31.1: Tape Diagrams and Equations

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Illustrative Mathematics
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Lesson

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

Exercise $\PageIndex{1}$: Which Diagram is Which?

Here are two diagrams. One represents $2+5=7$. The other represents $5\cdot 2=10$. Which is which? Label the length of each diagram.

Draw a diagram that represents each equation.

$4+3=7\qquad 4\cdot 3=12$

Exercise $\PageIndex{2}$: Match Equations and Tape Diagrams

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

$4+x=12$
$12\div 4=x$
$4\cdot x=12$
$12=4+x$
$12-x=4$
$12=4\cdot x$
$12-4=x$
$x=12-4$
$x+x+x+x=12$

Exercise $\PageIndex{3}$: Draw Diagrams for Equations

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

$18=3+x$
$18=3\cdot y$

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.

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:

$\begin{aligned} x+x+x&=12 \\ 3\cdot x&=21 \\ x&=21\div 3 \\ x&=\frac{1}{3}\cdot 21\end{aligned}$

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:

$\begin{aligned} y+3&=21 \\ y&=21-3 \\ 3&=21-y \end{aligned}$

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

Practice

Exercise $\PageIndex{4}$

Here is an equation: $x+4=17$

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

Exercise $\PageIndex{5}$

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

Complete the tape diagram so it represents the equation $5\cdot x=35$.
Find the value of $x$.

Exercise $\PageIndex{6}$

Match each equation to one of the two tape diagrams.

$x+3=9$
$3\cdot x=9$
$9=3\cdot x$
$3+x=9$
$x=9-3$
$x=9\div 3$
$x+x+x=9$

Exercise $\PageIndex{7}$

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

$x+9=16$
$4\cdot x=28$

Exercise $\PageIndex{8}$

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 $\PageIndex{9}$

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 $\PageIndex{10}$

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.

Figure $\PageIndex{6}$: 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)