# Self Check Exercise

## Overview

Students critique and improve their work on the Self Check, then work on more addition and subtraction problems.

Students solve problems that require them to apply their knowledge of adding and subtracting positive and negative numbers.

# Key Concepts

To solve the problems in this lesson, students use their knowledge of addition and subtraction with positive and negative numbers.

# Goals and Learning Objectives

- Use knowledge of addition and subtraction with positive and negative numbers to write problems that meet given criteria.
- Assess and critique methods for subtracting negative numbers.
- Find values of variables that satisfy given inequalities.

# Critique

# Lesson Guide

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

## Opening

# Critique

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

- Can you use a number line to help you understand the problem?
- Does choosing a value for the first number in the equation help you figure out what the second number could be?
- Do you think writing an equation is impossible? If so, explain why.
- Does writing a subtraction equation as an equivalent addition equation help you figure out what numbers could work in the equation?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will apply their knowledge of adding and subtracting positive and negative numbers to write problems that meet given criteria.

## Opening

Apply your knowledge of adding and subtracting positive and negative numbers to write problems.

# Revise Your Work

# Lesson Guide

Organize students into pairs to work on their revisions. Encourage students to incorporate ideas from their partner in their work.

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.

If students find it difficult to get started, try asking:

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

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.

ELL: When forming pairs, be aware of ELLs and ensure that they have a productive learning environment. Different ways to form partnerships include:

- Pairing ELLs with English speakers, so they can learn language skills.
- Pairing ELLs with students who are at the same level of language skills, so the partners can take active roles and work things out together.
- Pairing ELLs with students whose proficiency level is lower, so they play the role of the "supporter."

You can also pair students based on their math proficiency.

# 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 the number line. How far do you have to move to get to 6?

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

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

# Possible Answers

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

## Work Time

# Revise Your Work

Work with your partner to revise your work on the Self Check problem based on the questions posed and feedback from your partner.

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

# Extend Your Work

# Lesson Guide

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?

# Interventions

**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?

# Possible Answers

Answers will vary. Some possible answers:

- Not possible. The sum of two positive numbers is always positive.
- −5 + 2.5 = −2.5
- −1 + (−1.5) = −2.5
- 2.5 − 5 = −2.5
- −5 − (−2.5) = −2.5
- −1.5 − 1 = −2.5
- Not possible. A positive minus a negative is the same as a positive plus a positive, and the sum of two positive numbers is always positive.

## Work Time

# Extend Your Work

Use what you have learned to write an equation for these statements. If it is not possible to write such an equation, explain why.

- positive + positive = –2.5
- negative + positive = –2.5
- negative + negative = –2.5
- positive – positive = –2.5
- negative – negative = –2.5
- negative – positive = –2.5
- positive – negative = –2.5

# Evaluate a Statement

# Answer

- Marcus is not correct. Explanations will vary. Possible explanation: Marcus's claim can be disproved with a counterexample. If students need more explanation about the importance of using a counterexample, explain that a counterexample can be used when critiquing a mathematical statement.

For example, you cannot turn all the negative numbers in the first equation into positives to calculate the answer.

−1 − (−2) − (−3) = 4

1 + 2 + 3 = 6

## Work Time

# Evaluate a Statement

Marcus said, “To subtract negative numbers, you just turn all the negative numbers into positive numbers, and then add.”

- Is Marcus correct? Explain and justify your reasoning.

# Prepare a Presentation

# Preparing for Ways of Thinking

Look for students to present who can clearly explain adding and subtracting numbers and support their rules with numerical examples.

Students may also discuss the Intervention questions, prior misconceptions that were corrected in this lesson, methods that increased their fluency, or a better understanding of the characteristics of adding and subtracting.

Also look for students who approached the Challenge Problem in different ways.

# Challenge Problem

## Possible Answers

Answers will vary. Some possible answers:

- If
*x*= 5 and*y*= 3, then*x*+*y*= 5 + 3 = 8 and*x*−*y*= 5 − 3 = 2, so*x*+*y*>*x*−*y*. - If
*x*= 5 and*y*= −3, then*x*+*y*= 5 + (−3) = 2 and*x*−*y*= 5 − (−3) = 8, so*x*+*y*<*x*−*y*. - If
*x*= 5 and*y*= −3, then |*x*−*y*| = |5 − (−3)| = |8| = 8 and |*x*| − |*y*| = |5| − |−3| = 5 − 3 = 2,

so |*x*−*y*| > |*x*| − |*y*|.

## Work Time

# Prepare a Presentation

Explain what you learned from today’s problems about adding and subtracting positive and negative numbers. Support your thinking with examples.

# Challenge Problem

Find values of *x* and* y* that make each inequality true. If there are no such values, explain why.

*x*+*y*> x −*y**x*+*y*<*x*−*y*- |
*x*−*y*| > |*x*| − |*y*|

# Make Connections

# Lesson Guide

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

Make sure students explain their strategies and methods for adding and subtracting rational numbers in their presentations. You might want to have students write down the addition and subtraction rules in their own words. Have students ask questions and make observations as they view others' work.

ELL: Provide ELLs and other students with a sample or model for the concepts, strategies, and applications that will be addressed in the assessment, and the format you want them to follow. Be prepared to address and explicitly re-teach or review vocabulary, concepts, strategies, and applications.

## Performance Task

# Ways of Thinking: Make Connections

Take notes about your classmates' strategies for adding and subtracting rational numbers.

## Hint:

As your classmates present, ask questions such as:

- How do you know that it is impossible to write that equation?
- What strategy did you use to write the subtraction equations?
- Can you show me that the values you chose for
*x*and*y*make the inequality true?

# 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 students have learned about adding and subtracting 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.

**The most important thing I learned about adding and subtracting positive and negative numbers is …**