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.
\(4+3=7\)
\(4 \boldcdot 3=12\)
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.
 \(18 = 3+x\)
 \(18 = 3 \boldcdot y\)
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?
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:
\(\displaystyle y+3=21\)
\(\displaystyle y=213\)
\(\displaystyle 3=21y\)
We can use the diagram or any of the equations to reason that the value of \(y\) is 18.