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# Self Check Exercise

## Overview

Students critique and improve their work on the Self Check. They then extend their knowledge with additional problems.

Students solve problems that require them to apply their knowledge of multiplying and dividing positive and negative numbers. Students will then take a quiz.

# Key Concepts

To solve the problems in the Self Check, students must apply their knowledge of multiplication and division of positive and negative numbers learned throughout the unit.

# Goals and Learning Objectives

- Use knowledge of multiplication and division of positive and negative numbers to solve problems.

# Critique

# Lesson Guide

Students should look at the results of their Self Check and the questions under the Critique section.

## Opening

# Critique

Review your work on the Self Check problem and think about these questions.

- What are the rules for determining whether a product is positive or negative?
- What are the rules for determining whether a quotient is positive or negative?
- Will the value of
*x*that makes the inequality true be positive or negative? - If you change the inequality sign to an equal sign, what value of
*x*will make the resulting equation true? How can you use this information to help you find the possible values for*x*in the inequality? - How can you keep track of the values you try for
*x*and the results for each value? Can you make a table?

# Make Sense of Problems and Persevere

# Mathematical Practices in Action

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

Have students watch the video and listen to the dialogue between Karen and Maya. This video shows students engaged in Mathematical Practice 1: Make sense of problems and persevere in solving them. At first Karen thinks she can write an equation to solve the problem, but then Maya points out that they would need an inequality. Notice how Karen corrects herself as she makes sense of the problem and perseveres in solving it.

They also show how thorough they are in solving the problem by checking whether their solution is correct. Karen substitutes the value 12 for *x* in the original inequality to check whether $\frac{12}{3}>-\frac{1}{6}$. Since 4 is greater than $-\frac{1}{6}$, Maya and Karen conclude their solution is correct.

In the discussion, elicit that sometimes it takes a while to make sense of a problem, and it can also take time to go through a solution. The important thing is to keep working through it as Karen and Maya did.

ELL: When showing the video, monitor that ELLs are following the meaning of what is presented. If necessary, pause the video and allow them to ask clarifying questions. Alternatively, ask questions to check for understanding.

## Opening

# Make Sense and Persevere

Watch this video to see how Karen and Maya make sense of a problem and persevere in solving it.

- What things did Karen and Maya do to make sense of the problem?
- How did Karen and Maya show their perseverance in solving the problem?
- What things can you do to make sure you are making sense of a problem and persevering in solving it?

VIDEO: Mathematical Practice 1

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will apply their knowledge of multiplying and dividing positive and negative rational numbers to solve problems.

## Opening

Apply your knowledge of multiplying and dividing positive and negative rational numbers to solve problems.

# Revise Your Work

# Lesson Guide

Organize students into pairs to work on revising their work. Encourage students to incorporate ideas from their partner in their work. If several students in the class are struggling with the same issue, you could write a relevant question on the board. You might also ask a student who has performed well on a particular part of the task to help a struggling student.

While students work, note different approaches to the task:

- How do they organize their work?
- Do they notice if they have chosen a strategy that does not seem to be productive? If so, what do they do?

Try not to make suggestions that move students toward a particular approach to this task. Instead, ask questions that help students to clarify their thinking.

Questions to pose if students find it difficult to get started:

- What questions were you asked as feedback?
- How could you and your partner work together to address one of those feedback questions?

Select correct and incorrect solutions to be presented during Ways of Thinking. Choose students who used a variety of different approaches to explain their thinking.

# Mathematical Practices

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

Students might use the structure of each problem, as well as the relationship between the inequalities, to help them find solutions. For example, can finding a solution to −4*x* > −12 help us find a solution to −4*x* < −12?

# Interventions

**Student does not know the rules for determining the sign of a sum or product.**

- What did you learn in prior lessons that would be useful for this problem?
- Is the product of a positive number and a negative number positive or negative? What about the sign of a product of two negative numbers?
- Is the quotient of a positive number and a negative number positive or negative? What about the sign of a quotient of two negative numbers?

**Student has difficulty getting started.**

- Will the value of
*x*that makes the statement true be positive or negative? - Try guessing a number for
*x*and testing to see if it makes the statement true. If it doesn’t, how can you use the result to make another guess? - If the inequality were an equal sign, what value of
*x*would make the equation true? Now test values greater than this value and less than this value.

**Student works inefficiently.**

- What do you need to consider to make an “educated” first guess?
- How can you keep track of your guesses and the results? Can you make a table?
- Can your answer and work from one part help you find an answer to other parts?

# Answers

Answers will vary. The answer for each could be any number that satisfies the given inequality. Possible answers:

*x*= −10 or any other number*x*such that*x*< −3*x*= 10 or any other number*x*such that*x*> 3*x*= −$\frac{1}{10}$ or any other number*x*such that $-\frac{1}{4}<x<0$*x*= $\frac{1}{10}$ or any other number*x*such that $0<x<\frac{1}{4}$*x*= −10 or any other number*x*such that*x*< −6*x*= 10 or any other number*x*such that*x*> 6

## Work Time

# Revise Your Work

Revise your work on the Self Check problem with your partner. Be sure to keep in mind the Critique questions.

Find a value of *x* that makes each inequality true.

- −4
*x*> 12 - −4
*x*< −12 - 0 < −4
*x*< 1 - −1 < −4
*x*< 0 *x*÷ 3 < −2*x*÷ 3 > 2

# Extend Your Work

# Answers

Answers will vary.

- Any answer for which −$\frac{1}{2}$ <
*x*< 0. - Any answer for which 0 <
*x*< $\frac{1}{2}$.

## Work Time

# Extend Your Work

Find a value of *x* that makes each inequality true.

- −$\frac{1}{2}$ <
*x*< 0 - 0 <
*x*< $\frac{1}{2}$

# Make It True

# Lesson Guide

Look for students who apply properties of equality to find solutions to the equations, and students who use similar strategies to find solutions to the inequalities.

**[common error]** Some students may try to solve the inequalities in exactly the same way they would solve equations, which can lead to errors when multiplying or dividing by negative numbers. Look for students who use the solutions to the equation to help them choose appropriate test points for the “solved” inequality and set the direction of the inequality signs accordingly.

ELL: If you hear ELLs say the right things but use the wrong grammar structure, show signs of agreement and softly rephrase using the correct grammar and retaining the student’s words as much as possible.

# Mathematical Practices

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

Students reason abstractly as they consider the general meaning of an inequality (e.g., *xy* < *x*(*y* + 1) means that the product of two numbers, *x* and *y*, is less than the product of *x* and 1 more than *y*). Students think quantitatively as they think about and test specific values of the variables.

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

Students might use the structure of each problem, and the relationship between the inequalities, to help them find solutions. For example, if a pair of values for *x* and *y* does not make *xy* < *x*(*y* + 1) true, does it make *xy* > *x*(*y* + 1) true?

# Answers

Answers will vary.

- The inequality is true for any values of
*x*and*y*as long as*x*> 0. - The inequality is true for any values of
*x*and*y*as long as*x*< 0. - The equation is true if
*x*= 0.

# Challenge Problem

## Answers

- The inequality is true for any values of
*x*and*y*as long as*x*< 0 and*y*> 0, or*x*> 0 and −1 <*y*< 0, or*x*< 0 and*y*< −1. - The inequality is true for any values of
*x*and*y*as long as*x*> 0 and*y*> 0, or*x*< 0 and −1 <*y*< 0, or*x*> 0 and*y*< −1. - The equation is true for
*x*= 0, if*y*is not equal to 0 or −1.

## Work Time

# Make It True

Find values of *x* and *y* that make each inequality or equation true.

*xy*<*x*(*y*+ 1)*xy*>*x*(*y*+ 1)*xy*=*x*(*y*+ 1)

# Challenge Problem

Find values of *x* and *y* that make each inequality or equation true.

- $\frac{x}{y}<\frac{x}{y+1}$
- $\frac{x}{y}>\frac{x}{y+1}$
- $\frac{x}{y}=\frac{x}{y+1}$

## Hint:

- How can you use this information to help you find a value for x in the inequality?
- If you change the inequality sign to an equal sign, what value of x will make the resulting equation true?

# Make Connections

# Lesson Guide

Organize a whole class discussion to consider issues arising from the work students did to revise their answers. You may not have time to address all these issues, so focus the discussion on the issues most important for your students.

Have students share their work and talk about how they approached the problem. Students who had strategies that didn’t work should share how and when they realized their strategy didn’t work and what they did about it. Have students share how they addressed the intervention questions. Have students ask questions and make observations as they view each others’ work.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about the different approaches classmates used to find values for *x* and *y*.

## Hint:

As your classmates present, ask questions such as:

- How did you figure out that your value for
*x*would work? - If the first value you tried for
*x*did not work, how did you decide what value to try next? - Will the value for
*x*always be negative? How can you tell?

# 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 they learned in the unit.

## 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.

**The most important things I learned in this unit are …**