Additional Whole Number Operations Math 081 Additional Whole Number Operations © 2010 Pearson Education, Inc. All rights reserved.

Multiplying Integers We are learning to…multiply and divide integers by looking for patterns.

Multiplication of Integers -8 Evaluate: -2 + -2 + -2 + -2 = ________ Repeated addition is the same thing as:__________________ ...so -2 + -2 + -2 + -2 can be written:________ Evaluate: -4 + -4 + -4 = ______________ . ..so -4 + -4 + -4 can be written:________ Multiplication 4(-2) -12 Multiplication 3(-4)

Multiplication of Integers -35 Evaluate: -7 + -7 + -7 + -7 + -7 = ______________ ...Rewrite as a multiplication problem:_________ Evaluate: -12 + -12 = ______________ Evaluate: -1 + -1 + -1 + -1 + -1 = ______________ 5(-7) -24 2(-12) -5 5(-1)

Multiplication of Integers I know that: (positive number) × (negative number) = _______________________________________. The ______________________________ property of multiplication allows me to multiply in any order. So I also know that (negative number) × (positive number) = Negative Number Commutative Negative Number Evaluate the expressions below.

Multiplication of Integers Negative Negative Negative × Positive = _______________ and Positive × Negative = _______________ If the signs in a multiplication problem are different the solution is:_______________ I know that a: positive × positive = ______________ Now follow the pattern: (+)(–) = ______________ (–)(+) = ______________ (+)(+) = ______________ So… (–)(–) = ______________ I know that a: negative × negative = _________________________ If the signs in a multiplication problem are the same the solution is:_______________ Negative Positive Negative Negative Positive Positive Positive Positive

Which of the following statements is NOT true? A negative multiplied by a negative is a positive product. A positive multiplied by a negative is a negative product. A negative multiplied by a negative is a negative product. A negative multiplied by a positive is a negative product.

2.1 and 2.2 Dividing Whole Numbers Objectives 1. Write division problems in three ways. 2. Identify the parts of a division problem. 3. Divide 0 by a number. 4. Recognize that a number cannot be divided by 0. 5. Divide a number by itself. 6. Divide a number by 1. 7. Use short division. 8. Use multiplication to check the answer to a division problem. 9. Use tests for divisibility. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 8

Copyright © 2010 Pearson Education, Inc. All rights reserved. Just as 4  5, 4 × 5, and (4)(5) are different ways of indicating multiplication, there are several ways to write 20 divided by 4. 20 ÷ 4 = 5 Being divided Being divided Divided by Being divided Divided by Divided by Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 9

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Using Division Symbols Write the division problem 21 ÷ 7 = 3 using two other symbols. This division can also be written as shown below. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 10

In division, the number being divided is the dividend, the number divided by is the divisor, and the answer is the quotient. dividend ÷ divisor = quotient Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 11

Identifying the Parts of a Division Problem Parallel Example 2 Identifying the Parts of a Division Problem Identify the dividend, divisor, and quotient. 36 ÷ 9 = 4 b. quotient dividend divisor dividend quotient divisor Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1.5- 12

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 13

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Dividing 0 by a Number Divide. 0 ÷ 21 = b. 0 ÷ 1290 = c. d. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 14

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Changing Division Problems to Multiplication Change each division problem to a multiplication problem. a. b. c. 90 ÷ 9 = 10 becomes 3  9 = 27 or 9  3 = 27 becomes 8  7 = 56 or 7  8 = 56 becomes 9  10 = 90 or 10  9 = 90 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 15

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 16

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 5 Dividing 0 by a Number All the following are undefined. a. b. c. 25 ÷ 0 is undefined is undefined is undefined Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 17

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 18

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 6 Dividing a Nonzero Number by Itself Divide. a. 25 ÷ 25 = b. c. 1 1 1 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 19

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 20

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 7 Dividing Numbers by 1 Divide. a. 15 ÷ 1 = b. c. 15 72 34 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 21

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 22

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 11 Checking Division by Using Multiplication Check each answer. a. (divisor  quotient) + remainder = dividend (4  135) + 2 540 + 2 = 542 R2 Matches Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 23

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 11 Checking Division by Using Multiplication Check each answer. b. (divisor  quotient) + remainder = dividend (8  154) + 3 1232 + 3 = 1235 R3 Does not match original dividend. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 24

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 25

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 26

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 12 Testing for Divisibility by 2 Are the following numbers divisible by 2? 128 4329 Because the number ends in 8, which is an even number, the number is divisible by 2. Ends in 8 The number is NOT divisible by 2. Ends in 9 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 27

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 28

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 13 Testing for Divisibility by 3 Are the following numbers divisible by 3? 2128 b. 27,306 Add the digits. 2 + 1 + 2 + 8 = 13 Because 13 is not divisible by 3, the number is not divisible by 3. Add the digits. 2 + 7 + 3 + 0 + 6 = 18 Because 18 is divisible by 3, the number is divisible by 3. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 29

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 30

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 14 Testing for Divisibility by 5 Are the following numbers divisible by 5? 17,900 b. 5525 c. 657 The number ends in 0 and is divisible by 5. The number ends in 5 and is divisible by 5. The number ends in 7 and is not divisible by 5. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 31

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 15 Testing for Divisibility by 10 Are the following numbers divisible by 10? 18,240 b. 3225 c. 248 The number ends in 0 and is divisible by 10. The number ends in 5 and is not divisible by 10. The number ends in 8 and is not divisible by 10. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.5- 32

Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.4 Order of Operations Objectives 1. Use the order of operations. 2. Use the order of operations with exponents. 3. Use the order of operations with fractions bars. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 33

Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 34

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 1 Using the Order of Operations a. 18 + 20 ÷ 4 b. 3 − 24 ÷ 4 + 9 18 + 5 3 − 6 + 9 23 3 + (−6) + 9 −3 + 9 6 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 35

Evaluate: -14 -30 20 -8

Evaluate: 315 -105 3 -76

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 2 Parentheses and the Order of Operations a. −5(8 – 4) – 3 b. 3 + 4(4 – 9)(20 ÷ 4) −5(4) – 3 3 + 4(–5)(20 ÷ 4) −20 – 3 3 + 4(–5) (5) 3 + (–20) (5) −20 + (– 3) 3 + (–100) −23 –97 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 38

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 3 Exponents and the Order of Operations a. 52 − (−2)2 b. (−7)2 − (5 − 8)2 (−4) (−7)2 − (− 3)2 (−4) 25 − 4 49 − 9(−4) 21 49 − (−36) 49 + (+36) 85 Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 39

Copyright © 2010 Pearson Education, Inc. All rights reserved. Parallel Example 4 Fraction Bars and the Order of Operations Simplify. First do the work in the numerator. Then do the work in the denominator. −14 + 3(5 – 7) 6 – 42 ÷ 8 −14 + 3(– 2) 6 – 16 ÷ 8 −14 + (– 6) 6 – 2 4 −20 Denominator Numerator Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.4- 40

2.5 Using Equations to Solve Application Problems Objectives 1. Translate word phrases into expressions with variables. 2. Translate sentences into equations. 3. Solve application problems. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.8- 41

Copyright © 2010 Pearson Education, Inc. All rights reserved. As you read an application problem, look for indicator words that help you. Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 1.10- 42

Copyright © 2010 Pearson Education, Inc. All rights reserved. Translating Word Phrases into Expressions Write each word phrase as an expression Words Expression A 4 plus nine 7 more than 3 −12 added to 4 3 less than 8 7 decreased by 1 14 minus -8 4 + 9 or 9 + 4 3 + 7 or 7 + 3 −12 + 4 or 4 + (−12) 8 – 3 7 – 1 14 – (-8) Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.8- 43

Copyright © 2010 Pearson Education, Inc. All rights reserved. Translating Word Phrases into Expressions Write each word phrase as an expression. Words Algebraic Expression 3 times 4 Twice the number 5 The quotient of 8 and -2 30 divided by 15 The result is 3(4) 2(5) = Copyright © 2010 Pearson Education, Inc. All rights reserved. Slide 9.8- 44