# Ways of Thinking and Properties of Operations

## Overview

Students discuss as a class the important ways that listeners contribute to mathematical discussions during Ways of Thinking presentations. Students then use the properties of operations to find the value of each fruit used in equations.

# Key Concepts

Students use the properties of operations to find the value of each fruit used in different equations. By considering several equations, students can match each of the 10 fruits to the whole numbers 0 through 9. This work helps students see why representing unknown numbers with letters is useful.

# Goals and Learning Objectives

- Contribute as listeners during the Ways of Thinking discussion.
- Identify the whole numbers that make an equation true.
- Use the properties of operations, when appropriate, to justify which whole numbers represent unknown values.

# Participate in Mathematical Discussions

# Lesson Guide

During mathematical discussions, students will sometimes present their work to their classmates, and other times, listen to their classmates present. Emphasize that listeners’ contributions during mathematical discussions are equally as important as presenters’ contributions.

A good classroom math community focuses on the importance of learning, not simply on getting the right answer. Classmates help each other learn, but ultimately each student must take responsibility for his own learning.

Ask students:

- How can listeners contribute during presentations to improve the classroom math community?

Have students think about the question for a moment, and then ask them to discuss their ideas with their partners.

Ask students to share their ideas with the class. Record each idea. When all the ideas have been listed, review each one for clarity.

These hints for students list important ways listeners contribute to math discussions:

- Clarify—Tell the presenter when you do not understand. “I did not understand. What do you mean by…?”
- Critique—Challenge reasoning that is flawed. “How do you know that…?”
- Connect—Explore how different strategies result in the same answer. “What you said is like…”
- Compare—Describe similarities and differences. “Is that different from…?”
- Notice Structure—Ask whether a conclusion is always, sometimes, or never true. “Is your conclusion always true for…?”

Suggest these ways students can engage in their own learning while watching their classmates’ presentations:

- Listen carefully so that you understand the work and its relation to the mathematics in the lesson and unit.
- Notice how your way of thinking is related to your classmates’ ways of thinking.
- Ask questions.

## Opening

# Participate in Mathematical Discussions

Listening actively and asking questions during discussions are as important as the presentation.

To create a good math community, you should:

- Focus on learning, not getting the answer right.
- Help each other learn.
- Be responsible for your own learning.

View the Hints during Ways of Thinking to read questions you can ask the presenter. Pick one or ask a your own question.

Discuss:

- What can you as a listener contribute to improve your classroom math community?

Hint: Here are ways listeners contribute to math discussions:

- Clarify—Tell the presenter when you do not understand.

“I did not understand. What do you mean by…?” - Critique—Challenge reasoning that is flawed.

“How do you know that…?” - Connect—Explore how different strategies result in the same answer.

“What you said is like…" - Compare—Describe similarities and differences.

“Is that different from…?” - Notice structure—Ask whether a conclusion is always, sometimes, or never true. “Is your conclusion always true for…?"

# Properties of Operations as Fruit

# Lesson Guide

Have students look at the fruit equation and think about which property of operations the equation represents. Give them time to think about the problem on their own, and then have them discuss their ideas with their partner. Then ask students to share their ideas with the class.

Students might recognize that this equation represents the identity property of multiplication, which means that the coconut represents 1 and the apple is any number:

Identity property of multiplication: *n* = 1 × *n*

Another possibility is that the apple represents 0 and the coconut is any number:

Zero property of multiplication: 0 = *n* × 0

Both possibilities could be true when given only this equation.

This problem will give students some ideas about how to approach the Work Time problems. It connects with the work they have done on the properties of operations in the previous lesson.

## Opening

# Properties of Operations as Fruit

Discuss:

Which property of operations could this equation represent? How do you know?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explain how they know the value of each fruit.

## Opening

Explain how you know the value of each fruit.

# Identify the Value of Each Fruit

# Lesson Guide

Have students work on their own for several minutes before interacting with their partner. The time students spend working on their own will help them persevere through difficulty.

Students will use their notebook or hand draw and take pictures as they try different pairings between fruits and whole numbers. The general similarity of the shape of each fruit will make the fruit difficult for students to re-create precisely. This difficulty can help students see the usefulness of using letters (or simpler figures) to represent the fruits as they work though the problem.

If students are spending too much time re-creating the fruit shapes or tediously copying every fruit, it might be useful to ask:

- Is there another symbol you can use to represent each type of fruit?

Have partners discuss their work, and tell students to focus on explaining to each other how they know which whole number is represented by each fruit.

# Interventions

**Student does not try to rewrite equations.**

- How can you rewrite the addition equation for the 3 peaches?
- Can you change it into a multiplication equation?
- Are there other equations you can rewrite?

**Student gets stuck on one equation.**

- If you are not sure about that one, you can mark it with a question mark and try figuring out another equation.

**Student feels overwhelmed from too many equations.**

- Try focusing on one type of fruit and focus only on the equations that use that fruit.
- Wait until you have a guess for the values in those equations before looking at more equations.

# Mathematical Practices

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

Students will need to look for various entry points into this problem and persevere through difficulty. There are many equations to consider and many possible values that can work for a single equation, but students must find the value that works for all instances of each fruit.

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

Students’ primary task in this problem is to explain their pairings. To construct a cohesive argument, they need to make conjectures and follow the logic to verify the truth of their conjectures.

**Mathematical Practice 7: Look for and make use of structure.**

Students look for the pattern and structure in the equations and likely need to step back and shift perspectives to realize that a fruit could be paired with a different value.

# Answers

## Work Time

# Identify the Value of Each Fruit

- Find the whole number from 0 to 9 that each fruit represents to make the equations true.
- Explain your reasoning.

Ask Yourself:

- Which properties of operations are useful in identifying the value of each fruit?
- Does substituting each fruit with a simple symbol make it easier to work through the possible values for the different fruit?
- Would naming the equations make it easier to explain your reasons for identifying the values of the fruit?

# Prepare a Presentation

# Preparing for Ways of Thinking

Have students prepare a presentation even if they have not finished pairing all the fruits with whole numbers. Tell students to focus their presentation on their reasons for pairing the fruits with the values that they did figure out.

Encourage students to include in their presentation any mistakes they made along the way rather than only their final solution.

Look for students who have different explanations of the same parings and for students who have different pairings.

# Challenge Problem

# Possible Answers

- Answers will vary.

## Work Time

# Prepare a Presentation

Explain how you identified the value of each fruit.

- Write the equation(s) that helped you find a value for a fruit and explain why the equation(s) helped.
- State any properties of operations that you used to find the value of a fruit.
- Identify any mistakes you made and what you learned from them.
- Show which equations or symbols were difficult to determine the values.

# Challenge Problem

- Use the fruit to create equations for the other properties of operations.
- Challenge your partner to find the result and identify the property that you are representing.

# Make Connections

# Mathematics

Help students contrast different explanations for the same pairing of particular fruits with values. Ask students:

- What makes an explanation more convincing?

Facilitate a discussion for students who have different pairings to help the class understand why multiple pairings are possible and which equations eliminate possible pairings.

While you are facilitating the discussion, keep track of the types of questions students ask presenters. At the end of the discussion, thank students who asked presenters questions that are listed as hints as well as students who formulated their own questions.

Ask a few students who completed the Challenge Problem to present their equations representing the properties of operations.

## Performance Task

# Ways of Thinking: Making Connections

Take notes about how your classmates’ strategies for identifying the values of the fruit were similar to and different from yours.

As your classmates present, ask questions such as:

- Which equation was the most helpful in identifying the value of a fruit?
- Which was the hardest equation to figure out?
- Was it helpful to use other symbols in place of the fruit?
- Which properties did you use to identify the values of the fruit?

# Find Unknowns

# Mathematics

Have students read the summary of the math in this lesson. Ask students:

- What is an unknown quantity?
- How do the properties of operations help in determining the value represented by the fruit?
- What is a variable?
- Why are letters easier symbols to use than fruits?

Work with the students to add any additional details they thought were important to the summary.

# Additional Discussion Points

Additional things you might want to discuss are:

- Patterns in equations for which it was easier or more difficult to find the unknown values
- The types of equations that are useful to rewrite, such as repeated addition as multiplication, repeated multiplication as squaring, and equations with similar symbols on both sides of the equal sign

## Formative Assessment

# Summary of the Math: Find Unknowns

**Read and Discuss**

- One key concept in algebra is the idea of variables. A variable represents an unknown quantity in an equation.
- In this task, each fruit represents an unknown quantity.
- Look at this equation. You could express this equation as
*a*=*b*×*a*, where*a*is the apple and*b*is the coconut. - In future lessons, you will work with variables to represent unknown quantities.
- To solve equations with variables, use the same properties of operations that you use with numeric equations.

Can you:

- Explain your general strategy for finding the unknown values of the fruit?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to see what strategies students found useful in identifying the unknown values.

## 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.

**A strategy that I found useful to identify unknown values is…**