Lesson 1Tape Diagrams and Equations

Learning Goal

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

Learning Targets

I can tell whether or not an equation could represent a tape diagram.
I can use a tape diagram to represent a situation.

Warm Up: Which Diagram Is Which?

Problem 1

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

A tape diagram with 5 segments each labeled 2. The diagram label is empty. A second tape diagram has 2 uneven sized segments labeled 2 and 5. It also has and empty label.

Problem 2

Draw a diagram that represents each equation.

$4 + 3 = 7$
$4 \cdot 3 = 12$

Activity 1: Match Equations and Tape Diagrams

Match each equation to one of the tape diagrams.

$4 + x = 12$
$12 \div 4 = x$
$4 \cdot x = 12$

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

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

A tape diagram with 2 uneven segments labeled 4 and x. The diagram is labeled 12. The second diagram is also labeled 12 but has 4 even segments labeled x.

Activity 2: Draw Diagrams for Equations

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

$18 = 3 + x$
$18 = 3 \cdot y$

Are you ready for more?

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?

Lesson Summary

Tape diagrams can help us understand relationships between quantities and how operations describe those relationships.

Tape diagram A has 3 even segments labeled x and the diagram is labeled 21. Diagram B has 2 uneven segments labeled y and 3 and diagram is also labeled 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:

$x + x + x = 21$ $3 \cdot x = 21$ $x = 21 \div 3$ $x = \frac{1}{3} \cdot 21$

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.