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Copyright © 2021 Tang Math LLC. All rights reserved.

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Take out the numbers to make word problems make sense Learn More Try one tomorrow Primary (K-2) Older Students Explore ready-made problems Problem Banks More Problems This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

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Three (3) Read Protocol is a teaching strategy designed for sense-making of language-rich math problems.

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Solving Word Problems Using Tape Diagrams

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Solving Word Problems Using Tape Diagrams

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2021-2022 Archive problems 2020-2021 Archive problems
Word problems. Ugh!

How often have you heard that in your classroom. Heck...I certainly heard it uttered numerous times in my own classroom. In fact, there are many times in which I was the one doing the complaining.

Oh, how I wish I was taught how to use tape diagrams to solve word problems. Unfortunately, it was not until deep into my career as a math teacher that I learned about this awesome tool for solving word problems.

A tape diagram is a model that pictorially represents a word problem. Once the word problem is accurately modeled, the student has an easier time seeing which operations to use in solving the problem. The tape diagram does not do the calculations for the student, but it does make it easier for the student to see which calculations might be needed.

Tape diagrams are especially useful for modeling addition, subtraction, multiplication, division, fractions, and ratios/proportions.

8-step model method

Long ago I was introduced to the 8-step modeling method. I don't know for sure, but I think the eight steps were originally from this book by Bob Hogan and Char Forsten. As the name suggests, it is eight wonderful steps that guide the student to drawing a pictorial model of the word problem. As students read and re-read the problem, it becomes more of a story and less of a problem.

The eight steps:

1. Read the entire problem.

2. Turn the question into a sentence with a space for the answer.

3. Determine who and what is involved in the problem.

4. Draw unit bars of equal length.

5. Re-read each sentence one at a time and revise the bar(s).

6. Put the question mark in place.

7. Work computations to the side.

8. Write down your answer to the problem.

Addition

Let's dig into each of these eight steps using the following 2nd grade addition problem:

Brienne has 23 fewer pennies than Alonzo. Alonzo has 45 pennies. How many pennies do Alonzo and Brienne have altogether?

Step 1: Read the entire problem.

Students will find themselves reading the story numerous times. Each time will have a different purpose in the model drawing process.

Step 2: Turn the question into a sentence with a space for the answer.

Step 3: Determine who and what is involved in the problem.

Step 4: Draw unit bars of equal length.

Step 5: Re-read each sentence one at a time and revise the bar(s).

NOTE: There are often more than one way to revise the bars to match the story. For example, rather than shortening Brienne's bar, we could have lengthened Alonzo's.

Step 6: Put the question mark in drawing. Optional: Use letters to represent the unknown(s).

Step 7: Work computations to the side.

Step 8: Write down your answer to the problem.

As students continue using the eight steps to represent word problems, students will also begin to notice that word problems using addition and subtraction begin to have a common look about them.

Here is a wonderful online tool for practicing drawing models for addition and subtraction problems.

https://www.mathplayground.com/tb_addition/index.html

To help students remember the eight steps, consider prominently displaying this poster showing each of the steps.

http://bit.ly/wordproblem8steps

Students might also benefit from using a graphic organizer that shows each of the steps and provides workspace too.

http://bit.ly/8StepOrganizer

Just as addition and subtraction tape diagrams have common characteristics, multiplication and division tape diagrams will also look similar to one another.

Let's take a look at a multiplication problem. In this case all eight steps are completed at once.

At lunch recess, Karen passed out 3 times as many cookies as Vido did. If Vido passed out 35 cookies, how many cookies did they pass out altogether?

Take a moment to practice multiplication and division problems here.

https://www.mathplayground.com/tb_multiplication/index.html

Fractions

Ella has a collection of toys. After giving ⅗ of her collection to her younger brother, she had 6 toys left. How many toys did Ella give to her younger brother?

Try some fraction modeling here.

https://www.mathplayground.com/tb_fractions/index.html

Ratios and proportions

Lastly, let's look at a problem involving ratios.

In the orchestra at Willard Middle School, the ratio of girls to boys is 5 to 3. There are 28 more girls than boys in the orchestra. How many students are in the orchestra altogether?

The initial answer statement and model might look like this...

Once the problem has been modeled, we see that two units represents 28 students. The model with some worked filled in might look like this...

Practice modeling ratio problems here.

https://www.mathplayground.com/tb_ratios/index.html

Practice time

So, now you have seen an example from each of the variety of problems for which tape diagrams are particularly useful. Now it is time to practice!

Here are example problems taken from Eureka Math. Give them a try!

Nikil baked 5 pies for the contest. Peter baked 3 more pies than Nikil. How many pies did Peter bake for the contest? (G1 M6 L2)

Eighty-two students are in the math club. 35 students are in the science club. How many more students are in the math club than science club? (G2 M4 L16)

Mr. Nguyen fills two inflatable pools. The kiddie pool holds 185 liters of water. The larger pool holds 600 liters of water. How much more water does the larger pool hold than the kiddie pool? (G3 M2 L19)

Jayden has 347 marbles. Elvis has 4 times as many as Jayden. Presley has 799 fewer than Elvis. How many marbles does Presley have? (G4 M3 L12)

Mrs. Onusko made 60 cookies for a bake sale. She sold ⅔ of them and gave ¾ of the remaining cookies to the students working at the sale. How many cookies did she have left? (G5 M4 L16)

Mason and Laney ran laps to train for the long-distance running team. The ratio of the number of laps Mason ran to the number of laps Laney ran was 2 to 3. Mason ran 6 miles. How much further did Laney run than Mason? (G6 M1 L3)

Jillian exercises 5 times a week. She runs 3 miles each morning and bikes in the evening. If she exercises a total of 30 miles for the week, how many miles does she bike each evening? (G7 M2 L17)

Monica and Naomi had the same amount of money. After Monica spent $25 and Naomi spent $36, Monica had 2 times as much money as Naomi. How much money did each have at first? (G8...not from Eureka Math)

In a school choir, one-half of the members were girls. At the end of the year, 3 boys left the choir, and the ratio of boys to girls became 3:4. How many boys remained in the choir? (Algebra, Module 1, Lesson 25)

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Solving word problems using tape diagrams

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2021-2022 Archive problems 2020-2021 Archive problems Word problems. Ugh! How often have you heard that in your classroom. Heck...I certainly heard it uttered numerous times in my own classroom. In fact, there are many times in which I was the one doing the complaining.

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Introducing New and Improved Thinking Blocks ®

We redesigned Thinking Blocks and packed it full of new features!
read aloud word problems - visual prompts - better models - engaging themes - mobile friendly
Thinking Blocks works well on all devices.

Print Word Problems from Thinking Blocks

Addition and Subtraction Videos

Multiplication and Division Videos

Decimals and Percent Videos

Ratio and Proportion Videos

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Introducing New and Improved Thinking Blocks ® We redesigned Thinking Blocks and packed it full of new features! read aloud word problems - visual prompts - better models - engaging themes - mobile friendly Thinking Blocks works well on all devices.

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Cognitively Guided Instruction "CGI" Support

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Cognitively Guided instruction, or CGI, is based upon 30 years of research. It's a framework that helps teachers to understand how children's mathematical ideas develop, and provides an opportunity to build on the child's own thinking and understanding. CGI is not a curriculum. CGI is not a lesson. CGI is a framework, a way of thinking about how children learn math that is research-based. The central theme of CGI is that children come to school with mathematical knowledge. Children develop their math understanding by solving a variety of problems in a way that makes sense to them. The children build their own understanding and inform other children, and the teacher, as they share their thinking. As a result, children build connections and relationships among the mathematical ideas so that the math makes sense to them, and the math is meaningful in their lives.
Here are some videos of CGI in action with teachers from LAUSD: Kinder Video 3rd Grade Video 4th/5th Grade Video Videos from the Online course Navigating the CGI Math Classroom: The What and Why of CGI Multiplication and Division and CGI CGI and Place Value Supporting Teachers in CGI, Part 1 Supporting Teachers in CGI, Part 2
Resources and Links:

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Cognitively Guided Instruction "CGI" Support

# data[].attributes.attachment_link.description
Cognitively Guided instruction, or CGI, is based upon 30 years of research. It's a framework that helps teachers to understand how children's mathematical ideas develop, and provides an opportunity to build on the child's own thinking and understanding. CGI is not a curriculum. CGI is not a lesson.

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These pdf files are coded as follows: Easiest problems are coded with an shamrock Slightly more difficult problems are coded with a shamrock and a diamond More difficult problems are coded with a diamond Most difficult problems are coded with a heart.

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Build Math Minds 187 - Problem Solving Success: 3 Experts' Ideas from the 2025 Virtual Math Summit

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Students Collaborating

Student collaboration supports the Standards for Mathematical Practice as students should be able to "understand the approaches of others to solving complex problems and identify correspondences between different approaches" (MP 1) and "justify their conclusions, communicate them to others, and respond to the arguments of others" (MP 3).

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Student collaboration supports the Standards for Mathematical Practice as students should be able to "understand the approaches of others to solving complex problems and identify correspondences between different approaches" (MP 1) and "justify their conclusions, communicate them to others, and respond to the arguments of others" (MP 3).

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How many times have you watched a student read (...skim) a word problem and then immediately start computing the answer before you have even had a chance to give directions? No matter how much we talked about the importance of slowing down, actually THINKING about what the problem was saying and taking time to understand, I always have those students who just pull out the numbers, choose a random operation, and solve.

And let me tell you... it. drives. me. crazy.

This is about the time that I throw a new word problem on the board, take my favorite black sharpie and color over the numbers because I know numberless word problems are exactly what they need. So why does a word problem without numbers fix all my problems? Here's why.

When students see numbers in a word problem, they know that the expectation is to solve. They are "done" when they've added, subtracted, multiplied, or divided and they're left with an answer. When the numbers are removed from the problem and replaced with blanks, the focus immediately shifts away from solving, because there's nothing to solve! There is no operation to do because there are no numbers!

When students aren't being distracted by numbers and the pressure to solve, they have the opportunity to focus on the context. Numberless word problems force students to take time to truly make sense of the problem and determine what actions are being done in the problem. When they understand the actions in the problem, they can then translate that to an operation, or a math action.

All four operations are mathematical representations of a real life action (joining, separating, sharing, scaling, etc.). Unfortunately, students often miss this because they are so quick to just pull out the numbers and solve (or they have been taught to diminish word problems to a set of key words... but that is a whole new discussion for whole other day). When students work with numberless word problems frequently, they begin to see similarities in certain problems. They may notice that addition problems seem to be combining or joining two amounts while subtraction problems seem to be separating amounts or identifying the difference. These are the actions students need to understand in order to find success representing real life mathematically.

Numberless word problems allow for easy differentiation.

Many students struggle with word problems because of the numbers themselves. Some students are so defeated by seeing number like 1,236 or 0.05 or 7/12 in a word problem, that they don't have the motivation to even attempt the problem. Why would they? If they know they will struggle with the computation and the goal of the problem is to solve, then no matter how hard they work to understand the problem, they still have no chance to be successful.

When the numbers are removed, all students have the ability to work towards understanding of the problem because the numbers are not a limitation. They can think about the problem in terms of whatever numbers are comfortable to them. They can use problem-solving strategies to make sense of the problem without the pressure of computing "really hard numbers" in the end.

Students do need to know how to compute numbers, so after students have taken the time to understand the problem, you as the teacher can provide numbers that are appropriate for them to insert into the blanks and begin solving. I typically provide 3-5 sets of numbers for students to choose from to insert into the problem and solve.

Try a numberless word problem sort!

One of my favorite problem solving activities is a numberless word problem sort. It's not anything revolutionary. It is exactly what it sounds like... students sort a set of numberless word problems by the math action (operation) that they would use to solve the problem. It is such an easy activity but it is SO POWERFUL.

If you want students to practice computation, roll a dice or draw a number card and insert the numbers into the problem. Now students have taken time to understand the problem and they get computation practice. It's a win-win scenario! I highly recommend that every math teacher try this out with their students. It will likely reveal some misconceptions students might have!

Want to try numberless word problems in your classroom?

If you are an upper elementary teacher, check out this numberless word problem sort for FREE! This activity is powerful in helping students focus on the meaning of each operation.

If you are a 5th or 6th grade teacher, here is the set of numberless word problems I made for my students. For some reason, fractions tend to totally throw students off in word problems! It was encouraging for my students to see that they can understand these problems with or without fractions.

It also was really good practice for multiplying and dividing fractions because these operations tend to be a bit tougher to identify when you are working with fractions. Without the numbers, students were able to identify similarities and differences in the problems themselves instead of focusing only on the numbers.

So, what do you think? Have you given your students word problems without numbers?

How many times have you watched a student read (...skim) a word problem and then immediately start computing the answer before you have even had a chance to give directions? No matter how much we talked about the importance of slowing down, actually THINKING about what the problem was saying and taking time to understand, I always have those students who just pull out the numbers, choose a random operation, and solve.

And let me tell you... it. drives. me. crazy.

This is about the time that I throw a new word problem on the board, take my favorite black sharpie and color over the numbers because I know numberless word problems are exactly what they need. So why does a word problem without numbers fix all my problems? Here's why.

When students see numbers in a word problem, they know that the expectation is to solve. They are "done" when they've added, subtracted, multiplied, or divided and they're left with an answer. When the numbers are removed from the problem and replaced with blanks, the focus immediately shifts away from solving, because there's nothing to solve! There is no operation to do because there are no numbers!

When students aren't being distracted by numbers and the pressure to solve, they have the opportunity to focus on the context. Numberless word problems force students to take time to truly make sense of the problem and determine what actions are being done in the problem. When they understand the actions in the problem, they can then translate that to an operation, or a math action.

All four operations are mathematical representations of a real life action (joining, separating, sharing, scaling, etc.). Unfortunately, students often miss this because they are so quick to just pull out the numbers and solve (or they have been taught to diminish word problems to a set of key words... but that is a whole new discussion for whole other day). When students work with numberless word problems frequently, they begin to see similarities in certain problems. They may notice that addition problems seem to be combining or joining two amounts while subtraction problems seem to be separating amounts or identifying the difference. These are the actions students need to understand in order to find success representing real life mathematically.

Numberless word problems allow for easy differentiation.

Many students struggle with word problems because of the numbers themselves. Some students are so defeated by seeing number like 1,236 or 0.05 or 7/12 in a word problem, that they don't have the motivation to even attempt the problem. Why would they? If they know they will struggle with the computation and the goal of the problem is to solve, then no matter how hard they work to understand the problem, they still have no chance to be successful.

When the numbers are removed, all students have the ability to work towards understanding of the problem because the numbers are not a limitation. They can think about the problem in terms of whatever numbers are comfortable to them. They can use problem-solving strategies to make sense of the problem without the pressure of computing "really hard numbers" in the end.

Students do need to know how to compute numbers, so after students have taken the time to understand the problem, you as the teacher can provide numbers that are appropriate for them to insert into the blanks and begin solving. I typically provide 3-5 sets of numbers for students to choose from to insert into the problem and solve.

Try a numberless word problem sort!

One of my favorite problem solving activities is a numberless word problem sort. It's not anything revolutionary. It is exactly what it sounds like... students sort a set of numberless word problems by the math action (operation) that they would use to solve the problem. It is such an easy activity but it is SO POWERFUL.

If you want students to practice computation, roll a dice or draw a number card and insert the numbers into the problem. Now students have taken time to understand the problem and they get computation practice. It's a win-win scenario! I highly recommend that every math teacher try this out with their students. It will likely reveal some misconceptions students might have!

Want to try numberless word problems in your classroom?

If you are an upper elementary teacher, check out this numberless word problem sort for FREE! This activity is powerful in helping students focus on the meaning of each operation.

If you are a 5th or 6th grade teacher, here is the set of numberless word problems I made for my students. For some reason, fractions tend to totally throw students off in word problems! It was encouraging for my students to see that they can understand these problems with or without fractions.

It also was really good practice for multiplying and dividing fractions because these operations tend to be a bit tougher to identify when you are working with fractions. Without the numbers, students were able to identify similarities and differences in the problems themselves instead of focusing only on the numbers.

So, what do you think? Have you given your students word problems without numbers?

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The Power of Numberless Word Problems - Mix and Math

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Numberless word problems are simply contextual math word problems with the numbers removed. Learn why you should use numberless word problems in your 4th and 5th grade classroom and grab a free printable numberless word problem resource.

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K-2 Sense Making in Problem Solving (PD 12/9/21) 