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# Justifying Equations

# Simplifying Numerical Expressions

## Overview

Students use the distributive property to rewrite and solve multiplication problems. Then they apply addition and multiplication properties to simplify numerical expressions.

# Key Concepts

The distributive property is stated in terms of addition: *a*(*b* + *c*) = *ab* + *ac*, for all numbers *a*, *b*, and *c*. However, it can be extended to subtraction as well: *a*(*b* − *c*) = *ab* − *ac*, for all numbers *a*, *b*, and *c*. Here is a proof. (We have combined some steps.)

a(b − c) | Original expression |
---|---|

= a(b + (−c)) | Subtracting is adding the opposite. |

= a(b) + a(−c) | Apply the distributive property. |

= ab + a(−1 ⋅ c) | Apply the property of multiplication by −1. |

= ab + −1(ac) | Apply the associative and commutative properties of multiplication. |

= ab + −(ac) | Apply the property of multiplication by −1. |

= ab − ac | Add the opposite is subtracting. |

We can use the distributive property to make some multiplication problems easier to solve. For example, by rewriting $1.85 as $2.00 − $0.15 and applying the distributive property, we can change 6($1.85) to a problem that is easy to solve mentally.

6($1.85) | = | 6($2−$0.15) |
---|---|---|

= | 6($2) − 6($0.15) | |

= | $12 − $0.90 | |

= | $11.10 |

One common error students make when simplifying expressions is to simply remove the parentheses when a sum or difference is subtracted. For example, students may rewrite 10 − (6 + 9) as 10 − 6 + 9. In fact, 10 − (6 + 9) = 10 − 6 − 9. To see why, remember that that subtraction is equivalent to adding the opposite, 10 − (6 + 9) = 10 + [−(6 + 9)]. Applying the property of multiplication by −1, this is 10 + (−1)(6 + 9). Using the distributive property, we get 10 + (−6) + (−9) = 10 − 6 − 9.

# Goals and Learning Objectives

- Apply addition and multiplication properties to simplify numerical expressions.

# A Shopping Problem

# Lesson Guide

Have students watch the video in pairs and then discuss it, along with the remaining text in the Opening.

Cue students to “look out for” particular elements to support students’ understanding of the video’s contents. Provide a list of key ideas for students to review prior to viewing. Show the video as many times as needed.

ELL: When watching the video, be sure that ELLs can follow the explanations. Remind students they can “chunk” the video by pausing at key times to allow time to process the information. Have students view the video again if needed. Ask questions to check for understanding before moving on to the discussion. If you notice ELLs do not understand important parts of the video, show it one more time and ask students to identify all mathematical language.

# Mathematics

Many students have used the distributive property to distribute multiplication over subtraction in Grade 6. This lesson asks students to explain why we can do this. It is fine for them to simply say that it is because subtracting is the same as adding the opposite. However, you may want to show them a formal proof that $a\left(b-c\right)=ab-ac$ for any numbers $a$, $b$, and $c$:

$a\left(b-c\right)$ | $=$ | $a\left(b+\left(-c\right)\right)$ | Subtracting is adding the opposite. |
---|---|---|---|

_ | $=$ | $a\left(b\right)+a\left(-c\right)$ | Apply the distributive property. |

_ | $=$ | $ab+a\left(-1\cdot c\right)$ | Apply the multiplication property of $-1$. |

_ | $=$ | $ab+\left(-1\right)\left(ac\right)$ | Apply the associative property and commutative properties. |

_ | $=$ | $ab+\left(-\left(ac\right)\right)$ | Apply the multiplication property of $-1$. |

_ | $=$ | $ab-ac$ | Adding the opposite is subtracting. |

SWD: Allow students with language processing and/or attentional variability a preview of the video.

# Mathematical Practices

**Mathematical Practice 4: Model with mathematics.**

In the video, two girls use mathematics to solve an everyday problem involving finding the total cost of several items.

## Opening

# A Shopping Problem

Watch this video about how Karen and Lucy used the distributive property to solve a problem they encountered while shopping.

Here are the steps that Karen and Lucy followed to figure out the total price of the notebooks:

$\begin{array}{ccc}6\left(\text{\$}1.85\right)& =& 6(\text{\$}2-\text{\$}0.15)\\ & =& 6\left(\text{\$}2\right)-6\left(\text{\$}0.15\right)\\ & =& \text{\$}12-\text{\$}0.90\\ & =& \text{\$}11.10\end{array}$

Notice that Karen and Lucy distributed multiplication over *subtraction.*

- Can you explain why it is possible to use the distributive property in this way?

VIDEO: Girls Buying Notebooks

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will use the distributive property to simplify numerical expressions.

## Opening

Use the distributive property to simplify numerical expressions.

# Use the Distributive Property

# Lesson Guide

Have students work in pairs on the problems. Tell students that they do not need to show every single step, but they should show enough that their work is easy to follow. They should be able to explain their reasoning in their presentation.

Encourage students to rewrite the second factor as a sum or difference that is easy to calculate mentally.

ELL: For this task, encourage students to explain their ideas to one another. Math language must be used. Encourage the use of English without discouraging students from using their native language(s).

# Mathematical Practices

**Mathematical Practice 4: Model with mathematics.**

- Students mimic the modeling demonstrated in the Opening video.

**Mathematical Practice 6: Attend to precision.**

- Students must be careful as they carry out the steps in their computations to ensure they do not make errors. They also must be sure to include enough steps so others will understand their work.

# Interventions

**Student writes $6.12 as a difference rather than as a sum.**

- You wrote $6.12 as $7.00 − $0.88. That means you will have to find 7 ⋅ 0.88, which seems pretty hard.
- Is there another way to write $6.12 so you will have two easier products to compute?
- You can distribute multiplication over subtraction or over addition. Can you write $6.12 as a sum, rather than as a difference?

**Student makes computational errors.**

- Does your answer seem reasonable?
- Have you checked your work?
- Can you explain to me what you did, step by step?

**Student’s solution is difficult to follow.**

- Do you think someone else would be able to understand what you did?
- Can you add a few more steps so that your solution is easier to follow?

# Answers

Answers will vary. Possible answers:

Split $6.12 into a sum:

7 ⋅ $6.12 = 7($6 + $0.12) __ = 7($6) + 7($0.12) __ = $42 + $0.84 __ = $42.84 Split $2.95 into a difference:

12 ⋅ $2.95 = 12($3 − $0.05) __ = 12($3) − 12($0.05) __ = $36 − $0.60 __ = $35.40 Split $4\frac{1}{2}$ into a sum:

$-\frac{3}{4}\cdot 4\frac{1}{2}$ = $-\frac{3}{4}\left(4+\frac{1}{2}\right)$ __ = $-\frac{3}{4}\cdot 4+\left(-\frac{3}{4}\cdot \frac{1}{2}\right)$ __ = $-3+\left(-\frac{3}{8}\right)$ __ = $-3\frac{3}{8}$

## Work Time

# Use the Distributive Property

Use the distributive property, as Karen and Lucy did, to help you find each product. Use mental math when you can.

- 7 ⋅ $6.12
- 12 ⋅ $2.95
- $-\frac{3}{4}\cdot 4\frac{1}{2}$

## Hint:

Write the second factor in each problem as a sum or a difference and then apply the distributive property.

# Kevin’s Work

# Mathematics

**[common error]** Kevin’s problem addresses an error students often make when they simplify expressions. In the problem, Kevin simply removes the parentheses from 10 − (6 + 9) to get 10 − 6 + 9. Students should see that this cannot be correct because 10 − (6 + 9) means to subtract the sum of 6 and 9, which is 15, from 10. Kevin subtracted 6 and then added 9.

Students should remember that subtracting is the same as adding the opposite, so 10 − (6 + 9) = 10 + [−(6 + 9)], which is equivalent to 10 + (−1)(6 + 9). Applying the distributive property, we get 10 + (−1)(6) + (−1)(9), or 10 − 6 − 9.

# Answers

- 10 − (6 + 9) means to subtract the sum of 6 + 9 from 10. Kevin subtracted 6 and then added 9.
- The correct value is 10 − 15 = −5.

## Work Time

# Kevin's Work

Kevin simplified 10 − (6 + 9), but he made a mistake.

10 − (6 + 9) = 10 − 6 + 9

= 4 + 9

= 13

- What is Kevin’s error?
- What is the correct value of the expression?

## Hint:

If you work within the parentheses first and add 6 + 9 and then subtract this sum from 10, what answer do you get?

# Justify

# Interventions

**If student are having problems:**

- Remind them about the list of properties in Lesson 6.
- What is another way to write −(12 − 4)?
- How can you use the distributive property to rewrite the expression.

# Answers

10 − (6 + 9) Original expression = 10 + −(6 + 9) Subtraction is adding the opposite. = 10 + (−1)(6 + 9) Apply the property of multiplication by −1. = 10 + (−1)(6) + (−1)(9) Apply the distributive property. = 10 + (−6) + (−9) Apply the property of multiplication by −1. = 10 − 6 − 9 Adding the opposite is the same as subtracting. 5 − (12 − 4) Original expression = 5 + −(12 − 4) Subtracting is adding the opposite. = 5 + (−1)(12 − 4) Apply the property of multiplication by −1. = 5 + (−1)(12) − (−1)(4) Apply the distributive property. = 5 + (−12) − (−4) Apply the property of multiplication by −1. = 5 − 12 + 4 Subtracting is the same as adding the opposite.

## Work Time

# Justify

The table shows why 10 − (6 + 9) is equal to 10 − 6 − 9.

- Work on the table and fill in the missing justifications.
- Show that 5 − (12 − 4) can be rewritten as 5 − 12 + 4.

HANDOUT: Justifying Equations

## Hint:

To show that 5 – (12 – 4) can be rewritten as 5 – 12 + 4, remember that subtracting (12 – 4) is the same as adding the opposite of (12 – 4).

# Prepare a Presentation

# Preparing for Ways of Thinking

Select students who use the distributive property in different ways to present during Ways of Thinking. Select at least one student to present a solution for the Challenge Problem.

# Challenge Problem

## Answers

- Always positive.
- The sum of two negative numbers is negative, so
*b*+*c*is negative. The product of two negative numbers is positive, so*a*(*b*+*c*) must be positive.

## Work Time

# Prepare a Presentation

Use your work to show how you can use the distributive property to help you evaluate expressions with negative numbers.

# Challenge Problem

- For negative values of
*a*,*b*, and*c*, is*a*(*b*+*c*) always negative, always positive, or sometimes positive and sometimes negative? - Justify your answer.

# Make Connections

# Lesson Guide

Have students present their examples of how the distributive property can help you evaluate expressions. Other students should ask questions and point out and help correct errors. Remind students that the goal of this strategy is to simplify the calculation.

Questions to pose:

- Will using the distributive property make any multiplication problem easier?
- Can you think of a multiplication problem for which using the distributive property is not helpful?
- Can you think of another multiplication problem for which using the distributive property is helpful?

Students who present the Challenge Problem should be able to justify their answers.

SWD: Before students share, circulate and help them to identify ideas from their notes and thoughts that are appropriate to contribute during this portion of the Ways of Thinking discussion.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates’ use and explanations of the distributive property.

## Hint:

As your classmates present, ask questions such as:

- How did you figure out how to rewrite the second factor? What property did you use?
- Can you show where you used the distributive property to show that 5 - (12 - 4) can be rewritten as 12 - 5 + 4?

# Simplify Numerical Expressions

# Lesson Guide

Have students work individually on the problems.

# Answers

Answers will vary. Possible answers:

- $2-\left(\frac{1}{2}+\frac{3}{4}\right)$

$=\frac{8}{4}-\left(\frac{2}{4}+\frac{3}{4}\right)$

$=\frac{8}{4}-\left(\frac{5}{4}\right)$

$=\frac{3}{4}$ - $\frac{1}{3}\cdot \frac{1}{4}\cdot \frac{1}{5}\cdot \left(-3\right)\cdot \left(-4\right)\cdot \left(-5\right)\cdot \left(-6\right)$

$=\frac{1}{3}\cdot \left(-3\right)\cdot \frac{1}{4}\cdot \left(-4\right)\cdot \frac{1}{5}\cdot \left(-5\right)\cdot \left(-6\right)$

$=\frac{1}{3}\cdot \left(-1\right)\cdot 3\cdot \frac{1}{4}\cdot \left(-1\right)\cdot 4\cdot \frac{1}{5}\cdot \left(-1\right)\cdot 5\cdot \left(-1\right)\cdot 6$

$=\left(-1\right)\cdot \left(1\right)\cdot \left(-1\right)\cdot \left(1\right)\cdot \left(-1\right)\cdot \left(1\right)\cdot \left(-1\right)\cdot 6$$=\left(-1\right)\cdot \left(-1\right)\cdot \left(-1\right)\cdot \left(-1\right)\cdot 6$= 6

$-\frac{2}{5}\cdot \left(\frac{3}{7}\cdot 5\right)+\frac{6}{7}$

$=\left(-1\right)\frac{2}{5}\cdot \left(\frac{3}{7}\cdot 5\right)+\frac{6}{7}$

$=\left(-1\right)\frac{3}{7}\cdot \left(\frac{2}{5}\cdot 5\right)+\frac{6}{7}$

$=\left(-1\right)\frac{3}{7}\cdot \left(2\right)+\frac{6}{7}$

$=\left(-1\right)\frac{6}{7}+\frac{6}{7}$

= 0

2.75 ⋅ (−3.25) + 2.75 ⋅ (−6.75)

= 2.75 ⋅ (−3.25 + −6.75)

= 2.75 ⋅ (−10)

= −27.5

(4.8 + 0.36) ÷ (−6)

= (4.8 + 0.36) ⋅ (−$\frac{1}{6}$)

= (4.8) ⋅ (−$\frac{1}{6}$) + (0.36) ⋅ (−$\frac{1}{6}$)

= (−0.8) + (−0.06)

= −0.86

## Work Time

# Simplify Numerical Expressions

Simplify each expression:

- $2-(\frac{1}{2}+\frac{3}{4})$
- $\frac{1}{3}\cdot \frac{1}{4}\cdot \frac{1}{5}\cdot (-3)\cdot (-4)\cdot (-5)\cdot (-6)$
- $-\frac{2}{5}\cdot (\frac{3}{7}\cdot 5)+\frac{6}{7}$
- $2.75\cdot (-3.25)+2.75\cdot (-6.75)$
- $(4.8+0.36)\xf7(-6)$

# The Distributive Property

# Lesson Guide

Have students discuss the distributive property with a partner before turning to a whole class discussion. Use this opportunity to correct or clarify misconceptions. Make sure students discuss how the distributive property can simplify expressions.

## Formative Assessment

# Summary of the Math: The Distributive Property

**Read and Discuss**

**Distributive Property**

*a* ⋅ (*b* + *c*) = (*a* ⋅ *b*) + (*a* ⋅ *c*), where *a*, *b*, and *c* can be any numbers, including negative numbers.

Examples:

−5(2 + 3) = (−5 ⋅ 2) + (−5 ⋅ 3)

= −10 + (−15)

= −25

7 − (3 + 5) = 7 + (−1)(3 + 5)

= 7 − 3 − 5

= −1

## Hint:

Can you:

- Give an example that shows how you can use the distributive property to evaluate a multiplication problem?
- Describe how you can use the distributive property to evaluate expressions, particularly expressions with negative numbers?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of the class. Review the reflections to find out how students apply their math learning to real-world situations.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**An example of a situation in which I used something that I learned in math class is …**