# Properties of Operations

## Overview

The class reviews the properties of operations. The use of “ask myself” questions to make sense of problems and persevere is modeled. Students review things to do when they feel stuck on a problem. Finally, students use the properties of operations to evaluate expressions.

# Key Concepts

Students use the properties of operations to justify whether two expressions are equivalent.

# Goals and Learning Objectives

- To start to work on a problem, make sense of the problem by using “ask myself” questions.
- Persevere in solving a problem even when feeling stuck.
- Use the properties of operations to evaluate expressions.

# Properties of Operations

# Lesson Guide

Review the names of the properties of operations and have a brief discussion about them with students. The properties themselves should be familiar to students, but students may not be able to identify them by name. Do not go through each property at this time, because students will have access to the properties of operations through the Glossary during Work Time.

## Opening

# Discuss:

What do you remember about these properties of operations?

# “Ask Myself” Questions

# Lesson Guide

Ask students to silently read the instructions and example problem. Have one student explain in her own words what the problem is asking.

Tell students that today, and most days, they will work on problems on their own and then they will discuss their ideas with a partner.

# Teacher Demonstration

Demonstrate how to use “ask myself” questions to understand the problem:

- What is the problem asking me to do? The problem is asking if the two expressions are equivalent.
- What are the two expressions? Here are the expressions: 2(8 + 7) + 5 and 5(4 + 3).
- What does it mean if two expressions are equivalent? I know if two expressions are equivalent, the values of the expressions are equal.
- How can I find the value of each expression? I can use the properties of operations to evaluate the expression.
- What are the properties of operations? Hmm … I know most of them. I can look in the Glossary to read about them if I need to.
- Let me read the problem again. What does it mean to “use the properties of operations to show each step”? I guess that means I need to work though a set of steps to see if I can change one expression into the other. Maybe for each step, I write a property that allows it.
- Does the example show using the properties of operations for each step? I can use the example to help me understand.
- Why does the first line say “Distributive property”? The left side of the equation multiplies the quantity of 8 added to 7 by 2 and then adds 5. The right side of the equation shows the quantity 8 multiplied by 2 added to the quantity 7 multiplied by 2 and then adds 5. Oh, I see. The multiplier 2 was distributed to both the 7 and the 8 that were inside the parentheses.
- The next line does not even have a property of operation. It says “Multiplication.” Why is that? Oh, I see. The 2 times 8 and 2 times 7 are multiplied to change the expression into 16 + 14 + 5. And on the next line it says “Addition” because the numbers are added together to make 35.
- Wait, why does the next line show a new multiplication problem? It says the reason is multiplication, hmmm.
- We just found the result of the expression, so why would the example then change it back into a multiplication expression? Maybe I should go back to the problem to find out why. We have the two expressions. Oh, I see. The second expression uses multiplication. It multiplies the quantity of 4 added to 3 by 5. That is an easy expression because it is simply 5 times 7. Oh, that is what the example was doing, but it showed 7 times 5. Well, we know those expressions have the same result.
- What property is it that says 7 times 5 will have the same result as 5 times 7? If I do not know the property, I can look it up in the Glossary. It is a multiplication property, so I can search using the keywords “property” and “multiplication.” (Demonstrate for students how to access the Glossary and search using the keywords “property” and “multiplication.”)
- That search seems like it will show a lot of properties. Which one is it? I can look until I find the property that lets me switch the order of factors in a multiplication expression.
- Is it the associative property? That property is about parentheses, but this line of the expression does not have any parentheses.
- Is it the next property? Oh, yeah. Here the next property shown in the example is “Commutative property.” That property allows for switching the order of factors in a multiplication expression.
- Why does the last line of the example say, “Distributive property”? Hmm...It looks like the 7 was changed into 4 + 3. The multiplication is being distributed over addition.
- Does that mean the two expressions are equivalent? Yes, they are. The example changed the first expression into the second expression.

# Mathematics

After the demonstration, ask students:

- What questions can you ask yourself to make sense of a problem?
- What can you do if you get stuck on a problem?

First, have students spend a few minutes thinking about these two questions on their own. Then have students discuss their ideas with their partners.

Now have students share their ideas with the class. As with previous lessons, having students generate their own ideas allows them to take ownership of their own learning. If students are having trouble thinking of questions, give them a list of “ask myself” questions to refer to.

These Hints are ideas for students to use for “ask myself” questions:

- What does
mean?*_**_*_ - What is this problem talking about?
- What kinds of comparisons is the problem looking at?
- What are the numbers in the problem and what do they mean?
- What is the problem asking for?

These Hints are ideas for things students can do if they get stuck on a problem:

- Look at similar problems you have solved previously.
- Model the problem using counters or other materials.
- Sketch a diagram or other representation.
- Write what you do know.
- Write questions to ask later.
- Check other resources.

Record students’ ideas as two separate lists. After all ideas are listed, review each for clarity. Agreement is not necessary because having a diversity of ideas is useful. These lists can continue to grow throughout the year.

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Explain that it is important for students to make sense of a problem as the first step in working to solve it, and it is important for students to persevere in solving the problem even if they feel stuck.

## Opening

# "Ask Myself” Questions

When working by yourself, you should:

- Make sense of the problem.
- Use “ask myself” questions to see what you need to understand before starting.
- Persevere through difficulty by using what you do know.

Read the worked problem. Then watch your teacher give an example of how to use “ask myself” questions to understand the problem.

- Are these expressions equivalent?

2(8 + 7) + 5

5(4 + 3)

- Use the properties of operations to show each of the steps taken to determine if the expressions are equivalent or not.

Here are ideas for “ask myself” questions:

- What does
mean?*_**_*__ - What is this problem talking about?
- What kinds of comparisons is the problem looking at?
- What are the numbers in the problem and what do they mean?
- What is the problem asking for?

Here are ideas for what to do if you get stuck:

- Look at similar problems you have solved previously.
- Model the problem using counters or other materials.
- Sketch a diagram or other representation.
- Write what you do know.
- Write down questions to ask later.
- Check other resources.

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will justify equivalent expressions by using the properties of operations.

## Opening

Justify equivalent expressions by using the properties of operations.

# Justify Equivalent Expression for 3 • 2

# Lesson Guide

Have students work on both Work Time problems on their own for 5 minutes before working with their partner.

# Interventions

**Student does not start with “ask myself” questions.**

- What questions did you ask yourself before you started solving this problem?

**Student stops working when he or she gets stuck.**

- What can you do if you get stuck on a problem?

**Student changes the order of subtraction or division.**

- Why can you change the order of the numbers for addition and multiplication but not for subtraction and division?

**Student is confused about how to remove parentheses.**

- How do you know when to use parentheses and when you can remove parentheses?

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Students begin their work by first making sense of the problem by using “ask myself” questions. Then they persevere by using the ideas they generated for what to do when you get stuck.

# Possible Answers

$\left({2}^{2}\right)\left(4\cdot \frac{1}{4}\right)+2$ is equivalent to 3 ⋅ 2

= 4(4 • $\frac{1}{4}$) + 2 Square

= 4 + 2 Identity Property of Multiplication

= 6 Addition

= 3 × 2 Multiplication

$=\left({2}^{2}\right)\left(1\right)+2$ Inverse Property

## Work Time

# Justify Equivalent Expression for 3 • 2

- Are these expressions equivalent?

$\left({2}^{2}\right)\left(4\cdot \frac{1}{4}\right)+2$

3 ⋅ 2

- Use the properties of operations to show each of the steps taken to determine if the expressions are equivalent or not.

Hint:

- What questions did you ask yourself before you worked on this problem?
- Why can you change the order of the numbers for addition and multiplication but not for subtraction and division?
- How did you know when to use parentheses and when you can remove the parentheses?

# Justify Equivalent Expression for 58 • 31

# Lesson Guide

Have students work on both Work Time problems on their own for 5 minutes before working with their partner.

# Interventions

**Student does not start with “ask myself” questions.**

- What questions did you ask yourself before you started solving this problem?

**Student stops working when he or she gets stuck.**

- What can you do if you get stuck on a problem?

**Student changes the order of subtraction or division.**

- Why can you change the order of the numbers for addition and multiplication but not for subtraction and division?

**Student is confused about how to remove parentheses.**

- How do you know when to use parentheses and when you can remove parentheses?

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Students begin their work by first making sense of the problem by using “ask myself” questions. Then they persevere by using the ideas they generated for what to do when you get stuck.

# Possible Answers

- 58 × 31 is not equivalent to $1,500+240+\frac{1}{2}\left(58\right)$

58 × 31 = 58(30 + 1) Distributive Property

= (58 × 30) + (58 × 1) Distributive Property

= (58 × 30) + 58 Identity Property of Multiplication

= (30 × 58) + 58 Commutative Property of Multiplication

= 30(50 + 8) + 58 Distributive Property

= 1,500 + 240 + 58 Distributive Property and Multiplication

## Work Time

# Justify Equivalent Expression for 58 • 31

- Are these expressions equivalent?

58 ⋅ 31

$1,500+240+\frac{1}{2}\left(58\right)$

- Use the properties of operations to show each of the steps taken to determine if the expressions are equivalent or not.

Hint:

- What questions did you ask yourself before you worked on this problem?
- How did you know when to use parentheses and when you can remove the parentheses?

# Prepare a Presentation

# Preparing for Ways of Thinking

As students work, walk around the room and listen carefully to their reasoning. Watch for these things:

- Is the listener asking questions when the speaker is unclear?
- Do both partners explain their processes, even if they did similar things?
- Do students understand how to square a number?
- Do students follow the order of operations when evaluating expressions?

# Challenge Problem

# Answers

- Possible answers:

(3 + 9) − (2 × 3)

3(4 + 2) ÷ (3 + 0)

## Work Time

# Use properties of operations to justify equivalent expressions.

- Describe your strategies for identifying the relevant properties of operations.
- State which properties were more difficult to understand.
- Identify any mistakes you made and what you learned from them.
- Include any questions your partner asked about your explanation.

# Challenge Problem

- Write two other expressions that are equivalent to:

$\left({2}^{2}\right)\left(4\cdot \frac{1}{4}\right)+2$

# Make Connections

# Mathematics

For each problem, choose two student presenters who reached the same conclusion about the equivalence of the expressions but who did not break down the solution steps in the same way. Ask the class to compare the steps of the two solutions.

Spend a few moments discussing how the properties of operations are used in students’ justifications.

Ask a few students who completed the Challenge Problem to present the equivalent expressions that they created. Ask the students to explain how they knew that the expressions were equivalent.

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

Encourage students to talk about the way they made sense of the problems and any strategies they used to persevere when stuck on a problem.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about how your classmates used the properties of operations in their justifications.

As your classmates present, ask questions such as:

- How are the steps of these two different solutions similar and different?
- Why are these expressions equivalent even though they look different?

# What Is a Good Summary?

# Lesson Guide

Begin this activity with a brief discussion about the qualities of a good summary:

- It focuses on the important mathematics in today’s lesson.
- It uses a few sentences to explain the mathematics.
- It goes beyond the answers to the bigger ideas in the lesson.

Have each student write a summary of the mathematics in the lesson.

Then work together to write a class summary. When done, if the class summary is helpful, post it for students' reference.

SWD: Refer students to the Hints as a checklist of the information they need to include in their summaries. Consider creating a paper resource of the Hints for students to keep in front of them as they write their summaries.

# A Possible Summary

Properties of operations describe allowable changes to expressions that maintain equivalence. Equivalence means that the value of the expression stays the same. Each change should be shown and an operation or property of operations used to justify the step. Some properties are true for only some operations. For example, the commutative property and associative property work for addition and multiplication but not for subtraction and division.

# Additional Discussion Points

Additional things you might want to discuss are:

- The inverse property connects addition to subtraction and multiplication to division.
- The distributive property explains how to distribute a factor to both addends.

## Formative Assessment

# Summary of the Math: What Is a Good Summary?

A good summary clearly explains the important mathematics of today’s lesson.

- Write a summary about how to justify that two expressions are equivalent.

Hint:

- Check your summary. Do you explain what it means when expressions are equivalent?
- Do you define the properties of operations?
- Do you define
*justify*?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to learn whether students understand why the properties of operations are helpful.

## 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.

**Using properties of operations helps me…**