Add Integers

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 2−1. 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 .

Doing the Manipulative Mathematics activity "Addition of signed Numbers" will help you develop a better understanding of adding integers.We will model four addition facts using the numbers 5,−5and3,−3. 5+3−5+(−3)−5+35+(−3) Model: 5+3. Solution Interpret the expression. 5+3 means the sum of 5 and 3. Model the first number. Start with 5 positives.

Model the second number. Add 3 positives.

Count the total number of counters.

The sum of 5 and 3 is 8. 5+3=8

Model the expression.2+4

6 Model the expression.2+5

7 Model: −5+(−3). Solution Interpret the expression. −5+(−3) means the sum of −5 and −3. Model the first number. Start with 5 negatives.

Model the second number. Add 3 negatives.

Count the total number of counters.

The sum of −5 and −3 is −8. −5+−3=−8

Model the expression.−2+(−4)

−6 Model the expression.−2+(−5)

−7 and 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—either8 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. Model: −5+3. Solution Interpret the expression. −5+3 means the sum of −5 and 3. Model the first number. Start with 5 negatives.

Model the second number. Add 3 positives.

Remove any neutral pairs.

Count the result.

The sum of −5 and 3 is −2. −5+3=−2

Notice that there were more negatives than positives, so the result is negative. Model the expression, and then simplify: 2+(−4)

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

−3 Model: 5+(−3). Solution Interpret the expression. 5+(−3) means the sum of 5 and −3. Model the first number. Start with 5 positives.

Model the second number. Add 3 negatives.

Remove any neutral pairs.

Count the result.

The sum of 5 and −3 is 2. 5+(−3)=2

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

2 Model the expression: (−2)+5

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 53−37=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 - .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.

Simplify: ⓐ 19+(−47) ⓑ −32+40 Solution ⓐ 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 ⓑ The signs are different so we subtract 32 from 40. The answer will be positive because there are more positives than negatives−32+408 Simplify each expression: ⓐ 15+(−32) ⓑ −19+76 ⓐ −17 ⓑ 57 Simplify each expression: ⓐ −55+9 ⓑ 43+(−17) ⓐ −46 ⓑ 26 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 Simplify the expression: −31+(−19) −50 Simplify the expression: −42+(−28) −70 The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations. Simplify: −5+3(−2+7). Solution −5+3(−2+7) Simplify inside the parentheses. −5+3(5) Multiply. −5+15 Add left to right. 10

Simplify the expression: −2+5(−4+7) 13 Simplify the expression: −4+2(−3+5) 0

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. Evaluate x+7when ⓐ x=−2 ⓑ x=−11. Solution ⓐ Evaluate x+7 when x=−2

Simplify.

ⓑ Evaluate x+7 when x=−11

Simplify.

Evaluate each expression for the given values:x+5when ⓐ x=−3and ⓑ x=−17 ⓐ 2 ⓑ −12 Evaluate each expression for the given values: y+7 when ⓐ y=−5 ⓑ y=−8 ⓐ 2 ⓑ −1 When n=−5, evaluate ⓐ n+1 ⓑ −n+1. Solution ⓐ Evaluate n+1 when n=−5

Simplify.

ⓑ Evaluate −n+1 when n=−5

Simplify.

Add.

When n=−8, evaluate ⓐ n+2 ⓑ −n+2 ⓐ −6 ⓑ 10 Wheny=−9,evaluate ⓐ y+8 ⓑ −y+8. ⓐ −1 ⓑ 17 Next we'll evaluate an expression with two variables. Evaluate 3a+b when a=12 and b=−30. Solution

Multiply.

Add.

Evaluate the expression: a+2bwhena=−19andb=14. 9 Evaluate the expression: 5p+qwhenp=4andq=−7. 13 Evaluate (x+y)2 when x=−18 and y=24. Solution This expression has two variables. Substitute −18 for x and 24 for y. (x+y)2

(−18+24)2 Add inside the parentheses. (6)2 Simplify 36

Evaluate: (x+y)2 when x=−15 and y=29. 196 Evaluate: (x+y)3 when x=−8 and y=10. 8

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. Translate and simplify: the sum of −9 and 5. Solution The sum of −9 and 5 indicates addition. the sum of −9 and 5 Translate. −9+5 Simplify. −4

Translate and simplify the expression: the sum of −7 and 4 −7 + 4 = −3 Translate and simplify the expression: the sum of −8 and −6 −8 + (−6) = −14 Translate and simplify: the sum of 8 and −12, increased by 3. Solution The phrase increased by indicates addition. The sum of 8 and −12, increased by 3 Translate. [8+(−12)]+3 Simplify. −4+3 Add. −1

Translate and simplify: the sum of 9 and −16, increased by 4. [9 + (−16)] + 4 = −3 Translate and simplify: the sum of −8 and −12, increased by 7. [−8 + (−12)] + 7 = −13

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 $5 could be represented as −$5. 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. The temperature in Buffalo, NY, one morning started at 7degrees below zero Fahrenheit. By noon, it had warmed up 12degrees. What was the temperature at noon? Solution We are asked to find the temperature at noon. Write a phrase for the temperature. The temperature warmed up 12 degrees from 7 degrees below zero. Translate to math notation. −7 + 12 Simplify. 5 Write a sentence to answer the question. The temperature at noon was 5 degrees Fahrenheit.

The temperature in Chicago at 5 A.M. was 10degrees below zero Celsius. Six hours later, it had warmed up 14 degrees Celsius. What is the temperature at 11 A.M.? 4 degrees Celsius A scuba diver was swimming 16 feet below the surface and then dove down another 17 feet. What is her new depth? −33 feet A football team took possession of the football on their 42-yard line. In the next three plays, they lost 6 yards, gained 4 yards, and then lost 8 yards. On what yard line was the ball at the end of those three plays? Solution We are asked to find the yard line the ball was on at the end of three plays. Write a word phrase for the position of the ball. Start at 42, then lose 6, gain 4, lose 8. Translate to math notation. 42 − 6 + 4 − 8 Simplify. 32 Write a sentence to answer the question. At the end of the three plays, the ball is on the 32-yard line.

The Bears took possession of the football on their 20-yard line. In the next three plays, they lost 9 yards, gained 7 yards, then lost 4 yards. On what yard line was the ball at the end of those three plays? 14-yard line The Chargers began with the football on their 25-yard line. They gained 5 yards, lost 8 yards and then gained 15 yards on the next three plays. Where was the ball at the end of these plays? 37-yard line ACCESS ADDITIONAL ONLINE RESOURCES Adding Integers with Same Sign Using Color Counters Adding Integers with Different Signs Using Counters Ex1: Adding Integers Ex2: Adding Integers

Practice Makes PerfectModel Addition of Integers In the following exercises, model the expression to simplify. 7+4

11 8+5 −6+(−3)

−9 −5+(−5) −7+5

−2 −9+6 8+(−7)

1 9+(−4) Simplify Expressions with Integers In the following exercises, simplify each expression. −21+(−59) −80 −35+(−47) 48+(−16) 32 34+(−19) −200+65 −135 −150+45 2+(−8)+6 0 4+(−9)+7 −14+(−12)+4 −22 −17+(−18)+6 135+(−110)+83 108 140+(−75)+67 −32+24+(−6)+10 −4 −38+27+(−8)+12 19+2(−3+8) 29 24+3(−5+9) Evaluate Variable Expressions with Integers In the following exercises, evaluate each expression. x+8 when ⓐ x=−26 ⓑ x=−95 ⓐ −18 ⓑ −87 y+9 when ⓐ y=−29 ⓑ y=−84 y+(−14) when ⓐ y=−33 ⓑ y=30 ⓐ −47 ⓑ 16 x+(−21) when ⓐ x=−27 ⓑ x=44 When a=−7, evaluate: ⓐ a+3 ⓑ −a+3 ⓐ −4 ⓑ 10 When b=−11, evaluate: ⓐ b+6 ⓑ −b+6 When c=−9, evaluate: ⓐ c+(−4) ⓑ −c+(−4) ⓐ −13 ⓑ 5 When d=−8, evaluate: ⓐ d+(−9) ⓑ −d+(−9) m+n when, m=−15, n=7 −8 p+q when, p=−9, q=17 r−3s when, r=16, s=2 10 2t+u when, t=−6, u=−5 (a+b)2 when, a=−7, b=15 64 (c+d)2 when, c=−5, d=14 (x+y)2 when, x=−3, y=14 121 (y+z)2 when, y=−3, z=15 Translate Word Phrases to Algebraic Expressions In the following exercises, translate each phrase into an algebraic expression and then simplify. The sum of −14 and 5 −14 + 5 = −9 The sum of −22 and 9 8 more than −2 −2 + 8 = 6 5 more than −1 −10 added to −15 −15 + (−10) = −25 −6 added to −20 6 more than the sum of −1 and −12 [−1 + (−12)] + 6 = −7 3 more than the sum of −2 and −8 the sum of 10 and −19, increased by 4 [10 + (−19)] + 4 = −5 the sum of 12 and −15, increased by 1 Add Integers in Applications In the following exercises, solve. Temperature The temperature in St. Paul, Minnesota was −19°F at sunrise. By noon the temperature had risen 26°F. What was the temperature at noon? 7°F Temperature The temperature in Chicago was −15°F at 6 am. By afternoon the temperature had risen 28°F. What was the afternoon temperature? Credit Cards Lupe owes $73 on her credit card. Then she charges $45 more. What is the new balance? −$118 Credit Cards Frank owes $212 on his credit card. Then he charges $105 more. What is the new balance? Football A team lost 3 yards the first play. Then they lost 2 yards, gained 1 yard, and then lost 4 yards. What was the change in overall yardage over the four plays? −8 yards Card Games April lost 5 cards the first turn. Over the next three turns, she lost 3 cards, gained 2 cards, and then lost 1 card. What was the change in cards over the four turns? Football The Rams took possession of the football on their own 35-yard line. In the next three plays, they lost 12 yards, gained 8 yards, then lost 6 yards. On what yard line was the ball at the end of those three plays? 25-yard line Football The Cowboys began with the ball on their own 20-yard line. They gained 15 yards, lost 3 yards and then gained 6 yards on the next three plays. Where was the ball at the end of these plays? Scuba Diving A scuba diver swimming 8 feet below the surface dove 17 feet deeper; the pressure got to them and they rose five feet. What is their new depth? 20 feet Gas Consumption: Ozzie rode their motorcycle for 30 minutes, using 168 fluid ounces of gas. Then they stopped and got 140-fluid ounces of gas. Represent the change in gas amount 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: Monday−504Tuesday+142Wednesday−449Thursday+410Friday+369 What was the overall change for the week? −32 Stock Market During the week of June 22, 2009, the change in the Dow Jones Industrial Average each day was: Monday−201Tuesday−16Wednesday−23Thursday+172Friday−34 What was the overall change for the week?

Writing Exercises Explain why the sum of −8 and 2 is negative, but the sum of 8 and −2 and is positive. Sample answer: In the first case, there are more negatives so the sum is negative. In the second case, there are more positives so the sum is positive. 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?