# Lesson 1

Tape Diagrams and Equations

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

### 1.1: Which Diagram is Which?

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

2. Draw a diagram that represents each equation.

$$4+3=7$$

$$4 \boldcdot 3=12$$

### 1.2: Match Equations and Tape Diagrams

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

• $$4 + x = 12$$
• $$12 \div 4 = x$$
• $$4 \boldcdot x = 12$$
• $$12 = 4 + x$$
• $$12 - x = 4$$
• $$12 = 4 \boldcdot x$$
• $$12 - 4 = x$$
• $$x = 12 - 4$$
• $$x+x+x+x=12$$

### 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. $$18 = 3 \boldcdot y$$

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:

$$\displaystyle x+x+x=21$$

$$\displaystyle 3\boldcdot {x}=21$$

$$\displaystyle x=21\div3$$

$$\displaystyle x=\frac13\boldcdot {21}$$

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:

$$\displaystyle y+3=21$$

$$\displaystyle y=21-3$$

$$\displaystyle 3=21-y$$

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