Illustrative Mathematics Grade 7, Unit 6, Lesson 18: Subtraction in Equivalent Expressions

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Illustrative Math
Grade 7

Lesson 18: Subtraction in Equivalent Expressions

Let’s find ways to work with subtraction in expressions.

Illustrative Math Unit 7.6, Lesson 18 (printable worksheets)

Lesson 18 Summary

The following diagram shows how to re-write subtraction as adding the opposite and then rearrange terms in an expression.

Lesson 18.1 Number Talk: Additive Inverses

Find each sum or difference mentally.
-30 + -10
-10 + -30
-30 - 10
10 - -30

• See Video for Whole Lesson

Lesson 18.2 A Helpful Observation

Lin and Kiran are trying to calculate 7 3/4 + 3 5/6 - 1 3/4. Here is their conversation:
Lin: “I plan to first add 7 3/4 and 3 5/6, so I will have to start by finding equivalent fractions with a common denominator.”
Kiran: “It would be a lot easier if we could start by working with the 1 3/4 and 7 3/4. Can we rewrite it like 7 3/4 + 1 3/4 - 3 5/6?”
Lin: “You can’t switch the order of numbers in a subtraction problem like you can with addition; 2 - 3 is not equal to 3 - 2.”
Kiran: “That’s true, but do you remember what we learned about rewriting subtraction expressions using addition? 2 - 3 is equal to 2 + (-3).”

1. Write an expression that is equivalent to 7 3/4 + 3 5/6 - 1 3/4 that uses addition instead of subtraction.
2. If you wrote the terms of your new expression in a different order, would it still be equivalent? Explain your reasoning.

Lesson 18.3 Organizing Work

1. Write two expressions for the area of the big rectangle.
2. Use the distributive property to write an expression that is equivalent to 1/2(8y + -x + -12). The boxes can help you organize your work.
3. Use the distributive property to write an expression that is equivalent to 1/2(8y - x - 12).

Are you ready for more?

Here is a calendar for April 2017. Let’s choose a date: the 10th. Look at the numbers above, below, and to either side of the 10th: 3, 17, 9, 11.

1. Average these four numbers. What do you notice?
   • Show Answer

     The average is 10.

2. Choose a different date that is in a location where it has a date above, below, and to either side. Average these four numbers. What do you notice?
   • Show Answer

     The average is always the number in the middle.

3. Explain why the same thing will happen for any date in a location where it has a date above, below, and to either side.
   • Show Answer

     The date above is 7 less, the date below is 7 more, the date to the left is 1 less, the date to the right is 1 more. The average is then the number in the middle.

Lesson 18 Practice Problems

1. For each expression, write an equivalent expression that uses only addition.
   a. 20 - 9 + 8 - 7
   b. 4x - 7y - 5z + 6
   c. -3x - 8y - 4 - 8/7 z
   
2. Use the distributive property to write an expression that is equivalent to each expression. If you get stuck, consider drawing boxes to help organize your work.
3. Kiran wrote the expression x - 10 for this number puzzle: “Pick a number, add -2, and multiply by 5.”
   Lin thinks Kiran made a mistake.
   a. How can she convince Kiran he made a mistake?
   b. What would be a correct expression for this number puzzle?
   
4. The output from a coal power plant is shown in the table:
   Similarly, the output from a solar power plant is shown in the table:
   Based on the tables, is the energy output in proportion to the number of days for either plant? If so, write an equation showing the relationship. If not, explain your reasoning.

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