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2.3: Add Integers (Part 1)

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Learning Objectives
  • Model addition of integers
  • Simplify expressions with integers
  • Evaluate variable expressions with integers
  • Translate word phrases to algebraic expressions
  • Add integers in applications
be prepared!

Before you get started, take this readiness quiz.

  1. Evaluate x+8 when x=6. If you missed this problem, review Example 2.2.1.
  2. Simplify: 8+2(5+1). If you missed this problem, review Example 2.1.8.
  3. Translate the sum of 3 and negative 7 into an algebraic expression. If you missed this problem, review Table 2.2.3.

Model Addition of Integers

Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with integers.

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more difficult. This difficulty relates to the way the brain learns.

The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of 21. Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.

Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.

We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.

This figure has a blue circle labeled positive and a red circle labeled negative.

If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in Figure 2.3.1.

This figure has a blue circle over a red circle. Beside them is the statement 1 plus negative 1 equals 0.

Figure 2.3.1: A blue counter represents +1. A red counter represents −1. Together they add to zero.

We will model four addition facts using the numbers 5, 5 and 3, 3.

5+35+(3)5+35+(3)

Example 2.3.1: model the expression

Model: 5+3.

Solution

Interpret the expression. 5 + 3 means the sum of 5 and 3.
Model the first number. Start with 5 positives. CNX_BMath_Figure_03_02_025_img-01.png
Model the second number. Add 3 positives. CNX_BMath_Figure_03_02_025_img-02.png
Count the total number of counters. CNX_BMath_Figure_03_02_025_img-03.png
The sum of 5 and 3 is 8. 5 + 3 = 8
Exercise 2.3.1

Model the expression. 2+4

Answer

Ex 3.2.1.png

6

Exercise 2.3.2

Model the expression. 2+5

Answer

Ex 3.2.2.png

7

Example 2.3.2: model the expression

Model: 5+(3).

Solution

Interpret the expression. −5 + (−3) means the sum of −5 and −3.
Model the first number. Start with 5 negatives. CNX_BMath_Figure_03_02_026_img-01.png
Model the second number. Add 3 negatives. CNX_BMath_Figure_03_02_026_img-02.png
Count the total number of counters. CNX_BMath_Figure_03_02_026_img-03.png
The sum of −5 and −3 is −8. −5 + −3 = −8
Exercise 2.3.3

Model the expression. 2+(4)

Answer

Ex 3.2.3.png

6

Exercise 2.3.4

Model the expression. 2+(5)

Answer

Ex 3.2.4.png

7

Example 2.3.1 and Example 2.3.2 are very similar. The first example adds 5 positives and 3 positives—both positives. The second example adds 5 negatives and 3 negatives—both negatives. In each case, we got a result of 8 -- either 8 positives or 8 negatives. When the signs are the same, the counters are all the same color. Now let’s see what happens when the signs are different.

Example 2.3.3: model the expression

Model: 5+3.

Solution

Interpret the expression. −5 + 3 means the sum of −5 and 3.
Model the first number. Start with 5 negatives. CNX_BMath_Figure_03_02_027_img-01.png
Model the second number. Add 3 positives. CNX_BMath_Figure_03_02_027_img-02.png
Remove any neutral pairs. CNX_BMath_Figure_03_02_027_img-03.png
Count the result. CNX_BMath_Figure_03_02_027_img-04.png
The sum of −5 and 3 is −2. −5 + 3 = −2

Notice that there were more negatives than positives, so the result is negative.

Exercise 2.3.5

Model the expression, and then simplify: 2+(4)

Answer

Ex 3.2.7.png

2

Exercise 2.3.6

Model the expression, and then simplify: 2+(5)

Answer

Ex 3.2.6.png

3

Example 2.3.4: model the expression

Model: 5+(3).

Solution

Interpret the expression. 5 + (−3) means the sum of 5 and −3.
Model the first number. Start with 5 positives. CNX_BMath_Figure_03_02_028_img-01.png
Model the second number. Add 3 negatives. CNX_BMath_Figure_03_02_028_img-02.png
Remove any neutral pairs. CNX_BMath_Figure_03_02_028_img-03.png
Count the result. CNX_BMath_Figure_03_02_028_img-04.png
The sum of 5 and −3 is 2. 5 + (−3) = 2
Exercise 2.3.7

Model the expression, and then simplify: (2)+4

Answer

Ex 3.2.5.png

2

Exercise 2.3.8

Model the expression: (2)+5

Answer

Ex 3.2.8.png

3

Example 2.3.5: model the addition

Model each addition.

  1. 4+2
  2. 3+6
  3. 4+(5)
  4. 2+(3)

Solution

  1. 4+2
Start with 4 positives. CNX_BMath_Figure_03_02_035_img-01.png
Add two positives. CNX_BMath_Figure_03_02_035_img-02.png
How many do you have? 6. 4 + 2 = 6
  1. 3+6
Start with 3 negatives. CNX_BMath_Figure_03_02_036_img-01.png
Add 6 positives. CNX_BMath_Figure_03_02_036_img-02.png
Remove neutral pairs. CNX_BMath_Figure_03_02_036_img-03.png
How many are left? CNX_BMath_Figure_03_02_036_img-04.png
3. −3 + 6 = 3
  1. 4+(5)
Start with 4 positives. CNX_BMath_Figure_03_02_037_img-01.png
Add 5 negatives. CNX_BMath_Figure_03_02_037_img-02.png
Remove neutral pairs. CNX_BMath_Figure_03_02_037_img-03.png
How many are left? CNX_BMath_Figure_03_02_037_img-04.png
−1. 4 + (−5) = −1
  1. 2+(3)
Start with 2 negatives. CNX_BMath_Figure_03_02_038_img-01.png
Add 3 negatives. CNX_BMath_Figure_03_02_038_img-02.png
How many do you have? −5. −2 + (−3) = −5
Exercise 2.3.9

Model each addition.

  1. 3+4
  2. 1+4
  3. 4+(6)
  4. 2+(2)
Answer a

Ex 3.2.9a.png

Answer b

Ex 3.2.9b.png

Answer c

Ex 3.2.9c.png

Answer d

Ex 3.2.9d.png

Exercise2.3.10

Model each addition.

  1. 5+1
  2. 3+7
  3. 2+(8)
  4. 3+(4)
Answer a

Ex 3.2.10a.png

Answer b

Ex 3.2.10b.png

Answer c

Ex 3.2.10c.png

Answer d

Ex 3.2.10d.png

Simplify Expressions with Integers

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.

For example, if you want to add 37+(53), you don’t have to count out 37 blue counters and 53 red counters.

Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because 5337=16, there are 16 more negative counters.

37+(53)=16

Let’s try another one. We’ll add 74+(27). Imagine 74 red counters and 27 more red counters, so we have 101 red counters all together. This means the sum is 101.

74+(27)=101

Look again at the results of Example 2.3.1 - Example 2.3.2.

Table 2.3.1: Addition of Positive and Negative Integers
5 + 3 −5 + (−3)
both positive, sum positive both negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
−5 + 3 5 + (−3)
different signs, more negatives different signs, more positives
sum negative sum positive
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.
Example 2.3.6: simplify

Simplify:

  1. 19+(47)
  2. 32+40

Solution

  1. Since the signs are different, we subtract 19 from 47. The answer will be negative because there are more negatives than positives. 19+(47)=28
  2. The signs are different so we subtract 32 from 40. The answer will be positive because there are more positives than negatives 32+40=8
Exercise 2.3.11

Simplify each expression:

  1. 15+(32)
  2. 19+76
Answer a

17

Answer b

57

Exercise 2.3.12

Simplify each expression:

  1. 55+9
  2. 43+(17)
Answer a

46

Answer b

26

Example 2.3.7: simplify

Simplify: 14+(36).

Solution

Since the signs are the same, we add. The answer will be negative because there are only negatives.

14+(36)=50

Exercise 2.3.13

Simplify the expression: 31+(19)

Answer

50

Exercise 2.3.14

Simplify the expression: 42+(28)

Answer

70

The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.

Example 2.3.8: simplify

Simplify: 5+3(2+7).

Solution

Simplify inside the parentheses. −5 + 3(5)
Multiply. −5 + 15
Add left to right. 10
Exercise 2.3.15

Simplify the expression: 2+5(4+7)

Answer

13

Exercise 2.3.16

Simplify the expression: 4+2(3+5)

Answer

0

Contributors and Attributions

  • Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.

This page titled 2.3: Add Integers (Part 1) is shared under a not declared license and was authored, remixed, and/or curated by OpenStax.

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