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
Before you get started, take this readiness quiz.
- Evaluate when
If you missed this problem, review Example 2.13. - Simplify:
If you missed this problem, review Example 2.8. - Translate the sum of and negative into an algebraic expression.
If you missed this problem, review Table 2.7
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 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.
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 3.17.
Manipulative Mathematics
We will model four addition facts using the numbers
Example 3.14
Model:
Model the expression.
Model the expression.
Example 3.15
Model:
Model the expression.
Model the expression.
Example 3.14 and Example 3.15 are very similar. The first example adds positives and positives—both positives. The second example adds negatives and negatives—both negatives. In each case, we got a result of positives or 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 3.16
Model:
Model the expression, and then simplify:
Model the expression, and then simplify:
Example 3.17
Model:
Model the expression, and then simplify:
Model the expression:
Example 3.18
Modeling Addition of Positive and Negative Integers
Model each addition.
- ⓐ 4 + 2
- ⓑ −3 + 6
- ⓒ 4 + (−5)
- ⓓ -2 + (−3)
Model each addition.
- ⓐ 3 + 4
- ⓑ −1 + 4
- ⓒ 4 + (−6)
- ⓓ −2 + (−2)
- ⓐ 5 + 1
- ⓑ −3 + 7
- ⓒ 2 + (−8)
- ⓓ −3 + (−4)
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 you don’t have to count out blue counters and red counters.
Picture blue counters with red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because there are more negative counters.
Let’s try another one. We’ll add Imagine red counters and more red counters, so we have red counters all together. This means the sum is
Look again at the results of Example 3.14 - Example 3.17.
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. | |
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 3.19
Simplify:
- ⓐ
- ⓑ
Simplify each expression:
- ⓐ
- ⓑ
Simplify each expression:
- ⓐ
- ⓑ
Example 3.20
Simplify:
Simplify the expression:
Simplify the expression:
The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.
Example 3.21
Simplify:
Simplify the expression:
Simplify the expression:
Evaluate Variable Expressions with Integers
Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers when evaluating expressions.
Example 3.22
Evaluate
- ⓐ
- ⓑ
Evaluate each expression for the given values:
- ⓐ
- ⓑ
Evaluate each expression for the given values: when
- ⓐ
- ⓑ
Example 3.23
When evaluate
- ⓐ
- ⓑ
When evaluate
- ⓐ
- ⓑ
- ⓐ
- ⓑ
Next we'll evaluate an expression with two variables.
Example 3.24
Evaluate when and
Evaluate the expression:
Evaluate the expression:
Example 3.25
Evaluate when and
Evaluate:
when and
Evaluate:
when and
Translate Word Phrases to Algebraic Expressions
All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase the sum indicates addition.
Example 3.26
Translate and simplify: the sum of and
Translate and simplify the expression:
the sum of and
Translate and simplify the expression:
the sum of and
Example 3.27
Translate and simplify: the sum of and increased by
Translate and simplify:
the sum of and increased by
Translate and simplify:
the sum of and increased by
Add Integers in Applications
Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of could be represented as Let’s practice translating and solving a few applications.
Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.
Example 3.28
The temperature in Buffalo, NY, one morning started at below zero Fahrenheit. By noon, it had warmed up What was the temperature at noon?
The temperature in Chicago at 5 A.M. was below zero Celsius. Six hours later, it had warmed up What is the temperature at 11 A.M.?
A scuba diver was swimming below the surface and then dove down another What is her new depth?
Example 3.29
A football team took possession of the football on their In the next three plays, they lost gained and then lost On what yard line was the ball at the end of those three plays?
The Bears took possession of the football on their In the next three plays, they lost gained then lost On what yard line was the ball at the end of those three plays?
The Chargers began with the football on their They gained lost and then gained on the next three plays. Where was the ball at the end of these plays?
Media Access Additional Online Resources
Section 3.2 Exercises
Practice Makes Perfect
Model Addition of Integers
In the following exercises, model the expression to simplify.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
when
- ⓐ
- ⓑ
when
- ⓐ
- ⓑ
When evaluate:
- ⓐ
- ⓑ
When evaluate:
- ⓐ
- ⓑ
when, ,
when, ,
when, ,
when, ,
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate each phrase into an algebraic expression and then simplify.
The sum of and
more than
added to
more than the sum of and
the sum of and increased by
Add Integers in Applications
In the following exercises, solve.
Temperature The temperature in St. Paul, Minnesota was at sunrise. By noon the temperature had risen What was the temperature at noon?
Temperature The temperature in Chicago was at 6 am. By afternoon the temperature had risen What was the afternoon temperature?
Credit Cards Frank owes on his credit card. Then he charges more. What is the new balance?
Weight Loss Angie lost the first week of her diet. Over the next three weeks, she lost gained and then lost What was the change in her weight over the four weeks?
Weight Loss April lost the first week of her diet. Over the next three weeks, she lost gained and then lost What was the change in her weight over the four weeks?
Football The Rams took possession of the football on their own In the next three plays, they lost gained then lost On what yard line was the ball at the end of those three plays?
Football The Cowboys began with the ball on their own They gained lost and then gained on the next three plays. Where was the ball at the end of these plays?
Calories Lisbeth walked from her house to get a frozen yogurt, and then she walked home. By walking for a total of she burned The frozen yogurt she ate was What was her total calorie gain or loss?
Calories Ozzie rode his bike for burning Then he had a iced blended mocha. Represent the change in calories as an integer.
Everyday Math
Stock Market The week of September 15, 2008, was one of the most volatile weeks ever for the U.S. stock market. The change in the Dow Jones Industrial Average each day was:
What was the overall change for the week?
Stock Market During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was:
What was the overall change for the week?
Writing Exercises
Give an example from your life experience of adding two negative numbers.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?