Model Integer Subtraction

Model Integer Subtraction

Subtracting a Negative

Opening

Subtracting a Negative

Discuss the following with your classmates.

In this lesson, you will model subtraction of integers as “taking away.” In Lesson 4, you will look at subtraction in a different way—as the distance between two points.

  • Use the Hot Air Balloon interactive to model subtraction as taking away:
    • Start at 0 and add heat to air (for a positive number) or weight (for a negative number) to move the balloon to the first number (the minuend).
    • To subtract the second number (the subtrahend) from the first number, remove or take away units of heat or weights.
  • For example, to model −3 − (−7):
    • Start at 0 and add 3 units of weight to move the balloon to –3. To subtract –7, remove or take away 7 units of weight.

INTERACTIVE: Hot Air Balloon

Model Subtraction

Opening

Model Subtraction

Use the Hot Air Balloon interaction to model each problem and find the difference. Write your answer as an equation.

  • Subtract a negative from a negative.
    For example: −3 − (−7)
  • Subtract a negative from a positive.
    For example: 9 − (−9)
  • Subtract a positive from a negative.
    For example: −2 − 5
  • Subtract a positive from a positive.
    For example: 2 − 8

  • When you remove heat from air (subtract a positive number), what happens to the balloon?

  • When you remove weight (subtract a negative number), what happens to the balloon?

INTERACTIVE: Hot Air Balloon

On a Number Line

Opening

On a Number Line

Discuss the following with your classmates.

You can also use a horizontal or vertical number line to model subtraction of integers as taking away.

Just as with the balloon simulation:

  • Subtracting a positive number (heat) means moving in a negative direction (down or to the left).
  • Subtracting a negative number (weight) means moving in a positive direction (up or to the right).

Math Mission

Opening

Model integer subtraction on a number line and analyze guidelines for subtracting integers.

Subtract Integers on a Number Line

Work Time

Subtract Integers on a Number Line

Model each subtraction problem on the number line. Write your answer as an equation.

  • 7 − 4
  • 3 − 10
  • −5 − (−9)
  • 1 − (−5)
  • −2 − 3
  • −4 − (−4)
  • −7 − (−6)
  • Create your own equation by subtracting any two integers.

INTERACTIVE: Number Line Tool

Hint:

To figure out which direction to go in order to subtract the second number, think back to the balloon model. Think of subtracting a positive number as removing heat from air and subtracting a negative number as removing weight.

Subtraction as Addition

Work Time

Subtraction as Addition

Subtraction is the inverse of addition. So, subtraction problems can be written adding the opposite of the number. Subtracting a positive 2 is the same as adding a negative 2.

For example: 8 − (−2) = 8 + 2 = 10

Write these subtraction problems as addition problems:

  • 7 − 4 = 7 + ____
  • 3 − 10 = 3 + ____
  • −5 − (−9) = −5 + ____
  • 1 − (−5) = 1 + ____

Hint:

In the first problem, when you subtract 4 on the number line, you go 4 units to the left. What addition does this represent?

Use Reasoning

Work Time

Use Reasoning

Watch the video that shows Karen and Maya use reasoning as they discuss how to subtract positive and negative numbers.

  • What reasoning did Karen use to convince Maya that 6 − (−2) is not 4?
  • How did the number line help Karen explain her reasoning?
  • Can you make a generalization about subtracting integers?

VIDEO: Mathematical Practice 2

Hint:

Your generalization should involve changing a subtraction problem to an addition problem.

Prepare a Presentation

Work Time

Prepare a Presentation

Be prepared to share your generalization about subtracting integers. Support your generalization with examples.

Challenge Problem

Suppose n is an integer.

  • If 5 − n is negative, what do you know about the value of n?
  • If 5 − n is positive, what do you know about the value of n?
  • If −5 − n is negative, what do you know about the value of n?
  • If −5 − n is positive, what do you know about the value of n?

Make Connections

Performance Task

Ways of Thinking: Make Connections

Take notes about your classmates’ approaches to subtracting positive and negative numbers.

Hint:

As your classmates present, ask questions such as:

  • How did you know which direction to go for the second number?
  • Can you show me an example to help me understand your generalization?
  • Did you test your generalization to make sure it works for all the subtraction problems that you wrote as addition problems earlier in the lesson?
  • Why does subtracting a negative number always result in an answer that is greater than the first number?

Practice Subtracting Integers

Work Time

Apply the Learning: Practice Subtracting Integers

Solve the problems. Write each solution as an equation.

  • 123 − 4
  • 3 − 123
  • −5 − (−23)
  • 23 − (−15)

Subtract Integers

Formative Assessment

Summary of the Math: Subtract Integers

Read and Discuss

The value of ab is:

  • Positive if a > b (if a is to the right of b on the number line).

  • Negative if a < b (if a is to the left of b on the number line).

Any subtraction problem can be rewritten as an addition problem by adding the opposite: ab = a + (–b)

Hint:

Check your summary.

  • Explain what it means to subtract integers by taking away?
  • Describe how to subtract integers on a number line?
  • Explain how to subtract any two integers?
  • Explain the guideline for using addition to subtract any two integers?

Reflect On Your Work

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 place outside of the classroom where I have seen negative numbers is …