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Teaching Math Word Problems with Pictures

Help your students visualize their answers.

If you google word problem jokes, this one will pop up a lot:

“If you have 4 pencils and I have 7 apples, how many pancakes will fit on the roof?

Purple, because aliens don’t wear hats.”

It’s hilarious because it’s true! Many of our students see nonsense in word problems. They not only have to figure out what the word problem is asking them to do, but then they have to actually solve it. It’s a process. However, if we want them to internalize the concept, not just the numbers, pictures can help! Visuals can help students comprehend meaning when the words aren’t making sense to them. Don’t underestimate the power of teaching word problems with pictures. Here are a few easy ways to incorporate visuals into everyday math:

The power of “acting it out”

Before you go from manipulatives to drawing, try having students act out problems. If the problems involve eating, bring in food and have a student actually act like they are in the story problem. If the problem involves a specific number of boys and girls, have that many students get up and show the class what is going on in the problem. Taking this step will help students visualize the problem and think about the actions rather than just guessing if they should add, subtract, multiply, or divide.

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Getting ready to draw

When you’re ready to start solving word problems with pictures, follow Jan Rowe’s steps :

• Read the entire problem: Get all the facts and underline keywords.
• Answer the question: What am I looking for?
• Draw a picture or diagram: Visualize as a real-world situation.
• Solve the problem: Set up the equation and solve.
• Check your solution: Is this answer reasonable?

Drawing as a step of the process

Visual representations are a good starting point for word problems because it is an intermediate step between language-as-text and the symbolic language of math . Drawing lowers the affective filter because it can be less stressful. We aren’t stepping straight into that equation; we are just drawing to figure out what the question is actually asking us. Remind students that we are not in art class. It is perfectly OK for your math pictures to be just scribbles as long as you know what they represent. Another great perk of the “Draw a Picture” strategy is that you, as a teacher, can really step inside the student’s brain to see how their mathematical brain works. Encourage labeling so that you can catch misconceptions right away. These drawings can lead to great math talk conversations, which build that academic language that we all want to hear.

Jayden had two boxes of books with twelve books in each box. He gave four books to his sister. How many books did Jayden keep for himself?

Draw a picture:

Try photography instead of drawing

Story problems are grounded in reading. Photograph Math is an activity that can help visual learners grasp the math skill first, then add in the language. All you’ll need is a camera (or phone) to take pictures. Here’s how it works:

• Students brainstorm the ways in which, and the places where, they use math.
• They stage a photograph representing one of these ideas.
• Students write their own real-life word problems to go with their posed photograph(s).
• Students take a photograph of the menu inside the cafeteria. They write problems that help them figure out how much money they need to get specific lunches, snacks, or drinks. Extension: Think about how much you would need to get a snack every day, for a week, etc.
• Someone might take a photo of a number on a library book. Then try to figure out what value that number would have based on the number of digits and round the number to the nearest whole, tenths, or hundredth.
• Students take a photo of your stash of whiteboards markers. They could write problems about the amount of each color you have. They may choose to think about what happens if another teacher borrows a certain number of markers, etc.

Photograph math can help students to start thinking like a mathematician in all aspects of life. If students begin to see the math around them, then they can truly begin to comprehend the story problems that we present to them in class.

Read a math picture book

Remember when I said story problems are grounded in reading? Why not use actual stories? Children’s picture books can provide a rich context from which to begin mathematical investigations . As students see math concepts play out in stories and illustrations, they are engaged and better able to construct meaning. Seeing the problem-solving process in action through a book can serve as a model for students when they go off on their own to solve problems. Here is a list of 16 picture books about math .

Assign math as picture prompts

Consider starting each week with a relevant picture prompt. Sometimes, the picture is of a mathematical error we came across in real life. The challenge to students is to figure out the mistake. Other times, we simply provide an image and challenge students to come up with a story problem to accompany it. Giving students ownership of these problems can strengthen and extend their comprehension of story problems.

• Put up a sign of a price mistake in a store. Then write on the board: “Ms. Caudill saw this sign at Walmart this weekend and laughed. Why? How might you fix it?”
• Show students the following prompt: Write a story problem using the photo and the following word bank: apples, tomatoes, sweet potatoes, pound, gallon, farmer’s market.

The truth is, story problems are challenging for all students. We have to find ways to make the story problems relative to students’ lives if we want them to succeed.

How do you teach word problems with pictures? Share in the comments below.

Want even more strategies for teaching word problems check out 14 effective ways to help your students conquer math word problems ., want more articles like this make sure to sign up for my weekly newsletter for third grade teachers .

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14 Effective Ways to Help Your Students Conquer Math Word Problems

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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

Get step-by-step solutions to your math problems

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Strategy: Draw a Picture

Math Problem Solving Strategy: Draw a Picture to Solve a Problem

This is another free resource for teachers from The Curriculum Corner.

Looking to help your students learn to draw a picture to solve a problem?

This math problem solving strategy can be practiced with this set of resources.

Math Problem Solving Strategies

This is one in a series of resources to help you focus on specific problem solving strategies in the classroom.

Within this download, we are offering you a range of word problems for practice.

Each page provided contains a single problem solving word problem.

Below each story problem you will find a set of four steps for students to follow when finding the answer.

This set will focus on the draw a picture strategy for math problem solving.

What are the 4 problem solving steps?

After carefully reading the problem, students will:

• Step 1: Circle the math words.
• Step 2: Ask yourself: Do I understand the problem?
• Step 3: Solve the problem using words and pictures below.
• Step 4: Share the answer along with explaining why the answer makes sense.

Draw a Picture to Solve a Problem Word Work Questions

The problems within this post help children to see how they can draw pictures when working on problem solving.  

These problems are for first and second grade students.

Within this collection you will find two variations of each problem.

You will easily be able to create additional problems using the wording below as a base.

The problems include the following selections:

• Chicken and Cows – guess and check type problems
• Fruit Trees – multiple step simple addition and subtraction problems
• My Marbles – involves writing a simple fraction
• Art Box – simple subtraction
• Snowman – simple addition
• Cookies – multiple step problems
• Pillows & Buttons – simple multiplication (by 2, 5, 10)
• Chicken Nuggets – simple multiplication (by 2, 5, 10)
• Trading Cards – multiplication
• Flowers – multiplication

Extend the learning by encouraging your students to draw a picture and write an equation!

You can download this set of Draw a Picture to Solve a Problem pages here:

Problem Solving

You might also be interested in the following free resources:

• Addition & Subtraction Word Problem Strategies
• Fall Problem Solving
• Winter Problem Solving
• Spring Problem Solving
• Summer Problem Solving

As with all of our resources, The Curriculum Corner creates these for free classroom use. Our products may not be sold. You may print and copy for your personal classroom use. These are also great for home school families!

You may not modify and resell in any form. Please let us know if you have any questions.

Farm Fun Writing Word Problems Activity - The Curriculum Corner 123

Thursday 30th of January 2020

[…] Solving Problems Using Pictures (you will find word problems similar to the ones students might write within this set) […]

Strategy: Make a Table - The Curriculum Corner 123

Monday 27th of January 2020

[…] Draw a Picture to Solve a Problem […]

Strategy: Write a Number Sentence - The Curriculum Corner 123

[…] Drawing Pictures to Solve Problems […]

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Problem Solving Draw A Picture And Write An Equation

Problem Solving Draw A Picture And Write An Equation - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Polyas problem solving techniques, Solving linear equations using pictures by tiffany bryant, Problem solving strategies guess and check work backward, Multiplication arrays word problems, Algebra tiles and equation solving, Polyas problem solving techniques, Grade 3 supplement, K 12.

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Topics: Featured Math Concepts Conceptual Understanding Eureka Math Squared

From Read-Draw-Write (RDW) to Modeling–How Students Experience Problem Solving in Eureka Math²

by Great Minds

every child is capable of greatness.

Posted in: Aha! Blog > Eureka Math Blog > Math Concepts Conceptual Understanding Eureka Math Squared > From Read-Draw-Write (RDW) to Modeling–How Students Experience Problem Solving in Eureka Math²

Coherence is a key feature of the Eureka Math 2 ® curriculum. The problem-solving process employed in Grade Levels K–9 is a major part of that coherence. In Grade Levels K–5, students know it as the Read–Draw–Write (RDW) process. Starting in Grade Level 6 the process advances to Read, Represent, Solve, Summarize (RRSS) while maintaining the same foundational approach. Word problems become accessible as students cycle through reading and representing parts of a word problem to make sense of it. The process is simple, memorable, and powerful enough to support all students with language demands, sense-making, and algebraic thinking. This paper outlines the progression of problem-solving in Eureka Math 2 . We invite you to discover where your students fit into this progression, how you can support them with the work of prior grades, and build the foundation needed to problem-solve in subsequent grades.

Grade Levels K–2

The Read–Draw–Write process for solving word problems is introduced and intentionally taught step-by-step in Grade Levels K–2. You may not always recognize the developmental progression of this problem-solving process. After all, the first piece of the process is Read, a skill many of our youngest students have not learned yet. Let’s follow the journey our students take to problem-solving.

Kindergarten

One of the first experiences students have with problem-solving is counting. In the kindergarten Module 1, Topic G, Lesson 33, students use 10-frames and other problem-solving tools. Young students sometimes employ the Read-Draw-Write process without knowing it.

Are they counting a large set of objects or a small set?

Are the objects movable, or are they part of a picture?

Can they group the objects to count?

Even before students can technically read, they take in information and understand the task to get started.

The next step is Drawing. How can students organize the information they took in during the Read phase to solve the problem? For our young students, this may look like crossing out or drawing objects, it may be writing an equation or making tally marks, or it may be simply using their fingers to track their counting. Finally, students Write: they state or write the answer they have arrived at after contextualizing the problem. “There are 9 buttons!” or “I counted 25 erasers. I made 2 groups of ten and had five ones.”

Kindergarten students progress to more traditional problem-solving scenarios later in the year. In Grade Level Kindergarten Module 5 Topic A Lesson 2, students represent a math story by using pictures, number bonds, and number sentences.

There are some pigeons on our playground. Then, some more pigeons land on our playground. What could this math story look like?

The title Read–Draw–Write may be misleading. Understanding and recognizing the more subtle instances of this powerful problem-solving strategy when reading, drawing, and writing may not be applicable in the literal sense is important.

Grade Level 1

In Grade Level 1, RDW expands to include all addition and subtraction problem types. Most problems students were presented with in kindergarten were the dark-shaded problem types shown in the chart. Grade Level 1 signals the transition to working with all the problem types in the chart, including the more challenging types shown in white.

As students’ problem-solving development progresses, it is important to spend time crafting the problems. Problems should be varied in type, and the numbers in the problems need to be high enough so that RDW is necessary to find the answer. For example, in Grade Level 1, Module 2, Lesson 8, students see a Change Unknown problem.

The numbers in this problem are large enough to make mental math more challenging, and the problem type lends itself to drawing to understand what the question is asking. If we ask students to use RDW with a problem that is too “easy,” for example:

Kit is coloring a rainbow.

She has 3 markers and she gets 4 more.

How many markers does she have?

Students won’t see the value in problem-solving with RDW because they could do mental math or count on their fingers, which we want them to do in circumstances like this. Teaching first-grade students to discern when to use RDW is important, and asking them to “show your work” may not always be appropriate.

Comparison problem types frequently provide the productive struggle that leads to students seeing the need to use RDW. In Module 4, the tape diagram is taught and explored in the context of measurement. It begins with concrete cube trains (read) and moves to a tape diagram, a pictorial recording (draw), and then an equation and a statement (write).

Grade Level 2

With these added problem-solving complexities, Grade Level 2 students will find the part between reading and drawing the most challenging. Drawing a model that shows what is being asked in the problem and what helps students solve the problem needs to be the focus in this teaching. For example, in Module 4 Lesson 3, this problem requires two-steps to solve it. First, a student must understand that they need to find the total number of cars and trucks. Second, they must understand that they are only solving for the number of trucks. The round numbers in this problem are intentional because we want students to attend to the set-up of the problem rather than worrying about the calculations.

Further along in the Module, the numbers and the problem types increase in complexity. The focus remains on drawing or representing the problem, so students perform the correct operation with the correct numbers. Too often, students look at the numbers presented in a word problem and guess whether they should add or subtract them to get an answer. We hope to prevent that by teaching the RDW problem-solving process, as shown below.

Grade Levels 3–5

In Grade Levels 3–5, students use the RDW process to solve word problems involving all operations and to build multi-step problems. Lessons emphasize students developing strategies to address newer complexities of multiplication, division, and working with fractions in word problems.

Beginning in Grade Level 3, students are introduced to division through the context of word problems and encounter two types of division. In a partitive division problem, the number of groups and the total are known, but the number in each group is unknown. Partitive division asks, “How many are in each group?” In measurement division, the total and number in each group are known, but the number of groups is unknown. Measurement division asks the question, “How many groups are there?” Using the RDW process, students create a drawing that accurately represents the known and unknown number of groups and the number in each group to reveal the solution path.

In Grade Level 3, Module 1, Topic B, Lesson 9, students solve a partitive and measurement division problem using the RDW process. As they read and draw, they discern and represent what is known. Then, they identify the unknown and use division to solve it. In the problem shown here involving 24 desks, the number of groups is known. One approach is to begin by drawing 6 circles to represent the groups, then sharing the 24 desks between the groups. There is flexibility in the RDW process for students to draw and solve in a way that makes sense. In this case, a second student starts by drawing the 24 desks and knows they must form 6 equal groups. They can determine that forming six equal groups will result in 4 desks in each group.

How does thinking about what is known and unknown help you solve division word problems? Thinking about what is known helps me know if I should draw the number of groups or the number in each group. If the unknown is the number of groups, I can count the number of groups in my drawing to solve the problem. If the unknown is the number in each group, I can look at the number in each group in my drawing to solve the problem.

Later, in Lesson 18, students represent partitive and measurement division word problems with tape diagrams, rather than with equal groups. As a built-in feature of the RDW process, the teacher asks consistent questions: What is known? What is unknown? and How is it represented in the tape diagram? Students can consider those questions and rely on the familiar process of reading and representing the problem in chunks until identifying a path to solve it, even with the more abstract tape diagram. That process also reveals the difference between partitive and measurement division problems and deepens students’ understanding of problem-solving.

In Grade Level 4, word problem contexts expand to include multiplicative comparisons. There are multiple variations of the language in a problem that students encounter. Reading and drawing to represent the parts of the problem helps students distinguish between the known and unknown. In multiplicative comparison problems, the relationship between two quantities is often stated as “___ times as many as ___”, in other contexts, it is stated as “___ times as much as ___”, or even more specifically, “___times as heavy/long/tall as ___”. Both quantities may be given, and the total is unknown, or the total and one quantity are known, and the second quantity is unknown. Depending on the known and unknown quantities in the problems, students use multiplication or division to solve.

The RDW process supports students as they learn to interpret and evaluate multiplicative comparisons within word problems. For example, in the problem from Grade Level 4, Module 1, Topic A, Lesson 2 shown, students first Read and encounter comparison language in the phrase “4 times as many books.” Students typically represent the quantities with two separate tapes. The first tape shows 1 unit, the books that Ray reads. The second tape shows 4 units, or 4 times as many as Ray, the number of books Jayla reads. As part of the Draw step in the RDW process, students look at the tape diagram they’ve drawn and consider what it’s showing, what is known and unknown. This leads students to determine how to solve the problem. In this example, students recognize that they know the total of 4 units, so they can divide to find the unknown value of 1 unit.

How do you decide when to rename a product that Is a fraction greater than 1 as a mixed number? If it’s a word problem, thinking about what the product represents helps me decide when I should rename fractions greater than 1 as mixed numbers. If the answer doesn’t make sense as a fraction greater than 1, then I can rename it as a mixed number. Sometimes the question in the word problem helps me think about whether I should rename the fraction greater than 1 as a mixed number. When there isn’t a word problem or directions to write the answer as a mixed number, we can leave the product as a fraction greater than 1.

In Grade Level 4, Module 4, Lesson 33, students solve the problem: A kitten weighs 4 ⁄ 5 kilograms. A puppy is 6 times as heavy as the kitten. How many kilograms does the puppy weigh? Again, students can read the problem in chunks and draw a tape diagram to represent the weights of the kitten and puppy. The same process of reading and representing the problem supports problem-solving, even when one of the values is a fraction. However, with a fractional value, students consider whether to state their answer as a fraction greater than one or a mixed number. In the Write step, students return to the word problem to write an accurate solution statement and use the problem’s context to decide which number type makes sense to answer the question posed. As students engage in that decision-making process, they are reasoning abstractly and quantitatively and engaging in Standard for Mathematical Practice 2 (MP2). The Debrief discussion shown here emphasizes the role of the RDW process in deciding how to record an answer.

As the repertoire of problem types continues to expand along with number types, RDW continues to be a supportive process students use to read and understand a problem, create a drawing that clarifies the known, unknown, and solution path, and to state the answer in a way that makes sense.

In Grade Level 5, the complexity of word problems advances to multi-step (3+ steps) word problems. Students reiterate the RDW steps of reading a chunk and representing the information in a drawing multiple times. Students practice refining this process to create an accurate drawing and to use the RDW process efficiently in Module 3, Lessons 20 and 21.

In Lesson 20, through the RDW process, students explore multiple ways fractional amounts can be represented in tape diagrams. In reading and understanding the language of the problem, students determine whether it is a comparison problem, which helps them decide how many tapes to draw. Students also represent fractional amounts such as “ 2 ⁄ 5 of his money” and “ 1 ⁄ 3 of the remaining money.” When students draw step by step as they read, they first partition the tape diagram into 5 parts and label 2 parts to represent 2 ⁄ 5 . This leaves 3 parts unlabeled and 1 ⁄ 3 is simpler to identify.

Does the Read-Draw-Write process help us solve multI-step word problems Involving fractions? How? Yes. We read, draw, and write in chunks.When we learn new information, we pause to draw and then go back to reading and draw when we learn something new. We can write expressions or equations as we realize which operations we can use to find unknown information. Yes. The model I draw helps me decide which operation to use to find an unknown value. Each time we find new information by evaluating an expression, we can compare it to our model and ask, Does that make sense based on what I see in the model?

In Lesson 21, students encounter additional multi-step word problems with fractional amounts and take on more responsibility in determining how to represent and solve. When students use a self-selected method to solve a comparison word problem involving fractions, they model with mathematics and practice Standard for Mathematical Practice 4 (MP4). The key question of the lesson: Does the Read–Draw–Write process help us solve multi-step word problems? How? has them reflecting on the process throughout the lesson and finally in the Debrief, as shown. The flexibility of the process supports students as they read and return to modify their drawings multiple times. RDW also continues to aid students in identifying the known and unknown information and a solution path.

Representing comparisons and fractional amounts in Grade Levels 4 and 5 prepares students to visually represent ratios in Grade Levels 6 and beyond. The number choices and problem types continue to grow in complexity over the years. However, the RDW process remains a foundational tool for solving word problems.

Grade Levels 6–Algebra I

Much of the progression of mathematics in the middle school years includes algebraic understanding and representation, and the RDW routine progresses similarly. Instead of using drawing as the main problem-solving strategy, Eureka Math 2 Story of Ratios uses algebraic thinking as the main strategy. RDW progresses to RRSS: Read, Represent, Solve, Summarize. The representation can still be a drawing, but it could also be an equation with variables representing the unknown.

The Read portion of the routine is used to find the same information as in RDW, to identify what the problem is asking us to find and what we know. In the Represent portion of the routine, students choose the representation that makes the most sense to them, such as tape diagrams, a double number line, a table, or a graph, and then they share their representations with one another. Pairs then use their representation to solve the problem. After completing the problem, the teacher directs students to summarize their findings by asking questions such as, “Does my answer make sense?” and “Does my result answer the question?”

Have groups work to solve problem 3. Circulate as students work, and observe the strategies groups use as they read, represent, solve, and summarize. Encourage students to return to problems 1 and 2 if necessary, and ask the following questions to advance their thinking:

• What is the problem asking you to find? What do you know that is given in the problem? What do you know based on your chosen method of travel?
• What tool can you use to represent the problem?
• Does your model show what is known and whot is unknown? How can you improve your model?
• What units do you need to consider in the problem?
• Does your answer make sense?
• Does your result answer the question?

To continue practicing the routine, students choose from a list of transportation methods and determine how long it will take for that method to transport them to the moon. The teacher is given a list of possible questions to advance students’ use of the RRSS routine. Beyond the Read portion of the routine, questions such as “What tool can you use to represent the problem?” and “Does your model show what is known and what is unknown?” gently guide students through the Represent portion of the routine.

This lesson that specifies the use of the RRSS routine is rare. Generally, the routine is not called out in the problem prompts or lesson structures. Instead, the tool is part of the students’ toolkit, and students reach for it when they deem it appropriate. Because of this, teachers will likely need to post the steps to the routine, say the steps during think-alouds, and remind students during problem-solving activities. Consider using the advancing questions from Lesson 21 in other problem-solving situations.

The first instance of the routine in Grade Level 7 is in Module 1, Lesson 11, when students are further encouraged to represent the problem “…by using a tape diagram, an equation, a graph, a table, or any other model.” Advancing students’ thinking toward an algebraic representation, students use a double number line, then a graph, and finally an equation for the distance formula,  d=rt , in subsequent problems in the lesson. The RRSS routine is mentioned in Launch and in a Teacher Note but not called out further in the lesson guidance. Encouraging students to use RRSS as a tool in their kit will again be important for Grade Level 7 teachers.

There are many instances in Grade Level 8 when using the RRSS routine would aid students in problem-solving. However, the routine is not called out by name until Module 4, Lesson 10. In this Grade Level 8 lesson, students use linear equations to solve real-world problems. Students begin working with the routine as a class. Then, they use the routine with a partner and eventually work with it independently. While other representations may help students write the linear equation, representation of the problem with a linear equation is a specific task requirement. Students analyze the number of solutions to the equation in the context of the situation as they summarize the solution in the last step of the routine. Students recognize that the solution to the equation does not always answer the question from the problem, an additional complexity to problem-solving at this level.

The work with Read–Draw–Write and Read–Represent–Solve–Summarize engages students in many of the mathematical practices, but ultimately leads to the final goal, success with the modeling cycle (MP4) for problem-solving in high school mathematics. Math Practice 4 begins with, “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” 1 There are many ways to model the process of mathematical problem-solving. Most start with a real-world problem, iterate, formulating a solution strategy, performing some computations, interpreting the results, ensuring those results are accurate, and then reporting final results if they are validated.

The Algebra I Eureka Math 2 curriculum takes the modeling cycle to its fruition, giving students multiple experiences with real-world problems. The first instance is in Module 1, Lesson 7 titled Printing Presses. Students work in groups to find an entry point for solving a problem about printing presses. Students analyze a variety of solution paths, building connections between quantitative reasoning and the process of writing and solving an equation in one variable. Connections are made to the RDW and RRSS routines when the teacher asks, “What assumptions can be made? What information do we know remains constant?” and “What are the important quantities? How are these quantities related?” The answers to all of these questions result in ways to read and represent the information from the problem.

In Module 2, students watch a video showing a “regular” showerhead and a low-flow showerhead. To further develop the modeling cycle, instead of giving students the problem, students explore questions related to the problem before narrowing their focus to one question. Information is withheld to allow students to independently determine what is required to answer the question. Additional information is provided only after students decide what assumptions need to be made during the formulation and computation stages of the modeling cycle. Students eventually connect different solution paths to a system of linear equations and explore the effects of changing the assumptions within the context as groups interpret and verify their results.

In other lessons, Algebra I students analyze falling objects and projectile motion, maximize area, and determine how a search and rescue helicopter relates quadratic functions to the real world. Connecting linear and exponential functions to the real world involves lessons in which students work with world populations, temperatures of objects cooling over time, and invasive species populations. The final Module of Algebra I brings all the years’ learning together as students use the modeling cycle to analyze paint splatters, reflect on the role of a city planner, consider a financial deal proposed by a business owner, plan a three-dimensional model of the solar system, and plan a fundraiser that maximizes profit.

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Using Strip Diagrams as a Problem-Solving Strategy

There’s a lot to be learned from the data that comes from state testing if we use it correctly . Oftentimes, we’ll take a problem that students, as a whole, perform poorly on and we create a bunch of problems just like that one and “teach” them how to work that type of problem. In the end, that’s not an effective strategy, because they’ll probably never see a problem exactly like that one again. A more effective way to use the data is to analyze the wrong answers to determine underlying misconceptions that resulted in the wrong answers and provide students with strategies and tools to improve their overall mathematical reasoning. As far as strategies go, drawing strip diagrams is one of the most powerful strategies students can have in their toolbox. And I have some super clear examples to persuade you!

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

The test items in this post come from the 2021 State of Texas Assessments of Academic Readiness (STAAR) test. A wonderful organization called lead4ward analyzes the test each year and provides error analysis statistics.

Analyzing errors

Let’s get started! As you can see from this test item, only roughly half of the students in the state got the correct answer. Look at the most common wrong answer, choice H. Can you see the error that the students who chose that answer made? Take a minute to figure it out before you scroll down.

Look closely at the order of the numbers in the problem. The smaller number comes first in the problem. Students who chose H realized it was a subtraction problem, but took the numbers in order from the problem and subtracted them. So they did 379 – 514. Nine minus 4 is 5. Seven minus 1 is 6. They couldn’t do 3 minus 5, so they did 5 minus 3. The answer they got was 265. Your first thought might be, I need to make sure my students always know to subtract the smaller number from the bigger number. Except that’s not true. If the temperature is 18 degrees and the temperature drops 29 degrees…  We need to be extremely careful to not teach “rules” that expire .

The students who chose H lacked an understanding of the meaning of the numbers in the problem. The way we help them be more successful with problems like this is to give them tools to improve their comprehension of word problems and the numbers they contain. Enter strip diagrams.

Using drawings to describe problems

Drawing strip diagrams is a process that actually begins in Kindergarten and 1st grade when students should be drawing pictures to represent word problems. Students should understand that the numbers in a problem represent something—pizzas, soccer players, apples, money saved—and their drawings should include labels identifying what the numbers represent. At this point, the drawings don’t need to resemble strip diagrams. What’s important is the labeling. It could look something like what you see here.

Notice a couple of things. First, the boys and girls are represented by circles. Easy to draw and count. Students need to understand that these are math pictures, not art pictures. Next, and of critical importance, are the labels.

Beginning in 2nd grade, students can begin to draw more formal strip diagrams. Strip diagrams, also called tape diagrams, are often associated with Singapore Math. Char Forsten’s Step-by-Step Model Drawing is the book I learned model drawing from. Another great resource is Math Playground’s Thinking Blocks .

Modeling how to draw a strip diagram

Now let’s get back to that released test item I started this post with and see what the model-drawing process might look like.

This problem is a comparison subtraction problem. We always want students to draw the model with labels first. they will add the numbers in the next step.

Teacher: [Reads problem out loud]  What is this story about? (lions) How many lions? (2)  What does the problem tell us about the lions? (their weight)  Do we know their weights? (yes) Which lion weighs more? (the older lion)  What is the problem asking us to find? (the difference in their weights)  Huh, what does that mean?  (The older lion weighs more than the younger lion. The problem is asking how much more.)

NOTE: Notice that we didn’t talk about the numbers at all! The point of this discussion is to help students make sense of the numbers in the problem and verbalize what the problem is asking them to find.

Teacher: Drawing a model really helps me understand what math I need to do to solve a problem. Let’s draw a model to represent this problem. We know that the older lion weights more, so his bar should be longer, right? [draws and labels the older lion’s bar] That means the younger lion’s bar should be shorter. [draws and labels younger lion’s bar] And you guys told me the problem is asking for the difference.  [adds the difference with a question mark]

Now we can plug in the numbers from the problem.

Notice that what I’ve described is very scripted. I want students to hear my mathematical thinking, and I’m teaching them the mechanics of drawing the model. But it’s important to let students use the tool to solve problems. Think how the models would look for these variations of the problem:

There are two lions at the zoo. The weight of the younger lion is 379 pounds. That’s 135 less than the weight of the older lion. How much does the older lion weigh?

There are two lions at the zoo. The weight of the older lion is 514. That’s 135 more than the weight of the younger lion. How much does the younger lion weigh?

Remember, our goal is for students to be able to use strip diagrams to solve new types of problems, so once they understand and can use the model, we have to give them new types of problems to solve without scripted instruction.

More examples of strip diagrams

Let’s take a look at a few more problems from the same test. Each of these problems had pretty dismal results.

This first problem is what we in Texas call a  gridable . That means it’s not multiple choice—students have to write and bubble in their answers. A lot of times students will miss gridables due to calculation errors. But I’m pretty sure that’s not the case here. I doubt they miscalculated 4 x 5. What that means is that 38% of the 3rd-grade students in Texas did not recognize this as a multiplication problem. They likely added 4 + 5. Teaching keywords could be the culprit. Students see the word total in the problem, and they’ve been taught that  total means addition. Teaching keywords basically gives students permission to  not read and understand the problem—just find the keyword and plug the numbers into the operation. Not a sound problem-solving strategy. Instead, we see how a strip diagram could be drawn to represent the problem.

Here’s another multiplication problem. You can see from the error analysis that only 52% of the 3rd-graders correctly answered this problem. Answers B and D are calculation errors. Can you figure out the error these students made? Doing so can help you prevent these types of errors by addressing them in your instruction. Answer choice C results from adding the two numbers, not multiplying. Again, we see the keyword total in the question. Drawing a model would not only help the students visualize the problem as multiplication, but it might also prevent calculation errors. Students who are not confident with the standard algorithm could solve the problem with repeated addition.

Last one and it’s a doozy! Look at that error distribution. When it’s spread out like that, it usually means the kids just didn’t have a clue and guessed. There’s a lot going on here. How could we help students tackle a problem like this?

First, of course, is drawing a model. We see that this is a part/whole problem with three parts, one of which is missing. If you looked carefully at the wrong answers, answer choice F was adding all three numbers. Pretty hard to look at this model and think you’re supposed to add all three numbers.

Aside from model drawing, however, students should learn to write equations to match their models. That’s really the other thing that was hard about this problem. They weren’t asked to solve the problem, just find the correct way to solve it.

Final thoughts

Strip diagrams have to be presented to students as a problem-solving tool and they have to be used consistently. Yes, it takes longer for students to draw strip diagrams to represent their problems, but it should be an expectation. That means we probably need to assign fewer problems to allow students the time to draw their strip diagrams.

The labels are a must! If students can’t label their numbers, it’s a huge red flag. Work with those students in small groups to help them develop comprehension skills.

Students who say they know the answer without drawing a strip diagram should be reminded that we draw models when the problems are easy so we can use them as a problem-solving tool when the problems get harder. And if a large percentage of students can really solve the problems without drawing models are we challenging them enough?

So there you have it. Have I sold you on having students draw strip diagrams? Do you have tips of your own to share? I hope you’ll sound off in the comments.

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13 Comments

Strip/bar/tape diagrams are becoming one of my favorite strategies. The labels are crucial and I need to be better at expecting that. I am nearly to the part of 3rd grade where I will be teaching 2 step problems again and I think continuing the strip/tape/bar diagram representation is going to be a wonderful tool. Here’s hoping!

Thanks for your comment, Jessica! They are one of my favorites as well. So very powerful! Come back and let us know how it goes.

Any way we can show visuals for problems are going to support student understanding! Thank you for adding another tool to my teaching tool box! I teach 1st grade and I want to build my students’ confidence and understanding of math!

That’s wonderful, Suzanne! It’s so important that our students develop a positive math identity early.

Hi Donna, I am increasingly interested in the connections between literacy and math. Your post about how kids start by representing their mathematical thinking/problem solving with pictures, then labels, then drawing a model and later abstract number sentences. seems so similar to how young children write stories first with pictures, then we ask them to label and later on write sentences with words, punctuation etc. Do you know of any research linking the two? Thanks! Jennifer

You are so very correct! There are tons of professional books connecting the two. Solving word problems requires comprehension of the problem. Just like reading teachers tell their students to “make a movie in their head” when they read, I tell my math students the very same thing!

Thank you so much for breaking down strip diagrams. This is a tool I try to encourage my students to use, but I feel like I need to do more modeling for them after reading this. I also like how you pointed out we should analyze the best wrong answers for misconceptions.

I have been using strip diagrams to teach my 2nd graders for a few years. They always seem to struggle with understanding where to put the numbers when the story is comparing. How many more toys does Grant have than Amanda? So, I always pointed out that it’s a comparison, and that seemed to help. I am now a K-5 math interventionist and some upper grades teachers think it will be confusing to call that a comparison when they are not using >, <, = symbols. How can I help them understand those problem types, and is there harm in using the word comparison? I think they are still comparing – "how many more".

A strip diagram for comparison looks different. It’s one bar on top of the other. The difference is where the longer bar is shorter than the shorter bar. That might help with the confusion. Absolutely use the word comparison!

Hi Donna. I’m wondering how strip diagrams are any different than the part part whole organizer. They seem so similar to me. I’ve done a lot of work getting my students to use and understand the PPW as a tool. I like the idea of the strip diagram but fear I’ll confuse them if I introduce a new tool to use in place of a PPW.

They are very similar! Often students transition to strip diagrams in the intermediate elementary grades. No need to do both though.

Hi Donna! I started using tape diagrams with my students when my district adopted the Engage NY/Eureka Math Curriculum. I found that students have had great success in using this strategy. Thank you for this clear explanation on using tape diagrams during problem solving.

It’s such a powerful strategy! I’m glad to hear your students found success using it!

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