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?
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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.
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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.](https://cms-im.s3.amazonaws.com/rp8yLV7xgazfXg7SPdLVFvST?response-content-disposition=inline%3B%20filename%3D%226-6.6.A1.Image.Revision.05.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.A1.Image.Revision.05.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235351Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=01b1ea5abe143ec18cc2b6e70c910f365941538121e897c2c4e31ff136e82c9c)
- \(4 + x = 12\)
- \(12 \div 4 = x\)
- \(4 \boldcdot x = 12\)
- \(12 = 4 + x\)
- \(12 - x = 4\)
- \(12 = 4 \boldcdot x\)
- \(12 - 4 = x\)
- \(x = 12 - 4\)
- \(x+x+x+x=12\)
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.
![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.](https://cms-im.s3.amazonaws.com/WsojFDc2HtPMvST5i2sYQnUm?response-content-disposition=inline%3B%20filename%3D%226-6.6.A2.LessonSummary.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.A2.LessonSummary.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240726%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240726T235351Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f2ac3d6443bea62bc03bfc1a972ab6b9de313031c5f5d6a63bc7e7586048efde)
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.