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31.1: Tape Diagrams and Equations

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Lesson

Let's see how tape diagrams and equations can show relationships between amounts.

Exercise 31.1.1: Which Diagram is Which?

  1. Here are two diagrams. One represents 2+5=7. The other represents 52=10. Which is which? Label the length of each diagram.
clipboard_e8f443afe0e41dcb4ad3daed2ed627c27.png
Figure 31.1.1
  1. Draw a diagram that represents each equation.

4+3=743=12

Exercise 31.1.2: Match Equations and Tape Diagrams

Here are two tape diagrams. Match each equation to one of the tape diagrams.

clipboard_ebe00fefea010f9225c948fd40e8cf6d4.png
Figure 31.1.2
  1. 4+x=12
  2. 12÷4=x
  3. 4x=12
  4. 12=4+x
  5. 12x=4
  6. 12=4x
  7. 124=x
  8. x=124
  9. x+x+x+x=12

Exercise 31.1.3: Draw Diagrams for Equations

For each equation, draw a diagram and find the value of the unknown that makes the equation true.

  1. 18=3+x
  2. 18=3y

Are you ready for more?

You are walking down a road, seeking treasure. The road branches off into three paths. A guard stands in each path. You know that only one of the guards is telling the truth, and the other two are lying. Here is what they say:

  • Guard 1: The treasure lies down this path.
  • Guard 2: No treasure lies down this path; seek elsewhere.
  • Guard 3: The first guard is lying.

Which path leads to the treasure?

Summary

Tape diagrams can help us understand relationships between quantities and how operations describe those relationships.

clipboard_e5a6f330bf45b1967b61733757f9d9433.png
Figure 31.1.3

Diagram A has 3 parts that add to 21. Each part is labeled with the same letter, so we know the three parts are equal. Here are some equations that all represent diagram A:

x+x+x=123x=21x=21÷3x=1321

Notice that the number 3 is not seen in the diagram; the 3 comes from counting 3 boxes representing 3 equal parts in 21.

We can use the diagram or any of the equations to reason that the value of x is 7.

Diagram B has 2 parts that add to 21. Here are some equations that all represent diagram B:

y+3=21y=2133=21y

We can use the diagram or any of the equations to reason that the value of y is 18.

Practice

Exercise 31.1.4

Here is an equation: x+4=17

  1. Draw a tape diagram to represent the equation.
  2. Which part of the diagram shows the quantity x? What about 4? What about 17?
  3. How does the diagram show that x+4 has the same value as 17?

Exercise 31.1.5

Diego is trying to find the value of x in 5x=25. He draws this diagram but is not certain how to proceed.

clipboard_e2e8cd7d8fa5b6c3c3fefd7e1f85da113.png
Figure 31.1.4
  1. Complete the tape diagram so it represents the equation 5x=35.
  2. Find the value of x.

Exercise 31.1.6

Match each equation to one of the two tape diagrams.

  1. x+3=9
  2. 3x=9
  3. 9=3x
  4. 3+x=9
  5. x=93
  6. x=9÷3
  7. x+x+x=9
clipboard_ec74e2a540cf0ec1442e778024b058fc7.png
Figure 31.1.5

Exercise 31.1.7

For each equation, draw a tape diagram and find the unknown value.

  1. x+9=16
  2. 4x=28

Exercise 31.1.8

A shopper paid $2.52 for 4.5 pounds of potatoes, $7.75 for 2.5 pounds of broccoli, and $2.45 for 2.5 pounds of pears. What is the unit price of each item she bought? Show your reasoning.

(From Unit 5.4.5)

Exercise 31.1.9

A sports drink bottle contains 16.9 fluid ounces. Andre drank 80% of the bottle. How many fluid ounces did Andre drink? Show your reasoning.

(From Unit 3.4.5)

Exercise 31.1.10

The daily recommended allowance of calcium for a sixth grader is 1,200 mg. One cup of milk has 25% of the recommended daily allowance of calcium. How many milligrams of calcium are in a cup of milk? If you get stuck, consider using the double number line.

clipboard_e2c0fc2c560563fbe2eede46518e2e930.png
Figure 31.1.6: A double number line with 2 tick marks at either end of the line. The top number line is labeled calcium in milligrams and the tick marks are labeled 0 and 1200. The bottom number line is not labeled and the tick marks are labeled 0 and 100 percent.

(From Unit 3.4.2)


This page titled 31.1: Tape Diagrams and Equations is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Illustrative Mathematics via source content that was edited to the style and standards of the LibreTexts platform.

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