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6-8 Accelerated 7 Unit 3 Lesson 3

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K-5 Math •6-8 Math 9-12 Math

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Accelerated 6 •Accelerated 7

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Lesson 3

3.1 Warm-up 3.2 Activity 3.3 Activity Student Lesson Summary Glossary

Unit 3, Lesson 3

Reasoning about Equations with Tape Diagrams

Accelerated 7

3.1

Warm-up

Find Equivalent Expressions

Select all the expressions that are equivalent to . Explain how you know each expression you select is equivalent.

Are You Ready for More?

3.2

Activity

Matching Equations to Tape Diagrams

Standards Alignment

Building On

Addressing

Building Toward

7.EE.2

Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that "increase by 5%" is the same as "multiply by 1.05."

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7.EE.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

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Match each equation to one of the tape diagrams. Be prepared to explain how the equation matches the diagram.

Tape diagram A, 3 parts, 5, x, x, total 19. — A

Tape diagram B, 3 parts, x, x, 5, total 19. — B

Tape diagram C, 6 parts, 2, x, x, x, x, x, total 19. — C

Tape diagram D, 2 parts, x + 5, x + 5, total 19. — D

Tape diagram E, 5 parts, x + 2, x + 2, x + 2, x + 2, x + 2, total 19. — E

Are You Ready for More?

3.3

Activity

Drawing Tape Diagrams to Represent Equations

Standards Alignment

Building On

Addressing

7.EE.3

Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making 2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

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Draw a tape diagram to match each equation.
Use any method to find values for and that make the equations true. Explain your reasoning.

Student Lesson Summary

We have seen how tape diagrams represent relationships between quantities. Because of the meaning and properties of addition and multiplication, more than one equation can often be used to represent a single tape diagram.

Let’s take a look at two tape diagrams.

Tape diagram, one large part labeled 26, four small equal parts labeled x, total 46.

We can represent this diagram with several different equations. Here are some of them:

, because the parts add up to the whole.
, because addition is commutative.
, because if two quantities are equal, it doesn’t matter how we arrange them around the equal sign.
, because one part (the part made up of 4 ’s) is the difference between the whole and the other part.

Tape diagram, four equal parts labeled, x + 9, total 76.

Here are some equations that represent this diagram:

, because multiplication means having multiple groups of the same size.
, because multiplication is commutative.
, because division tells us the size of each equal part.

Glossary

equivalent expressions

Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.

For example, is equivalent to .

When is 3, both expressions equal 21.
When is 10, both expressions equal 70.
When is any other number, both expressions still have equal value.

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Standards Alignment

Building On

6.EE.4

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

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Addressing

Building Toward

Are You Ready for More?

Building Toward

7.EE.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

View Full Standard