## 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.

- $4+3=7$

- $4 \boldcdot 3=12$

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

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.

- $4+3=7$

- $4 \boldcdot 3=12$

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

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

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

- $12 - 4 = x$
- $x = 12 - 4$
- $x+x+x+x=12$

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

- $18 = 3+x$
- $18 = 3 \boldcdot y$

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?

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.