cylinders) and peel off the label to show the rectangular face of a cylinder.  It can be tricky for students to see how a rectangle fits on a cylinder’s net.

2. Digital Activities

Our 7th Grade Surface Area Digital Activities do a fantastic job of breaking down the nets (see below). Our 8th Grade Surface Area Digital Activities provide great application opportunities.

3. Change Up the Writing Utensil

Using formulas over and over again can be pretty repetitive. Allowing students to practice using dry erase markers or chalk outside is a great way to break it up.

4. Assess Creatively

This performance task activity is a great way to apply surface area. 

5. Match Nets to Shapes

This Surface Area and Nets Cut and Paste (no time to cut and glue? Just make it a matching activity) is perfect for students to practice matching nets with their corresponding 3D shape.

Common Misconceptions

There are quite a few, so let’s dig in:

• Assuming the base of the figure is the face is at the bottom. Make sure to emphasize that the base must be congruent and parallel for prisms and cylinders. 
  • I love this simple idea from a member of our curriculum team, Ashleigh, “Students assume the figure is always sitting on its base, so I would hold up a triangular prism (this one was hardest for them). I would lay it in my hand where it was “sitting” on the triangle. They would tell me it was a triangular prism. Then I would flip it over where it was laying on one of the rectangular faces and ask them what it was again. They would say it was still a triangular prism. This helped them see that no matter which way it was facing or which shape was on the bottom didn’t matter.” 
• Using the formula, students will plug in a length or width of the base instead of the total area of the base. Our curriculum writers, Reagan and Asheigh, recommend always solving for the area of the base first and separately and clearly label it a B. 
• Students think that height is always a vertical measurement. Height is perpendicular to the base no matter where the base sits.
• With triangular prisms, there are so many heights and so many bases. This can be so confusing! Avoid using the words “height” or “base” without saying “height of the triangle” or “height of the prism.” Base is now and forever “area of the base” and always solved first separately. 
• When working with cylinders, students may confuse the diameter and the radius. Radius, as a word, is shorter than diameter which supported my reminder that the radius is half of its diameter.

What tips do you have for teaching surface area?

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For Texas Teachers

The STAAR (State of Texas Assessment of Academic Readiness) is moving online with a variety of new open-ended question types. Maneuvering the Middle has designed a bank of questions to prepare middle school math teachers for these changes.

This post is part 2 in our series about the STAAR redesign. To learn about all of the changes, go back and read last week’s post. 

Today we are going to be chatting about our STAAR Question Bank and how you can use it in your math classroom. 

Using the Question Bank

Every classroom is different – different students, different learning styles, and different class lengths. We sought to create a resource that can be flexible for teachers to use based on their students’ needs. 

Recommendations:

• Gradually work through the questions together in class – Don’t assign students 20 STAAR new question types for homework. Testing can be stressful for students, and their confidence needs to be built not in isolation. 
• Allow questions to encourage discussion and helpful clarification – Assigning the question bank as an independent assignment can result in missed opportunities for students to ask about the functionality of the online assessment.
• Select and use the questions your students need the most – The resource can still be effective even if your students don’t work on every single problem. 

Classroom Routine Idea

I would recommend spending around 10 minutes on preparing students for the new question types by following this routine.

• Project a question and provide a paper copy to students (makes a great warm-up)
• Allow students a few minutes to work through the question on their own. Circulate and observe.
• Come back together to discuss solutions, strategies, and misconceptions. Provide feedback to students. 

The STAAR Question Bank is intended to look exactly like what they will see on the STAAR test, but it doesn’t have the technology functionality. Fortunately, TEA has provided a STAAR online practice test (just click the green “Sign In” button at the bottom of the screen). Choose a day before STAAR to allow students to take a practice test where they can play with the functionality of the new question types and you can circulate and answer questions.

Strategies and Support for New Question Types

This is not exhaustive, but here is a list of things I would point out to students about the new question types. 

1. Equation Editor 
   • Allow students time in class to look at examples of keypads and ask any questions about unfamiliar keys.
   • Ask questions such as – How would you handle a fraction? An equation or inequality? An exponent?
   • When a question requires a fraction or exponent, pull up an equation editor sampler question to look at how certain functions appear.
2. Graphing
   • In most instances, students need to be able to identify correct points and not actually create the line aspect of the graph themselves; so in your practice, focus on discrete points and being able to test (x, y) values that are true for their scenario.
   • After they have chosen points, ask – does the resulting line or shape make sense/seem reasonable? (Example: the slope should be negative but their points created a positive slope)
   • Students will need to carefully observe the scales of the x and y-axis before creating the graphs.
3. Number Line
   • Highlight the differences in the button options (open, filled, left, right) as the buttons might all look similar and details can be easy to overlook.
   • Remind students that there are two steps to creating a number line – selecting a button and then dragging the circle to the correct point on the number line. 
   • After the students create the number line, have them select a point in the solution set to check their answer.
4. Inline Choice
   • Encourage students to re-read their final selections as sometimes choosing an option mid sentence can interrupt their train of thought.
5. Hot Spot
   • Be sure they read to understand exactly what detail they are looking for (is it one point, two?).
   • These are like multiple choice questions, but instead of A, B, C, D, they are choosing points on the graph.
   • Remind students that they are often selecting more than one single point. 
6. Drag and Drop
   • Read to see if options can be used more than once.
   • Treat these like multiple choice questions and apply any same strategies – process of elimination or  “plug in” choices to see if true 
7. Match Table Grid
   • Take time first to read columns and understand what they are identifying, sorting, or classifying each statement by to provide guidance in how they read through each example
   • Highlight that an answer must be selected in each row and not to leave a row unmarked
8. Multiselect 
   • How many should I choose? Which are obviously incorrect? 

STAAR Test Prep Unit + Question Bank

The STAAR Redesign Question Bank serves as a gradual practice where kids are becoming familiar with the question types over time. In addition, they are benefitting from spiral review over the readiness standards. 

The test prep units, instead, provides a more intensive and immersive content review organized by unit concepts.

Teachers could combine the test prep unit together with the STAAR question bank. In both resources, there is more content than there are days to prepare, so teachers will need to prioritize what to focus on. But hey, that is what data is for! 

What questions do you have about the STAAR Redesign Question Bank?

The post Utilizing Our STAAR Question Bank appeared first on Maneuvering the Middle.

For Texas Teachers

If you teach middle school math in the great state of Texas, then this is for you! The STAAR (State of Texas Assessment of Academic Readiness) has made some pretty significant changes starting in the 2022-2023 school year which means your students will be taking a different test than in years prior. Let’s check out what the TEA STAAR redesign has in store for you.

This is probably not your first time to hear this information. I imagine you will be attending (or have already attended) some professional developments regarding STAAR changes, but this post will serve as a general synopsis of what the changes are and how Maneuvering the Middle will aid in your preparation. Come back next week where I will give more tips on how to use this new resource effectively with your students. 

Why is STAAR changing?

STAAR is changing to better align with how students are learning in the classroom. According to TEA, “Classroom practices that over-use multiple choice questions… can get small, short-term gains on STAAR, but evidence has shown they don’t lead to high performance or long-term mastery.”

What is Changing?

House Bill 3906 (first time I have reference a specific law on this blog, I believe) requires these changes to improve the STAAR (not all of the changes are listed):

• 75% multiple choice question cap
• Transition to 100% online testing

75% multiple choice question cap means there will be 25% (or more) free response type of questions that will be NEW question types. This means your students are moving away from filling in the griddable and will need exposure to the new question types. This will allow students more ways to show their understanding. 

• Equation editor – Students can write responses in the form of fractions, expressions, equations or inequalities
• Graphing – Students select points, draw lines, drag bar graphs and more to create different types of graphs
• Number Line – Students select a point, an open or closed circle, and a direction arrow to demonstrate a solution set on a number line
• Inline Choice – Students select the correct answer(s) from a drop-down menu
• Hot Spot – Students respond by selecting one or more specific areas of a graphic
• Drag and Drop – Students evaluate given options (words, numbers, symbols, etc) and chooses which response(s) to drag to a given area (a diagram, map, chart, etc)
• Match Table Grid – Students match statements or objects to different categories presented in a table grid
• Multiselect – Students can select more than one correct answer from a set of possible answers

If you are feeling a little anxious reading this list, you aren’t alone. Let’s take a collective deep breath together… ahhhhhhh. (Did you read that as screaming? No? Me neither.)

Source: TEA Website

Maneuvering the Middle’s STAAR Question Bank

Our curriculum team has been working hard behind the scenes to prepare a question bank with a variety of practice problems presented in the new question type formats.  Your students get exposure to the new question types while reviewing the year’s content all at once. Win – win! 

While reviewing the content is of primary importance to encourage students’ success in answering questions correctly, we believe practicing the new question type formats achieves two main goals:

1. To relieve students’ anxiety that inevitably comes when seeing something presented in a new way. 
2. To increase success on all question types – even multiple choice formats. Many of the new question type formats require deeper thinking that can translate to success on the multiple choice questions.

How did we create a Question Bank?

When creating this STAAR Question Bank, we researched and became familiar with the new question types based on the released materials and practice tests.

Every readiness standard was studied and their historical trends were analyzed. Based on our findings, we asked ourselves, “What question types would the standard naturally lend itself to?”

• Open-ended questions or multiple choice questions with a single value answer could be assessed as an equation editor on the new test
• Vocabulary concepts, comparisons, or procedures could be assessed as inline choice
• Sorting or classifying concepts might be assess as a match table grid
• Questions that lend themselves to multiple follow-up questions, observations or more than one correct representation/answer might easily be asked as multi-select 

Grab Your STAAR Question Bank Resource

Next week will be talking about HOW to use the STAAR Question Bank in your classroom, so be sure to come back!

Texas teachers, how are you feeling about these changes to the TEA STAAR redesign?

The post TEA STAAR Redesign appeared first on Maneuvering the Middle.

We are happy to share that we have the perfect FREE activities (that’s right, plural!) for your students! Download your free resources below and read our best tips on teaching circles. 

Tips for Teaching

Before I jump to the activity list, let me please share some tips for teaching the various intricacies regarding circles. 

Vocabulary and Formulas

• Vocabulary is key! I never tire of saying that. Circles have their unique set of terms (radius, diameter, circumference), so embrace it. Create arm movements, put them on your word wall, or create and reference an anchor chart that has all of the parts of a circle labeled. 
• Label, label, and then label some more! One criteria for success I had in my geometry units was to always label the shapes. If a circle is provided, label the parts! If a model with a diameter is given, require students to draw and label the radius before moving on to solve the problem. 
• d=2r and r=d/2 is something I made students write on every problem. Problems generally give diameter when asking for the area or radius when asking for the circumference, so the first step would be to find the measurements actually needed to solve. It helped bypass the mistake of just plugging in any ol’ number into the formula. 
• Pizzas are circles! The crust represents the circumference (which they both conveniently start with C) and the sauce is the area.
• Write the relevant formulas! Not only does this help students internalize the various formulas, but it also helps students practice substitution. 
• Be sure to highlight that the radius and diameter pass through the center of the circle. This will be an important distinction as students see different segments in a circle in high school geometry. 

More Tips

• Draw your shapes! If no model is provided, require your students to draw a circle and label the given parts. This helps contextualize the problem! 
• Consider purchasing sewing measuring tapes in addition to rulers for your classroom. They make measuring the circumference of a circle easier than using a string that will then have to be measured too. Not to mention, they are easier to store! You can also print your own paper rulers that do the exact same thing but will be limited by the length of a standard 8.5×11 piece of paper. Lastly, places like IKEA, HomeGoods, and TJ Maxx have long paper rulers too!
• Sir Cumference and the Dragon of Pi is a book that I have not personally read, but I have seen recommended by many math teachers. Middle schoolers are not too cool for story time.

Helpful Videos

What is Pi? – This video comprehensively explains pi and just about everything else regarding circles. I would fast forward to 3:20 for an excellent visualization of 3.14 diameters going around a circle’s circumference or you can watch the clip below.

Area of Circles – This video may be the best explanation I have ever seen of how the formula for area of a circle is derived. The visuals are not something you can recreate without the help of video editing, so I highly recommend showing it to your students before you cover calculating the area of circles.

Circumference of a Circle – This video explains circumference and solves a few circumference problems. 

Free Activities

Maneuvering the Middle Pi Day Activities – Choose from a variety of activities in this freebie, or use all 3 if you think that pi deserves more than just a day of celebration!

• Pi Day Discovery Activity – Students can work collaboratively to explore the relationship between the circumference and diameter of circles by measuring mini pies. 
• Pi Day Two Truths and a Pi – This play on two truths and a lie has students finding which statement is false about various measurements related to circles. 
• Pi Day Riddle Activity – This self-checking activity asks students to calculate circumference and area of circles. The riddle is a nice touch!

Make Bubbles

Grab a jumbo container of bubble solution, mix in food coloring, and allow students to blow a bubble on paper. Once the bubble pops, students measure the diameter and from there can calculate the area and circumference.

Go Outside and Run Idea

Have an especially active class? This idea is for you! Go outside and form a giant circle with your class. Choose two runners. One runner will be running the diameter of the circle while the other is running the circumference. The idea is to see how many times the diameter runner can go back and forth before the circumference runner makes it back to their original spot. Assuming the runners’ speeds are fairly similar, it should be a little more than 3 times. Repeat with different students to get all of the wiggles out.

Measure Circles

Circles are everywhere! Tops of tupperware lids, paper plates, old CDs, cups, and Hula hoops. Ask students to bring something circular from home to class and now you have 30 hands-on, practice problems for students to complete.

Need more Circle Practice?

• 7th Grade CCSS Plane Geometry and Similarity Unit
• 7th Grade Plane Geometry and Similarities Activity Bundle
• Circumference and Area of Circles Find Someone Who Activity
• Circumference and Area of Circles Digital Math Activities
• Circumference and Area of Circles 4 Unique Mazes
• Intro to Circles Exploration Activity

The post Pi Day Activities appeared first on Maneuvering the Middle.

We encounter math on a daily basis, but it can be a challenge for students to connect what they learn in class to the outside world. Here are 4 (update: 5!) ways teachers can engage students by making math relevant to their lives.

LISTEN ON: APPLE PODCAST | SPOTIFY

1. Share Your Enthusiasm

The key to modeling what it looks like to be a math person is to share your excitement! Excitement is contagious. In my classroom, I loved teaching ratios and proportions! As a class, students performed so high on the concepts. While I can’t be certain it was caused by my enthusiasm, it sure did help!

2. Promote Problem Solving

A second way to make math relevant is to promote problem solving in your classroom! (We talk about problem solving more in depth here.)

“When are we ever going to use this?” used to scare me. I used to think students were asking as a ‘gotcha,’ but I think most students genuinely want to know. They are curious how the unfamiliar concept connects to their life.  

So, I love to set the stage at the beginning of the year. I say that math is all about learning to problem solve. Problem solving is a skill that is limitless in where it will take you. Regardless of your future profession, you will be required to problem solve, and persevering through a math problem gives you the confidence and grit to do so in the real world.  

Make this a mantra in your class and continue to reinforce it all year long.

3.  Tailor Curriculum to Students’ Interests

This study shares that when the curriculum (in this case, Algebra 1) is personalized to students’ interests, they are more successful.

“In the study, half of the students chose one of several categories that interested them — things like music, movies, sports, social media — and were given an algebra curriculum based on those topics.  The other half received no interest-based personalization… Walkington found that students who had received interest-based personalization mastered concepts faster.” 

While changing your entire curriculum and/or rewriting problems may not be something you can realistically manage, consider Walkington’s approach. “We picked out the students who seemed to be struggling the most in Algebra I, and we found that for this sub-group of students that were way behind, the personalization was more effective.”

So this may be something that you consider as you write future problems or consider future projects. What are your struggling students genuinely interested in? How can you include that in your math class? Can your classroom economy be related to an interest? What about the names of your groups? 

One quick win comes to mind. When I was teaching small groups, I had 3 students who needed a little extra incentive to stay engaged. They loved soccer, so we made everything soccer related. As they got problems correct, they scored “goals,” counters were soccer balls, and all word problems were changed on the spot to be soccer themed

4.  Teach Students to Ask the Questions

In the book Quality Questioning, the author breaks down the importance of the questions we ask in the classroom and the responses we accept from our students.  One of the key things they mention is teaching students how to ask questions on their own and providing them the opportunity to do so.  

This easy lift is a great way to engage students. 

• It extends students’ thinking
• Makes for great math discourse
• Any student can participate
• Allows for students to flex their creativity muscles
• Students make interesting connections

There are a few options here. 

1. When presenting a word problem, cover up the question. Typically, a world problem gives information and then asks a question. Instead cover up that last question, and ask students to come up with a question. 
2. Put up a graph, a table, a picture of a price at the grocery store, a receipt, whatever you can find that has some numbers of it, and ask students: “What could the question be?”

Try it with your class at the beginning of your rates or proportionality unit. What could the question be? And then ask again at the end of the unit and see what your students have learned. 

Another idea could be to simply take your receipt from your latest gas purchase, project it, and ask students: what could the question be? 

5. Include Projects to Your Scope and Sequence

Project Based Learning is popular for a reason! Students take more ownership in their learning, and experience first hand how math can help solve real-world problems. Maneuvering the Middle’s projects are perfect for this. 

When students ask, “When will I ever use this?” then it may be time to start a project. Here is a snippet of what our projects ask students to solve:

RATIONAL NUMBERS + LINEAR RELATIONSHIPS

• 6th graders research and calculate the costs of flying or driving to various destinations. Grab it here.
• 7th graders will calculate the cost of traveling to various National Parks and calculate the percent change in park and gas prices. Grab it here.
• 8th graders will plan a vacation and apply discount options to their vacation expenses to explore the effect on the linear relationship. Grab it here.
• Algebra 1 students will use and represent linear relationships to help them plan a vacation on a budget. Grab it here.

FINANCIAL LITERACY

• 6th graders plan a career fair and compare the lifetime earnings of various careers. Get it here.
• 7th graders calculate household incomes and analyze the best cities to live in based on earnings. Get it here.
• 8th graders calculate and plan saving for college. Get it here.
• Algebra 1 students find and use an exponential function to predict the rising cost of college. Get it here.

What are some of the ways you make math relevant to your students?

The post 4 Ways to Make Math Relevant appeared first on Maneuvering the Middle.

Factoring Trinomials When A>1

I hesitated to put a>1 in this blog post since it isn’t necessarily a special case, but I considered that you probably would teach this AFTER you teach factoring trinomials when a=1.

Let’s dive in! 

Double check that there isn’t a GCF that can be factored out the a, b, c terms. 

The AC method is a great way to teach your students how to factor. Explaining this method through text is hard to follow (I tried), so again, I am going to pass it to Sara to teach you how to factor trinomials when a>1.

Insert video here

If you love this video, then check out All Access, our curriculum membership that includes ~3 videos for every single lesson! She is also using the Student Handouts that are available in our Factoring Polynomials Unit.

Using the Box Method

The box method is excellent, and if you taught it when factoring trinomials when a=1, then there isn’t much new to cover. This video shows how to do this, but I also included a few graphics for you to reference. 

Factoring with Difference of Squares 

I love difference of squares! We like to start by explaining how difference of squares exists. Let’s take x^2 -16. You can still ask your students, “What multiplies to -16 that also adds to 0?” Remember that b=0 in a difference of squares polynomial.

Forgive my repetitiveness, but remember that we still have to check to see if there is a Greatest Common Factor!

When I taught Algebra 2, we did Around the World or Head to Head Challenges using squares and square roots. I wanted students to internalize squares and square roots (for a multitude of reasons) but it served this skill very well. You may consider putting up an anchor chart so students have a visual.

Perfect Square Trinomials

By the time you are teaching perfect square trinomials, it is likely that students may have already factored a few perfect square trinomials. In fact, you don’t actually have to teach perfect square trinomials as a special case – it is helpful for students to recognize patterns, absolutely! Students can still factor these trinomials using the methods already taught!

Teaching students to recognize the form is helpful and will increase their proficiency in taking the square root of numbers. 

How do you teach factoring trinomials and polynomials?

The post Factoring Polynomials with Special Cases appeared first on Maneuvering the Middle.

This makes factoring polynomials so important for students to conquer. As a student, I remember only learning to guess and check, and lucky for us now, there are better ways!

Introducing Factoring with Algebra Tiles

I learned this method from Making Math Moments. You can watch the full video here or read on. Before even introducing factoring polynomials, ask your students to represent a trinomial using Algebra tiles like this one below: 

Before proceeding, make sure your students understand that the area of the x tile is x*1. 

Then ask your students to arrange the tiles in a rectangle using all of the pieces. Let them solve this puzzle – they may leave pieces out or create a shape that isn’t a rectangle.

Then ask them to figure out what the dimensions (the length and the width) of the rectangle would be. You haven’t explicitly taught factoring at this point. You are letting students explore and come up with their own patterns using hands-on practice. Here is the solution:

Maneuvering the Middle’s Factoring Polynomials Modeling Activity is the perfect way to introduce or practice this skill. You can find it in our Factoring Polynomials Activity Bundle.

Start with the Greatest Common Factor

Students won’t get far without mastering factoring out the GCF. 

Start by teaching students to write all of the factors of each term and circle/highlight the factors that they each have in common. Then whichever factors are not in common, will be the terms that remain.

This idea is from our curriculum writer, Reagan. “When my kids struggled with factoring, it was usually because they didn’t have multiplication facts memorized.  When students struggled with this, we taught them to use their calculator to type y = #/x and look at the table to easily see all the factors.”

Factoring Trinomials When A=1

Keeping students’ work organized is key when teaching this skill, which is why I highly recommend showing students to organize their work using a sum and factors table. In the example above, you can see that the b term is the sum of 4 and 6 while simultaneously, the c term is the product of 4 and 6. Simply put, I would ask my students when trying to factor: 

• What multiplies to c (24) that also adds up to b (10)?

If we couldn’t think of it off the top of our heads, we would make a table (which is especially helpful when you have a  negative b or negative c term.)

Watch the video to see how Sara teaches this:

If you love this video, check out All Access for more student videos!

Using The Box Method

The box method is also a great tool to familiarize your students with factoring. The same thinking of “what are the factors of c that will also add to b?” is required, but your students will practice more with finding the GCF of two terms which will set them up nicely when a>1. This silent video explains how to do this quite well.

You can also follow the visuals here:

If you are looking for some additional resources to use with your students, then check out the links below:

• CCSS Factoring Polynomials Unit
• TEKS Factoring Polynomials Unit
• Factoring Polynomials GCF Digital Activity
• Factoring Polynomials a=1 Digital Activity
• Factoring Polynomials Activity Bundle
• Factoring Polynomials Digital Activity Bundle

Make sure to come back next week when we talk about Factoring When a>1 and Difference of Squares and Perfect Square Trinomials.

How do you teach factoring trinomials?

The post Teaching Factoring Trinomials appeared first on Maneuvering the Middle.

Today I will share some tips for teaching the complexities of this grid and some engaging activities that you and your students will love.

Three Tips for Teaching

The most common misconception you will see is students moving up and down on the y-axis before moving right or left on the x-axis. There are many memory tricks like “you have to crawl before you can climb” or “you have to cross the street before you can get on the elevator” to help students plan their steps. Because graphing on the coordinate plane doesn’t require “showing work” like setting up a proportion, I have no problem asking students to annotate the coordinates. I will model and require students to write a tiny right or left arrow over the x-coordinate and a tiny up or down arrow over the y-coordinate every single time they encounter a set of coordinates. I also ask students to label their graphs with “x-axis” and “y-axis.”

Don’t overestimate students! It can be easy to think your 6th graders can graph on all 4 quadrants on day 1. Start by just graphing in Quadrant I on the first day. Then move to graphing on all 4 quadrants the second day. By the third day, you will be more successful graphing rational numbers.

Graphing on the axes can be particularly challenging. When I saw students mix up coordinates, it was usually because one of the coordinates was 0. Tip: remind students that if there is a 0 for that coordinate, then it won’t show up on that axis. For example, (0, 4) means that it cannot end up on the x-axis because it has a 0 for the x-coordinate. It has to end up in the y-axis. 

Coordinate Plane Activities

Coordinate Plane Unit – This 6th grade unit does an excellent job scaffolding instruction. Plus, student handouts, homework, a study guide, and an assessment are done for you!

Demos Coordinate Plane Activities – Desmos really delivers on this skill. I’ve linked the entire scope of their coordinate plane practice, but this Mini Golf Marble Slide is especially useful in plotting points, while incorporating error analysis. Battleboats is a play on Battleship, and I can visualize the engagement!

Coordinate Plane Digital Activities – Do you need practice for all of the 6th grade CCSS coordinate plane standards – introducing the parts of the graph, graphing, reflections, and distance? These digital activities cover everything, come with a 2-question exit ticket per skill, and include 16 total activities.

Stock the Shelves Online Game – I came across this website from a member of our Maneuvering the Middle VIPs facebook group. This is online practice for graphing integers on all 4 quadrants. I like that if a student is incorrect, they have to keep trying before they can move on to the next problem.

Design a Dorm Room Performance Task – This Coordinate Plane Performance Task can be a typical assignment, group project, or an extension.  It requires students to think outside of the box as they solve real-word and mathematical problems by graphing on the coordinate plane. 

Create a Coordinate Plane on your Floor – Our MTM team member and current teacher Marissa is a big fan of this activity. She pushes all of the student desks to the edge of her classroom, and uses painters’ tape to create a giant coordinate plane. Students are then asked to walk the graph as they would plot points (start at the origin, walk the x-axis and then move vertically along the y-axis). In this activity, she has students practice reflecting over the axes using a partner to represent their reflection. 

Coordinate Plane Scavenger Hunt – Students will move around a coordinate plane map of the city using the clues provided at each station. Hands-on and interactive!

Coordinate Plane Battleship – Students just need coordinate planes. Have them mark various points as their battleships. You call out various coordinates. Students say ‘hit’ when you say a coordinate that has one of their battleships on it. The student that still has a battleship at the end of the activity wins!

Flyswatter Game – The flyswatter game was perfect when I ended a lesson early or if I needed to inject some energy into my morning classes. Simply provide 2 students a flyswatter and give them locations to smack like:

• The origin
• The y-axis
• The x-axis
• The various quadrants
• (5, -3) and more

What coordinate plane activities do your students love?

The post Coordinate Plane Activities to Try appeared first on Maneuvering the Middle.

Math Activities to Survive December

Business as Usual – Perhaps, you have timed it perfectly; the end of the unit or your midterm falls on the last day before the break. Bravo! You are a planning wizard. 

1. Projects

If you haven’t heard, Maneuvering the Middle launched projects for 6th, 7th, 8th, and Algebra 1! These projects are flexible in nature and can span 3-7 days. They are purposeful, standards-based and make a great alternative to an assessment.  

RATIONAL NUMBERS + LINEAR RELATIONSHIPS

• 6th graders research and calculate the costs of flying or driving to various destinations. Grab it here.
• 7th graders will calculate the cost of traveling to various National Parks and calculate the percent change in park and gas prices. Grab it here.
• 8th graders will plan a vacation and apply discount options to their vacation expenses to explore the effect on the linear relationship. Grab it here.
• Algebra 1 students will use and represent linear relationships to help them plan a vacation on a budget. Grab it here.

FINANCIAL LITERACY

• 6th graders plan a career fair and compare the lifetime earnings of various careers. Get it here.
• 7th graders calculate household incomes and analyze best cities to live in based on earnings. Get it here.
• 8th graders calculate and plan saving for college. Get it here.
• Algebra 1 students find and use an exponential function to predict the rising cost of college. Get it here.

2. Winter Solve and Color Freebie

If you need a day or two for your students to complete something calmly, but also keep it math related, check out our winter solve and color freebie.

Click here to get it!

• 6th – Ratio Application
• 7th – Proportional Relationships
• 8th – Non-Proportional Relationships
• Algebra 1 – Writing Linear Functions

These concepts are typically taught in the Fall and will be great for review.

*** Fun tip: Project a fireplace, turn on some tunes, provide some colored pencils, and walk around blissfully as the fire hypnotizes your students into a peaceful calm. 

3. Cookie + Dessert Recipe

This is a project that requires little planning and has a festive energy, but also engages students in a unique way. You can also give students a choice! Print off a variety of dessert recipes and allow students to pick what they would like to “make.”

Then students have to determine how much they need to make (ex: enough for the class, enough for the whole school, enough for the staff). Students have to use rational operations to calculate how much of each ingredient they will need.

You can go even further and have them shop for all of the ingredients using the various curbside or delivery options available. Texan here to recommend HEB.com.

4. Shop for Friends and Family

Let students go shopping! Give students a budget for how much they can spend to buy gifts for their friends and family. Give students a specific website to stay on like target.com and have them record their total. To spice it up, provide coupons and BOGO opportunities.

5. Jigsaw to Review

If you are preparing for a midterm, then a jigsaw may be right for your students! If you aren’t familiar with a jigsaw, essentially students become an expert in a specific math skill and then come back together with students who became experts on other math skills. Then, they teach each other their respective skills. If you need more ideas for test review, check out this test review post.

Just For Fun

While these are not December activities that are math specific, there are sometimes opportunities to do something winter themed and festive.

6. Gingerbread Houses

At my last school, we used the last few days of the semester to do celebratory type activities. I would ask students to save their milk cartons (they act as the base) from lunch for the week leading up to the activity, and have  students bring in various candies, frosting, and graham crackers. I would use butcher paper to cover my tables, we would decorate for an hour and then students would take their gingerbread houses home. 

Tip: You need plastic knives for spreading frosting, paper plates as a base, and gallon-sized plastic bags to transport.

7. Holiday Cards

This is the perfect activity for your class after a midterm or on an adjusted bell schedule class period. Students can make holiday cards for custodial staff, cafeteria workers, or administrators. You could also adopt a nearby assisted living facility! 

8. Make Snowflakes

I used this activity for our adjusted bell schedule (30 minute classes) after our benchmarks. Students needed a brain break, and I needed something for them to do. This is the video I used to make colossal snowflakes that were hung in the gym for the winter dance. 

No matter what you choose to do to make it the end of the semester this December, remember that students are jazzed for an upcoming break and to use that energy to create excitement for math. What December math activities are you planning on implementing?

The post Math Activities to Survive December appeared first on Maneuvering the Middle.

Make Percents Visual

Percents are perfect for visual representation! There is no better way to do this than using a tape diagram. In fact, if you teach your students to set up a part/whole = %/100 proportion, then a tape diagram is just a picture of that exact proportion . The best part of the tape diagram visual is that students instinctively are better able to estimate answers based on using benchmark fractions like 50%. This modeling percent activity is perfect for practicing those tape diagrams!

This Interactive Fraction Model is an awesome tool for exploration! When the numerator and the denominator are manipulated, the model, decimal, and percent are changed. You can also limit the number of denominator options or adjust how the model looks. 

Some questions and ideas for your class: 

• What do you notice about the percent when the numerator is greater than the denominator?
• What fractions give you repeating decimals?
• Give students a percent, and ask them to discover the equivalent fraction.

Bring a king size Hershey bar to class.  Discuss what size it is and then draw the connection to a percent bar (aka tape diagram).  Ask students to describe how many rectangles would be one-half, one-fourth, etc. Then connect this to 50%, 80%, etc.

What percent of Manhattan is Central Park? As an adult, I estimate percents more than I actually calculate or find percents, so I thought this lesson was a great way to get students to begin estimating percents. The video introduces the question, “What percent of Manhattan is made up of Central Park?” Students will have to discuss and ask questions about what information that they still need in order to solve. 

Percent Application

Percents are everywhere. Introducing percents was met with a lot of buy-in from students; they have some prior experience with percents in their grades or at the store. Let’s capitalize on that interest by using as many real-world opportunities as possible. 

Financial Literacy + Percent Project – This project is a homerun! Students are tasked with helping their “clients” who work remotely find the best place to live based on their income, financial goals, and lifestyle. Students will use finding percents to make decisions about the best use of their client’s income. 

Posing Percent Problems on Desmos – “Students apply what they’ve learned about increasing and decreasing by a percentage to generate and answer questions about the society in which we live.” This activity is more paper-based than what we traditionally see from Desmos, but it puts the responsibility on the student to determine the question. For example, a student is provided facts about wage gaps, and then asked to come up with the question and the answer based on that wage gap fact. There is opportunity for some interesting responses.

Percent of Change Class Demonstration – Check out what one of our teachers had to say about this activity:

“Today’s lesson is always one of my absolute favorites. We do the shopping activity for percent change…It not only gives the kids a bunch of practice with percent change, but gets them asking so many big questions! “Why would magazines get so expensive? Don’t they want us to read and learn?” “Why aren’t coffee prices higher? Are the coffee workers making enough money?”

General Tips

There are several ways to determine the percent of a number, including an equation, a proportion, or a tape diagram (percent bar).  I suggest teaching all of the methods to see what clicks the best with your students. Additionally, I would assign some problems to only set up the problem without solving. 

Consider using a similar problem and similar information to show students how to set up problems and solve for different parts of the equation/proportion.

• Example 1: A carnival is made up of 80 booths. Of those 80 booths, 14 sell food. What percentage of the booths sell food? (finding the percent)
• Example 2: At a second carnival, food booths make up 40% of the total booths. There are 16 food booths. How many total booths are at this carnival? (finding the total)
• Example: At a third carnival, there are 75 booths. 30% are food booths. How many food booths are there? (find the part)

When annotating word problems, I would require my students to write a fraction bar with 100 under any percent that they came across. This reminded students that we were going to use 100 in our solving.

FDP Everyday – Students calculate the number of days they have been at school over a total (it could be the month, number of days until the next break, or total number of school days there are) and then calculate the decimal and percent.  It’s a perfect warm-up or extension activity. 

If you have a Flocabulary Account, your students will enjoy this Percent rap! 

What finding percents activities do your students enjoy? 

The post Finding Percents Activities appeared first on Maneuvering the Middle.

What is/are the Order of Operations?

Order of Operations is the rule provided to standardize how to solve any problem involving multiple operations.

You have probably seen popular memes like the one below where you can find a spirited debate in the comments. In fact, showing this to students would be a great hook! “By the end of today, you will be able to tell these people what is ACTUALLY the correct answer!” (Though there are arguments for both!)

Start Simple by Scaffolding

Like most (all?) math skills, order of operations benefits from scaffolding. It is actually the perfect skill to scaffold – you are in direct control of how many steps are included in each problem! Start with a two-step problem, move to a three-step problem, then continue until your students have mastered multiple-step problems. 

Depending on the grade level you are teaching, you will either be working with whole numbers, possibly integers, and finally rational numbers. If you teach 7th grade, then you will be covering everything! Start with whole numbers before moving to problems that include integers and rational numbers.

I love this 6th grade Maneuvering the Middle Student Handout 2 problem. Each problem is just asking for what the first step would be. Brilliant!

Showing Your Work Tips

I can’t think of a math skill where students want to try to do it all in their heads more than order of operations. Not rewriting the problem after each step makes students prone to error. Here are three ideas that make showing work fundatory (fun + mandatory):

1. Don’t ask for the final answer.  Instead, ask what would the problem look like after they “insert an operation”
2. Tell students that you aren’t grading their final answer at all – just their work! This will make students who are okay with losing credit for not showing their work think again since they won’t even get credit for their correct answer. 
3. Try a round table activity! A group of 2-4 students will work on a problem together rotating through the problem after each step. Teamwork requires every step to be written down.

Make it Visual

As you can see, my PEMDAS poster lived on my wall year around. And you can have it for free! Click here to get your free PEMDAS poster. 

Encourage students to write down the acronym vertically down their page and cross off the letter/operations as they solved the problem. Students can be enormously successful using this strategy! When I modeled order of operations problems, I highlighted or underlined the step I was completing. This helps students too!

Try These Activities

All of these activities were a hit in my classroom! Students really can’t practice a skill too much.

• Order of Operations Solve and Color – This was a perfect extension activity.
• Order of Operations Scavenger Hunt – Scavenger hunts were a favorite in my classroom since they are self-checking and allow students to move around.
• Order of Operations Digital Math Activity – Perfect for an in-class assignment while I worked with small groups.

How do you teach order of operations?

The post Mastering Order of Operations appeared first on Maneuvering the Middle.

Does this sound familiar? You just sit down with your small group when you see a raised hand. You vacate your small group table to check on this student when they point to their first completed problem and ask, “Is this right?” You can’t fault this student for wanting feedback, but there has got to be a better way!

Benefits of Self-Checking

Self-checking procedures and activities will give students the autonomy to take control of their learning. Not only are they taking more ownership of their math understanding, but they are now able to work at their own pace. (Reminder to provide extension opportunities!) They are able to get the feedback they need to move on.

Like in the example above, students just want to know if they are doing it right. Who wants to complete an entire worksheet of problems to find out at the end that they had made a critical error over and over again that could have been corrected after the first problem? On the other hand, there are also students who won’t ask if they are solving correctly, so they could end up in the same boat of making a repeated mistake too.

For the teacher, giving students the ability to check their own work in real-time, gives you the opportunity to circulate to look for common misconceptions, facilitate stations, or host a small group. 

Simple Ideas for Self-Checking

These are a non-exhaustive list of self-checking ideas that I have tried in my own classroom. They are all very tweakable!  Find out what works best for you and your students. 

• Mixed answer key – This was my go to! I would simply project or write the answers to all of the problems in a random order on the board. Students would solve a problem and look up on the board to make sure that they saw their answer somewhere. To be extra clear, these answers were not numbered with the problem number. 
• Formative – We have a blog post from 2020 all about Formative that I recommend taking a look at. This technology is gold! You can post an assignment and students can submit answers and receive immediate feedback. Win-win!
• Hidden answer key – I would create an exemplar answer key with all of the work displayed, make copies of it onto a bright color cardstock, and then put the copies into a folder that each table group shared. After every 2 or so problems, students were able to check their work and answers against my answer key. I found that the answer keys being placed in the folder prevented students from being tempted to copy while also giving them the opportunity to check their work against my work. That way if the student did miss a problem, they were able to look at my work to figure out their mistake which was perfect before a test.
• Odd answers Only – Similar to textbooks, I would just write the answers to odd problems on the worksheet right next to the corresponding question.
• Post the answer key to Google Classroom or Schoology for them to check after a certain timeframe. 
• Mazes or Scavenger Hunts – These activities are self-checking by nature. Scavenger hunts are a student favorite! And we have this incredible self-checking Corn Maze Activity freebie.

Best Practices for Self-Checking

Before you begin implementing this routine, it is important to explain the why to your students. If you can instill the purpose of self-checking their work (see any of the benefits above), students will be less likely to take advantage. 

Remind students that they won’t have the answer keys on assessments or in the real-world, and that the work that they do in class prepares them for both of those times. 

Lastly, give students steps for what to do when they get the answer wrong. Do they move on? Rework the problem? Get help from a neighbor? Mark it wrong? That will be up to you and depend on the assignment. 

One last reminder is that a self-checking routine doesn’t replace teaching students what to do if they do not know how to solve a problem. One of the downsides of providing a self-check is for students who need help to not ask for help or that they just write down the answer. This How to Get Help Flowchart freebie is a great jumping off point. 

Do you have a self-checking routine in your classroom? What self-checking activities have you tried?

The post Self-Checking Activities in Math appeared first on Maneuvering the Middle.

We are so excited to announce our amazing middle school math projects for 6th, 7th, 8th grade math and Algebra 1 are complete and ready for your classroom! 

Benefits of Implementing Projects in Math

There are numerous benefits to students working through the open-ended nature of projects in math. This list is non-exhaustive, and I would love to hear additional benefits in the comments below. 

You get to see different students shine – This is exciting to see! The students who may not excel in a traditional math setting can truly step up to really surprise you when given the opportunity to complete a challenge creatively. 

Students are able to use different parts of their brain – Projects, by nature, involve synthesizing and analyzing information in a way that does not always happen in a standard instructional setting. Not only that, but students will practice skills like orally presenting their findings or displaying their work in a visually appealing way.

Students problem solving in real-world scenarios – When students ask, “When will I ever use this?” then it may be time to start a project. Here is a snippet of what our projects ask students to solve:

Rational Numbers + Linear Relationships

• 6th graders research and calculate the costs of flying or driving to various destinations. Grab it here.
• 7th graders will calculate the cost of traveling to various National Parks and calculate the percent change in park and gas prices. Grab it here.
• 8th graders will plan a vacation and apply discount options to their vacation expenses to explore the effect on the linear relationship. Grab it here.
• Algebra 1 students will use and represent linear relationships to help them plan a vacation on a budget. Grab it here.

Financial Literacy

• 6th graders plan a career fair and compare the lifetime earnings of various careers. Get it here.
• 7th graders calculate household incomes and analyze best cities to live in based on earnings. Get it here.
• 8th graders calculate and plan saving for college. Get it here.
• Algebra 1 students find and use an exponential function to predict the rising cost of college. Get it here.

All of our projects are jam packed with everything you need to implement a project from start to finish.

Projects are an alternative to tests – Students take so many tests in a year. Why not replace an assessment with a project that does the same thing?

Projects are great for that interim time –  Do you have 4 days before winter break that you aren’t sure what to do with? Is Thanksgiving Break coming up, and starting a new unit doesn’t make sense? Projects are your solution!

Projects help create a positive classroom environment – You can read more about how to build a classroom environment conducive to projects here, but in my personal experience, my classroom felt more joyful when projects were taking place.

Have you implemented projects in your classroom? What benefits have you and your students experienced by working on projects in your classroom?

The post The Benefits of Math Projects appeared first on Maneuvering the Middle.

Model by Using Algebra Tiles

Are you surprised I am starting with Algebra tiles? They are foundational for concrete understanding! You can learn more about using Algebra tiles in your classroom by grabbing our free Getting Started with Algebra Tiles guide, which can be found by checking out our post on Solving Equations. This Modeling Equations with Variables on Both Sides activity is a great way to practice solving equations with Algebra tiles. You just need some Algebra tiles (or your students can draw them!).

Student Teacher

Put students into pairs and show an equation on the board. Have one student instruct the other on how to solve as the student listening writes each step and solution. Then, show a new equation and have students switch roles. This gives students a chance to teach and reinforce what they remember about linear equations. I love this activity because it is simple and it makes every student explain their thinking. You, as the teacher, can circulate listening to each pair.

Round Table

Give students individual white boards and have them work in teams of 2-4.  With one equation written on the board, the first person will solve step one.  The second person will complete the second step in solving and the third will complete the next step. Keep rotating until the problem is solved and the last person checks the solution.  Have groups hold up their boards when they are finished.  If you want something like this, we have this Solving Equations with Variables on Both Sides Round Table available! 

Board Races

After students have had time to practice, implement “Board Races.” Two students will come up to the board and race to solve an equation shown on the board. The person who solves it correctly first stays up at the board for the next equation with a new competitor. I like to have the students who aren’t at the board working the equations on notebook paper to help check the solutions. An element of competition makes repetitive practice more fun! For race type activities, I like to have teams compete (boys v. girls or one side of the room v. the other side of the room).

Digital Activities 

I love our digital activities! This linear equations set of digital activities includes simplifying expressions, solving one and two-step equations, solving multi-step equations, and equations with variables on both sides, making it the perfect review before a linear equations test. 

What are some fun ways that students practice solving linear equations in your class? You can check out how to turn any worksheet into an activity which can easily be used for linear equations too!

The post Linear Equations Activity Ideas appeared first on Maneuvering the Middle.

Sidenote: I have to admit that when I first taught the distributive property, I focused solely on the procedure. I had students draw two arrows from the term outside of the parentheses to the two terms inside and called it a day. This is how I was taught. I tell you this to remind you that there are more ways to teach a skill than the way you may have learned it. 

Let’s get to it!

Introducing the Distributive Property

By the time you are teaching the distributive property, students are familiar with order of operations. If you give them a problem like 3(8+2), they will jump at the chance to solve. Students will add 8+2 to get 3(10) and then multiply to get 30. 

You can have students discover that the distributive property allows for another way to solve.  3(8+2) = 3(8) + 3(2) = 24+6 = 30. Proving to students that a property works instead of just telling them a property works will always earn a thumbs up from me.

Introduce Variables

Students may be wondering, “Why can’t I just keep following the order of operations? Why do I need to use the distributive property?” This is a fair question, and a perfect opportunity to introduce the distributive property using variables.

Let’s look at: 3(2x+4).

Ask students if 2x+4 can be combined or added together. No, they do not have the same variable.

Combining like terms only applies to addition and subtraction. You can multiply and divide terms that do not have the same variable or exponent, so you can use the distributive property to simplify this expression further. 

3(2x+4) is like having 3 groups of (2x+4). I encourage you to use Algebra tiles as every possible opportunity, so here is a beautiful visual.

Use the CRA Method

Just like I explained in the previous post, Simplifying Expressions by Combining Like Terms, the Concrete, Representational, and Abstract Framework will help your students develop a solid understanding of the distributive property. 

Area Models with the Distributive Property

As you can see in the “representation” column above, teaching and requiring students to use an area model for distribution, especially when you are teaching the distributive property in Algebra 1 can be extremely helpful. An area model will set them up for success when they are multiplying polynomials and factoring trinomials. 

Distributing with a Negative

The most common error you will see regarding the distributive property will be related to signs. Those sneaky little negatives can get lost pretty easily. I have some ideas that I have not used in my classroom (in full transparency), but came to mind when writing this blog post. 

Since multiplying by a negative, always results in the opposite, you can teach students that if there is a negative outside the parentheses, then it will always change the inside signs to the opposite. (Reviewing integer rules will reinforce this, but sometimes economy of language wins). 

I actually think that this is best introduced using a visual. Here is how I would show it and ask students to show their work using this method too.

Other Tips

There are so many ways to introduce this topic! Here is another idea from our Student Handouts below.

 Saying “I want one drink, 2 slices of pizza, and one ice cream cone” 4 separate times isn’t efficient.  “I want 4 drinks, 8 slices of pizza, and 4 ice cream cones” makes a lot more sense.

Angie, our amazing editor (and so much more), found this test problem on a New York 7th grade state assessment. This problem could definitely benefit from sketching a square and labeling each side. This would be an excellent problem to talk about why the wrong answers are wrong.

That about covers it! How do you teach the distributive property?

The post The Distributive Property appeared first on Maneuvering the Middle.

Vertical Alignment

As always, I like to take a look at the standards. Simplifying expressions starts as early as 6th grade in some states and continues to be used through… all future math.

In my experience, students need a thorough review of the basics of simplifying expressions each year, so these tips are for 6th grade teachers, 7th grade teachers, 8th grade teachers, Algebra teachers, and so forth.

Ideas for Hooks

As you introduce or review simplifying expressions, grab your students’ attention with a hook. Help them relate to the content with a real word example! Here are a few ideas for simplifying expressions hooks that I have used or have heard other teachers use.

• Watch this MadTV clip with your students. Ask them – what would be an easier way to communicate his order? 
• Ask students to group like items. Display a picture with different objects and ask how they would group them. You can see an example of this on one of our 7th grade student handouts below. Connecting this to ordering food in the drive-thru is a real-life example that students can probably connect. The drinks are usually grouped in a drink carrier, the fries are all in one bag, and the chicken nuggets are typically in another bag. Combining like items is all around us!

• Write down various animals on notecards (1 animal per notecard). Give each student a notecard and ask them to group themselves accordingly. Don’t give them any additional instructions. After they make groups, ask them why they chose their groups. Ideally, students will have found “like” animals, giving you a perfect segway into like terms.

Instructional Ideas

Anchor Chart  – The basis of simplifying expressions is thoroughly understanding that a “like term” is a term with the same base (or variable) and the same exponent (or power) and that combining like terms is adding and subtracting them. Because this is so important, I recommend creating an anchor chart with some examples and non-examples of like terms. In addition, there are lots of vocabulary terms that will be sticking around for a long time – coefficient, variable, and term – so be sure to have those words and definitions posted for easy reference. 

Use the CRA Framework – It’s no secret that we are fans of the CRA framework (you can see the many posts we have written here and here), and simplifying expressions by combining like terms is another perfect skill that fits into this method. 

• Concrete – Use Algebra tiles to group like terms. You can learn more about how to use Algebra tiles in our Getting Started with Algebra Tiles Guide. You can grab it here. 
• Representational – Students can draw shapes or tallies to represent the terms. 

• Abstract – When students are ready, they can combine like terms using the rules that they now have conceptual understanding of.

Simplify the Expressions by Annotation

Once students are working in the abstract part of the framework, you will want students to use a method for grouping to keep their work organized and create barriers for simple errors. Here are some popular ideas:

• Underline or draw various shapes around like terms including the sign right before the term
• Highlight using different colors for different like terms
• Rewrite on different colored sticky notes and then rearrange

• Use a T-chart. Personally, this is what I used with my students since I found that sometimes the shapes could make the signs hard to read.

Other Tips

• You can have students put an exponent of 1 on terms like x.
• Physically show your students the difference between 2x and x^2 using algebra tiles. 
• Easy activity – write a bunch of different terms on your whiteboard and play the flyswatter game! Say a term, and students have to hit a like term. 
• Integer fluency is pretty key when combining like terms. If it has been a while since they have practiced, go over some of the basics before jumping into combining like terms.

Simplifying expressions is truly the foundation of algebra – you have to simplify before you can solve equations. Make sure to come back next week where we talk about simplifying expressions with the distributive property. 

The post Tips for Teaching Simplifying Expressions appeared first on Maneuvering the Middle.

Vertical Alignment

Exponent Tips

It is important with both exponents and scientific notation that students understand that they show a different way to represent a value.

Before even showing an exponent, start by showing the expanded form like  7*7*7*7*7. You can start by asking students:

• Is this the most efficient way to write this?
• Would 7*5 give you the same result? Why or why not?

Tip: I have learned the hard way to NEVER use 2^2 in any of your early examples because it will just confuse students into thinking you multiply the base and exponent.

Easy, no-plan activity idea: A fun way for students to practice exponents is by using concentric circles. The inside circle is the base and the outside circle is the exponent. Assign students numbers 1-10 and have them rotate to a new partner each round. Students pair up, write down the exponent form, the expanded form, and then calculate the standard form. Keep rotating until your time is up. 

Laws of Exponents

The laws of exponents are so fun! I love how students can build on their previous knowledge to come up with the laws themselves. For example: 

On a Facebook thread, I recently saw a teacher say, “When in doubt, expand it out.” If a student forgets a law, all they need to do is expand it and calculate to discover the law again. That is something a student is more likely to do if you are modeling it consistently.

Because the laws are so accessible, this content really shines as a discovery-based lesson. Your students could also participate in a jigsaw. Each group becomes experts at their assigned law, then they present the law, the proof, and examples to their peers. 

If you go the traditional teaching route, I recommend splitting this skill up over at least 2 days. Maneuvering the Middle 8th grade curriculum covers multiplying/dividing like bases, power to power, and product to power on day one. Negative and zero exponents are covered on day two.

I highly recommend an anchor chart with all of the laws for easy reference. Sometimes in my last class on a Friday, my brain needed to look at an anchor chart to give it the boost it needed (and I am the teacher).

Scientific Notation

Like I said before, scientific notation is just a different way to represent a value. Here is a great way to introduce why we might use scientific notation. Write down the mass of Earth and Mars on your whiteboard or project it. Make sure students will have to copy it down themselves since that is part of your point. 

Start by asking students to read the numbers to you. You will get some funny responses. Then ask students to add them or subtract the masses. As students write and count all of the zeros and inevitably miscalculate or miscount the number of zeros, you can introduce why we used scientific notation. (Less room for error, more efficient) Scientific notation is similar to typing TTYL instead of typing “talk to you later.”

Tips for Scientific Notation

Avoid using right or left when describing the direction to move the decimal. Instead, emphasize that smaller numbers will have negative exponents and larger numbers will have positive exponents. This re-enforces the negative exponent law.

Speaking of exponent laws, scientific notation operations reinforce the laws of exponents. If you look at the vertical alignment, scientific notation only shows up in 8th grade (in TEKS and CCSS), so at least, it reinforces other important concepts that students will use in Algebra 1 and 2.

I have never (and will never) teach Science, but it did occur to me to look up the Texas Science standards, and take a look at this chemistry standard – 

“C.2(G)  express and manipulate chemical quantities using scientific conventions and mathematical procedures, including dimensional analysis, scientific notation, and significant figures”

An opportunity for cross curricular?! Wahoo! If this is something that has peaked your interest, here is a NASA themed exploration lesson with resources for practicing scientific notation. This demos activity is also a great science based activity.

What tips do you have for teaching exponents and scientific notation?

The post Teaching Scientific Notation and Exponents appeared first on Maneuvering the Middle.

• Does Maneuvering the Middle have pre-assessments?
• Should teachers give pre-assessments?
• If yes, what are some best practices?

It is likely that your school already has something in place like MAP or benchmark testing. In those cases, I would say that your pre-assessment is good to go. If you’re interested in other approaches to pre-assessments, then keep reading. 

What is a Pre-Assessment?

Pre-Assessments are typically a test students take at the beginning of the year or unit. The data can help teachers make decisions about lessons, pacing, and differentiation.

Pros of Math Pre-Assessments

Pre-assessment data can provide teachers with the information they need to make strategic decisions about the support students will require. Perhaps, across the board, students performed low on proportional relationships, a teacher may add extra days to their unit on proportions. 

According to the Tar Heel State Teacher, pre-assessments provide students with evidence of their growth and prime students for their upcoming learning. I completely agree! Keep reading to see this in action. 

Cons of Math Pre-Assessments

Students are over tested. According to the Washington Post, “the average student in America’s big-city public schools takes some 112 mandatory standardized tests between pre-kindergarten and the end of 12th grade – an average of about eight a year.” 

Your school may already require multiple benchmarks in addition to the state tests. Does testing foster a love of learning? A pre-assessment would easily eat up at least one class period. Time is precious! 

If your students are taking one big, comprehensive pre-assessment at the beginning of year, that is an exorbitant amount of data to comb through. A question to ask yourself is, “Will I actually look at this data to make future decisions about instruction?” If I were being 100% honest, this would be overwhelming for me. And is it fair to tailor instruction of a unit covered in the spring based on something students did in August?

If pre-assessments are given at the beginning of the year, then your data will reflect the “summer slump.”

Math Pre-Assessment Recommendations

Ask yourself, “Why do I want to pre-assess?”

If you are pre-assessing because you think this is something you should do or something your students expect to do, then you are likely better off spending that class time in other ways.

Avoid one large pre-assessment. Shoot for a short pre-assessment at the beginning of a unit with content specific to the unit you are about to begin. The timeliness will ensure that you can actually use what you observe. 

• Marissa McCarthy, an All Access 6th grade teacher, only pre-assesses on 3 units – decimals, fractions, and the coordinate plane because those are covered in 5th grade and built upon in 6th grade. She gives students 4 problems at the beginning of the unit. At the end of the unit, she gives the exact same problems. “Students love seeing that they went from 1/4 to 4/4. You can use Maneuvering the Middle’s editable unit test to do this!” Or our end of year assessments to pull questions from.

Consider only pre-assessing previously taught content – not new to the grade level content! For example, in 7th grade, probability is introduced. If students are assessed on probability, they are more likely to miss questions due to lack of prior knowledge. However, a short fraction assessment that covers simplifying and multiplying fractions would do the trick. This data could tell me if I need to spend an entire lesson reviewing fraction multiplication or if I can jump straight into probability.

Short on time? Perhaps, you have 45 minute class periods, and you do not have the time to devote to extras. You can completely avoid pre-assessments by using existing data. Use the state assessment data that already exists! We work so hard to prepare students for a state assessment that we almost never look at that data. If you can, review the data reports of your incoming students (this may be trickier if your students come from a different school). 

• Don’t spend too much time on the knitty gritty – look at overall trends. 
• Find where a misunderstanding may impact your grade level content.
• You can even use this data to decide if you need to include pre-assessments for future units. 

There is no one right way to pre-assess or collect data from your students. Do you give pre-assessments?

The post Are Math Pre-Assessments Necessary? appeared first on Maneuvering the Middle.

When your students aren’t where you want them to be, it can be easy to just keep trucking along. Ignoring the unfinished learning in your classroom is a sure way to be frustrated when your students aren’t making the gains you are hoping for. 

Keep reading if you want to learn more about creating a sustainable plan for tackling the unfinished learning in your classroom so you can serve students well, without sacrificing your sanity and mental health. 

What is unfinished learning?

Concepts or skills that have not yet been mastered – can be from yesterday or years ago.

For example, your students may have never mastered squares and square roots. Where would this unfinished learning be present? Pythagorean theorem.

Unfinished learning can be addressed through planning, formatively assessing, and differentiating. 

Step 1: Planning

Determine what the standard is asking students to do and how they are expected to do it. Consider the potential challenges students may face. If you haven’t taught this content before, you may need to ask a fellow teacher what common misconceptions students may face tackling that specific skill. 

For example, in teaching the Pythagorean Theorem, you may spend much of your lesson on solving for the missing side using the formula, only to realize that students can’t differentiate between the hypotenuse and the legs.

When presenting the concept, provide multiple ways of solving. This is obviously easier said than done. I love the CRA framework, so be sure to check out the linked post for more context. You can provide multiple ways to solve by:

• Making the math concrete (introduce hands-on opportunities like manipulatives)
• Giving math context (create some real-world connections)
• Visualizing the concept (drawing pictures)

Step 2: Observation through Formative Assessments

Formative assessments do not need to be formal. Using what you already have is effective and efficient! Here are some simple ways to assess:

White boards are a favorite of mine! By observing through formative assessments, we can use our time and energy to address any misconceptions early on.

Step 3: Adapting using just-in-time supports

What is a Just-in Time support? It is a support provided based on a demonstrated need.

How do we adapt just in time?

When observing your students’ work, there are two outcomes. Either students are correct and can continue working, or students demonstrate a misunderstanding and it is time to act.

If you already have task cards on your agenda, you don’t have to change it. The rest of your class works on task cards in groups; the students who demonstrated a misconception join you at your small group table where you have an extra set of task cards already organized from easiest to most difficult. As you work with your students, start by asking guiding questions. 

• “What do you know?”
• “What do you need to know?”
• “What can we do to get started?”

Swapping the values is another quick way to differentiate for students that are demonstrating a misunderstanding. Let’s say that your students are working on surface area. You don’t want to spend lots of time readdressing how to multiply fractions or decimals. You are working on surface area after all. Swap the numbers! That way you can focus on what students need to understand to calculate the surface area of different 3D figures instead.

Error analysis is a great way to create an extension for your students, as well as address misconceptions. For students who got it when formatively assessed, you could push them to analyze how the incorrect answers were constructed. For students who need additional support, you can share the correct answer and scaffold your question in a small group.

Making small adaptations to an activity allows us to support students with unfinished learning, without the pressure to reinvent the wheel. The process outlined is ongoing – daily, weekly, all year long, so don’t get discouraged. 

All Access is a great tool for tackling unfinished learning. Since the stress of planning is done for you, you can use your energy to implement these tips.

How do you combat unfinished learning in your math classroom?

The post A Plan to Unfinished Learning appeared first on Maneuvering the Middle.

• Can x be 3.9999?
• Can x be 4?
• Can x be -100? 

Below, I have outlined a few ideas and things to consider when planning to teach one- and two-step inequalities.  

Let’s take a look at the standards, shall we?

Many of my recommendations for solving one or two-step inequalities are the same as solving one and two-step equations. This post talks about how using algebra tiles is my favorite practice. 

Something I did not learn until I was teaching is the why behind changing the sign when you divide or multiply by a negative number. As

You can do whatever you want to an inequality as long as it is done to both sides, it will remain a true statement.

This is true of inequalities with one exception. If you multiply or divide by a negative number, the inequality becomes untrue.

While this mistake still persisted, I found that my students did much better than previous groups in remembering to flip the sign.

Able to Read an Inequality Statement

Another big misconception that I have found to be true is the “alligator.”  Sure, “the alligator eats the bigger number” works when you are comparing 3 and 7, but what about when it’s 3 and -7 or 4x and 6?   Students need to be able to read an inequality statement and explain what it means in terms of the numbers around it.  They need to feel confident choosing a number for x to test the inequality.  

I have found the students struggle identifying “<” as less than and “>” as greater than, so when they do solve an inequality and they are left with x<4, they don’t actually know what that symbol means. 1<4 is something that a student can say but replace one of the numbers with a variable and the inequality becomes unclear. This is definitely a trick, but showing students how the less than symbol can be slightly rotated to be an L will allow them to actually state, “x is less than 4.”

Using Number Lines

The graph can seem like one more thing to do, but the solution(s) and the thinking lies within the graph.  The number line representation is a perfect visual to actually get students talking about the potential solutions.  Some questions to get students thinking:

• What is a value of x that is in the solution set?  Is there another?
• Why is _____ not a solution to inequality?  What other numbers would not be a solution?
• Describe the process for graphing the solution.
• Given the graph, what inequality statement best describes it?

When we are given a graph and are trying to figure out what inequality is the match, I avoid the trick that the arrows should match the inequality sign. Instead I might ask:

• What are the potential solutions? 
• Are the potential solutions greater than or less than the given amount?
• When I substitute a solution into X, is the original inequality true?

Common Misconceptions

1. Forgetting to change the inequality sign when dividing by a negative number
2. General confusion when the answer is “4 > x” or written with the constant on the left
3. Unclear of what number is considered a solution
4. Unable to make connections between the inequality and the actual solution on the number line
5. Writing inequality statements from situations when the terms “less than” or “greater than” are not included

Anchor Chart Ideas

Anchor charts are fabulous ways to showcase the content in a visual manner for students to reference.  They can easily be created before the lesson or as you are teaching, depending on the content.  This example includes the emphasis on vocabulary, as students tend to struggle with writing inequalities.

The real-life example that I think students have the most understanding of is related to movie ratings and/or height requirements for roller coasters. These very real-life experiences will help students understand the terms minimum means greater than or equal to. The minimum height you can be to ride this roller coaster is 48 inches. That means you can be 48, 49, 50 … inches to ride. Similarly, a person must be the minimum age of 17 to purchase an R-rated movie ticket.

Maximum was also another term that confused my students. Again, relating it to something that students experience daily seemed to do the trick. I would use seat belts as an example. I would ask students how many seat belts were typically in a car (or their car). Then I would say that the maximum number of people that could safely ride in said car was, let’s go with 5, which means 5 people or fewer could ride in the car. Maximum meant less than or equal to.

IDEAS FOR STRUGGLING STUDENTS

1. Bring out the manipulatives – algebra tiles, pattern blocks, etc – have students group like items
2. Break down the steps into a simple checklist
3. Go back to positive whole numbers to see if students are struggling with the concept or the mathematical skills
4. Give students a possible solution and ask them to work backwards

Hopefully, this gives you some ideas for teaching one- and two-step inequalities or even insight as to what knowledge your students are coming with.  I would love to hear other great activities or ideas you have used! Feel free to share in the comments.

Be sure to check out these resources for inequalities:

• 6th Equations and Inequalities Unit (TEKS)
• 6th Equations and Inequalities Unit (CCSS)
• 6th Grade Equations and Inequalities Activity Bundle
• 7th Grade Equations and Inequalities Unit (TEKS)
• 7th Grade Inequalities Unit (CCSS)
• 7th Grade Inequalities Activity Bundle

How do you teach inequalities?

The post Teaching One- and Two-Step Inequalities appeared first on Maneuvering the Middle.

The Standard

Why the struggle?

Solving equations is very revealing. If your students struggle with integer operations, then it will show up again when solving equations. If your students struggle with rational number operations, then it will show up again when solving equations.

We asked our Facebook group what students mostly struggle with, and here are some of the responses:

• Combining like terms incorrectly
• Dividing/multiplying a negative coefficient. Students trying to add instead.
• Not distributing when they should. Example: 5 – 2(x + 2) = 10 simplifying to 3(x +2) = 10  
• Students completing inverse operations on the same side of the equal sign instead of combining like terms (see below for picture)

Does this sound familiar?

Use Algebra Tiles (No, really, use them!)

I was resistant! Manipulatives can cause undue stress (tiny items + 30 children); it is a lot! However, I think that if I had used algebra tiles in my classroom, students would have been way more engaged! I occasionally drew a model to demonstrate solving equations, but then Noelle showed me all the ways algebra tiles could be used to combine like terms, distribute, and solve two-step equations (even quadratics). I was on board!

Before using algebra tiles to introduce students to solving equations, use algebra tiles to demonstrate combining like terms. Combining like terms can be challenging for students, but when you ask students to combine all of the long green tiles (x), the long red tiles (-x), the small yellow tiles (+1), and the small red tiles (-1), it provides excellent concrete practice. It will also assist with zero pair understanding, which will be essential to solving equations with algebra tiles.

What would you rather combine as a student? The terms or the algebra tiles?

Let’s move on to solving equations. Look at the following example:

Before exposing your students to two-step equations with variables on both sides, let them practice with one-step equations first. Scaffolding is key. I chose this example to show you how versatile algebra tiles are (variables on both sides and negative values)

To isolate the variable, students would need to recognize that they needed to get rid of positive 2.  To do this, they would add two negative tiles to make a zero pair. Adding negative 2 to one side would mean they would add to the other side to keep the equation balanced.

After students have removed the two zero pairs on the left side of the equation, they might be a little stuck. They are exploring, after all. Remind students that we want all variables on one side, since we want to find out what one x is equal to. Students would then add a negative x to both sides to create another zero pair.

Lastly, to get the solution for one x, students would divide the remaining red tiles among the two x tiles: X = -4.

Students are actually solving for x by undoing the problem, by using inverse operations (adding -2 to both sides and dividing by 2), and by keeping the equation balanced (they are adding tiles to both sides). All the procedural steps that might mean nothing to students in a traditional problem have meaning when students have been exposed to practicing with algebra tiles.

Remember that algebra tiles (like most manipulatives) exist to make the math visual. They provide conceptual understanding. Eventually, students will move into the algorithm. When students are exploring, make sure all of the solutions are integers (you can’t break the tiles into pieces).

To learn more about students making the jump from concrete to abstract, please check out our posts about the CRA Framework – Part 1 and Part 2. 

Another tip I recommend is to have students write out what is happening as they are using algebra tiles to solve an equation. If they are combining 3 green tiles with two red tiles, then they would need to write 3x-2x with evidence of only 1x remaining.

If you don’t have access to algebra tiles, students can use this website.

Be sure to grab our Getting Started with Algebra Tiles freebie to learn more about using algebra tiles to tackle simplifying expressions, the distributive property, solving linear equations, adding and subtracting polynomials, multiplying and dividing polynomials, and factoring polynomials.

Helpful Tips

These teacher tips from our Math Teacher VIP Facebook Group might help your students.

• Draw a line to separate the two sides of the equation.
• Do Undo Line – this is another strategy that can help students.
• Color-coding to help with combining like terms
• Making sure to actually say (and make students say), “2 times x equals 5” as opposed to “2x = 5.”

For more resources, check out these units and bundles.

What additional tips do you have? Do you use algebra tiles in your classroom?

The post Solving Equations in Middle School Math appeared first on Maneuvering the Middle.

We will be attempting to cover everything that will set you up for success in your first year. This is the last post in our series, so be sure to go back and read the previous three posts to get caught up.

• Part 1 – How to Teach Middle School Math as a New Teacher
• Part 2 – Cultivating a Strong Classroom Culture 
• Part 3 – What To Do When Everything is Urgent and Important 
• Part 4 – Our Favorite Practices (You are reading this one now!)

Today’s post will be all of the little things that will set you up for success. I am calling them best math practices, but since I have no scientific data to back them up, I will refer to them as favorite practices too. Let’s do it!

Work Out Every Problem Before You Teach the Lesson

This is probably my #1 best math practice. I like to do this for three reasons. 

1. As I work through problems that I will use for my direct instruction, partner work and independent practice,  I am prepared with the misconceptions my students may encounter. By mentally scripting what I might say or ask to combat these misconceptions, I am more prepared to teach. If while working through some problems, I am not really sure why [blank] is the next step, I can watch a video to help my explanations be clear and concise. 
2. When I am circulating while my students are working, I have my worked out answer key in hand, so I can check student progress. Instead of just looking at final answers to monitor progress, I like to quickly look at the students’ work too. If their work matches my work, I can give them a thumbs up, and move on to the next student. This idea also reinforces to students that their work is just as, if not more, important than the answer. In addition, if I do see an incorrect answer, I can look at their work, compare it to mine, and find the error faster. 
3. My first principal, Luz, told me this before my first year. It will take students 3 times longer than you to complete a problem. By working out every problem, I was able to assess whether I would need more work to fill up the class period … which leads me to my next point.

Always Have More Prepared

Your students will have varying math skills and speed levels. Some students will accurately complete all of their work before another student finishes the first problem.

Another best math practice is to provide meaningful work for students to work on after they have completed their assigned work. If these students need a challenge, make sure that it is rigorous enough to keep them engaged (don’t give them fluency practice) but not so challenging that they need your help. 

My go-to would be to ask them to write a test question based on what they learned that day, complete with answer choices and an answer key. Our digital activities are a great supplement too!

Assign Seats

Often I see teachers recommend to abandon seating charts at the beginning of the year, so you can learn who is friends with whom. This is not my advice! 

While you may not know your students well enough to create an informed seating chart, creating a seating chart that will help you learn names! I would start with alphabetical order because it helped me learn first and last names and made attendance easier.

Here is my rationale for starting with a seating chart. You can always give students seating choice after they have earned it. It is always easier to loosen up an expectation than try to wrangle students back in after they aren’t meeting your expectation. 

In addition, students without friends in that class or who are new, will feel more comfortable and safe in a classroom with a seating chart. You can read more advice about seating charts here.

Don’t Talk Over Students

There are entire books written about classroom management, so advice in this department cannot really be summarized in one paragraph in a blog post. I am going to pick the tool that packs the biggest punch. Don’t talk over students.

When giving instructions (not direct instruction), get your students’ attention, stand still, and wait. For example, it is time for students to go from working in stations back to their desks to start their exit ticket. 

*Attention Getter*

Teacher is standing still, squared up, and facing a majority of students. 

“I am waiting for all eyes to be on me. Thank you, Gabriel. Thank you, Max.”

“Most voices are turned off. Thank you. I am waiting on two more.”

*Teacher turns body and makes eye contact with the two students who were talking.*  (non-verbal redirection)

Students are still talking. *Teacher walks over to them.* (proximity) 

Students are now silent, and the teacher gives clear and concise directions. Teacher stops and waits if there are any interruptions.

When you give a direction while students are talking, you are communicating to them that what you have to say is not that important and they have a choice whether they need to listen to you or not.

Be Flexible

Being in the classroom for any length of time will result in a variety of mishaps – fire drills in the rain, copies running out, technology rendered useless since the internet is down, vomiting by both students and teachers, and a mouse running around. (These are examples from my classroom. Yes, the mouse visited my classroom during state testing, so that was …fun.) 

You can’t really get too worked up when something chaotic happens. Your students will follow your lead, so take a deep breath, problem solve, and make the best of it.

Build Relationships

Students will work harder for teachers they like and feel like them. Here are some ways to build relationships with your students on a daily basis.

• Use an individual’s name (so learn those names quickly )
• For every correction or redirection a student needs, be sure to praise them two times
• If a student invites you to a game or performance, go! If they went out of their way to invite you, then they love you and want you there.
• Be consistent. You can’t treat every student the exact same, but you have to hold all students to the same bar. 

Become an All Access Member

Finally, my best math practice is to become an All Access member! It bears repeating that the best thing you can do as a first year math teacher is find a reliable, standards-based curriculum. You will be spinning so many plates; don’t add curriculum writing to your very full scope of work.

Veteran teachers, what are some of your best math practices? New teachers, what questions do you have?

The post Best Math Practices for New Teachers appeared first on Maneuvering the Middle.

We will be attempting to cover everything that will set you up for success in your first year. This is part 3 in our series, so be sure to go back and read the last two posts to get caught up.

• Part 1 – How to Teach Middle School Math as a New Teacher
• Part 2 – Cultivating a Strong Classroom Culture 
• Part 3 – What To Do When Everything is Urgent and Important (You are reading this one now!)
• Part 4 – Our Favorite Practices

Today’s post will be all about how to overcome being an overwhelmed teacher and how to manage your to-do list. Let’s do it!

Feeling Overwhelmed

In my first year teaching high school math, I remember receiving a weekly update email from my principal on a Sunday night with many deadlines and tasks (on top of planning and preparing content for two preps), and I started to panic. How was I supposed to get all of this done? 

My husband, Taylor, asked me, “How do you eat an elephant?” I looked at him with a this-is-not-helpful stare. His response? “One bite at a time.”

In your first year of teaching math (or teaching at all), you will feel overwhelmed with the sheer number of tasks ahead of you. It is inevitable. The only way to overcome this sense of overwhelm is to start. Not sure of where to start? Make a list of everything you have to do for the next day and start with the easiest task. Use that momentum to knock out the next task and then the next. When you are feeling overwhelmed, any progress (even progress on peripheral tasks) will help you overcome your stress. 

Plan Your Work then Work Your Plan

When I have a written plan (in this case, a to-do list), I am efficient and I can knock things out. When I sit down to work with no plan, I am literally and figuratively listless.

No minute of your planning period should be wasted. Especially since planning periods tend to be commandeered by meetings or (possibly, but hopefully not) covering other classes. When you do have a full planning period ahead of you, take advantage of every second.

So how do you take advantage of every second?

• Make sure students are gone. That means that when the bell rings you are dismissing students, not asking them to start cleaning up.
• Know ahead of time what you need to accomplish during that planning period. This process is unique to each person, but personally, I liked to dedicate a day to the same tasks. Monday was for planning for the next week, Tuesday was for making copies and answer keys, Wednesdays was for grading and entering grades into the computer, Thursdays and Fridays were for miscellaneous or unfinished tasks. 
• Close your door. This sounds unfriendly, but if you are often interrupted or slow to recover after an interruption, then this gesture will keep visitors at bay. You can always chat with teachers in the workroom. 

If you want to read more about saving time during your planning period, check out these posts:

• Teacher Planner Printables (comes with free planner pages)
• 6 Time Saving Tips for Your Planning Period

Focus on What is Most Important

Teachers don’t just teach. Teachers do everything. They host clubs, they organize fundraisers, they plan field trips, they coach other teachers, and so much more. Nothing makes an overwhelmed teacher feel more overwhelmed than taking on more.

You may be tempted to take on an extracurricular or host an after-school club. If your heart is set on that, then go for it!

However, I do believe your first years in math should be dedicated to familiarizing yourself with your content and developing strong mathematical practices.  Give yourself permission to decline taking on additional roles, so you can participate in math professional developments or stay updated with new pedagogical practices. Being a lifelong learner of math and instructional practices is the sign of a great teacher!

If you haven’t read part one of this series, How to Teach Middle School Math as a New Teacher, you can read more about getting to know your math content.

Done is Better than Perfect

Since you will have a very full plate, sometimes we have to finish a task at a B- level. We want to strive for our personal A+, but when we have 900 things to do, done is better than perfect. 

If a B- has the same result as an A+, then don’t waste your time making it an A+. For example, bulletin board displays. Some teachers go all out with a theme and change the look each month – good for them, but not for me. I put one bulletin board display up in August, and only switched up the student work when it was required. 

Another example was something I saw an Algebra teacher do at my school. Since typing formulas and various mathematical symbols into the computer slowed her down, she would hand write problems for her students to complete. She would handwrite the problems, scan it to email it to herself, and then make copies. Would the worksheet look better printed? Sure. Did it matter? No. Done is better than perfect. 

Become an All Access Member

The best thing you could probably do as an overwhelmed teacher is to find and use a reliable, standards-based curriculum. If you can save yourself the time that it takes to scour the internet for worksheets or starting from scratch, you will already be so many steps ahead!

Veteran teachers, how do you tackle your to-do list? New teachers, what questions do you have?

The post Are You An Overwhelmed New Teacher? appeared first on Maneuvering the Middle.

Welcome, New Math Teacher! If you have made it to this blog post, you are most likely about to enter your first year of teaching or your first year of teaching math. (And if you are a veteran math teacher, we would love for you to share your tips in the comments :))

We will be attempting to cover everything that will set you up for success in your first year. If you haven’t read part 1, do that first. 

• Part 1 – How to Teach Middle School Math as a New Teacher
• Part 2 – Cultivating a Strong Classroom Culture (You are reading this one now!)
• Part 3 – What To Do When Everything is Urgent and Important
• Part 4 – Our Favorite Practices

Today’s post will be all about cultivating a strong math classroom culture. Let’s do it!

Be Proactive

A strong classroom culture is predicated on students knowing what is expected of them and rising to meet that bar.  This means you will need to communicate what you want students to be doing 100% of the time they are in your classroom. 

I can talk at length about routines and procedures and we have 4 blog posts dedicated to them, so I won’t belabor those systems anymore. Please be sure to check them out though because they are pretty comprehensive.

• 20 Must Teach Routines and Procedures
• 15 More Must Teach Routines and Procedures
• How to Teach Routines and Procedures that Stick
• Routines and Procedures for the First Day of School

Create a Safe Space

I had a student who would pick up her bell ringer from the door and stare at it for 5 straight minutes. This student was not off task in the slightest; she had her supplies, she came to class early, and she wasn’t chatting with classmates. She would just stare at her paper. After inquiring about her inactivity, she revealed to me that she feared making an error. 

This took me back to my math experience as a student; I couldn’t think of a single instance that I volunteered to answer by raising my hand. I was so scared to be wrong that I avoided eye contact any time a teacher asked a question. 

Creating a safe space for making mistakes is vital to your students’ success in math. 

I will point you to Jo Boaler’s research on how mistakes make our brains grow which is a fascinating read. This quote really nails it:

So how can you encourage mistake making? Start with yourself.

If a student points out an error, celebrate it: Wahoo! I made a mistake! My brain is growing!

If a student asks you a question that you do not know the answer to, admit to that. Great question! I am not sure I can answer that right now, but I will find out. 

If a student shares something incorrect, tell them that their brain just grew instead of telling them their answer is incorrect and then give them a moment to try again.

After I learned that the student was fearful of making an error, I started to hand-write (before making copies) “You don’t have to be right, but you do have to try!” on my bell ringers.

Lean into the Productive Struggle

If you are teaching middle school (particularly math) then you should prepare yourself for this second point… teach students to work through hard things, also known as productive struggle. 

This is still a battle I fight daily.  The first thing I begin teaching my students to say is, “Mrs. Brack, will you clarify this part to me?” or even simply, “I need help.”  “I don’t get it” suggests that you are passive in the understanding of a problem. 

You can respond with, “I don’t get it yet.” Or you can come up with a norm (see next point) for how to combat passive learning.

Then, you need to shift those “I don’t understand” type responses to a specific question about the content. Or by asking, “What do you understand?” We want them to make a connection to something they already know and then construct a plan based on that.

This flowchart (that you can grab for free below) is something I would point to when my students were stuck.

Click here to get your own copy of my How to Get Help in Math Flowchart.

Create Norms as a Class

This goes hand-in-hand with being proactive. With your students, come up with 3 or more “norms” or practices that will hold everyone accountable to learning math. To start the brainstorming process, you can ask students and yourself, “What needs to be in place for everyone to be successful in our classroom?” Students may respond by saying:

• Don’t give up. 
• Ask for help when stuck.
• Respect each other.
• Mistakes help us learn.

After you and your students have brainstormed a list of norms, post these norms on your classroom walls and refer to them daily. Students are bought in because they have a say in the very foundation of your classroom culture.

Start With Yourself

As a new math teacher, you don’t know what you don’t know. Even veteran teachers are lifelong learners of instructional math practices. Get comfortable asking for help from a trusted colleague, or even the brilliant teachers that are readily available in many Facebook groups.

Join All Access

An All Access membership will save you time and energy in your first year of teaching. I wish All Access existed my first year teaching – I believe it would have made for a less stressful year!

“Maneuvering the Middle gave me my life back and I know my students are getting what they need!” – All Access Member

Click here to learn more about All Access.

When it comes to teaching math, many things have changed. If you are relying solely on how you learned it growing up, then you are likely missing out on some great strategies. The way that I taught students to divide fractions during my first year teaching is nowhere near how I would approach teaching dividing fractions today.

Veterans, what tips do you have for new math teachers? New math teachers, what questions do you have?

Editor’s Note: Maneuvering the Middle has been publishing blog posts for teachers for nearly 6 years. This post was originally published in June of 2017. It has been revamped for accuracy and relevancy and to include a Good Morning Teacher podcast episode.

The post Tips for a New Math Teacher appeared first on Maneuvering the Middle.

Let me be the first to offer my congratulations! I hope that you will find our blog useful, so please stay a while. 

We will be covering everything that will set you up for success in your first year. Be sure to check back to read:

• Part 2 – Cultivating a Strong Classroom Culture
• Part 3 – What To Do When Everything is Urgent and Important
• Part 4 – Our Favorite Practices

Today’s post will be all about what to spend your time and energy on this summer. You have a huge task ahead of you and still a month before you begin. How should you use this time before you? Let’s dive in!

What To Focus on This Summer 

Most advice (including my own) around being a first year teacher is about building relationships with students and creating consistent routines and procedures for your classroom. But if you are reading this in July, you don’t have a classroom of students to start working on this yet. So what to do while you are anxiously waiting for the start of the year? 

Get to Know Your Content

This was a hard reality I discovered when I started teaching math: being a good math student does not always make a good math teacher. Math was always my best subject, so when it was time to reach struggling students, I felt stuck. I had learned math this one specific way and it worked for me, so I lacked additional tools that could help me help my students.

You don’t know what you don’t know, but you can assume that the way you may have learned a math skill is not the only way that you can teach it. So it is time to educate yourself! 

Before I get into specific math skills, I want to recommend reading about and implementing the Concrete, Representational, Abstract framework in your daily lesson plans. This study is short and does a great job of summarizing the CRA framework. We also have two posts: here and here.

The CRA framework helps guide students from the hands-on materials (Concrete), to the drawings and models (Representational), and finally, to the equations and algorithms (Abstract). By using the CRA framework, you are setting students up to understand the WHY behind the procedures of various skills which develops their conceptual understanding, builds math confidence, and gives students multiple methods for solving a problem. (Hint: you don’t need a huge supply of math manipulatives to use the CRA framework – check out our favorite math manipulatives here and how to get them on a budget.)

Let’s talk about what you are teaching! If you know what grade you are teaching, then I would start by looking at the standards for that grade level. You can find Common Core standards by Googling “CCSS + math + grade level” and by clicking here for the TEKS (if you are in Texas). If your state does not teach Common Core, I would search your state + grade level + math standards. 

Don’t expect to read your grade level’s math standards and instantly understand what you are reading and supposed to teach. But do read through the standards and make some observations. Here are some guiding questions:

• Are any topics or skills mentioned more often than others?
• Are there any words or phrases that I am unfamiliar with?
• Are there any skills that I only know one method of solving?

For the last question specifically, if you noticed a few or many skills that you only know one way of solving, use the internet to learn more. Before teaching 6th grade math, I knew exactly one way to solve a proportion: cross multiply and divide. I now know about 16 ways to solve a proportion: finding the scale factor, bar models, double number lines, scaling, graphing, and so forth.  This is something that you can expect to do throughout the school year. A good rule of thumb is to come ready to teach with at least two different ways a student can solve. 

Now I am going to make a few plugs

You can grab our middle school math + algebra 1 pacing guides by clicking below. This will help you get an idea of how our Maneuvering the Middle curriculum is organized.

Math teachers are incredibly helpful and we have almost 10k in our Maneuvering the Middle VIP Facebook Group. Join us, read and learn, and post your own questions. 

Join All Access! I truly believe if I had access to this curriculum in my first year of teaching math, I would have been less stressed and 100% more confident to tackle these standards. You can learn more about All Access here (like how it has everything you need and more to teach math), but we hear from teachers and administrators all the time about how our student video library helps new teachers learn and internalize the content.

What to Spend Your Money On 

If you can help it, try to resist the temptation to start spending money on your classroom until you have done some research.

First, find out if you have a budget from the school. I have been given $50 to unlimited funds (within reason) to spend on my classroom. Unfortunately, (and might I add, wrongly) some teachers receive zero dollars for their classroom. If you see something that you MUST have, at least hang onto the receipt since it may be reimbursable. 

Next, find out what the school already provides (or doesn’t provide). Your school may keep the teacher workroom stocked with #2 pencils, so you may never need to make that purchase. You may find out that you have a budget of $200 for the year, but the school doesn’t provide copy paper, so you may want to use that $200 on a paper supply.

You may want to purchase classroom posters and decorations! I get it, but it can get expensive, so be sure to check out how I decorated my classroom on a serious budget. All you need is colored cardstock and a printer.

Whether your school is paying or you are paying out of pocket for your classroom, I would recommend checking out our list of essential classroom supplies. This list even includes how necessary I think the item is. 

Lastly, remember that asking for donations from parents at Open House is totally acceptable. 

If you are a new math teacher, let us know how you are spending your summer. Veteran teachers, any additional advice you would include for the summer before school starts? Be sure to come back next Tuesday for more new math teacher tips.

The post How to Teach Middle School Math appeared first on Maneuvering the Middle.

Teachers in our Facebook Group shared how they used the student videos to meet the needs of their students and I was blown away. What is most fascinating is how teachers use them in ways we didn’t even think of when we were in the planning stages of that project. Teachers are so creative!

FOR ABSENT STUDENTS

“I use them for when students are making up missed assignments or in ISS.” -Jennifer

“They watch vocabulary and intro to the lesson and then one or two problems per video then do a few on their own. Videos are great for students that are absent or need to review the concept.” – Courtney

FOR TEACHERS

“They work great when you have a sub, so you do not get behind in your curriculum. These videos are great tools for new and seasoned teachers.

Teachers on my team have watched the videos, especially if they are in a new grade level!!! – Courtney

“I watch them sometimes to make sure that I’m teaching the concept to the depth the students need.” -Jennifer

FOR REFERENCE LATER

“I post them to Google Classroom, so if students are absent or need to revisit the material, they are there!” – Amy

“I post them in Google Classroom. Students can use them if they are out or if they need more instruction.

I think it would work great for a flipped classroom. You know multiple voices, word choices, explaining the same thing only benefits the kids.” – Karen

IN CLASS 

“I have found that using the videos frees me up to pull small groups, provide feedback while students are applying the content, etc.

The biggest impact is allowing me the time to work with students more. To proactively intervene as needed.” -Amber

“I play the first video or two for each set of notes in class. I can circle around and check homework/attendance/make sure every child has a pencil and is writing. I then usually work through the example problems then it’s off to stations in independent practice and small groups with me.” – Brittany

“I use them (6-8) in class for my SpEd students to take notes. I will stop the video and we have discussions or have turn and talks. I love the videos because they chunk lessons that our other text (Big Ideas) lump together.” – Ann

FOR FLIPPED OR SELF-PACED CLASSROOMS

“I use the videos for every lesson to partially flip my class. There isn’t enough time to do all the lesson problems and have students practice within a class period. Students watch the videos and complete those problems as homework. The next day, I do the rest of the problems and students do the “homework” page in class. Works pretty well. I teach 6-8.” -Carla

“I have a self-paced classroom. I upload all instruction videos and assignments for a unit so that students are able to work at their own pace. I hold small groups at the same time for my struggling students so they are able to ask questions and get the face-to-face instruction they may require. My higher level students love it because they aren’t held up by questions and small talk that cause notes to go on forever.” -Emilie

GENERAL

“They allow the kids to hear from a different person besides you. She presents them in different ways.” -Sandra

“Parents of my students have watched them to help their children study for tests, complete study guides, etc.”  -Marissa 

If you are thinking that you would like your students to benefit from these math videos and all of these incredible ideas, then click over to learn more about All Access.

There you have it! This list is in no way exhaustive, so I would love to hear – how do you use our student math videos in your classroom?

The post How Teachers Use Our Math Videos appeared first on Maneuvering the Middle.

Standards

Comparing and ordering rational numbers is predominantly a 6th grade skill in both Common Core and TEKS. It will also support students locating and plotting rational numbers on the coordinate plane.

1. 6.NS.7 Understand ordering and absolute value of rational numbers.
2. 6.2D Order a set of rational numbers arising from mathematical and real-world contexts (Readiness Standard)

Released Staar Question

Looking at a test question isn’t a way to “teach to the test,” but rather, a way for me to make sure that I am teaching the complexity of the standard. By looking at the test question above, it helps me to understand that:

• Students will need to order a set of 5 (or possibly more) rational numbers
• These rational numbers will vary in type (fractions, mixed numbers, improper fractions, whole numbers, and decimals) so I can assume that test questions can also include percentages
• It also helps me recognize that I need to make sure students pay attention to the order in which the question is asking
  • Greatest to least – which can also include words like “descending”
  • Least to greatest – which can also include words like “ascending” 

Convert to the Same Form (fraction, decimal, percent)

Before you can jump into ordering rational numbers, you will need to spend a few days on converting between fractions, decimals, and percentages. 

But I wouldn’t start by ordering a set of numbers that include fractions, decimals, and percentages. Start by scaffolding! Order a set of decimals, then a set of percentages, and lastly a set of fractions before introducing a mixed bag of rational numbers. 

For decimals, I suggest asking students to line up the decimal point and compare the digits going from left to right. 

Have students add zeros to numbers after the decimal point so no student confuses a number with more digits as greater.

For percentages, I would use the same method. Remind students where the decimal point is in a percentage if it is not visible.

For fractions, this is where I don’t have strong opinions. Perhaps because I have not seen one method to be more successful than others. Butterfly method, finding a common denominator, converting fractions into a decimal, reasoning through it – I think each student will gravitate toward their favorite based on their past experiences. 

Once students have mastered ordering from the same type of rational numbers, have students order a small set of 3 rational numbers by converting them to the same form. In my experience, students (and me, the teacher) like to convert most rational numbers to decimals. I think this is because they are the easiest to compare and then order.

Use a Number Line

When negative numbers are included, a number line is an absolute must. The number line gives us context! A quick sketch of a number line with a zero in the middle immediately serves as a reminder that the numbers with the least value will be the furthest left and the numbers with the greatest value will be the furthest right!

In fact, a number line was a requirement for work shown when we ordered numbers in class. A dry erase marker and a desk allowed for lots of space for converting between the fractions, decimals, and percentages and placing them on the number line.

Quick Hits

• Have a half day or need an extension activity? Have students make flashcards of benchmark fractions: ¼, ½, ¾, ⅕, ⅖, ⅗, ⅘. Once my students could ace this in a flyswatter challenge, I would add ⅛, ⅜, ⅝, ⅞. If students had these benchmark fractions memorized, it lightened the mental load for ordering rational numbers.
• Make ordering numbers hands on! This can be done by using these cards. Print a set, laminate, and then have a living number line using the sticky side of painter’s tape (see the idea here). 

• Get students moving! Give each student a card with a rational number. Tell them to get into groups of 4 and stand in order from least to greatest. Then tell them to find a new group with 3 people and stand in order from greatest to least. Go around the room and listen to their reasoning. You can do this as many times as you want using a different number of students and in a different order. You can finish the activity by having all the students stand in ascending or descending order.

How do you teach comparing and ordering rational numbers?

The post Ordering Rational Numbers appeared first on Maneuvering the Middle.

All of the ideas in this post come from Project Based Teaching, a book that we are reading to learn more about implementing projects in our math classrooms this year. Click the link here to grab your copy – a must for any teacher’s shelf.

Make sure to scroll down to grab a “50 Ways to Use Math in the Real-World” printable!

And make sure to grab our “50 Ways to Use Math in the Real World” printable for your classroom. This printable can be used as a jumping off point for projects, classroom decoration, or a resource to answer the question, “When are we ever going to use this (math skill)?”

And we are very excited to announce that two projects will be added to All Access in time for Back to School!

Update October 2022 – Math projects are now available for purchase! You do not have to be an All Access member to implement these standards-based projects in your classroom.

Click here to learn more about each project!

The standards-aligned projects focus on content covered in the fall semester with the second project covering content in the spring semester.

Each project is flexible in nature and includes teaching slides, warm-ups, exit tickets, student recording sheets and a project overview to help you structure class time and implement project components as smoothly as possible. Here is a little sneak peak of the 7th grade project.

Where to Find Good Projects

Borrow and adapt – Don’t start from scratch, especially if this is your first rodeo.  Editing and adjusting an existing project is a great place to start. Check out www.bie.org. This project library allows you to filter by subject and grade level. 

• Here are more websites with existing projects: 
  • www.enablingfuture.org
  • www.iearn.org
  • www.k12science.org/mterials/k12/technology/real-time-data
  • https://learn.outofedenwalk.com/

Remodel – Evaluate previously taught units to see which unit may lend itself to being taught using a project instead. Since you are familiar with the content, you will be able to focus on the execution. Consider the units in which students weren’t as engaged; a project may be the perfect way to turn that around.

Listen and codesign with students – “Student questions offer a renewable source of project inspiration.” Consider surveying them about their interest and what they would like to learn to get their ideas. Once you have identified the problem your students want to tackle, you can incorporate the required academic standards. 

Use current events – Headlines in recent events offer perfect examples of using data and statistics in favorable or unfavorable ways. 

Connect to popular culture – What are your students interested in? Is there a movie, book, or video game that has captured their attention that you can capitalize on? 

Respond to real requests – Does the community have needs that your students could meet using your subject matter?

Build on your passions – While students’ passions are important, so are yours! For me, I love architecture, so I always made sure to include a project on designing a home for a client and building a scaled model using what we knew about ratios and scale factor.

All Access – In each grade level, we’ve added two real-world projects (to our already amazing curriculum and videos) intended to be engaging opportunities for extension and application of skills! Each project can serve as an assessment of students’ understanding while encouraging inquiry and critical thinking amongst your students. Coming in August 2022 – so exciting!

What Makes a High Quality Project

While you are creating or editing an existing project, you may be wondering what makes a good project? What components are necessary to make sure that students are thinking through, analyzing, and solving problems?

• Challenging problem – The problem needs to grow students’ thinking muscles. Too hard and students won’t sustain; too easy and students will disengage.
• Sustained inquiry – Students “need to be asking questions, conducting research, carrying out investigations, and weighing evidence to arrive at answers.”
• Authenticity – The project must connect to the real world. 
• Student voice and choice – Students are driving the learning. 
• Critique and revision – Students are constantly giving and receiving (from peers and teachers) feedback to improve their project.
• Public product – Students will present their project to a public audience – an extension of the classroom. This increases the engagement of the students and the quality of the projects. 

Preparing Your Resources

While you are planning projects, don’t forget to prepare and secure what is needed for students to execute successfully. Don’t forget to reserve items and spaces to keep your students engaged and excited about the project. Here is a non-comprehensive list to get your started:

• Technology – apps, software, tools
• Content Experts – mentors, panel members, clients or product users
• Connecting to other subjects – can you go cross curricular? 
• Community or school resources – video cameras, science labs, art studios 

Planning projects can be so fun! What projects do you want to try in your math classroom?

The post Planning and Designing PBL Projects in Math appeared first on Maneuvering the Middle.

Maneuvering the Middle Curriculum Works

The reason we can proudly promote our resources as the best thing for students (and teachers) is because we have first hand experience with how effective Maneuvering the Middle curriculum is.  When I used our 6th grade TEKS curriculum, the number of students reaching “Meets Grade Level” increased by 15% from the previous year (when I was using a different resource). 

We receive emails and comments from teachers daily who share that their students are growing. 

I love this testimonial because it shows some celebratory data (93% is amazing), but also because Hayley states that our materials help her students love math. 

Standards Aligned

If you are in a Common Core State or teach in Texas, then you can be confident that by using our curriculum, you will be covering each and every standard. If you are like me, you read a standard and wonder, “What does that even mean?” You can rest assured knowing that our curriculum team (shoutout Kim, Noelle, Ashleigh, and Sara) have done the work to make sure that the standard is taught with the proper scaffolding to reach the depth and complexity that is required.

All of our curriculum teachers were once teaching this material in their own classrooms, so they know that it isn’t simple or easy to get everything covered in a school year.

In addition, schools that use Maneuvering the Middle across all grade levels will see how the content builds upon itself. There are no holes in the content from one grade level to another. 

Saves Your Time and Energy

When you are a teacher who can focus on the teaching part of your job and not the curriculum creation part of your job, you can be more effective. Let’s be real – writing curriculum is its own job! To be a teacher and a curriculum writer means you are working 2 jobs and being a teacher is hard enough!

My favorite principal said she would always prefer a rested and happy teacher over a tired and stressed teacher with a great lesson plan. 

When the curriculum is taken care of, you can use your time and energy for all of the other aspects of the job – building relationships, providing feedback, remediation and intervention, instruction, implementing routines and procedures, running copies and preparing classroom materials, and (insert 1000 more tasks here). 

Consistent Routine + Lots of Variety

Maneuvering the Middle offers student handouts, video lessons, study guides, assessments, digital activities, and more. Not to mention, our hands-on activities include:

• Solve & Colors and Mazes – for fluency practice
• He Said, She Said and Find It, Fix Its – for error analysis
• Card Matching and Cut and Pastes – for tactile learning

We offer a lot! These resources are consistent across the 4 different grade levels, so students know what to expect and can focus on learning the math – not a new routine or activity.

Because there is so much variety, you can truly pick and choose what works best for your students. Personally for me, cut and paste activities were not my jam, so I skipped those. Even without those activities in my rotation, I NEVER lacked for anything. I would turn to a digital activity instead!

When I say that I never lacked for anything, I mean it! Check out everything that is included in All Access.

Works for Different Classrooms and Teaching Styles

Maneuvering the Middle works in so many different types of classrooms! If you join our Facebook Group, you can ask how teachers use the curriculum in their classroom and you will find that no two teachers use it the same way.

Teachers use our curriculum in flipped classrooms, with direct instruction, and self-paced learning classrooms. Our materials are used in homeschool settings, hybrid classrooms, virtual classrooms, and in-person classrooms. Teachers who coteach or teach intervention use our materials too. You can find it in public schools, private schools and charter schools! 

Our video library makes all of those different teaching methods even more accessible for teachers and students.  

It may seem like we are tooting our own horn! We are really proud of what we have created. We know our curriculum doesn’t work without amazing teachers, so thank you teachers for all that you do!

Why do you love Maneuvering the Middle curriculum?

The post What Makes Maneuvering the Middle Different appeared first on Maneuvering the Middle.

However, it is a little tricky! Be sure to read “How to Teach Integer Operations” where we cover the specifics on adding and subtracting integers, as well as some common misconceptions to avoid. 

Multiplying and dividing integers is taught as early as 6th grade (in Texas) but primarily introduced in 7th grade. Multiplying and dividing integers extends into rational numbers which means there is a lot to cover!

Tip #1: Start By Using Models or Manipulatives

“Tricks” definitely have their time and place in math, but building conceptual understanding is key for students to build that mathematical fluency. Since math is always progressing and building on itself, tricks can often be mixed up with other tricks. Anyone who has taught integers or fraction operations probably has experienced this first hand. (Keep, change, flip or keep, change, change?)

Modeling why like signs result in a positive answer and why unlike signs result in a negative answer is more likely to stick than having students only copy down the “rules.” Though I do think it is helpful to do that too! In fact, I kept an anchor chart with all of the rules posted throughout most of the school year.

Here is a helpful Google Slide Deck that I recommend using to introduce WHY a positive times a negative results in a negative product and WHY a negative times a negative results in a positive product. I used a number line to introduce this concept, but counters can work too.

Here is the slide deck for you to copy and use in your classroom! The animations are included, so you just have to click your mouse to make the arrows move.

If you prefer counters, here is how I would teach positive times a negative.

And a negative times a positive. One thing to note is since you can’t take away 4 groups of 2, you can introduce zero pairs. With the zero pairs, you now have 2s to “take away.”

And lastly, a negative times a negative.

Tip #2: Ask them to Come Up With the Rules Using Patterns

The great thing about math is that the rules are always supported by patterns.

When using the above table (a snippet from a student handout), it is important to start in the top left corner where students are familiar with those facts. Allowing students to see the patterns that create the rules really makes the content stick.

You could give students a list of numbers like the one below and ask them to make observations about what they see. If a student can’t remember a rule, they can recreate a list of multiplication facts, and then synthesize the rules on their own.

• -5*(3)= -15
• -5*(2)= ?
• -5*(1)= -5
• -5*(0)= 0
• -5*(-1)= 5
• -5*(-2)= ?
• -5*(-3)= ?

Tip #3: Other Fun Ideas for Practice

• Give students a value they are trying to reach.  Provide sticky notes or cards marked with a variety of integers. Students match integers to equal the given value. Similar to using counters, this allows for students to practice their fluency but also to be flexible problem solvers.
• Our MTM Activities – Entire bundle, speed dating, scavenger hunt and multiplying rational numbers digital activity, dividing rational numbers digital activity
• Playing War using this idea from Mrs. E Teaches Math.
• Make sure to include opportunities for real-world situations. Money, football gains and losses, temperature, and sea level are things that give these numbers context. Context provides students with opportunities for application as well as verifying if an answer makes sense. 
• Make sure to buy and laminate these number lines. I would give my students dry erase markers to write on them.

Tip #4: Putting It All Together

In my experience, students are doing well and seem to really grasp when they are completing operations in isolation. Asking students to complete work where they are switching between the 4 operations or using them together can cause students to confuse what they have learned.

When that happens, ask students to draw a picture of what is happening. Send them back to the models because that is what they are there for – to make sense of the math. Don’t put the counters or number lines away just because you have already taught the models. 

Since many students are resistant to draw models when they think they’ve “got it,” I would remind students that drawing the models accounted for half of the work/grade/completion. Once students mastered the four operations on a summative assignment, I would give them permission to use the algorithm only.

What tips do you have for teaching multiplying and dividing integers?

The post Teaching Multiplying and Dividing Integers appeared first on Maneuvering the Middle.

While the standards for middle school math do not include modeling specifically (those can be found in 4th and 5th grade), middle school students can benefit from growing their conceptual understanding.

Start with Whole Numbers

The basis of modeling fraction multiplication uses arrays. Before you jump into using arrays with fractions, start with modeling multiplication using whole numbers and arrays. Have students create their own arrays too.

Anytime I am introducing a new concept, I connect it to something they already know how to do. This builds students’ confidence and creates more buy-in which means my job as teacher is a lot more enjoyable  

Build the Conceptual Understanding Using Reasonableness

Before introducing the models, I want students to continue to build their reasonableness. Start with something that students are very familiar with like pizza.  

Asking questions like –

• “If I have 1 pizza, and I eat ½ of it, did I eat more or less than 1 pizza?”
• “If I have 3 pizzas, and I burned ⅔ of them, how many pizzas did I burn?”
• “If I have 1½ of a pizza left and you ate ½ of that half, did you eat more or less than 1 pizza?”

One of the 6th grade TEKS standards is developing the understanding that multiplying a whole number by a fraction less than 1 will result in a product less than the original whole number, and multiplying by a fraction greater than 1 will result in a product greater than the original whole number. Using reasonableness can help support this standard and a conceptual understanding of multiplying fractions.

Modeling a Whole Number by a Fraction

4 times ⅖ can be interpreted as 4 groups of ⅖ or ⅖ + ⅖+ ⅖+ ⅖. This is the easiest to model. Draw a picture or use pattern blocks to demonstrate this concept for students.

It gets a little trickier when you have ⅖ of 4. Sure you will get the same answer, but students do need to understanding that this means taking a ⅖ part of 4 wholes. If you haven’t checked out Mix and Math yet, Brittany is the queen of hand-on teaching in upper elementary math. Check out this video to see how she uses pattern blocks to teach these concepts.

Modeling a Fraction by a Fraction

If you need a refresher (or didn’t learn it this way yourself) this video explains how to multiply a fraction by a fraction using models. Again, this array will be more effective if you have introduced arrays with whole numbers first. 

• Brownie pans or rice krispie pans are my go-to representations of this skill. 
• Fraction Multipliers are manipulatives that support this skill. These manipulatives are very  effective in a small group or intervention classroom.

Modeling Multiplying Mixed Numbers

In the past, I have taught students to convert mixed numbers into improper fractions and then to use the algorithm. This totally works! However, there are so many steps and so many places to make mistakes. 

Here are a few tips to make this more manageable. Try using an area model to keep students’ work more organized. 

You can always have students make the problem simpler in order to estimate.

Let’s say the problem is 3 ⅜ times 5 ⅚. You could think of it as 3 ½ times 6. This becomes 3 times 6 plus 1/2 times 6. (Think: 34*5 is the same thing as 30 times 5 plus 4 times 5.) Therefore, 18 + 3 = 21. The answer to 3 ⅜ times 5 ⅚ is going to be verrrryyy close to 21. And since we rounded each factor up, I know my answer should be less than 21. 

At the very least, you could have your students multiply the whole numbers to check that their answers are reasonable. 

Additional Tips

• Give students the models and ask them to come up with the the number sentence that best represents it
• Remember that modeling isn’t just an introductory step. If you only use them to introduce the concept and never revisit, neither will your students.
• There is nothing wrong with the algorithm! Teach it, use it, and model it!

How do you teach multiplying fractions?

The post Multiplying Fractions with Models appeared first on Maneuvering the Middle.

If you are interested in incorporating more projects in your classroom, then grab the book and follow along.

Update October 2022 – If you want to include projects in your class, but aren’t sure where to start, then grab our Math Projects!

Math projects are now available for purchase! You do not have to be an All Access member to implement these standards-based projects in your classroom.

Click here to learn more about each project!

Why Projects?

The world is turning to collaborative projects. At Maneuvering the Middle, everyone works on teams to complete various projects. None of our work is done in isolation. 

Why Culture Matters

”Positive culture doesn’t get built with a one-day team builder. It’s an ongoing effort to create an inclusive community of learners.”

Classroom culture is necessary in order for students to feel safe making mistakes and asking questions that are required during Project Based Learning. We have a whole post about how to make your classroom feel positive and safe, so be sure to check it out. 

4 Strategies for Building Strong Culture

After reading through the strategies for building strong classroom culture for Project Based Learning, I was so relieved to read nothing brand new. If you are a teacher reading a blog post about teaching, you are probably already doing many of these things!

1. Beliefs and Values

In this section of the book, many teachers that the authors have interviewed share examples of how they show students their core values and beliefs. Examples include: asking students for feedback about what they like or would change about class so students feel like their voice matters, reminding students that they are all capable of solving big problems, and providing projects that are relevant to students’ lives.

2. Shared Norms

“Norms … are shared agreements about how classmates and teachers treat one another and what they value as a community of learners.”

I agree that the most valuable norms are the norms that students have a say in. Creating buy-in is crucial in any classroom! These should be posted and repeated often by both teachers and students. Here are examples of norms I used in my classroom:

• Everyone contributes
• Respect each other
• Share the mic
• Follow directions 
• Good attitudes only

I love how Todd Finley establishes norms in his classroom. He has students brainstorm examples of actions that have impeded learning. Here is an example from his classroom:

• Example: If students laugh when I make a mistake, I don’t want to participate. 
• Norm to counteract: We learn from mistakes. 

3. Physical Environment

Your classroom environment also contributes to a strong classroom culture. Make sure students have access to technology and other tools that will help support their learning. Here are 4 ideas that Project Based Teaching recommends considering:

1. Flexible seating and arrangements that support working in partners and groups
2. A project wall: “By dedicating a bulletin board… to the project currently underway, you create a central location to manage information, highlight upcoming deadlines and milestones, remind students of the driving question, capture need-to-knows, and point to resources.” A digital space could serve this purpose too. 
3. Sentence starters: Since student voice is so important in project based learning, it is important to provide support for ALL students. These sentence stems can help students engage with each other productively and appropriately. 

4. Evidence of the messy middle: Keep rough drafts and unfinished work visible. It provides opportunities for questions, discussion, and feedback.

Routines and Habits for a Student Centered Classroom

We are no stranger to routines and procedures! Routines and procedures increase learning time. Here are the routines and procedures that are necessary for PBL (or truly any classroom):

• Active listening – all voices in a group are heard and valued
• Providing feedback – how to give and receive feedback in a way that is helpful and kind
• Morning meetings – check in time with class that helps strengthen relationships
• Thinking routines – like “think-pair-share”
• Closers – ending class with shout outs or to state a class norm

Starting Small

If you are interested in a PBL classroom, start small with a small team building challenge. 

• Community Counts is a great place to start – it helps building relationships between students and teacher and student to student

“If you build that culture early, students will be ready to tackle longer, more content-heavy projects later in the school year.”

To learn more about incorporating project based learning into your classroom, check out Project Based Teaching. 

Do you use project based learning in your classroom? What are ways you build your classroom culture?

The post Strengthening Classroom Culture Through Projects appeared first on Maneuvering the Middle.

Here are my 7 tips/ideas for teaching volume in a way that students will get the most from it.

Start with Area

While most scope and sequences require mastering area (or surface area) before moving onto volume, the connection between area and volume cannot be ignored.

In fact, in all of the formulas for volume, the area of the base is abbreviated as ‘B’. And while this formula chart does give the formula for the various shapes’ areas, it does that separate from the volume formulas. 

Encourage students to shade the base in every problem and then sketch the base separately with its dimensions. This helps students to actually calculate the area of the base instead of  using only one of it’s measurements, which was often the most common misconception I witnessed. 

This Desmos Activity really helps make the connection between the area of the base and the height.

Write down the Formula + Draw the Shape

You don’t have to provide formulas when you first introduce volume! Have students make observations and predictions. For example, using a rectangular prism and a triangular prism with the same height and base measurements, ask – “how many triangular prisms could fit inside the rectangular prism?” You can do this for a cone and a cylinder too! Unifix cubes are the perfect manipulatives to use for students to understand how volume is measured in a rectangular prism. 

After you have introduced the formula for a shape, require students to write down the formula for the shape as part of their work. This will prevent students from leaving off multiplying by ⅓ for the volume of a cone or 4/3 for a sphere. I would even make this mistake on purpose, so students would shoot up their hands and say, “Ms. Brack, you forgot to multiply by ⅓!”

In addition, if a given problem lacked a figure, I required students to draw the figure and label the sides with the given measurements. 

Use the Exact Formula Chart They Will See on Testing Day

Set your students up for success for testing day! Provide students with the EXACT formula chart they will see when they take their state test. I made a class set on bright neon cardstock that I would laminate. It stayed out throughout the entirety of my Geometry Unit. 

During a geometry lesson,  I would always instruct my students to silently point to the formula we would need on their formula chart.  I would do a quick sweep to make sure students were on the right track. Don’t assume your students will understand how to find what they need from a formula chart. 

Start with an Exploratory Activity

This is pretty obvious, but volume is so hands-on! It is physically all around us. 

You can do this in a variety of ways: calculating the volume of various items like – tissue boxes, cereal boxes, oatmeal canisters, saltine boxes etc. A sleeve of crackers is the perfect cylinder and is a perfect way to show how the volume is the area of the base times height.

Our Exploration Activity helps students explore the relationships between cylinders and cones using play dough or dried beans. You can find it in our 8th grade volume activity bundle.

This is a free activity that sounds like a perfect way to include a project for rectangular prisms. Students are given a single piece of paper and instructed to create the largest possible box. 

Maneuvering the Middle Activities

Our activity bundles are engaging and hands on! With all of the shapes that are covered in middle school math, it’s nice to know that every prism, cylinder, cone, and sphere will receive practice.

• 6th Grade Geometry Activity Bundle
• 7th Grade Volume Activity Bundle
• 8th Grade Volume Activity Bundle

Other Tips

Use fabric measuring tapes over rulers. They are less expensive, take up less space, and students will not be tempted to sword fight. 

If you have access to unifix cubes, give students a specific volume and have them work backwards to create as many rectangular prisms with different measurements that satisfies that volume. 

What tips do you have for teaching volume?

The post Volume Activities and 7 Teaching Tips appeared first on Maneuvering the Middle.

Why Students Need to Solve for Y

While “solving for y” is not mentioned in any specific standard, it is necessary for so many other standards: 

1. A.2(B) write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y – y1 = m(x – x1), given one point and the slope and given two points
2. A.5(C) solve systems of two linear equations with two variables for mathematical and real‐world problems.
   

First off, slope-intercept form (y=mx+b) makes graphing possible. Slope-intercept form makes identifying your slope and y-intercept instantaneous. This question was pulled from a previous STAAR test:

This problem requires a student to convert standard form 8x+3y=15 to slope-intercept form which means they must solve for y. (They could also use -A/B which supports the steps needed to isolate y anyway!) This problem also requires point-slope form and then converting back to standard form – jeez! 

Slope-intercept form also supports solving systems of equations by graphing. Solving for y will support students using substitution to find a solution too. 

Essentially, students must be flexible in converting between the various types of equations: standard form and slope-intercept form.

Furthermore, in Quadratics, converting standard form to vertex form requires solving for y too. Similar to slope-intercept form, vertex form gives information about the function (like the vertex and the direction it opens) that cannot be determined in standard form. 

Teach Literal Equations First

Literal equations are equations with only letters. Typically, students are solving for an assigned variable. It can be extremely tricky for students to wrap their heads around this which is why I love how our Algebra 1 Solving Equations Unit covers it.

Maneuvering the Middle’s student handouts set students up to solve various step equations with numbers and then follow the same steps with variables only. (Need more ideas for solving equations?)

Start From Scratch

When students see something foreign, they assume that it is all brand new information, so start at the beginning:

• Review order of operations backwards
• Review what the inverse operations are
• Review what makes something “like terms”
• Review that if there is not a sign before a number, it is positive
• Review integer operations rules
• Review solving for a single variable when there are numbers present

Once this is ingrained in their head, then make the jump to solving for y when the variable x is present. Students get stuck usually because teachers assume that students know and understand more than they do. 

Try the X-citing Move + the Great Divide

If you are teaching converting from standard form to slope-intercept form, the steps will often be the exact same to isolate y. This mnemonic device is a way to help your students remember the steps required. Mrs. Newell’s math blog has other great ideas, so be sure to check it out. “X-citing move” reminds students to move x to the other side of the equal sign using inverse operations. “The great divide” reminds students to divide by y’s coefficient. Sometimes it is the simplest methods that make the biggest impact!

Substitution, Graphing, and Elimination

With so many ways to solve systems of equations or inequalities, it is important to teach when you apply different methods. Or better yet, have students make observations for what method works best and why. Solving for y supports solving systems by graphing or substituting.

• Substitution – an equation is already solved for a single variable
• Elimination – both equations are already in standard form and don’t require lots of manipulation to eliminate both of the x’s or y’s
• Graphing – one or both of the equations are already in slope-intercept form

What would you teach students regarding this problem?

• Students may want to graph – the answer choices have graphs, the numbers are friendly, but the equations are in standard form requiring the extra step of converting to slope-intercept form.
• You could substitute the answer choices’ solutions into the standard form equations to see whether the solution provides true statements.
• You could use elimination, but all of these require multiple steps. 

I think that is what makes this a good problem – students will have to weigh the pros and cons of each method with their confidence in solving. 

How do you teach solving for y?

The post How to Teach Solving for Y appeared first on Maneuvering the Middle.

Regardless of your answer, I think most math teachers would agree that multiplication fluency helps students see patterns, relationships, and supports more complex mathematical processes. If you are covering middle school standards, you don’t have a lot of time to devote to teaching or reteaching multiplication, but there are ways to support your students without adding too much work to your already heavy workload.

Be Strategic

There are so many websites and apps devoted to practicing multiplication facts. Check out some of our recommendations down below.

However, if you just assign these to students with zero direction or instruction, chances are their fluency won’t improve.  If these students have made it to middle school without mastering multiplication facts, then just giving them more of the same probably won’t move the needle to mastery.

Drill and kill is not known to improve fluency. One of the practices I took part in was printing a multiplication chart off for every student to put inside their math folder. For students who needed it, they had it there to use. I wanted my students to access the grade-level content, and if they could do that with a multiplication chart, then wonderful!

Brittany, from Mix and Math, says this about multiplication charts, and I whole-heartedly agree: “Giving students no supports to access grade-level work while they are still working to build their fluency with multiplication would be like sending the patient home from the hospital to heal without crutches. We want students to have some way to work with more advanced math concepts, even though they aren’t where we’d like them to be with multiplication yet. Consider giving students partially filled multiplication charts so that they are only using them for facts that they personally struggle with.”

I grabbed a free, colorful multiplication chart from Hannah at Math, Kids, and Chaos.

Capitalize on Friendly Facts

Generally speaking, students who haven’t mastered all of their multiplication facts usually have mastered their 0s, 1s, 2s, 5s, and 10s. These facts are actually a hop-skip-and-a-jump from facts like their 6s and 9s.

Teaching 6 Times Tables

Let’s take 6×6. If a student can solve 5×6=30, then adding another group of 6 will help students get to 36. Students need a solid understanding that 5×6 means 5 groups of 6 and that 6×6 would mean to add another group of 6 to 30.

Teaching 9 Times Tables

While I learned my 9 times tables by using the finger trick, I think using your 10s times table is actually best! Take 9×7. 10×7=70 which means 10 groups of 7 is 70; 9 groups of 7 would be 1 group of 7 less than 70. 70-7=63.

When To Implement Multiplication Practice

This can be tough because there are so many other standards to devote time to. So when do you devote time teaching multiplication fact fluency?

• Beginning of the school year warm ups – provide students with some multiplication facts practice via a simple Mad Minute style worksheet or a blank or partially blank multiplication chart. This strategy works because students are usually still easing back into math after a long summer, so every student will benefit. You can walk around and gauge which students will need extra support. 
• Stations – One of the ways to have stations that make planning as brainless as possible is to have one station always devoted to some type of fact fluency. I love Math Dash Ninjas for this because it is super engaging for students and you can easily differentiate the type of fact fluency that students are practicing. 

• Lastly, Fact Fridays is a great way to use 10 minutes of class practicing multiplication fact fluency. Kahoot or Quizziz have so many prebuilt games devoted to fact practice.
• Intervention Class – If multiplication fact fluency is something the majority of your intervention students struggle with, then I highly recommend diving deep into multiplication concepts at the beginning of the school year. How does one do that? First I would read this post and watch this video on Subitizing Cards where you can also grab a free download.

Websites and Apps for Multiplication Practice

These websites were recommended in our Facebook Group – Maneuvering the Middle VIPs. 

• Reflex Math – Requires student logins, but the program is adaptive and engaging. 
• Xtra Math – Requires student logins, but you can sign up to receive weekly progress reports of student progress.
• Transum Tablemaster – Free and requires no login. Practice is specific to one set of facts at a time. No mixed practice. 

Teaching multiplication facts an be tricky! How do you improve multiplication fluency with your middle school students?

The post Teaching Multiplication Facts in Middle School appeared first on Maneuvering the Middle.

Let’s see how important math vocabulary is to understanding and solving this problem. Could you solve this problem? I covered up a vital piece of information needed to solve this problem to emulate what a student might experience without knowing the vocabulary necessary to solve.

Model Using the Math Language

If we want students to use the words we are teaching, we need to practice using it ourselves. Vocabulary requires exposure. Using the words as frequently as possible, students will hear  the words as frequently as possible, increasing their comfort with the words.

If a student uses a vocabulary word incorrectly, then make sure to correct it. “Bottom number” is a “denominator.”

As teachers, we can jump ahead to the solving of a problem, but using the STAAR test question above, we should start by asking students – “What does surface area mean?”

Annotating word problems or questions is also a way to practice math vocabulary. Anytime we read the word “percent,” we wrote “/100” to remember that percent meant “out of 100.”Let’s look at another math vocabulary rich problem: 

Here are some questions you can ask to practice that math vocabulary. 

• What makes a number an integer?
• Is -53 an integer?
• What does absolute value mean?

Get Ahead By Previewing Vocabulary

If you are a Texas teacher, you can use this excellent document that will show you which vocabulary words are new to the grade level as well as words from previous grade levels. This is a great place to start for a word wall. 

• 6th grade 
• 7th grade
• 8th grade
• Algebra 1

Previewing the vocabulary for an upcoming unit is a great place to start when teaching math vocabulary. In my experience, students copying definitions killed the energy in class, but offering students a “kid-friendly” definition that you referenced daily and had them practice (using some of the ideas in this post) was much more successful. 

Display a Word Wall

Update 7/28/2023: Maneuvering the Middle now has a Middle School Math + Algebra 1 Word Wall.

As you can see in the video below, our Word Wall includes 190 essential math terms, their clear-cut definitions, and their visual representations.

We’ve included Spanish translations for all terms and definitions, ensuring a supportive and accessible learning experience for English Language Learners.

They were designed to be minimal prep and flexible to customize the formatting to suit your students’ unique needs.

Word walls are a vital part of any math classroom. You can learn more about word walls in this post.  If students are taking a brain break and staring off into space, they are likely staring at some math content. To have the most useful word wall, make sure words include a short definition, picture, and can be visible from the furthest spot in the classroom.

My word wall was constantly building. The wall started with 3 words in unit 1 and eventually built to just under 100 by the end of the school year. I purchased my sixth grade TEKS word wall here. 

Pointing out the addition of new words to the word wall and where students can access help if needed lets students know that the word wall is for their use! It is meant to be used!

Provide Opportunities to Use the Words in Context

When asking students questions, prompt the response to include vocabulary in their answer. This is the lowest lift, but it is so effective! Use a turn and talk and a cold call to get every student responding.

• Instead of: how do we divide fractions?
• Try: Using the word reciprocal, explain how we divide fractions.
• Instead of: What sides of the triangle are congruent?
• Try: Using the word congruent, describe what you notice about the sides of this triangle.

I read that you need to use a new word about 10 times before you remember it! Teaching math vocabulary is something that you build into your instruction.

Fun Practice for Spiraling Definitions

To keep vocabulary and definitions fresh, use any of these activities in the last few minutes of class:

• Flyswatter Games – If you want students to get familiar with your word wall, use the Flyswatter Game.  This is a very engaging review game. If you are like me and don’t bother to cover up anything in your room before a test, this will help remind students where to look when they are stuck.  Two students face off with fly swatters in hand.  You give them a prompt such as “2, 4, 6, 8” are examples of ______” And the first student to swat the word ‘multiples’ earns their team a point.
• Flashlight Game -This game is great for those last few minutes of class as a sponge activity.  Turn off the lights and use a flashlight to point to a word on the wall.  Students can then shout out an example, the definition, or even a counter-example. 
• Guess the Word – I played this in a PD, and immediately implemented it in my classroom. One student stands with the white board behind them facing the rest of the classroom. You write (or have a slide deck prepared) a vocabulary word behind the student. Students in the classroom take turns giving the students hints to what vocabulary word is written behind them. You see how many words the student can guess in a given amount of time. 
• Quizziz or Kahoot – Both have a vast library of vocabulary rich games.

How do you teach math vocabulary to your students?

The post Teaching Math Vocabulary that Sticks appeared first on Maneuvering the Middle.

1. Make your own or use MTM

Have students collect data about something they are interested in and put the data in whatever display you are teaching over the course of the unit: dot plots, box plots, histograms, circle graphs, etc.  Here are some ideas: 

• Sports data – Students pick a team and create a histogram or box plot to represent the data of the athletes’ heights, years playing, or whatever applicable data point students find interesting. Display everyone’s graphs and complete a gallery walk where students use Post-it Notes to write down questions and observations. 
• Real-time data – Using a similar format, ask students to track their own usage data over different forms of technology over the course of the week. 
• Data from their own lives – Students can use information from their classmates, friends, or themselves.  Maneuvering the Middle’s performance task is perfect for this. (Find it inside this 6th Grade Statistics bundle.) Students take a survey of the class and calculate the interquartile range, mean, median, and mean absolute deviation. They are then asked to graph this data in various ways. 

• We also have an Introduction to Statistics Sampling Activity included in our 7th Grade Data and Statistics activity bundle.  Students take data from their class like physical attributes and interests and compare it to the 100 People: A World Project. 
• Infogram is a perfect tool to use for students to display the data they have collected.

2. PBS Learning

If you haven’t had an opportunity to explore PBS Learning Media, check it out! Everything is free and there are many real-world examples and videos complete with lesson plans. Filter by subject, topic, or lesson type. These lessons serve as great application opportunities to whatever you are learning in class.

Here are 3 activities that I found compelling:

• Describing Distributions Using Shoe Sales Forecast 
• Measures of Center Activity Using Highest and Lowest Paid Jobs  
• Scatter Plots Using Arm Span 

3. He Said, She Said Freebie

I love He Said, She Said activities! Our Data & Statistics He Said, She Said activities are a hands-on way for students to analyze others’ work and apply their knowledge about box plots and two-way tables. Students are asked to justify their answers, so it pushes students to use those higher-level thinking skills. He Said, She Said activities are so versatile and can be implemented 100 different ways. 

• Use them in pairs or in groups of 3-4. With many group activities, I liked to start students off working collaboratively, and then give them 5 or so minutes at the end to work independently. This allows them to test their ability, and for me to see who still needs support.
• Use them as a review. Box plots were a skill that my students needed lots of practice on, so these activities provided extra opportunities for students to ask questions.
• Use them as an extension. Students finished early? He Said, She Said cards can easily be hung around the classroom or put on a ring for students to grab when they have completed their assigned work for the day.

4. Desmos

I love how interactive Desmos activities are. There is something for every grade level on a range of statistics skills. 

• Overview of Statistics Activities Organized by Topic – Start here to see everything Desmos has to offer for data and statistics.
• Dot Plot Activity
• Scatter Plots Activity
• Mean Absolute Deviation

5. Digital activities

Digital activities have so many pros. Easy to assign to remote learners, scaffolded questions, and complete with a short formative assessment. They also make for a no-brainer station or rotation. 

• 6th Grade Statistics Bundle or individually: Box Plots, Histograms, Measures of Center, and Measures of Variability
• 7th Grade Statistics Bundle or individually: Comparing Box Plots, Population Inferences, Samples and Populations, and Comparing Dot Plots
• 8th Grade Data and Scatter Plots Bundle or individually: Intro to Scatter Plots, Two-Way Tables, Scatter Plots and Trend Lines, and Scatter Plots and Predictions

If you want access to all of the resources shared today, be sure to check our All Access membership!

There you have it! 5 ways for your students to engage with data and statistics. What data and statistics activities do you love?

The post 5 Statistics Activities for Middle School appeared first on Maneuvering the Middle.

What is the CRA?

I like this definition best. “The CRA is a three-stage learning process where students learn through physical manipulation of concrete objects, followed by learning through pictorial representations of the concrete manipulations, and ending with solving problems using abstract notation.”

ROUTINES AND PROCEDURES

How do we set students up for success when using manipulatives? 

• Tell students the purpose of using manipulatives. Explain how it is foundational to their success in math and how it is something you want to be able to trust your students with. 
• Tell students WHAT TO DO. Telling students what to do is better than telling them what not to do. Example: manipulatives stay on the dry erase mat versus don’t throw manipulatives.
• Have students clear their desks of computers, binders, or anything else that can obscure your view of how the manipulatives are being used.
• Assign a materials manager as a group role. This student is responsible for making sure their group is using the manipulatives correctly and that they all get returned.
• Build in time for cleanup! If you are using manipulatives or any extra type of supply, give yourself lots of extra cushion at the end of class for clean up. Make sure to use a timer to encourage speed. 
• If you want to minimize the number of manipulatives out at a time or you don’t have a whole class set of manipulatives, use a station or small group table for the manipulatives. This will narrow down the materials to manage to one area of the classroom and a smaller set of students.
• Remember, the more practice students have using manipulatives, the better they will get at cleanup and using them responsibly.

BEST PRACTICE #1: LET STUDENTS EXPLORE

There are 2 things you will have to balance as you model for students – the conceptual understanding piece with the procedures of using the manipulatives, but also I would encourage adequate exploration time. Exploration time can be really meaningful if you have strong and planned questions to guide your students’ thinking. 

In this post, I wrote about not taking opportunities away from students to think. Sometimes demonstrating exactly what to do removes any of the thinking for students. It makes them follow a procedure to the T, which doesn’t allow for flexibility in their thinking. 

Here is an example of how you can give students time to grapple with the content without modeling exactly what to do for them.

Let’s consider adding integers. Conceptually, students will need to understand that one side of the counter and its color represents negative and the other side represents positive. Students will also need to understand negative and positive numbers value in relation to zero. Perhaps, students can discover zero pairs without your explanation?

Now this will not always work with more complex situations, but I think with proper scaffolding and questioning, you should give students the opportunity to try something before showing them the exact procedure for working with the concrete representations.

BEST PRACTICE #2: Pair Manipulatives and Representations with the Abstract

“Explicit instruction that involves the use of manipulatives should also include the presentation of the numerical problem (Miller, Stringfellow, Kaffar, Ferreira, & Mancl, 2011).”

This study makes a great point. Students need to connect the manipulatives to the abstract as they are working, which means as they are working with the manipulatives, they need to be writing down what they are doing as they solve.

For example, when a student is using algebra tiles, connecting that the physical removal of two tiles from each side of the equation is the same thing as writing down “-2” on both sides of the equal sign will build that deeper understanding. We want students to understand WHY they are doing something.

Be sure to grab our freebies for Getting Started with Algebra Tiles.

The best part of the CRA method is how students will internalize the process. Some students will arrive at the abstract and never look back. Other students will draw models to support their thinking, while some students will request using a set of manipulatives well after you have taught the algorithm. That is the beauty!

Continue to spiral these concepts throughout the year using the different parts of the framework. If you go over the concrete and the representational on one day and then spend the entire year using the abstract to solve equations, then chances are your students will do the same thing too. 

WHEN AM I GOING TO HAVE TIME TO DO THIS?

If you are already feeling very stretched with your scope and sequence, here are my thoughts: 

Many times our scope and sequences allow some time for reteaching when the students do not master the concepts. Consider that by spending time building the conceptual understanding using concrete and pictorial representations, that you may not need the extra time for reteaching.

You can teach the concept in tandem using the manipulatives. If you are already going to spend time teaching math, you may as well introduce tools that will support student understanding. 

“Students who use concrete materials develop more precise and more comprehensive mental representation, often show more motivation and on task behavior, understand mathematical ideas, and better apply these ideas to life situations.” 

“Students are more apt to gain and retain an understanding of math concepts when they are taught using CRA.” 

These two quotes from this study are the positives seem to outweigh the negative of being a little bit behind. Developing these foundational skills will only benefit students in the long run.

What are your strategies for teaching math concepts?

The post Strategies for Teaching Math Concepts appeared first on Maneuvering the Middle.

The CRA framework is an instructional strategy that stands for concrete, representational, and abstract; it is critical to helping students move through their learning of math concepts. 

To fully understand the idea behind CRA, or concrete representational abstract, think about a small child learning to count. They may learn counting to 10 by memorizing a song. Then, these numbers are used to teach them to count. First, they may be counting blocks by pointing or moving the object. Then, they may be able to count dots on a page or something on paper that represents that same concept. Lastly, they may be able to then look at a picture with five dots and not need to count it, they have learned what five “looks like”. This process takes place with all different math concepts and new learning.

Looking for more math intervention ideas?

This is part 3 of our Instructional Design series. We’ve covered how to unpack math standards and how to use questions to push students to think more critically. Make sure to grab our lesson planning freebie down below!

What is the concrete representational abstract framework?

The CRA framework helps us to present this learning in a specific method to help tie all the connections together. There have been numerous studies measuring the effect of using the concrete representational abstract (CRA) sequence for students at risk of failure. You can read more here, here, and here. 

This teaching method breaks down the concept (examples: multiplication, subtraction with regrouping, subtracting integers) in a methodical process in which students move from one phase to the next.  

Sometimes CRA is referenced as a sequence, so moving from one phase of understanding to the next. But more so, it is called a framework, which shows that all three of the representations can be used simultaneously. If we were teaching solving equations, students would actually use the algebra tiles, draw a picture of the model, and then write it mathematically at the same time.

Concrete

When students have a concrete understanding of a mathematical concept, they are utilizing manipulatives to demonstrate the math.  For example, when my son was learning to count, I wanted him to physically touch or move the items he was counting. In doing this, he was developing one-to-one correspondence.  He was internalizing that each item represents one.

When our middle school students are learning to solve equations, we want them using algebra tiles to physically remove manipulatives from both sides of the equation.  This allows students to internalize that an equation must remain balanced.

There are many reasons why teachers might skip this crucial step:

• Practice with concrete manipulatives takes time and energy. It is not easy to facilitate with 30+ students.  
• It can be expensive to purchase manipulatives.
• It can be frustrating for students who seem to already grasp the concept.  

Why it is beneficial:

• Students are using their kinesthetic learning style.
• Students are engaging with the content in a deeper way, and abstract concepts like “isolating the variable” are given meaning.
• This foundation is formidable so that they can move to conceptual understanding and aren’t just following a list of steps with no meaning.
• Research shows that it is essential for students who have a shaky math foundation.

Representational

The representational part of the framework refers to students drawing a representation of those concrete materials. 

In the case of my son above, that would be counting dots on a page or drawing eight tally marks to represent the number 8.  

In the case of our middle schoolers and solving equations, this would look like drawing the algebra tiles, creating a key, and demonstrating how the process works by crossing off the tiles or grouping as they solve.  

We see this quite a bit in middle school.  From integer counters being sketched to a graph that represents a linear relationship, our standards include many representational aspects. 

Why We Might Start Here:

• This is a little easier to execute in a classroom full of students.
• The standards include the representational language. 
• It also leaves some gaps for students who need to understand why you are crossing out the positive integer circles.

Abstract

Math can be a very abstract subject.  When you think about solving an equation and finding the value of x, why are we doing this?  Why do we need to know this? The value of x is abstract. It is always changing. It doesn’t actually mean anything without context.

I think this is where it is really difficult for students who struggle.  When we don’t provide a context for learning and for the application of the abstract, it just seems too confusing.  I think it also warrants the phrase, “I don’t get it.” Many times students don’t even know what to ask because the content means nothing to them.

This is when math becomes too procedural and you risk students understanding the process, going through the motions, and yet lacking number sense.  When I was a student, I was taught solely in the abstract. It wasn’t until I was a teacher trying to make sense of models before I had several AHA! moments. Why didn’t I learn this when I was in school??

Best Practices

Shelley Grey says, “When you teach a math lesson, make it your goal to incorporate concrete, representational and abstract into the same lesson. This way you can be certain that you are differentiating for all your students, regardless of where they are in their understanding.

And it does naturally lend to differentiation — you are scaffolding, using different parts of the brain, and then eventually allowing students to work with a method they feel comfortable with. 

You as the teacher can make sure to use different methods when you are tackling problems. Just because you taught the algorithm for multiplying fractions already doesn’t mean you can’t draw a model to solve at a later date. This ensures that students don’t forget how to use the models themselves.

How do you teach difficult math concepts?

The post Concrete Representational Abstract Sequence appeared first on Maneuvering the Middle.

Higher level thinking is a goal for many teachers. Students retain knowledge when they have thought deeply about it. And in order to get our students to think at a deeper level, we have to be asking questions that require them to access those parts of the brain. 

Encouraging higher level thinking is so interconnected to your classroom culture, the rigor of your content, and then of course the types of questions you are asking. Then we get into the challenges of collecting your students’ thinking – are they answering verbally or on paper? Can you give feedback quickly? 

It makes sense that would be the case. If the whole entire goal of education is to provoke thought, then of course there would be so many facets of instruction connected to it. 

This post is part 2 in our Instructional Design Series. Check out Unpacking Math Standards and the CRA framework. And if you haven’t grabbed our lesson planning template yet, make sure to grab it down below.

LISTEN ON: APPLE PODCAST | SPOTIFY

THE PROBLEM

The purpose of questioning in our classrooms:

1. Questions give us feedback on what and how our students are processing and internalizing their learning.
2. Questions also can be used to guide student thinking. 

If you are asking questions throughout class and then students are struggling on the assessment, it’s possible that your questions in class are not aligned to that which a student is expected to do on the assessment.

Peter Lijedahl touches on this a bit in his book Building Thinking Classrooms, where he talks about “studenting”. Doing behaviors that look productive, “filling in notes or mimicking the teacher” but they aren’t actually reasoning their way through the problem.

We can make some small adjustments to the questions and interactions in our classroom to build those thinking muscles. 

Building a Strong Classroom Culture

For students to think and answer questions at a higher level, we want students to be comfortable with making mistakes. 

Phil Daro (co-author of the CCSS) says that teachers put too much emphasis on the answers; answers are part of the process, but they are not the only learning outcome. That wrong answers are part of the learning outcome. 

What if you framed wrong “answers” as “discoveries”? If students reached a wrong answer, you instead talked more about why that approach doesn’t work instead of how to get the right answer.

The Questions You Are Asking

If we are focusing on higher level thinking using higher level questions, we have to focus on the types of questions we are asking. 

To do this, I am going to refer to Bloom’s Taxonomy which is a hierarchy of learning objectives that rank lower order thinking skills to higher order thinking skills. 

Now remember this is a pyramid, so those lower levels have a purpose and provide a foundation. These are the easy questions to ask, the ones that just roll off the tongue and don’t really require much forethought.

In math it sounds like – “What do we do next?” “What is the solution?” “What operation will we use?”

The higher level thinking questions are those that take a little more forethought and planning. These are the questions that require students to analyze the relationships in a process or make an evaluation on how to approach the problem based on given information.

One of the easiest ways to elicit higher level thinking is to teach students multiple approaches to a specific concept and then have students evaluate and justify why one process may be better than another based on given information.

Let’s try a non-example in a typically procedural type of skill – solving equations. 

Teacher: 4x-2=18 What do we need to do first?

Student: Add two to both sides.  [Teacher adds 2 to both sides]

Teacher: 4x=20 And what do we do next?

And here is a way to teach the same problem using a mix of higher and lower order thinking skills.

This is where I would make a plug to use algebra tiles for this concept because students can visualize and explore how solving equations actually works with concrete models. When students move to the algorithm, their responses will be richer because the abstract is now concrete. 

Not every skill or concept warrants a question in the highest order of Bloom’s Taxonomy. As you scaffold instruction, the types of questions you ask in Bloom’s hierarchy will move up. 

You can find many question stems for math with a simple google search, so I took a problem from a He Said, She Said Activity from our Proportional Relationships bundle.

Wait Time & Eliciting Answers

In my fifth year of teaching, our campus read the book Quality Questions which helped me to learn about the technical term of “wait time” and that there were different wait times. 

Wait time is the amount of time we sit in silence after a question has been posed. Then, there is actually a second wait timere which is the amount of time that you wait after a student responds. 

The idea behind wait time is that we are giving time for students to think and that as soon as we accept an answer or acknowledge that the answer is correct, then everyone else stops thinking.

Here are a few ideas to incorporate this into your classroom:

1. This can be that they have to write something down, show a finger for a multiple choice response, or they share their answer with a partner while you circulate.
2. I like to give students a little bit of heads up if I want them to share, known as “warm calling”.  You ask a question, give a minute of silent think time, ask students to share with their partner, and then circulate. I listen to things I want the whole class to hear. If I hear a great response from a student, I let them know that I would like for them to share, so they aren’t caught off guard. Then when we come back together as a class, I have that student respond. 
3. You can do a fist to five. Typically, fist to 5s are used at the end of the lesson for students to gauge how well they feel about the lesson on a scale from 0 (did not understand) to 5 (completely understand). In the case of answering a question, you can require all students to show their fist to five to assess how well they think they can answer the question. It actually gives you some information – which students are completely lost, which students are ok, and which students think they have exemplary responses. I would call on a mixture of 4 and 5s while making sure my fist and 1 students are listening in. Then I will go back to a fist or 1 student and ask for them to put the response in their own words. 

Note: I didn’t do this for every single question I asked. Who has the time? I reserved these techniques 2-3 times a lesson for questions that warranted more time and thought.

Two Quick Hits

This first idea comes straight from Building Thinking Classrooms. Before you model a skill, first give students the problem and ask for them to try it on their own. Give them just enough background knowledge to get them started. 

You can do this for many math skills. Estimating square roots, operations with rational numbers, geometry skills, solving equations. It doesn’t have to be done daily, and use your discretion for when this approach may not benefit students. 

My second quick hit is something that I picked up in a training that made a huge difference in how I ended my lessons.  I adjusted my language from, “Are there any questions?” to “What questions might someone have about what we learned today?” 

• The former resulted in silence. And made me think as a teacher that I was good to go. 
• The latter resulted in students trying to think of something that wasn’t covered. In addition, students who may not want to admit that they have a question or don’t understand something, would be more willing to inquire. 

How do you incorporate higher level thinking in your teaching?

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Unpacking math standards helps teachers build the foundation for what content they are teaching to students. Subsequently, unpacking math standards provides a framework to create actionable lessons. Essentially, unpacking math standards helps us answer these two questions:

1. What are students learning?
2. How will they learn it?

This is part 1 of our Instructional Design series. Come back to read more about higher level thinking and the CRA framework. And make sure to grab our free lesson planning template down below!

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WHY DO WE LESSON PLAN?

Much like creating a to-do list to alleviate the mental load of upcoming deadlines, lesson planning keeps all of our thinking about a topic or skill in one place where it can be shared, referenced, and revised. 

When we are prepared intellectually to deliver a lesson, we allow for real-time adjustments and instruction to the depth the standards require. In addition, the physical plan allows a space for collaboration and feedback.

Not ready to lesson plan quite yet? Need the bigger picture?

Grab our FREE Middle School Math + Algebra 1 Pacing Guides.

MISCONCEPTIONS ABOUT MATH STANDARDS

• Standards do not need to be taught sequentially or in isolation. They do need to be organized with purpose though!
• Standards are not so rigid that there is only one way for them to be taught. 
• Standards are not to be interpreted in a vacuum. The more math teachers you can have in the process, the more likely you will have a full picture of the scope of the standard.

HOW TO UNPACK MATH STANDARDS

Let’s talk about how we go about unpacking math standards using a 6th grade TEKS standard. 

• 6.7D: generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties
• This comes from the strand 6.7 “The student applies mathematical process standards to develop concepts of expressions and equations”

Step 1 – Read the standard (and related standards) in its entirety

• There is a related standard that asks students to determine if two expressions are equivalent using concrete models, pictorial models, and algebraic representations. (6.7C) 
• This is helpful because I can support my instruction of that initial standard with concrete and pictorial models. It gives me more information about how I can teach it.

Step 2 – Create a T chart listing the knows and the dos 

• The Dos are the verbs. What the student is expected to do.
• Examples include: apply, solve problems, represent, determine, calculate, predict, write, model, compare, convert, and describe. 
• Make sure that whatever the student is doing in class is the same as the verb in the standard or that the lesson will eventually support it.

• It is crucial to understand what the verb means. Oxford dictionary defines generate (in the math context) as producing (a set or sequence of items) by performing specified mathematical or logical operations on an initial set. The lesson should be focused on the student creating equivalent forms of the expression using the properties.
• The Knows are the content. The knowledge that the student must have to execute the verb.  Typically, this is going to come from the nouns. So in our case, students will need to know how all of the properties function in order to generate equivalent expressions. 
• One other best practice to unpacking math standards is to determine if the standard is more conceptual, procedural, or application. If the standard is very procedure heavy, I am going to want to make sure that there is some conceptual understanding introduced and application type of problems too. We want our lessons to be well rounded.
• If you have done this and are still unsure what the standard means, then I recommend looking at a state test question to see how the standard is assessed. 

Step 3 – Look at the vertical alignment

Texas teachers, grab yours here. I like to read the math standards that are connected from the previous year to see what my students should know already or more likely, what I will need to refresh them on before they can be successful with the new standard. 

Step 4 – Write a learning target or lesson objective

Lesson objectives should be measurable and they should be student friendly. Different schools follow different guidelines – your school might use “I CAN statements” or “student will be able to” language. My preference is below:

Remember that whatever question, problem, or activity your students are completing, it should support your learning target or objective.

Step 5 – Write Big Ideas and Essential Questions

Big ideas are the concepts that transcend your unit and connect the content to their real lives and make it relevant. Perhaps students won’t walk away from your data and statistics unit remembering exactly how to find the median, but they will take the big idea that data can be represented graphically in order to solve problems and draw conclusions. 

Essential questions support your unit’s big ideas. These are recurring questions that can be asked all unit long that are designed for students to have that lightbulb moment. They allow open ended discussion from your students and to stretch the thinking of your students. Use them at the beginning of your unit to pique student interest and/or at the end of the unit to summarize or synthesize their learning over the last few weeks. 

How do you unpack math standards? What is your lesson planning process?

The post Unpacking Math Standards When Lesson Planning appeared first on Maneuvering the Middle.

Math manipulatives are some of the best ways to introduce a new mathematical concept and are the foundation of the C-R-A. They help to form a solid mathematical foundation as students move from concrete understanding to representational to abstract.  By providing the opportunity to concretely work with the manipulatives, students are able to develop a conceptual understanding rather than just memorizing procedures. Today, I want to share what I consider essential math manipulatives.

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MUST-HAVE FOR MATH

Two-Color Counters:

Math you will use them for: integer operations, dots as a real-life dot plot, probability experiments, and stacking to model the volume of cylinder

Budget solution: Buy two colors of sticky foam (you can get at Hobby Lobby, glue them together and cut into tiny squares. They do not have to be circles for most applications.

Individual number lines:

Math you will use them for: modeling integer operations, ordering numbers, inequalities

Budget solution: Grab our vertical number line freebie, print and laminate. The set I bought from Amazon lasted about 5 years. I hole-punched each one and stored them on a small hook at my small group table. 

Algebra tiles:

Math you will use them for: Algebra tiles are a lot like two-color counters; in that they can be used time and time again.  Solving equations is the first thing that comes to mind, but also consider using them when teaching properties of operations, specifically the distributive property, and even when incorporating the area model. 

Budget solution: Again, I think purchasing some foam, putting on some Netflix and cutting for a few hours can save you money.

3D shapes:

I firmly believe that 3D shapes are a necessity!  If I could manufacture my own, I would make them about 18 inches tall.  Sure, there are online models, but it is really nice when students can physically touch all the faces, vertices, edges, etc.  If you are on a budget, just bring some from home! 

Budget solution: Bring items from home. An oatmeal container is your cylinder, a rubix cube is your cube, a tissue box is your rectangular prism, an ice cream cone is your cone.

NICE TO HAVE MATH MANIPULATIVES

XY coordinate pegboard:

This might be my new favorite!  I am not sure how long they have been around, but they are new to me.  These are excellent for making slope more concrete and for comparing graphs.  Beware that the pegs are included, but rubber bands are not…maybe that was intentional.   

AngLegs:  

Another really cool and new(er) math manipulative to help students understand triangles and all the different angles involved.  This sure beats the string and three points. 

Geo Reflectors and Patty Paper:  

An oldie-but-goodie manipulative!  The Geo Reflectors are awesome for reflections, and the patty paper makes translations a bit more tangible.  

If you are working in small groups the great news is that you don’t need a class set of each of these.  Of course that is nice, but with school budgets what they are, you can totally get away with a small quantity that you add to over time.  

Fraction bars or circles:

I personally prefer the bars, but the circles are fine, too.  These make so much sense when learning about equivalent fractions, how to rename fractions, and how to simplify fractions.  Once students are really good with the models, it makes the connection to the algorithm so much easier! 

Pattern blocks:

I would love to know how you incorporate pattern blocks!  My friend, Brittany from Mix and Math, demonstrated how she uses them with multiplying fractions, and my mind was blown!  Amazing! 

Dice:

Dice obviously are super helpful with probability, but I also love how they can generate various numbers and you can then do things with the numbers.  One day, students were in partners, each one rolling die to create fractions. From there they would simplify the fraction or rename it. Easy, quick, hands-on practice! 

Spinners:

Perfect for probability, data, and statistics.  If you don’t own physical spinners, consider using this site to utilize digital ones on a device. 

NOT MATH SPECIFIC BUT HIGHLY RECOMMEND

While I believe that manipulatives are amazing and make math come alive, I would be remiss to not include things I used in my classroom daily to either aid using the manipulatives or just to make math more hands-on.

Dry-erase pockets, Small Whiteboards, Dry erase markers  

These dry erase pockets (or whiteboards) are perfect to spice up the typical paper and pencil practice.  Plus, you save on paper! Win-win!

For students, I recommend using the skinny expo markers. Students write big, so the skinny point helps them use their writing space efficiently. They are smaller, which makes them easier to store and basically impossible to flip. 

Colored card stock:  

Make your task cards or anything else you don’t want to print off year after year by laminating. Also, an MTM user recently shared that her and a few members on her team worked together to print and laminate every single activity. They put it all in a large tub and they work together to share it throughout the math department. Talk about teamwork!

STICKY NOTES:

Another handy and easily accessible office supply that can be used in the math classroom!  We have a list of 15 ways to use sticky notes for math practice here.

BUTCHER PAPER:

This is probably available for free in the workroom.  It is nice to have for students to work together on large pieces of paper! An easy way to make any boring worksheet interactive is to cover tables with butcher paper, provide markers, and give students problems to work with.

WAYS TO PAY

Manipulatives are not cheap. Take into consideration that you may need 30+ sets of something and it can really add up. Here are some suggestions:

1. Buy a set for just you or your small group before you commit to a class set
2. Create an Amazon wishlist and share on FB or with parents. One thing I have realized is if parents don’t realize there is a need, they can’t help! If your school has a PTA, ask if they can cover the cost. 
3. Donor’s Choose
4. Ask around! My fellow 5th grade teacher had collected many manipulatives since she had been teaching about 10 years longer than me. She was always happy to share when I asked. Maybe you will discover an old closet filled with items you can use.

What other math manipulatives do you consider essential? 

The post Essential Math Manipulatives appeared first on Maneuvering the Middle.

Math Stations, when done correctly, can solve many classroom challenges. Math stations can meet the needs of various learners, provide practice on spiraled concepts, and it puts the responsibility of learning on the students. Seems like a win-win! In addition, technology has made personalized learning easier than ever!

You may be eager to try math stations (or math centers), but are unsure of where to start. Hopefully this post will give you a little confidence to try them out.

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Stations Wouldn’t Work in My Classroom

If you just thought those words to yourself, then let’s get all of the roadblocks to stations out and in the open:

• You may not think you have the bandwidth to create or find multiple activities for a single lessons
• Your class periods are too short
• Your students will be off task
• You don’t have the space or your class sizes are too big
• You might not know where to even begin – how do I start this? 

Starting Small with Math Stations

In that case, I will remind you to start small. No teacher should commit to an entirely new approach to teaching without dipping their toes in the water first.

Small Groups: Start with just pulling a small group while the rest of the class works on something else. Get used to the practice of using data to lead small group instruction or intervention to students who need additional support. Not sure where to start with small groups? This post and this post are great places to start.

Weekly Stations: Introduce stations as a once a week activity for the entire class period. For example, every Friday is Station Friday. On that day, students get to practice what they learned that week with stations. Whether you have 40 or 90 minute class periods, students can work on what they learned from Monday to Thursday. Station 1 is Monday’s skill, Station 2 is Tuesday’s skill and so on. You can make your small group one of your stations that every student visits, or pull specific students for small groups while they are participating in stations.

Keep Transitions Easy: Students don’t have to actually move. If students out of their seats or managing a transition sounds overwhelming, then perhaps, the tub or folder rotates instead. 

Keep the Station Group Small: Maybe you have 30 students but only 3 activities for stations. Ten kids in a station will not work, right? Well, you can always have 6 stations with repeated activities. You get to keep the groups small but not double the amount of work to plan. 

Once you and your students are more comfortable with the processes put in place, you can consider these ideas.

Framework for Stations

The framework of M.A.T.H. is a great jumping point for a math teacher. It’s memorable for both students and teachers.  

• M – Meet the Teacher
• A – At your Desk or Assignment
• T – Technology
• H – Hands On

You may be familiar with student choice boards or playlists that would work in a station setting. In fact, one of our MTM teachers reached out this morning and shared her playlist using our All Access videos and MTM materials. It was phenomenal to see her utilize the All Access resources and allow her to provide small group instruction and support.

Best Practices for Math Stations

• Visuals and timers are your friend! Set a big, loud timer that all students can see and hear to keep everyone on track. 
• If you are having students move, practice! Set a timer for one minute and set high expectations for how you want this done. 
• Simplify your groupings at first. Groups can be homogeneous or heterogeneous learners. The groups can stay the same for an entire unit or few months. Since I had tables, my groups were their table groups. On the other hand, I am reading Building Thinking Classrooms, and the author, Peter Liljedahl, encourages teachers to group students randomly.
• Change it up, but also keep the routine. For example, there is always a tech station, but what program they use changes each week.
• Have a plan for students needing help or checking their answers. If necessary, provide students with something self-checking or an answer key that they can look at. Maybe they input their solutions into a Form or scan a QR code to check their work. 
• Think about the amount of time it will take to complete the activities and try to make them equal or longer than the length of the station. What you don’t want is one station that takes 5 minutes and another station that takes 20 minutes to complete. 
• The Maneuvering the Middle activities that best lend themselves to stations are: card sorts or card matches, She Said He Said, Find It Fix It, a Maze or a Puzzle Train. You can find our activity bundles here.

All of our hands-on activities, digital activities, student video library, and standards-based curriculum are included in All Access. Implementing Math Stations is attainable when you aren’t creating all of the materials from scratch!

Have you started implementing math stations in your classroom?

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Should teachers assign homework? Should you assign homework to your students? The answer to that question is dependent on a variety of factors, so let’s dive in. 

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What is the purpose of homework?

What is your purpose behind assigning homework? Here is a quick brainstorm of how you might answer this question:

• Homework is required.
• I need a specific number of grades.
• Students need to practice.
• You believe homework builds a habit of responsibility.

The next question to ask yourself is: “Is my current situation working well?” For example, if you assign homework daily and only 50% of students complete it, then you may need to reevaluate. 

Academic Pros of Homework

Homework has many benefits. Even as a student, I remember working on my math homework, and having some aha! moments. Many teachers depend on homework because their class periods are so short. Homework allows for students to practice what they learned in class when class time doesn’t allow for it.

Flipped classrooms depend on students watching videos at home. While they aren’t working problems independently, they are still learning at home. This allows them to do the majority of their work in class, removing the barrier of trying to practice something you don’t understand with no assistance.

Lastly, our brain is a muscle that does grow as we continue to use it. If learning an instrument or playing a sport requires practice, then so does math. 

Social-Emotional Pros of Homework

Homework isn’t just about knowledge. Homework can build a variety of other valuable habits – responsibility, ownership of their learning, and time management. 

If my students weren’t taking class work seriously, all I had to say was, “Whatever isn’t completed in class will be homework,” and students QUICKLY got back on track. Incentivizing students to use their time wisely in class can help students stay on task. 

Lastly, in some cases, homework allows parents to see what their kids are learning and their child’s academic strengths/weaknesses. In years that I didn’t assign homework (when I had 90 minute classes), parents reached out often to ask what students were working on since they never saw homework. 

Academic Cons of Homework

You probably don’t need me to list them because you already know! All those amazing homework pros that were listed above become moot if students don’t actually do it. Homework isn’t actually practice or an indicator of what students know because it can be copied from a friend or apps like Photomath make it super easy to cheat. 

Not to mention, some students would rather just take a zero than complete the work, so now you have missing grades to deal with. And for the students who do complete their homework with fidelity, well, they can be practicing it incorrectly without immediate feedback. Which is why I highly recommend something that is self-checking like a riddle or mixed answer key.

Social Emotional Cons of Homework

While research shows that there is a correlation between completing homework and academic success, it does not show that students do better because they do their homework. Cathy Vatterott, an education professor at the University of Missouri-St. Louis, stated “Correlation is not causation. Does homework cause achievement, or do high achievers do more homework?”

Some parents and teachers argue that students have already spent 8+ hours at school. Students benefit from resting, playing, and spending time with their families. The whole child should be considered. 

Assign Homework, but Do It Purposefully

According to this recommendation, homework should follow the 10 minute rule.  Multiply the grade level you teach by 10 and that is how many total minutes a student should have of homework of all subjects for one night. If you teach 6th grade, students should have 60 total minutes of homework a night. 

With this recommendation in mind, you have to consider the varying abilities of your students. A 10 question assignment may take one student 10 minutes to complete while it may take another student 1 hour to complete.

Which leads to my next point, it has to meet students’ needs.  Online math homework, which can be designed to adapt to students’ levels of understanding, can significantly boost test scores according to this study. 

These 5 questions from Edutopia give a great framework to help guide what type of homework you assign to your students. 

1. How long will it take to complete?
2. Have all learners been considered?
3. Will an assignment encourage future success?
4. Will an assignment place material in a context the classroom cannot?
5. Does an assignment offer support when a teacher is not there?

If you decide that homework is beneficial to your students, here are 5 best practices for implementation:

1. Give less homework more frequently
2. Ensure that students are practicing what they just learned
3. Provide feedback as quickly as possible
4. Explain to students the purpose of homework and how it will be evaluated

If you need Independent Practice (whether that is homework or in class practice), All Access has you covered! Each lesson comes with an aligned Independent Practice.

Many middle schools specifically are moving towards a model (or already have) that allows for a tutorial or advisory period. Utilize that time period and teach students to do the same. 

Hopefully, some of these thoughts will help you to weigh your options and come to a conclusion that meets both your students’ needs and your philosophy and approach to teaching. Let us know in the comments – do you assign homework?

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We are so excited to announce the launch of our All Access membership community for middle school math and algebra 1 teachers! Our math curriculum is receiving some fun changes!

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Some Context

Our MTM team worked hard to create resources that would be helpful for teachers and students when the education world changed in an instant.

• In the immediate shutdown, we put together 13 free videos for any math teacher to use in their classroom. Thousands of teachers used these videos.
• We then created additional videos and practice pages that we thought would be useful in a pinch. 
• Then, last fall we worked to transition many of our paper-based activities into activities that could be used digitally. 

However, we knew that we wanted to be able to offer something all-inclusive.

Teachers Don’t Need to do it all

Let’s disregard the myth that teachers have to do (and be excellent at) everything in order to be considered a great teacher. There are so many different facets of teaching (organizing, planning, execution, relationship building, and at least 100 more); it is unrealistic to think that a teacher should be excellent in all of them. 

What if you could be the best teacher you can be and make an impact on students because your energy is focused on execution, intervention, and student relationships?  Not constantly lesson planning and spinning your wheels trying to pull parts of a lesson from various websites and textbooks.

What is All Access?

That is where we want to help! What exactly is All Access? 

All Access is an all-inclusive membership for middle school math teachers. We have resources for grades 6-Algebra 1 and one of the facets is our standards-based math curriculum. All Access is replacing the math curricula in our shop.

It is ready-to-go math resources with our new growing student video library! 

Let me break that down for you:

• Ready-to-go math resources: that includes our units (guided notes, independent practice, and assessments), our hands-on collaborative activities — like scavenger hunts, task cards, our find it and fix it activities, all of those fun things, and our digital resources that are designed for use with Google Slides and PPT. 
• New growing student video library: We are working on a comprehensive video library that will begin rolling out in September of 2021. I wish I could say they will roll out immediately but we really want feedback from you and want to take our time to get them right. We know that video lessons allow classrooms to be more flexible, create opportunities for teachers to flip their classrooms, and generally allow teachers to be in more than one place at a time – but who has time to create these on their own? Let us do it for you!
• We will continue to expand our offerings based on your feedback.

i want to find out more

We would love to invite you to become a founding member of All Access! You can become a member by clicking here. We will drop in a video demo soon!

If you have questions, feel free to reach out to our MTM team by clicking here. Rachel will take care of you!

What if I already own the math curriculum? If you would like to give All Access a try, we are excited to offer all existing curriculum customers the opportunity to transition to MTM All Access for 50% off for the life of your membership. This discount will automatically apply in your cart. All you need to do is log in to your account and add the membership to your cart! If you choose to upgrade to All Grades, then the discount will vary based on the number of curricula you own.

We hope to see you in All Access and look forward to supporting you this year!

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Vertical Alignment

Before we jump to that, let’s take a look at the standard and how it progresses through middle school, and then take a look at some STAAR test question examples. I have highlighted some helpful pieces.

Real Number System Test Questions

Strategy #1 – Vocabulary

Vocabulary is crucial when teaching the real number system. Luckily, the content scaffolds by grade level.

• 6th Grade: whole, integer, rational
• 7th Grade: natural, whole, integer, rational
• 8th Grade: natural, whole, integer, rational, irrational

Since the progression of standards is pretty clear, each subsequent year a student must learn one brand new word. Although the vocabulary is important, I think students need to see examples more than they need to memorize the exact definition. 

Bright idea! One way to push students’ learning is to ask them to come up with definitions based on observing numbers that have already been classified. Check out the example below.

Some thought provoking questions might be:

• What differences do you see between the numbers inside integers and whole numbers? 
• What is different about the fractions classified as whole numbers versus the fractions classified as rational numbers?

Strategy #2 – Visuals

Notice that in each grade level standard, the term “visual representation” is used. In addition, in each test question example, there is a venn diagram of sorts. This means that students will not be expected to classify a number in the real number system without a venn diagram present to guide them, so make sure you are modeling with one, and students are practicing with one.

Check out the one I made! Hint: Washi tape helps with straight lines.  If you use Post-it Notes, the anchor chart can be interactive and reused each class period.

Students need to be taught how to use the venn diagram. Don’t assume (like me) that it is intuitive. I would start by using a similar venn diagram that is not related to math. You can steal this example if you would like. (Warning: You may be concerned about students’ geography if you use this example.) 

You can ask these types of questions:

• If someone is from Texas, can you assume they are also from the United States?
• If someone lives in Oklahoma, where would you place them on the diagram? 

If students are struggling to use the venn diagram to understand the relationships between sets of numbers, then try exposing them to the funnel example. Using the number 17: 17 would be dropped into the natural number funnel thus falling through the whole, integer, rational, and real number funnels. The number -17 would be dropped into the integer funnel and thus continue into rational and real numbers. Then you would explain that -17 is an integer, a rational number, and a real number, but you typically call numbers by the funnel that it is dropped in.

Strategy #3 – Simplify Before You Classify

I saw this idea on a Middle School Math Facebook group, and it is so clever and catchy! Teach students to simplify before they classify in the real number system. Fractions like 16/4 are a great example of this. If students are familiar with the definition of rational numbers, they may think, “16/4 must only be a rational number because rational numbers are numbers that can be written as fractions.” That student is technically right, 16/4 is rational, but that is not all. If you teach students to simplify before you classify, a student would simplify this to 4, thus changing its classification.

You see this with square roots too. Many irrational number definitions include the phrase “square roots,” so a student might incorrectly classify the square root of 100 as irrational. Simplify before you classify!

Strategy #4 – Make It Interactive

Since this skill requires very little computation, this is an opportunity to engage students in something hands-on. Here is what I have done:

• Flyswatter Game (ideally after students have shown mastery, so they aren’t just swatting uncontrollably)
• Card sort
• Post-it Notes – have students write down a bunch of different types of numbers on individual Post-it Notes and then swap with a partner. Then they have to categorize the numbers from their partner.
• Grab a 6th grade or 8th grade activity bundle that includes classifying the real number activities.

Classifying numbers in the real number system can be really engaging. It also would provide a math-win for some of my struggling students. How do you make classifying numbers engaging?

The post Teaching the Real Number System appeared first on Maneuvering the Middle.

Dividing fractions doesn’t have to be scary!  I was shocked to learn that this study determined that students’ knowledge of fractions and division uniquely predicts their achievement in high school years later.

Make sure to grab your free how to teaching fraction division guide below

Check out the player to hear the interview, or you can read the transcript that has been edited for succinctness below.

Listen On: APPLE PODCAST  | Spotify

Who is our expert on fractions?

To connect with Brittany and learn more, you can follow her on Instagram @mixandmath or check out her website Mix and Math. 

Brittany Hege is a math educator who has worked with students and teachers in both upper elementary and middle grades. She holds a master’s degree in Elementary Mathematics Education and is passionate about helping teachers grow their understanding of the math they teach. Brittany believes in the power of experiencing math through hands-on work and uses her platform, Mix and Math, to equip upper elementary teachers with the knowledge, resources, and confidence to inspire a generation of empowered math learners.

Why are students and teachers fearful of fractions?

Similar to multiplication facts, fractions are where students decide if they love math or hate math. Typically, fractions are taught very procedurally. There are many rules and procedures to memorize, and when students struggle, they lose their confidence. 

Teachers carry their own learning experience with fractions into their teaching. Just like students, teachers can also fear fractions.  And if we have taught fractions, and the lesson bombed, it reinforces the idea that fractions are something to just get through. 

How do students overcome this fear of fractions?

It starts with teachers. Confidence in teaching a concept comes from understanding the concept. When teachers take the time to really understand fractions, it grows their confidence and then builds students’ understanding. 

Teachers have to make fractions concrete by making fractions hands-on.

• What does this fraction actually mean?
• What does it look like in real life?
• How do we operate in a concrete way?

What is the progression of fractions in elementary school?

Fraction understanding begins in 3rd grade. In 4th grade, students start operating with fractions, but the standards focus on conceptual understanding, not the algorithm. In 5th grade, students should be able to understand fraction division and what it looks like in the real world. By 6th grade, students are expected to divide fractions by using the standard algorithm.When students come into middle school with gaps in their knowledge of fractions, teachers typically drill and kill the procedures because they feel the urgency to fill the gaps so that they can move onto their grade-level content.

Why should we integrate models into our teaching?

Models help students visualize; models show what a fraction looks like and what the operation actually means. 

What does 2.5 times ¼ actually mean? If students don’t have context or hands-on experience, they don’t realize that this number sentence means two groups of ¼ and ½ of ¼. 

For a teacher who did not learn it that way, it can be intimidating. The best thing a teacher can do for their students is to grow their own content knowledge and to grow their own understanding.

We cannot change everything about the way we teach in one year.  Fractions would be a good place to start.  Get into a community like a Facebook Group and ask other teachers how they would teach it. 

What are the two different types of division?

The two types of division are fair share and measurement.

Fair share (or partitive) division means to divide an amount into a given number of groups, and we want to know how much is in each group.

Whole Number Example: I have $12, and I want to give 3 kids the same amount. How much money does each kid get?

Fraction division example: I have ¼ of a cookie and I want to share it with 2 friends. What fraction of the cookie does each friend get?

Measurement (or repeated subtraction) division means to divide evenly into groups of a given size.

Whole number example: I have $12, and I want to give each kid $4, how many kids can I give money to? 

Fraction division example: I have 2 cups of flour. Each cupcake requires ¼ cup of flour. How many cupcakes can I make? So we are measuring it out – ¼ cup and another ¼ cup and another until we reach 2 cups of flour. 

Measurement division is typically how we see fractions divided in middle school, but students may not understand that that is a division problem. Students also cannot determine if their answer is reasonable because they do not know the action behind it. In elementary, students learn that division makes numbers smaller, so it can be confusing to students when 2 divided by ¼ equals 8. 

DIVE DEEPER BY GRABBING YOUR HOW TO TEACH DIVIDING FRACTIONS FREEBIE

For a better viewing experience, please download the guide and open in Adobe Reader or Preview.

To connect with Brittany and learn more, you can follow her on Instagram @mixandmath or check out her website Mix and Math. 

How do you teach dividing fractions?

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Number sense, specifically middle school number sense, is an intriguing topic in the world of education. As a math teacher, I was amazed at how some students would compute numbers in their head while others liked to work out problems by hand. Let’s break down what number sense is, how you can teach it, and discuss 4 strategies to use with your students to develop this life-long skill. 

LISTEN ON: APPLE PODCAST  | SPOTIFY

Books Mentioned on Good Morning Teacher: About Teaching Mathematics | Quality Questioning | Developing Numerical Fluency Understanding | Making Number Talks Matter

What is number sense?

Is number sense a skill that is taught or a skill that is innate to strong math students? Let’s think abstractly – are you a born leader? Or can you learn to be a leader? 

Number sense is the ability to be flexible and fluid with numbers.

• As a young elementary math student, understanding that since 5 is half of 10, 50 must be half of 100. 
• Being able to add 58 + 14 by breaking the 14 into 12 and 2, so that you are adding 60 + 12 quickly in your head.
• It is understanding that if you have half of an object, and you take half of that, you naturally have a fourth of the original object.

Number sense is not rote memorization or the ability to navigate an algorithm quickly. 

Should I teach number sense?

Simply put – YES!

Number sense is another tool to add to our students’ problem-solving toolkit. At the end of the day, we are teaching students to think. We can build this number sense slowly over time by being intentional in our questioning and the way we model our thinking and strategies. 

4 strategies for building MIDDLE SCHOOL number sense

Model different methods

It is easy to model the method that you are most familiar with. I was hesitant to teach decimal multiplication using partial products or an area model; I was more familiar with the standard algorithm. Instead, I took the opportunity to teach multiple methods, and allowed students to work with the method that felt the most comfortable for them. It was interesting to see how the same student would use different methods depending on different factors. By teaching different methods, you are allowing middle school students to determine which method works best for the problem at hand thus strengthening their math reasoning. 

When I taught Algebra 1, one of my favorite units was on polynomials. As a student, I learned the acronym FOIL stood for…first, outside, inside, last…but I had no clue as a student what I was doing.  Then as a teacher, I learned to double distribute and how to use the box method (which is just an area model). 

Marilyn Burns says in her book, “When children think that there is one right way to compute, they focus on learning and applying it, rather than thinking about what makes sense for the numbers at hand.”

Discuss computing strategies

This method can fit inside any lesson. If you are covering division of fractions, you can ask students how many halves are in six wholes and to explain their thinking.  After a student responds with their process for solving, an additional questioning strategy would be to not acknowledge whether the student is correct or not, and ask for ANOTHER student who solved it differently to explain their thinking. Middle school number sense can be strengthened by not acknowledging what is correct right away, but by giving students ample time to share many different ways to solve. When students know that you are more interested in HOW you arrived at the answer than whether the answer is right, they will think more creatively about ways to solve.

Utilize friendly numbers and estimation

Teaching students to utilize friendly numbers to estimate solutions is another number sense strategy. In the real world, people often estimate when they are working with numbers. (Maybe not engineers or accountants). Sixth and 7th grade students are asked to apply percents to problem situations and frequently those examples include calculations involving money. Teaching and showing students how to utilize friendly numbers (½, 1, 2, 5, 10, 25, 50, 100) to get you close to the actual solution is one way to incorporate number sense. 

Parts and Wholes

There are so many part and whole relationships in middle school math. Think: fractions and decimals, ratio and proportional reasoning, and percentages. When we build that understanding of a part to whole relationship between numbers or representations, they are able to apply it to other mathematical concepts. Consider questions like “if half a cake feeds 4 people, how many people will a whole cake feed?”

When can I incorporate number sense?

Low lift opportunities in your normal instruction, incorporate:

• wait time (for your feedback response)
• asking for multiple strategies
• shifting your line of questioning to incorporate estimation

For higher lift opportunities, try:

• Daily warm up – pose questions that would solicit different strategies for solving
• A formal number talk – this structure focuses on using mental math to develop an understanding of numbers and operations. Present a problem to students, give them time to solve mentally, and collect responses focusing on HOW they arrived at their answer.  In middle school, you might ask for the sum of 15.6 and 8.6. See how many different methods students used mentally to arrive at their answer.

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This post is sponsored by Post-it® Brand. All opinions and ideas are my own.

Math Specific 

1. Real number system diagram

Whenever I taught the real number system, I struggled with a way to make it hands-on. Using Post-it® Notes would have been a perfect solution! After defining a whole number, integer, rational number, and irrational number, students practice placing the types of numbers in the appropriate spot on the diagram. This would make a perfect ‘living’ anchor chart that could continue being added to as students encounter more types of numbers.

2. Combining like terms

Students really struggle with this concept, and I think Post-it® Notes can help make this process more organized. When introducing students to combining like terms, teach students to write down each term and its preceding sign onto a different Post-it® Note. Students can rearrange and group like terms together. Instant activity with hardly any prep!

3. Substitution

Post-it® Notes can provide a clear visual for what is happening when a value is being substituted (or an equation is being checked) in an expression. Using a Post-it® Note, cover the variable in the original expression with the substituted value and have students calculate.

4. Order of operations

This idea could serve as a brain teaser or extension idea for your early finishers. Post a few complex “order of operation” problems and use Post-it® Notes to serve as an answer bank where the answer choices will only be used one time. I like that students can interact with the problem. In my example, I used a red Post-it® Note to signal that those are numbers that shouldn’t move.

5. Ordering numbers

This idea is as simple as it sounds. You or your students write down a variety of rational numbers on Post-it® Notes (you could even color coordinate the types of numbers – green for whole numbers, pink for integers, yellow for rational numbers) and students place them on a number line. 

6. Dot Plots

Dot plots are perfect for Post-it® Notes. Provide each student with a Post-it® Note that will act as a dot, and ask a statistical question like, “What size shoes are worn in Mrs. Brack’s 5th period class?” My Data and Statistic units were always more successful if I incorporated students’ specific data into my examples. Students come up and place their Post-it® Note dot in the appropriate spot. They help keep the size of the ‘dots’ consistent, so that students can have an accurate picture of data that skews right or left or is symmetrical. And they are sticky enough to be reused by the next class period which is a total win.

7. Graphing points

Write down different parts of the graph (y-axis, origin, quadrant IV, etc) and a variety of coordinate points on individual Post-it® Notes and distribute them to students. Project a coordinate plane and have students take turns placing the parts of the graph and points in their respective location.

General Classroom

8. Thankfulness Campaign

This was a schoolwide initiative that brought so much joy to our school but could also be done in an individual classroom. During homeroom, students wrote down one thing they felt thankful for on an individual Post-it® Note. They were collected over the course of several weeks, and eventually went on to create a Post-it® Notes mural in one of the main hallways. 

9. Summarizing a daily lesson

Post-it® Notes provide the perfect amount of space to force students to synthesize their learning from that day into a bite-sized chunk. As teachers, we might equate copious notes as higher-level learning when, in actuality, the opposite can be true.

10. Bookmark for interactive notebook

A Post-it® Note is a perfect bookmark for interactive notebooks. They can be repositioned daily, and the variety of colors can indicate important sections of the notebook – vocabulary, previous units, etc.

11. Visual indicator a student needs help

If you have a student who will not ask questions during a lesson (but frequently has questions), a Post-it® Note could provide a perfect solution. Provide a few and come up with a system that makes sense for you and your classroom. For me, the student would write her question down, hand it to an assigned buddy, and that buddy would ask the question for her. We came up with that system one-on-one when we brainstormed ways for her to take more ownership in her learning and for me to better clarify misunderstandings in the moment. 

12. Daily must-do list

I would be remiss to not include the power of a Post-it® Note when it comes to writing down my must-dos each day. This list helps me quell that overwhelmed feeling when I look at a VERY long list of to-dos. I take the 3-5 most important and urgent action items and put them on my must-do list. The day instantly feels more manageable!

This list just scratches the surface of all the ways Post-it® Notes can be used in a math classroom setting. How do you use Post-it® Notes in your classroom?

The post USING POST-IT® NOTES IN MATH appeared first on Maneuvering the Middle.

Let’s talk about digital activities – specifically, free digital math activities! Maneuvering the Middle has digital math activities for 6th grade, 7th grade, 8th grade, and Algebra 1. Let’s talk about what digital activities are, why we love them, and some ideas on how you can use them. Most importantly, you can try them out in your classroom with a freebie down below!

UPDATE: ALGEBRA I DIGITAL ACTIVITIES ARE NOW AVAILABLE!

LISTEN ON: APPLE PODCAST  |  SPOTIFY

What is a digital math activity?

Digital math activities are designed for use in Google Slides and Google Forms, but you do not have to be a Google school or use Google classroom to make these work.  In fact, you can use them with any LMS; your students just have to have access to either Google Slides or PowerPoint. 

The digital activities are made up of two parts. First, the interactive activity portion; students either drag and match, insert shapes and lines, or type their responses. Then, the formative assessment portion is a quick, two question Google Form quiz that assesses if a student understands the concept. 

We released these resources back in 2019 (pre-COVID) to provide a way for teachers to use technology and still have high-quality problems and scaffolding of a specific concept.  We wanted to include higher-level thinking skills and applications that require students to make those connections between what they are learning and the real world.

Why should you try them out?

You might be wondering – are these digital activities for me? Which is why we want to give you a free set to try! Free digital activities to make your life easier!

This way you can try them out with students and see if they are a good fit! If you love the freebie, you can find more digital math activities here. 

Here’s why you should try them:

• They work in all classroom settings as long as your students have access to a device like a Chromebook or iPad.
• If you are in-person, use them as paperless practice. You could even have students collaborate by working on the exact same file.
• If you are in a hybrid classroom, they are a great fit for that at-home day.
• If you are fully virtual, they can easily be broken up and shared as individual slides or they can be shared as a weekly assignment. They are ready to go and easy to share, making the virtual component just a little lighter. 
• They provide variety. Not only are they visually appealing, but they won’t overwhelm students. There is nothing like opening an assignment and realizing you have 30 problems to solve or questions to answer.
• They are more interesting than a worksheet. And we all know that we are trying our best to keep our students engaged. Cue this meme:

Anytime you can bring some creativity and an interactive element without losing the rigor (or even building up to the rigor) and it’s ready to go — shoot, sign me up!

How other teachers are using them and why they love them?

I love hearing how other teachers are using them in their unique circumstances and wanted to share with you — especially if you aren’t a math teacher, then you may be able to apply these ideas to an activity in your content area.

Shawna from our MTM FB group says, “We are all remote. I use them for a collaboration day every week. Students get their own copy in GC (Google Classroom) but are in breakout rooms to work together on them.”

Angelique shared, “1) I use digital activities for group work and each group is assigned 1 problem they will present to the class. 2) I use them as a competition to see which breakout room will return to the main room first. I have found this has kept all of my classes on task and focused.”

Marissa says, “Since we are on a hybrid schedule, I use them for all my students at home days. I either check one slide of the digital activity for accuracy or they get a completion grade for doing all slides… I pair them with the Google Form exit tickets and use those as a little formative check.”

And Brina shared, “I’ve been using digital math activities in my skills practice choice boards. I try to give students 3-5 choice practice activities over whatever skill we are doing in class. This way they can practice the skill by picking the activity that they are most interested in… It has been helpful since we have gone back and forth between remote and face to face a few times this year.”

If you want to learn more about digital activities (or really any math topic for that matter), join our Facebook group of brilliant math teachers!

Don’t forget to grab our free digital activities! If you are already using our digital math activities, I would love to know how you are using them, so comment below or let us know on our FB page or Instagram.

The post Free Digital Math Activities appeared first on Maneuvering the Middle.

HOW TO TEACH SLOPE

Here are a few TEKS to show you some of the vertical alignment of this skill(s). 

It is always more helpful for me to see a test question than a standard, so I pulled a few questions from the 2019 STAAR to show the progression of slope from middle school to algebra. This is a great method to exercise when thinking about vertical alignment. For today’s purposes, I focused on proportional relationships.

6th GRADE

7th GRADE

8th GRADE

Algebra 1

Strategy #1: Vocabulary and Types of Slope are Key

Each grade level will focus on a different vocabulary word according to the standards, but it is important to connect these concepts to what students have already been taught in previous years. Whether it is unit rate, constant of proportionality, slope, or rate of change, it is important to articulate that all of these terms essentially mean the same thing and to remind students that they can and will see these words used interchangeably. 

As 7th graders are expected to begin graphing slope using slope intercept form, I found this idea particularly brilliant (from one of the amazing teachers in our Facebook group):

y = mx + b —> b is where you begin (y-intercept) and m is what moves (the slope)

I would also like to add that while I was never shown Slope Dude or Mr. Slope Man, it would have been incredibly helpful to help me remember the difference between an undefined slope and a zero slope. Ms. Gertson, the Algebra teacher at my school, had a huge anchor chart of Mr. Slope Guy in her classroom.  

Strategy #2: Scaffold

As teachers, we can look at math concepts through the lens of full understanding and underestimate how challenging it can be for a student who is seeing the concept for the first time.  

Identifying the type of slope, calculating slope from two points or a line on the graph, and graphing an equation all require multiple days on your scope and sequence.  My recommendation is to:

• Day 1: Start with graphing unit rates as y=mx and introduce the different types of slopes – positive, negative, undefined and zero.
• Day 2: Have students practice finding slope from the graph of the line using rise over run.
• Day 2 or 3: Introduce the slope formula. Have students determine the slope from two points and/or from the graph. 
• Day 4: Use real-life applications (like salary + commission or a cell phone plan with a data usage rate + processing fee) to connect to slope-intercept form. A teacher in our Facebook group said she makes the connection to transformations of functions since slope intercept form is a translation of y=mx. 

Another teacher said, “I do the same, making sure they understand the relationship between tables, graphs, and equations with unit rates and then extending it with an initial/starting value. (We practice) a lot of word problems and applications before we introduce slope and formulas…. to (improve) conceptual understanding.”

The best part is that we have done this for you – check out our Linear Functions Unit  for Algebra 1.

Are you an 8th grade teacher? Grab Linear Relationships Unit for 8th Grade.

Strategy #3 Use Technology for Real-World Application

There is something that helps math click when students can see how manipulating the equation of the graph will impact the actual line of the graph itself in real-time.  We love and recommend Desmos all of the time. Here are two activities that are sure to help students make those connections. 

Marble Slides: “In this delightful and challenging activity, students will transform lines so that the marbles go through the stars. Students will test their ideas by launching the marbles and will have a chance to revise before trying the next challenge.”

Land the Plane: “In this activity, students practice finding equations of lines in order to land a plane on a runway. Most of the challenges are well-suited to slope-intercept form, but they are easily adapted to other forms of linear equations depending on the goals of an individual class or a student.”

Grab these Winter Solve and Color freebies that include rate of change and slope!

What are some ways you teach slope? Tell us what grade you teach and how you discuss slope with your students. 

Looking for more great content? Maneuvering the Middle resources are now for Algebra 1 TEKS. Check it out here.

The post How to Teach Slope appeared first on Maneuvering the Middle.

Teaching Functions in Algebra 1

Standards

Here are the standards that best describe what we are going to focus our time on today. 

• A.12(B) evaluate functions, expressed in function notation, given one or more elements in their domains
• A.12(A) decide whether relations represented verbally, tabularly, graphically, and symbolically define a function

Here’s what those standards look like as 2019 and 2018 STAAR test questions.

Vertical Alignment

In 8th grade, students will begin to identify functions in ordered pairs and with graphs; it is a readiness standard, so it is tested more heavily in 8th grade than in Algebra 1. You can see an example of am 8th grade STAAR test question below.

I think vertical alignment is one of the things teachers can overlook the most when lesson planning. If you are unsure, look up what students have already been exposed to before introducing a topic. I like using this document here.

Identifying Functions

If you have taught how to identify a function before, you are probably familiar with the definition of a function —

“A function is a rule that assigns each input exactly one output. They occur when every x-value is associated with exactly one y-value.”

You are also probably very familiar with the vertical line test.

When I was a student, I learned to just use the vertical line test. If given a set of ordered pairs, I would quickly sketch it to see if it passed the vertical line test. I had NO understanding of why this worked and what made a function an actual function. 

It wasn’t until I taught functions that I came across a concept that helped me understand the WHY behind the vertical line test. If you have a graph with time on the x axis and distance on the y axis, a vertical line would represent someone or something at a particular moment in time being in more than one place at once, which is not possible. A horizontal line would represent someone or something not moving over a period of time which is possible.

Here are other strategies to help students conceptually understand:

• Vending machine example: Buttons are the input. Drink is the output. A1 will give you a coke. A2 could give you a coke. But A3, can’t give you a Sprite or a Coke. 
• Speed dialing on a phone example: X is the number you push, and y is the person that is called. Your phone is functioning when you hit a 3 and it calls your mom only, or if you pressed 4 and it called your brother. The phone is NOT functioning if you would hit a 3 and it sometimes goes to your mom and sometimes goes to your brother. However, you can program two numbers to go to the same person. 
• Multiple students can be the same height but one student cannot be multiple heights.

Evaluating and Graphing Functions

Once students have a firm grasp on functions and relations, evaluating and graphing functions come a little more naturally. Students have been substituting and graphing in all four quadrants since 6th grade.  Spend time reviewing the order of operations and how to graph on a coordinate plane. Never assume students already know how to do that – you will be surprised by the misconceptions! However, function notation will be new to students. From my experience, students pick up that f(x) is the new y fairly well!

Use Technology for Real-World Application

And because nothing makes math come to life more than our favorite web-based program, here are a few Desmos links.

• Guess My Rule: Students are introduced to the concept of a function by using input-output pairs in a table. They explore different rules, some of which are functions and some of which are not.
• Card Sort – Functions: In this activity, students sort graphs, equations, and contexts according to whether each one represents a function.

Pacing

Properties of functions – identifying, evaluating, and graphing are big skills that need their own day. Each of these skills requires a myriad of formats for students to comprehend — set of ordered pairs, mapping diagrams, a graph, a table, or a real life example.  Students need exposure, so do not plan on flying through these standards in a few days. Since functions are foundational in Algebra, extra time spent setting the stage will not go to waste. 

What I also love about this particular skill is that it softly introduces all of the types of functions students will be exposed to over the course of the year — linear, quadratic, and exponential. If students feel comfortable evaluating or describing these types of functions early on, they will be more inclined to find the axis of symmetry (for example) later in the year. 

Further reading for Algebra 1: How to Teach Domain and Range | How to Teach Solving Equations

UPDATE: ALGEBRA I DIGITAL ACTIVITIES ARE NOW AVAILABLE!

The post Teaching Functions in Algebra 1 appeared first on Maneuvering the Middle.

This is a thank you for all the hard work you have done during this pandemic. We thought a freebie that was…

• TEKS and CCSS aligned
• Vertically aligned from 6th grade through 8th grade
• Perfect for in person or digital use

…would satisfy the needs of most teachers. How about a math task that can also be used in a variety of ways? 

3 Ways to Use MATH PERFORMANCE Tasks for Middle School 

1. As a Group Project

Do your students beg for projects? Mine sure did. Any opportunity to collaborate or change up the routine was always met with such enthusiasm from my sixth graders.  These tasks can be the basis of an exciting project! Students can complete the math task and present the project via FlipGrid or by creating a Google Slides presentation. Since the digital world does not seem to be going away, this could be an opportunity to teach students how they can work together remotely. 

2. As an Assessment

These math performance tasks cover the following standards in their depth and breadth which makes them the perfect assessment tool.

Common Core State Standards

(Standards have been edited for brevity)

CCSS.MATH.CONTENT.6.RP.A.3 – Use ratio and rate reasoning to solve real-world and mathematical problems

CCSS.MATH.CONTENT.7.RP.A.1 – Compute unit rates associated with ratios of fractions measured in like or different units.

CCSS.MATH.CONTENT.7.RP.A.2.C – Represent proportional relationships by equations. 

CCSS.MATH.CONTENT.8.F.B.4 – Construct a function to model a linear relationship between two quantities. Interpret the rate of change and initial value of a linear function in terms of the situation it models.

Texas Essential Knowledge and Skills

6.4(B) apply qualitative and quantitative reasoning to solve prediction and comparison of real‐ world problems involving ratios and rates

7.4(A) represent constant rates of change in mathematical and real‐world problems

8.4(B) graph proportional relationships

8.4(C) use data from a table or graph to determine the rate of change or slope and y‐intercept in mathematical and real‐world problems 

8.5(I) write an equation in the form y = mx + b to model a linear relationship between two quantities

Ideally, you could have this math task self-grade by using Go Formative or a variety of other tech tools. 

3. As an Extension or Extra Credit

If you are looking for something for the students who are always asking for more, here you go! (Those students do exist, I promise!) Give one of these math performance tasks to your early finishers at the beginning of your Proportional Reasoning Unit. Explain that this is something they will be expected to complete as they acquire the skills to do so. Perhaps, it is extra credit, or it could even replace a student’s lowest grade in the grade book. Use it knowing that it will push your students to think critically! (It isn’t just busy work.)

The possibilities with this printable + digital freebie are endless. How do you use math performance tasks in your classroom?

The post Free Math Performance Tasks for Middle School appeared first on Maneuvering the Middle.

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A.2(A) determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real‐world situations, both continuous and discrete; and represent using inequalities 

Looking at former test questions helps me envision how standards can be tested. Here is an example from the 2017 Released STAAR test:

A.6(A) determine the domain and range of quadratic functions and represent it using inequalities

This test question example from the 2018 Release STAAR test: 

 Let’s jump into some ways that will help your students master domain and range!

1. Vocabulary is Key

This concept introduces new vocabulary that is necessary for using the skill. Make sure students have a solid understanding before moving forward. Don’t have them just copy it down. When was the last time you remembered something after writing it down one time? Encourage them to say it to a partner. Have students write a left to right arrow whenever they see the word domain and a vertical arrow whenever they see range. Make this sticky!  

I relied heavily on reading Math Equals Love when I taught Algebra 1 and 2. Sarah taught me to use the mnemonic tool DIXIROYD (which she credits to a unknown blog reader) originally to help students remember that Domain is the set of all Input values, X values, and the Independent variable, while the Range is the set of all Output values, Y-values, and the Dependent variable. You know a mnemonic device is good when you rely on it just as much as your students.

Update 7/28/2026: Maneuvering the Middle now has a Middle School Math + Algebra 1 Word Wall.

As you can see in the video below, our Word Wall includes 190 essential math terms, their clear-cut definitions, and their visual representations.

We’ve included Spanish translations for all terms and definitions, ensuring a supportive and accessible learning experience for English Language Learners.

They were designed to be minimal prep and flexible to customize the formatting to suit your students’ unique needs.

2. Scaffold over a Few Lessons

As teachers, we are all guilty of jumping to an example that students are not ready for, which can cause them to become overwhelmed and shut down. Or worse, they develop false confidence and do an entire set of problems incorrectly.  I thought domain and range was really intuitive, so I tried to cover everything  in one class period and my students were lost. 

This skill requires scaffolding. I recommend starting with finding domain and range from tables and mappings first. Move to determining the domain and range from word problems or equations by creating an input and output table. Finally, move to graphs in this order — discrete graphs, graphs with endpoints, and then graphs with arrows. Interval notation can be tricky for students, so make sure to review  inequality symbols too.

3. Tools are Your Friend

I combed through the 2017 and 2018 STAAR tests and out of all the domain and range problems, 80% used a graph. Students have to be able to determine the domain and range by looking at a function on a graph. I recommend having students annotate the graph by use of colored pencils or highlighters. I have seen this done two ways. 

1. Highlight the interval of the domain and range on the x and y axis.
2. Draw a box. Though I think this makes infinity less clear. 

Lastly, I used this foldable from Math Equals Love after my initial domain and range lesson bombed, and it was much more successful. 

Ready to teach it!? You can find our Properties of Functions Unit here

If you need hands-on Properties of Functions Activities, you can find them here (+Domain and Range)!

4. Use Technology for Real Time Application

We, like many teachers, are big fans of Desmos. A great lesson for introducing domain and range can be found here. For additional practice, I recommend this lesson.  Even if your students do not have access to technology, there is nothing stopping you from projecting these activities and leading a demonstration or discussion with your students. I firmly believe that the more students can manipulate a graph and see its effects, the more memorable or “sticky” that concept becomes!

UPDATE: ALGEBRA I DIGITAL ACTIVITIES ARE NOW AVAILABLE!

I love reading your comments about how you teach math. How do you teach domain and range in your classroom?

The post Teaching Domain and Range in Algebra 1 appeared first on Maneuvering the Middle.

1. Make a Plan for Managing Algebra Tiles

Routines and procedures are king! Algebra tiles are tiny pieces of plastic that are bound to end up on the floor or mixed with other sets, even with the most well-intentioned students. These series of questions will help you prepare answers and procedures for your students as you set the stage for managing algebra tiles.

• What is the procedure for grabbing a new set of algebra tiles? Can I use them without permission or do I need to ask first?
• Where do I get them from? Do I get them for my whole table? Can my set be mixed up with others? 
• When can I use them? Can I use them for homework? Can I take a set home?
• What are the consequences for using the tiles inappropriately, leaving them on the floor, and/or throwing them in the air?
• When do I clean up? Where do I return them? 

By thinking through these questions and explicitly instructing your students, you will be much more likely to experience success!  Your future self will thank you. 

2. Get Familiar with How to Use THEM

If you are like me, you didn’t learn math concepts with algebra tiles when you were in school, and might have taught for several years before giving them a try. Before implementing them with students, familiarize yourself with them. Play with them on your own.  Without a pencil or paper, solve problems from your curriculum using only the manipulative. 

Whenever I needed to practice teaching a lesson I wasn’t confident in, I would pop into my coworker’s classroom and ask if I could explain the concept to her. My coworker was a self proclaimed “not a math person,” and would ask excellent questions that helped me clarify my lesson and prepare for student misconceptions. 

Download our Algebra Tiles Starter Guide to see how you can use them to teach simplifying expressions, distributive property, solving linear equations, adding and subtracting polynomials, multiplying and dividing polynomials, and factoring polynomials. 

3. Don’t Just Use THEM for One Day

I love manipulatives, but they can be hard to manage on a daily basis.  Sometimes, I found that I would introduce a topic with a manipulative, check the “used manipulatives” box, and then wonder why it wasn’t as beneficial as I had hoped. While I had good intentions, I wasn’t serving the students who could benefit from more concrete learning and practice on a more regular basis.

The purpose of using tiles (or any manipulative) is that it allows for conceptual math to become concrete. If you are not familiar with the concrete representational abstract sequence, we wrote a blog post about it here.

Essentially, in the case of solving equations, the concrete representational abstract sequence would look like this:

• Use the tiles (concrete) to build their conceptual understanding
• Draw the tiles (representational) to show the process of solving an equation
• Solve an equation (abstract)

While some students might love using algebra tiles and want to continue using them indefinitely, there will be problems where they are not the best choice (example: rational numbers). However, if algebra tiles support the students’ processing of the problem, then I think students should be able to use them whenever they would like. Allow tiles to be a method for solving at any point in the unit or school year.

Are algebra tiles new to you? Do you have them in your math closet but don’t see the value in using them? Do you need a few examples of how to use them?  We have you covered!

Our Getting Started with Algebra Tiles Guide is going to walk you through the process step-by-step with the different ways you can use algebra tiles along with some of the language you can use with your students. 

Interested in more about algebra tiles? Check out our post on how to solve equations using algebra tiles. 

UPDATE: ALGEBRA I DIGITAL ACTIVITIES ARE NOW AVAILABLE!

The post Getting Started with Algebra Tiles appeared first on Maneuvering the Middle.

Find a small set here or a large set here.

You can read about how we use algebra tiles to teach solving equations here.