problem solving tape diagrams

Tape Diagrams in Math: A Problem Solving Strategy

What is your go-to strategy for teaching word problems? I have always encouraged my students to model, model, model when solving word problems. Modeling is a way for students to make sense of a problem and catch mistakes before they happen. Sometimes, what seems like a good answer just doesn't work out once it's drawn as a model.

Until recently, I really didn't care what kind of model my kids used. Drawings, tally marks, symbols: anything was fair game. But as we've worked our way through multiplication and into division, I've found myself returning over and over again to the tape diagram. It's a nearly fail-proof way to work through a story problem and my students have become problem solving masters!

So, what exactly is a tape diagram?

A tape diagram is a drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length model."

In practice, a tape diagram in math looks like this...

Tape diagrams have become my go-to model when teaching math. They work for any type of word problem, but I especially love them for solving multiplication and division.

how do you solve a tape diagram?

To solve problems using tape diagrams, students must first ask, "Do I know the whole amount?" If it's supplied by the problem, fill it in. If not, put a question mark at the bottom of the diagram. Then look for other information and fill that in. The question mark always represents whatever piece of information is missing.

Here is an example of how I use it to teach division problems:

Students will quickly begin to see a pattern... the bottom number is always the product of the top two numbers.

Therein lies the beauty of this model:

It is self-correcting. If the top two numbers, when multiplied, do not equal the bottom, you've done something wrong.

When teaching with this model, it's a good idea to have students write all four possible equations. So in the above example, we would also write ? x 4 = 32 and 32 ÷ ? = 4. This is important because it reinforces the concept of inverse operations and fact families. It also gives students a tool to use when the encounter problems with a missing factor or divisor.

Once my students are able to interpret story problems and solve using tape diagrams, I teach them to analyze a tape diagram and write their own story problems to go with it. This requires high-order thinking and really develops their mathematical minds.

how to use tape diagrams

To use tape diagrams with your students, I suggest following this instructional sequence...

Teacher supplies the story problem, draws the tape diagram, and models how to solve.

Teacher supplies the story problem, draws the tape diagram, and students help solve.

Teacher supplies the story problem, students help draw the diagram, and students solve on their own.

Teacher supplies the story problem, students draw and solve alone.

Teacher draws a tape diagram and students create a story problem to go with it.

Along the way, you'll want to move from including one equation that represents the problem to showing all four possible equations. This process, from introduction to proficiency, might take several weeks depending on the skills of your students. But it will pay off in the long run when your students become experts at solving word problems !

If you're interested in teaching your students to use this strategy, you may want to check out this resource:

Problem Solving with Tape Diagrams: A Model for Multiplication and Division

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the video needs someone explaining the proses of the tape diagram

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The Mathematics Master

How to Use Tape Diagram for Problem Solving

September 19, 2023

Imagine having a simple yet powerful tool at your fingertips to unravel math mysteries! That's where tape diagrams step into the spotlight.

Think of them as your math buddy – a visual representation resembling a snippet of tape that opens up a world of number connections. Whether you’ve heard them referred to as strip diagrams, bar models, fraction strips, or length models, they all point to the same idea. With tape diagrams, you’re not just solving math problems; you’re crafting a visual journey that untangles the knots of mathematical concepts . These diagrams are like keys to unlock the doors of word problems, helping you quickly organize information and share your brilliant mathematical insights.

What are Tape Diagrams?

Imagine a visual aid that effortlessly guides you through mathematical puzzles – that’s the magic of tape diagrams. These rectangular wonders, resembling strips of tape, are here to simplify your computations. Tape diagrams are your trusty sidekick if you grapple with word problems built on ratios.

Every student’s toolkit should include the skill of creating and using tape diagrams. Why? Because these diagrams hold the key to understanding mathematical concepts and depicting relationships within problems. With tape diagrams, you’re solving problems and bringing mathematical ideas to life. They become your artistic canvas, allowing you to showcase your problem-solving strategies and share your mathematical thoughts.

• Mastering Tape Diagrams

Tape diagrams might sound complex, but fear not! Let’s walk through the steps to craft these problem-solving marvels. Think of tape diagrams as your creative toolkit – they slice daunting tasks into bite-sized graphical goodness. Whether wrestling with tricky word problems or tackling equations head-on, tape diagrams are your reliable companion, making math a visual adventure. So, let’s dive into the artistic journey of creating tape diagrams, one step at a time.

Step 1: Analyze the Problem

Take a moment to read and dissect the problem statement or equation. This initial step is like getting to know the characters before writing a story – it’s essential! By absorbing the problem’s essence, you’ll grasp the scenario and envision pathways to crack it wide open.

Step 2: Gather Your Tools

Now that you’re on a first-name basis with the problem, it’s time to dissect it further. Think of it as breaking down a complex puzzle into its elemental pieces. What parts of the problem are like the corner pieces of a jigsaw puzzle, crucial for completing the whole picture? Identify these components, the building blocks that will populate your tape diagram.

Step 3: It’s Time to Draw the Tape Diagram

With your mental toolkit assembled, it’s time to create your masterpiece – the tape diagram. Envision a long, rectangular canvas resembling a piece of tape. Divide this canvas into smaller sections using neat little boxes. These sections will represent the components you identified in Step 2. Depending on the complexity of the problem, your diagram might have just a couple of sections or a handful. There’s no hard and fast rule, but aim for at least two sections to give your diagram a solid foundation.

Step 4: Solution

Your tape diagram is like a map guiding you to the treasure of the solution. It’s time to put your diagram to work. Solve the problem using the insights your diagram provides. Label the sections, incorporate your gathered information, and let the magic happen.

Putting It into Practice

Now, let’s bring all this theory to life with an example. Imagine a problem where you need to divide your allowance among different expenses. This is where your tape diagram steps in, carving out sections for each expense and revealing the mathematical treasure of how much to allocate where.

Tape diagrams might be simple in concept, but they’re versatile and mighty in practice. They’re not just sketches on paper but your companions on your mathematical journey, making complex problems a visual delight. So, the next time math gives you a puzzling stare, remember that tape diagrams are your secret weapon – transforming numbers into captivating stories of logic and creativity.

Example: Create a tape diagram that represents these equations.

( i ) \(4 + 8 = 12\)

( ii ) \(9 + 6 = 15\)

( i ) Crafting a tape diagram for \(4 + 8 = 12\) involves making four equal cells and adding eight more.

Another option for visual representation could be a complete rectangle divided into two segments, depicted below. It’s worth noting that the rectangle representing \(8\) is longer than the one corresponding to \(4\).

( ii ) Constructing a tape diagram to visualize \(9 + 6 = 15\) involves crafting nine identical cells and adding six more.

Another way to depict this would be a complete rectangle divided into two sections, as presented below. It’s important to observe that the rectangle representing \(9\) is longer than the one representing \(6\).

Tape Diagram for Multiplication and Division

Tape diagrams can also serve as a visual aid for word problems involving multiplication and division.

Examples: Draw a tape diagram that represents these equations.

( i ) \(3 × 9 = 27\)

( ii ) \(2 × 4 = 8\)

( i ) In the case of \(3 × 9 = 27\), \(3\) and \(9\) are called factors. To illustrate this with a tape diagram, let’s create nine equal cells, with three inscribed within each cell as a demonstration.

( ii ) To represent \(2 × 4 = 8\), sketch four identical cells and jot down the number \(2\) within each cell. This serves as an illustrative tape diagram for multiplication.

In essence, the tape diagram stands as an efficient tool for cracking mathematical puzzles. Compared to rote memorization, which can fall flat if grounded in understanding, the beauty of the tape diagram lies in its adaptability. It empowers students not only to master the “how” but also the “why” – fostering a genuine grasp of concepts. By embracing these visual aids, students embark on improved comprehension. As they unravel problems with the guidance of tape diagrams, they enhance their problem-solving prowess and nurture a deeper connection with the world of mathematics.

What are alternate terms for tape diagrams?

Tape diagrams often go by other names, such as strip diagrams, bar models, length models, or fraction strips. These terms are commonly used interchangeably to refer to this visual representation tool.

Why is it called a tape diagram?

A tape diagram earns its name from its resemblance to a strip of tape. This visual tool aids in tackling a variety of calculations, including ratios, addition , subtraction, and, most notably, multiplication . Its simplicity and effectiveness make it a versatile asset in mathematics.

What advantages do tape diagrams offer?

Tape diagrams shine by preventing students from memorizing concepts they may need help understanding. They encouraged a deeper understanding of mathematical ideas and proved to be particularly effective in successfully solving word problems in mathematics.

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Student-Centered Math Lessons

Using Tape Diagrams to Solve Problems

Welcome to Part 3 of our Problem Solving Strategy series! Today we are diving into helpful tape diagrams to solve problems. 

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.

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Tape Diagrams

Introduction.

In tape diagrams, rectangles are used to show the parts of a ratio visually. They require careful drawing because they are a visual model. They simplify and deconstruct complex mathematical word problems . They are shown as a strip of paper or a piece of tape. 

Every learner needs to be able to draw and use a tape diagram to solve mathematical problems. It helps students learn arithmetic ideas and portray relationships in a math problem. Students can use it to communicate how they approach solving problems and organize their thoughts. Students can use these illustrations to solve any word problem.

We will learn more about tape diagrams in this article, including how teachers and students use them to solve mathematical problems that involve addition, subtraction , multiplication , division, ratio , fractions , and equations.

What are Tape Diagrams?

A tape diagram is a rectangular illustration that resembles a piece of tape and has sections to help with computations. It is a frequent graphic tool for resolving mathematical word problems with ratios as their basis. A strip diagram, a bar or length model, or a fraction strip are additional names for tape diagrams that are frequently used.

Each student must be able to create and use a tape diagram to solve mathematical problems. It helps students understand mathematical concepts and illustrate relationships in mathematical problems. Students can utilize it to share their ideas on any problem and how they would go about addressing it. 

What are the advantages of using tape diagrams?

The tape diagram has the advantage of preventing students from memorization of ideas they do not comprehend. The tape diagram is used to solve a word problem in mathematics successfully. The table below shows examples of mathematical problems for which we typically use tape diagrams.

Steps in Creating a Tape Diagram

A tape diagram is simple to make if you follow a few simple steps. It uses a tape piece broken into more manageable rectangular sections to show difficult tasks graphically. Any difficult word problems are simplified with tape diagrams. To draw a tape diagram, adhere to these easy steps.

Step 1: Read and analyze the problem.

To understand the problem, reading the provided problem statement or math equation is essential. You will then comprehend the presented situation and the ways to solve it. There are many different sorts of equations and problems in arithmetic; therefore, it is essential to understand them before you try to answer them by creating a diagram.

Step 2: Gather information essential for the diagram.

After reading the problem description, the next step is to segment the problem to identify its simplest solution. By breaking the word problem down into smaller parts, you can see which parts are necessary for the diagram and which parts have already been resolved in the given data, saving you from having to do the work again.

Step 3: Draw the tape diagram.

The next step is to draw a long, rectangular segment that resembles a piece of tape and then divide it into several parts using small boxes. Depending on the problem statement, you will draw a different number of cells. There is no max limit, but you must construct at least two sections.

Step 4: Solve the problem.

Solving the problem determines whether the tape diagram is accurate by completing the provided task. Labeling the diagram and using the information provided helps to solve the entire mathematical equation.

For example, let us use a tape diagram to illustrate and solve the problem below.

Problem:  There are six sheets of red colored paper and four sheets of green colored paper. How many sheets of colored paper are there? 

The given problem requires us to identify the total number of sheets of colored paper.

Red colored paper: 6 sheets

Green-colored paper: 4 sheets

Therefore, there is a total of 10 sheets of colored paper, 6 + 4 = 10.

How Do Tape Diagrams Work?

A tape diagram is a rectangular illustration that resembles a piece of tape and has sections to help with computations. The following examples use tape diagrams for addition, subtraction, multiplication, division, ratio, fractions , and equations.

Tape Diagrams for Addition and Subtraction

Tape diagrams are used for addition and subtraction as a visual tool to solve problems. The following are examples of how to show equations using tape diagrams.

Create a tape diagram that represents each equation.

( a ) 3 + 9 = 12

( b ) 10 + 4 = 14

(a ) To create a tape diagram for 3 + 9 = 12, we must create 3 equal cells and 9 more. 

An alternative illustration would be a whole rectangle with two parts, as shown below. Notice that the rectangle for 9 is longer than that of 3. 

(b ) To create a tape diagram for 10 + 4 = 14, we must create 10 equal cells and 4 more. 

An alternative illustration would be a whole rectangle with two parts, as shown below. Notice that the rectangle for 10 is longer than that of 4. 

A florist chose seven red and five white flowers to make a bouquet. How many flowers did he use for the flower arrangement?

The problem requires us to find the total number of flowers the florist used for the flower arrangement. The following is the given information:

Let us use a tape diagram to visualize the problem.

Using an equation, we may represent the solution to the problem as 7 + 5 = 12. Therefore, the florist used a total of 12 flowers in his bouquet.

Draw a tape diagram that represents each equation.

( a ) 11 – 4 = 7

( b ) 9 – 6 = 3

( c ) 15 – 5 = 10

( a ) To create a tape diagram for 11 – 4 = 7, we must create 11 equal cells for the minuend and 4 equal cells for the subtrahend. A tape diagram is shown in the figure below for the equation 11 – 4 = 7.

( b ) To create a tape diagram for 9 – 6 = 3, we must make 9 equal cells for the minuend and 6 equal cells for the subtrahend. A tape diagram is shown in the figure below for the equation 9 – 6 = 3.

( c ) To create a tape diagram for 15 – 5 = 10, we must create 15 equal cells for the minuend and 5 equal cells for the subtrahend. A tape diagram is shown in the figure below for the equation 15 – 5 = 10.

Tape Diagram for Multiplication and Division

Tape diagrams can also be a visual tool for multiplication and division-related word problems. 

( a ) 2 × 6 = 12

( b ) 5 × 4 = 20

( c ) 7 × 8 = 56

( a ) For 2 × 6 = 12, both 2 and 6 are called factors . Make 6 equal cells with the number 2 written in each one as an example of a tape diagram for this.

( b ) For 5 × 4 = 20, draw 4 equal cells with the number 5 written in each one as an example of a tape diagram for this.

( c ) For ( c ) 7 × 8 = 56, draw 8 equal cells with the number 7 written in each one as an example of a tape diagram for this.

Francis has 32 candies and plans to give them to his 4 friends. Use a tape diagram to illustrate and solve the number of candies each person receives.

This requires the process of division to find how many each of Francis’ friends will receive. So, we have 32 ÷ 4 = 8. The tape diagram below is an illustration of the problem.

Tape Diagrams for Fractions

Use a tape diagram to solve 45 of 35.

In this example, since the fraction’s denominator is 5, it tells us that we must divide our tape diagram into 5 equal parts.

By the process of division , we can have 35 ÷ 5 = 7. Hence, we may label each cell as 7. 

Since we need to find 4 parts of the whole, we must multiply 7 by 4. 

Therefore, 45 of 35 is equal to 28.

Stephanie is planning a party. She purchased a total of 3 dozen balloons. ¼ of these balloons are red, and the rest are blue; how many of each color did she purchase?

Let us start by determining the total number of balloons Stephanie purchased. Since she has three dozen balloons, we must multiply 3 by 12 since there are 12 balloons for every dozen. Hence, Stephanie purchased a total of 36 balloons.

The tape diagram that must be created has 4 equal parts since it is given from the denominator of the fraction ¼. Hence, 36 ÷ 4 = 9. So, let us label each cell in our tape diagram as 9.

Since ¼ of the balloons are red and ¾ are blue balloons, then we have the following solution:

Red balloons : 9 × 1 = 9

Blue balloons: 9 × 3 = 27 

Stephanie bought 36 balloons, 27 of which were blue and 9 red.

Tape Diagrams for Ratio

Another illustration of a ratio is a tape diagram. Every part in the diagram that is the same size has the same value.

The ratio of pencils to pens in Nathalie’s collection is 3: 7. If there are 18 pencils in her collection, how many are pens?

The ratio of pencils to pens in Nathalie’s collection is shown in the diagrams below.

Since it is given that there is a total of 18 pencils, we can have 18 ÷ 3 = 6. The quotient 6 will be used as a factor to get the total number of pens.

Thus, 7 × 6 = 42. 

With a ratio of pencils to pens of 3: 7, Nathalie has 18 pencils and 42 pens.

Diana uses a paint mixture of 8 cups yellow and 12 cups pink. Find the ratio of yellow paint to pink paint in the mixture.  

The tape diagram below illustrates the paint mixture ratio

The ratio of the number of cups of yellow paint to the pink paint is 8: 12. However, 8 : 12 can still be simplified as 2 : 3 since both numbers have 4 as their common factor.

8 ÷ 4 = 2 12 ÷ 4 = 3

Therefore, the ratio of the number of cups of yellow paint to pink paint, which Diana uses in her mixture, is 2 : 3.

The tape diagram below shows that for every 2 cups of yellow paint, 3 cups of pink paint are used.

A box of black and blue marbles has a ratio of 3 to 8. If you have 15 black marbles, how many marbles in total do you have?

Let us start by finding the total number of blue marbles in the box. If the ratio is 3: 8, and you have 15 black marbles, then we can compute for the factor which makes 3 to 15. Hence, we have,

Thus, we must multiply 5 by 8 to find the total number of blue marbles. So, we have,

Since we have 18 black and 40 blue marbles, the total number of marbles in the box is 55 ( 15 + 40 = 55 ).

Tape Diagrams for Equations

For each of the diagrams below, write an equation and indicate the length of each diagram. 

( b )  

( a ) Equation: 3 + 4 = 7

The total length of the diagram = 7

( b ) Equation: 3 × 7 = 21

The total length of the diagram = 21

Samantha is trying to solve for the value of m in 6×m= 42. Use a tape diagram to represent the equation and find the value of m.

To illustrate the problem using a tape diagram, we will use a rectangle with a total length of 42; it must be divided into 5 equal cells or parts.

We may find the value of m using the division process; we have 42÷6=7. 

Therefore, m = 7. 

A technique for quickly solving mathematical problems is the tape diagram. The tape diagram has the benefit of preventing students from memorization of ideas they don’t comprehend. A rule that is remembered won’t help a student unless they know how to apply it or adapt it to other circumstances. The tape diagrams, on the other hand, are simple visual aids that help students comprehend a topic better. By applying them, they will improve as problem solvers.

The following are the four easy steps to follow in using a tape diagram to answer mathematical problems.

Step 1: Read and analyze the problem. Step 2: Gather information essential for the diagram. Step 3: Draw the tape diagram. Step 4: Solve the problem.

Frequently Asked Questions on Tape Diagrams (FAQs)

What is meant by tape diagrams.

A tape diagram is a rectangular illustration that resembles a piece of tape and has sections to help with computations. It is a frequent graphic tool for solving mathematical word problems with ratios as their basis. 

A tape diagram is a technique for quickly solving mathematical concepts. The tape diagram has the benefit of preventing students from memorization of ideas they do not comprehend. A rule that is remembered won’t help a student unless they know how to apply it or adapt it to other circumstances. The tape diagrams, on the other hand, are simple visual aids that help students comprehend a topic better. By applying them, they will improve as problem solvers.

What are other terms for tape diagrams?

A strip diagram, a bar or length model, or a fraction strip are additional names for tape diagrams that are frequently used.

Why do we call it a tape diagram?

A tape diagram is a visual representation that looks like a piece of tape and is used to help with ratio calculations, addition, subtraction, and, most frequently, multiplication.

What are the steps in creating tape diagrams?

A tape diagram is simple to make if you follow a few simple steps. It uses a tape piece broken into more manageable rectangular sections to show difficult tasks graphically. Any difficult word problems are simplified with tape diagrams. 

Why do teachers use tape diagrams?

At its foundation, a tape diagram is a straightforward solution to a challenging mathematical problem. Teachers frequently use it to improve the way they instruct their students. The diagram helps teachers by enabling them to incorporate a fun component into learning. Students who struggle with math can more easily determine which computations are required to answer word problems using tape diagrams. Additionally, tape diagrams can be utilized to create more effective learning settings where students can discover various solutions to challenging situations. They are straightforward to use for fractions, ratios, multiplication, division, addition, and subtraction.

Why do students need to learn tape diagrams?

These diagrams can be used by students of all grade levels to solve mathematical problems and equations. Students are transitioning to new grades and working with advanced arithmetic benefit greatly from knowing tape diagrams.

What are the benefits of using tape diagrams?

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Course: 6th grade   >   Unit 1

Ratios with tape diagrams.

• Ratios with tape diagrams (part:whole)
• Equivalent ratio word problems
• Simplify a ratio from a tape diagram
• Equivalent ratios with equal groups
• Ratios and double number lines
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6.1.3: Reasoning about Equations with Tape Diagrams

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Let's see how equations can describe tape diagrams.

Exercise \(\PageIndex{1}\): Find Equivalent Expressions

Select all the expressions that are equivalent to \(7(2-3n)\). Explain how you know each expression you select is equivalent.

• \((2-3n)\cdot 7\)
• \(7\cdot 2\cdot (-3n)\)

Exercise \(\PageIndex{2}\): Matching Equations to Tape Diagrams

• Match each equation to one of the tape diagrams. Be prepared to explain how the equation matches the diagram.
• Sort the equations into categories of your choosing. Explain the criteria for each category.
• \(2x+5=19\)
• \(2+5x=19\)
• \(2(x+5)=19\)
• \(5(x+2)=19\)
• \(19=5+2x\)
• \((x+5)\cdot 2=19\)
• \(19=(x+2)\cdot 5\)
• \(19\div 2=x+5\)
• \(19-2=5x\)

Exercise \(\PageIndex{3}\): Drawing Tape Diagrams to Represent Equations

• \(114=3x+18\)
• \(114=3(y+18)\)
• Use any method to find values for \(x\) and \(y\) that make the equations true.

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To make a Koch snowflake:

• Start with an equilateral triangle that has side lengths of 1. This is step 1.
• Replace the middle third of each line segment with a small equilateral triangle with the middle third of the segment forming the base. This is step 2.
• Do the same to each of the line segments. This is step 3.
• Keep repeating this process.

• What is the perimeter after step 2? Step 3?
• What happens to the perimeter, or the length of line traced along the outside of the figure, as the process continues?

We have seen how tape diagrams represent relationships between quantities. Because of the meaning and properties of addition and multiplication, more than one equation can often be used to represent a single tape diagram.

Let’s take a look at two tape diagrams.

We can describe this diagram with several different equations. Here are some of them:

• \(26+4x=46\), because the parts add up to the whole.
• \(4x+26=46\), because addition is commutative.
• \(46=4x+26\), because if two quantities are equal, it doesn’t matter how we arrange them around the equal sign.
• \(4x=46-26\), because one part (the part made up of \(4x\)’s) is the difference between the whole and the other part.

For this diagram:

• \(4(x+9)=76\), because multiplication means having multiple groups of the same size.
• \((x+9) \cdot 4=76\), because multiplication is commutative.
• \(76\div 4=x+9\), because division tells us the size of each equal part.

Glossary Entries

Definition: Equivalent Expressions

Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.

For example, \(3x+4x\) is equivalent to \(5x+2x\). No matter what value we use for \(x\), these expressions are always equal. When \(x\) is 3, both expressions equal 21. When \(x\) is 10, both expressions equal 70.

Exercise \(\PageIndex{4}\)

Solve each equation mentally.

• \(\frac{1}{3}x=6\)
• \(-\frac{1}{2}x=-7\)

(From Unit 5.5.1)

Exercise \(\PageIndex{5}\)

Complete the magic squares so that the sum of each row, each column, and each diagonal in a grid are all equal.​

(From Unit 5.2.2)

Exercise \(\PageIndex{6}\)

Draw a tape diagram to match each equation.

• \(5(x+1)=20\)
• \(5x+1=20\)

Exercise \(\PageIndex{7}\)

Select all the equations that match the tape diagram.

• \(35=8+x+x+x+x+x+x\)
• \(35=8+6x\)
• \(6+8x=35\)
• \(6x+8=35\)
• \(6x+8x=35x\)
• \(35-8=6x\)

Exercise \(\PageIndex{8}\)

Each car is traveling at a constant speed. Find the number of miles each car travels in 1 hour at the given rate.

• \(135\) miles in \(3\) hours
• \(22\) miles in \(\frac{1}{2}\) hour
• \(7.5\) miles in \(\frac{1}{4}\) hour
• \(\frac{100}{3}\) miles in \(\frac{2}{3}\) hour
• \(97\frac{1}{2}\) miles in \(\frac{3}{2}\) hour

(From Unit 4.1.2)

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At Great Minds ® , we’re committed to ensuring our curricula are aligned to the latest research on how students best learn to read, write, and build knowledge.

Explore webinars, blogs, research briefs, and more to discover how we incorporate this important body of research.

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SOLVING WORD PROBLEMS with tape diagrams

This free webinar led by the teacher-writers of  Eureka Math /EngageNY Math explains how the tape diagram model supports solving word problems.

During this webinar, you will:

• Learn how the tape diagram model supports understanding of both the text and the operations of word problems.
• Explore part-whole relationships.
• Practice modeling single- and multi-step word problems
• Learn methods for scaffolding as students encounter increasingly complex problems.

NOTE: The webinar content is focused on grades 3-6, but educators and administrators from  all grade-bands are encouraged to watch.

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