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.

    Two tape diagrams. Tape diagram on the left, 5 equal parts labeled 2. Total, blank box with dotted sides. Tape diagram on the right, 2 parts, labeled 2, 5. Total, blank box with dotted sides.
  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.

Two tape diagram. Tape diagram on the left, 2 parts labeled 4, x. Total, 12. Tape diagram on the right, 4 equal parts, labeled x. Total, 12.
  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. \(18 = 3 \boldcdot y\)


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.

Two tape diagrams, labeled A and B. Tape diagram A, 3 equal parts labeled x, x, x. Total, 21. Tape diagram B, 2 parts, labeled y, 3. Total, 21.

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.