# 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?

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. Draw a diagram that represents each equation.
1. $4+3=7$
1. $4 \boldcdot 3=12$

## 1.2: Match Equations and Tape Diagrams

Here are two tape diagrams. Match each equation to one of the tape diagrams. 1. $4 + x = 12$
2. $12 \div 4 = x$
3. $4 \boldcdot x = 12$
1. $12 = 4 + x$
2. $12 - x = 4$
3. $12 = 4 \boldcdot x$
1. $12 - 4 = x$
2. $x = 12 - 4$
3. $x+x+x+x=12$

## 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. $18 = 3 \boldcdot y$

## 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:

$$x+x+x=21$$ $$3\boldcdot {x}=21$$ $$x=21\div3$$ $$x=\frac13\boldcdot {21}$$

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:

$$y+3=21$$ $$y=21-3$$ $$3=21-y$$

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