# Reviewing The Properties Of Addition

## Overview

Students review the properties of addition and write an example for each. Then they apply the properties to simplify numerical expressions.

# Key Concepts

The properties of addition:

- Commutative property of addition: Changing the order of addends does not change the sum. For any numbers
*a*and*b*,*a*+*b*=*b*+*a*. - Associative property of addition: Changing the grouping of addends does not change the sum. For any numbers
*a*,*b*, and*c*, (*a*+*b*) +*c*=*a*+ (*b*+*c*). - Additive identity property of 0: The sum of 0 and any number is that number. For any number
*a*,*a*+ 0 = 0 +*a*=*a*. - Existence of additive inverses: The sum of any number and its additive inverse (opposite) is 0. For any number
*a*,*a*+ (−*a*) = (−*a*) +*a*= 0.

These properties allow us to manipulate expressions to make them easier to work with. For example, the associative property of addition tells us that we can regroup the expression $(\frac{3}{11}+\frac{4}{9})+\frac{5}{9}$ as $\frac{3}{11}+(\frac{4}{9}+\frac{5}{9})$, making it much easier to simplify.

Students must be careful to apply the commutative and associative properties only to addition expressions. For example, we cannot switch the −7 and 8 in the expression −7 − 8 to get 8 − (−7). However, if we rewrite −7 − 8 as the addition expression −7 + (−8), we can swap the addends to get −8 + (−7).

# Goals and Learning Objectives

- Understand the properties of addition.
- Apply the properties of addition to simplify numerical expressions.

# Review Properties of Operations

# Lesson Guide

Review the properties with students or have them read and discuss them with a partner. Have students work in pairs to write one numerical example of each property. Encourage students to include negative numbers in their examples. Discuss some of the examples as a class.

ELL: As in other lessons, when academic language is reviewed (or introduced, for that matter), show it in writing and leave it in a place that is visible so that all students can refer to it. Make sure students copy academic language into their Notebook.

# Mathematics

Students will know the commutative and associative properties from earlier grades. The existence of additive inverses is new. It states that the sum of any number and its additive inverse (opposite) is 0. Be sure students understand that these properties apply to all numbers, including negative numbers.

## Opening

# Review Properties of Operations

Discuss the following with your classmates.

- Review the properties of operations.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will use the properties of operations to evaluate numerical expressions with negative numbers.

## Opening

Use the properties of operations to evaluate numerical expressions with negative numbers.

# Use Properties of Operations

# Lesson Guide

Have students work individually on the problems and then compare answers with a partner.

ELL: While listening to their responses, give students advance notice they will be presenting their work on a specific problem during Ways of Thinking. This will give them ample time to prepare a thoughtful response.

Select a couple of different methods of explaining that *b* − *a* is the opposite of *a* − *b* for students to share during Ways of Thinking.

# Answers

- Answers will vary. Any two different numbers will provide a counterexample. For example, −3 − 1 = −4, while 1 − (−3) = 4.
- −4.5 − 2.5 = −7

2.5 − (−4.5) = 7

The statement is true for these particular values. - Yes. Explanations will vary. Possible explanation: Change the subtraction to addition, since subtraction is adding the opposite:
*a*−*b*=*a*+ (−*b*)*b*−*a*=*b*+ (−*a*)

The two are opposites of each other if they add together to get 0, so add them together: (*a*+ (−*b*)) + (*b*+ (−*a*)).

Apply the associative property of addition: =*a*+ (−*b*+*b*) + (−*a*)*b*and −*b*are additive inverses, and additive inverses add to 0: =*a*+ 0 + (−*a*)

Apply the additive identity property of 0: =*a*+ (−*a*)*a*and −*a*are additive inverses, so they add to 0: =0

Since (*a*−*b*) + (*b*−*a*) = 0,*b*−*a*is the opposite of*a*−*b*.

## Work Time

# Use Properties of Operations

- Give an example to show that subtraction is not commutative. That is, find two numbers
*a*and*b*such that*a*−*b*≠*b*−*a*. - A student claimed that
*b*−*a*is the opposite of*a*−*b*. Check whether this statement is true for*a*= −4.5 and*b*= 2.5. - Is the statement “
*b*−*a*is the opposite of*a*−*b*” true for all values of*a*and*b*? Explain.

# Justify Steps

# Lesson Guide

Students should work individually and then compare answers with a partner.

# Mathematics

[common error] Watch for students who try to switch the order of numbers on either side of a minus sign. Remind them that the last task showed that subtraction is not commutative. Students must rewrite the subtraction as an addition before they can apply the property.

# Mathematical Practices

**Mathematical Practice 2: Reason abstractly and quantitatively.**

Students must understand and apply the properties of addition to problems.

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

Students must carefully apply the addition and subtraction rules and the properties. In Ways of Thinking, they must be able to explain and justify their steps to other students.

# Interventions

**Student switches the order of numbers on either side of a minus sign.**

- What does the commutative property of addition tell us?
- Does this property apply to subtraction?
- How can you write a subtraction as an equivalent addition?

# Answers

$4.36+1.85-4.36$ | Starting expression |
---|---|

$=4.36+1.85+\left(-4.36\right)$ | Change the subtraction to addition, since subtracting is the same as adding the opposite. |

$=4.36+\left(-4.36\right)+1.85$ | Apply the commutative property of addition. |

$=0+1.85$ | Apply the additive inverse property. |

$=1.85$ | Apply the additive identity property of 0. |

## Work Time

# Justify Steps

Justify each step for evaluating this numerical expression using one of the properties of operations.

Step | Justification |
---|---|

$4.36+1.85-4.36$ | Starting expression |

$=4.36+1.85+\left(-4.36\right)$ | Subtracting is the same as adding the opposite. |

$=4.36+\left(-4.36\right)+1.85$ | |

$=0+1.85$ | |

$=1.85$ |

# Evaluate the Expression

# Lesson Guide

Have students work individually and then compare answers with a partner.

# Mathematical Practices

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

Students must carefully apply the addition and subtraction rules and the properties. In Ways of Thinking, they must be able to explain and justify their steps to other students.

# Interventions

**Student makes computation errors.**

- Did you write out all of your steps?
- Let’s simplify step by step. What should we do first?
- The properties apply to addition only. It might be easier to work with the expression if you change all of the subtractions to additions first.
- Which numbers in the expression would be easy to combine? How can you use properties to move those numbers together?

# Answers

Steps may vary. Possible sequence and justifications:

$-3\frac{3}{8}+\left(4\frac{5}{6}+1\frac{3}{8}\right)-\left(-\frac{1}{6}\right)$ | Starting expression. |
---|---|

$=-3\frac{3}{8}+\left(4\frac{5}{6}+1\frac{3}{8}\right)+\frac{1}{6}$ | Change the subtraction to addition, since subtraction is the same as adding the opposite. |

$=-3\frac{3}{8}+\left(1\frac{3}{8}+4\frac{5}{6}\right)+\frac{1}{6}$ | Apply the commutative property. |

$=\left(-3\frac{3}{8}+1\frac{3}{8}\right)+\left(4\frac{5}{6}+\frac{1}{6}\right)$ | Apply the associative property. |

= −2 + 5 | Complete the addition. |

= 3 |

## Work Time

# Evaluate the Expression

Evaluate the expression by combining terms. Justify your steps using one of the properties of operations.

- $-3\frac{3}{8}+(4\frac{5}{6}+1\frac{3}{8})-(-\frac{1}{6})$

## Hint:

Numbers with the same denominator are easy to combine. Can you use the properties of operations to rearrange the expression so that terms with like denominators are together?

# Kevin’s Work

# Lesson Guide

Have students work individually and then compare answers with a partner.

SWD: When participating in a whole class discussion, Ways of Thinking can be intimidating for students with language based learning vulnerabilities and/or learning challenges. Have students work on the speaking and listening skills implicit to this portion of the lesson. Supports for students during this portion of the lesson include:

- In small groups or partners, give students a few minutes to discuss their ideas, the questions posed, and what has taken place during the lesson.
- Conference with individual students prior to the discussion to ascertain what they might be able to successfully contribute to the discussion. Students should rehearse their contribution and/or write notes for themselves to refer to when they speak. This will support students with expressive language difficulties and/or students who are anxious or reluctant to participate in class discussions.

# Mathematical Practices

**Mathematical Practice 3: Construct viable arguments and critique the reasoning of others.**

In Kevin’s problem, students must identify and correct the error.

# Answers

- Kevin tried to apply the associative property to regroup an expression with subtraction. He should have rewritten the subtractions as additions first, since the associative property only applies to addition.

(14 − 27) − 27= [14 + (−27)] + (−27)

Now Kevin can apply the associative property.

= 14 + [(−27) + (−27)] = 14 + (−54) = 14 − 54 = −40

## Work Time

# Kevin’s Work

Here is how Kevin evaluated an expression from his homework.

(14 − 27) − 27 = 14 − (27 − 27) = 14 − 0 = 14

His answer is incorrect.

- Find his mistake and correct it.

## Hint:

It looks like Kevin tried to use the associative property of addition. Review the property. Did Kevin use it correctly?

# Prepare a Presentation

# Preparing for Ways of Thinking

Choose students who demonstrate the properties of operations when evaluating expressions to present in Ways of Thinking. Students’ presentations should make clear connections between their work for evaluating expressions and the properties of operations used to justify each step.

# Challenge Problem

## Answers

- $-6\u26054=-6+\left|4\right|=-6+4=-2$$5\u2605-1=5+\left|-1\right|=5+1=6$
- No. For example, $5\u2605-1=5+\left|-1\right|=5+1=6$ , but $-1\u26055=-1+\left|5\right|=-1+5=4$ .
- No, only numbers less than or equal to $0$ have inverses. If $a$ is positive, then $a\u2605b=a+\left|b\right|$ will be greater than or equal to $a$ because $\left|b\right|$ cannot be negative. Therefore, $a\u2605b$ can never be $0$ no matter what $b$ is.

## Work Time

# Prepare a Presentation

Explain how the properties of operations help you evaluate expressions. Use your work to support your explanation.

# Challenge Problem

Suppose there is a new operation, $\u2605$, such that for any numbers $a$ and $b$,

$a\u2605b=a+\left|b\right|$

For example, $3\u2605-7=3+\left|-7\right|=10$ .

- Find $-6\u26054$ and $5\u2605-1$ .
- Is $\u2605$ a commutative operation? That is, does $a\u2605b=b\u2605a$ for any numbers $a$ and $b$? Explain.
- For the operation $\u2605$, does every number $a$ have an inverse? That is, for any number $a$, is there a number $b$ such that $a\u2605b=0$ ? Explain.

# Make Connections

# Mathematics

Presenters should justify their steps with properties or rules and explain how they decided what to do. Choose presentations that demonstrate the application of properties in different ways, and have students discuss whether or not the methods are equivalent. Discuss Kevin’s mistake and have a student show the correct way to simplify the expression.

Have students who solved the Challenge Problem present and explain their solutions during Ways of Thinking as well.

## Performance Task

# Ways of Thinking: Make Connections

Take notes on how your classmates used the properties of operations to simplify expressions with negative numbers.

## Hint:

As your classmates present, ask questions such as:

- Can you explain that last step one more time?
- Why did you change all the subtractions to additions?
- Why did you switch the order of 4 5 6 and 1 3 8

# Properties of Operations

# A Possible Summary

The commutative property of addition says you can change the order of numbers that are added without changing the sum. The associative property of addition says you can change the way numbers that are added are grouped without changing the sum. The inverse property of addition says the sum of any number and its opposite is 0. The additive identity property says the sum of 0 and any number is that number.

The addition properties can be used to help simplify expressions. For example, you can use the commutative and associative properties to move terms that are easy to combine together and then add them. The additive inverse property lets you combine terms that are opposites to get 0. You have to be careful to only apply these properties to terms that are added. If an expression has subtractions, you need to change all the subtractions to additions before applying these properties.

## Formative Assessment

# Summary of the Math: Properties of Operations

Write a summary on how to use the properties of operations to simplify expressions with negative numbers.

## Hint:

Check your summary.

- Do you describe each of the addition properties of operations?
- Do you discuss how the properties can be used to rewrite and evaluate an expression?
- Do you explain how the addition properties of operations work with negative numbers?

# Write Equations

# Lesson Guide

This task allows you to assess students’ work and determine what difficulties they are having. The results of the Self Check will help you determine which students should work on the Gallery and which students would benefit from review before the assessment. Have students work on the Self Check individually.

SWD: Provide clear feedback to students as they attempt to solve problems or articulate concepts. This type of feedback guides students explicitly as they develop their thinking about mathematics.

# Assessment

Have students submit their work to you. Make notes on what their work reveals about their current levels of understanding and their different problem-solving approaches.

Do not score students’ work. Research shows that scoring is counterproductive, as it will encourage students to compare their scores and will distract them from finding out what they can do to improve their understanding of the mathematics.

Share with each student the most appropriate Interventions to guide their thought process. Also note students with a particular issue so that you can work with them in the Putting It Together lesson that follows.

# Interventions

**Student has trouble getting started.**

- Pick any negative number and imagine it on a number line. How do you move on the number line when you add a positive number?
- Pick any number and imagine it on a number line. How far do you have to move to get to 6?

**Student does not explain why writing an equation is impossible.**

- How do you know it is impossible to write the equation?
- Can you use the rule of subtraction to help you explain why the equation is impossible to write?
- Can you refer to the number line to help you explain why the equation is impossible to write?

**Student does not have a strategy for writing the additions and subtractions.**

- Can you visualize how you move on the number line for a problem of this type?
- What direction do you move for the first number? What direction do you move for the second number? How can you choose the two numbers so the end of the second arrow ends at 6?

**Student thinks adding always gives a greater result than subtracting.**

- Are there numbers you can substitute for
*x*and*y*so that*x*+*y*is negative? What is the value of*x*−*y*in this case?

# Possible Answers

Please note: Students can use positive/negative fractions and/or decimals to satisfy the equations.

Answers will vary. Possible answers:

- −3 + 9 = 6
- Not possible. The sum of two negative numbers is always negative.
- 10 − 4 = 6
- −3 − (−9) = 6
- Not possible. Subtracting a positive number is the same as adding a negative, and the sum of two negative numbers is always negative.
- 5 − (−1) = 6

## Formative Assessment

# Write Equations

Complete this Self Check by yourself.

Write an equation in the given form. If it is not possible to write such an equation, explain why.

- negative + positive = 6
- negative + negative = 6
- positive − positive = 6
- negative − negative = 6
- negative − positive = 6
- positive − negative = 6

# 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 what questions students still have about positive and negative numbers.

## 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.

**One thing I still wonder about adding positive and negative numbers is …**