Download Multiply and Divide Integers with Order of 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 = ________
Multiplication
Repeated addition is the same thing as:__________________
4(-2)
...so -2 + -2 + -2 + -2 can be written:________
-12
Evaluate: -4 + -4 + -4 = ______________
Multiplication
Repeated addition is the same thing as:__________________
.
3(-4)
..so -4 + -4 + -4 can be written:________
Multiplication of Integers
-35
Evaluate: -7 + -7 + -7 + -7 + -7 = ______________
5(-7)
...Rewrite as a multiplication problem:_________
-24
Evaluate: -12 + -12 = ______________
2(-12)
...Rewrite as a multiplication problem:_________
-5
Evaluate: -1 + -1 + -1 + -1 + -1 = ______________
5(-1)
...Rewrite as a multiplication problem:_________
Multiplication of Integers
I know that:
(positive number) × (negative number) =
Negative Number
_______________________________________.
Commutative
The ______________________________
property of multiplication
allows me to multiply in any order.
So I also know that (negative number) × (positive number) =
Negative Number
_______________________________________.
Evaluate the expressions below.
Multiplication of Integers
Negative
Negative
Negative × Positive = _______________
and Positive × Negative = _______________
Negative
If the signs in a multiplication problem are different the solution is:_______________
I know that a:
Positive
positive × positive = ______________
Negative
(+)(–) = ______________
Now follow the pattern:
Negative
(–)(+) = ______________
Positive
(+)(+) = ______________
So…
I know that a:
Positive
(–)(–) = ______________
Positive
negative × negative = _________________________
Positive
If the signs in a multiplication problem are the same the solution is:_______________
Which of the following statements
is NOT true?
A.
A negative multiplied by a
negative is a positive product.
B.
A positive multiplied by a
negative is a negative product.
C.
A negative multiplied by a
negative is a negative product.
D.
A negative multiplied by a
positive is a negative product.
2.1 and 2.2 Dividing Whole Numbers
Objectives
1.
2.
3.
4.
5.
6.
7.
8.
9.
Write division problems in three ways.
Identify the parts of a division problem.
Divide 0 by a number.
Recognize that a number cannot be divided by 0.
Divide a number by itself.
Divide a number by 1.
Use short division.
Use multiplication to check the answer to a division
problem.
Use tests for divisibility.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 8
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.
Being divided
20 ÷ 4 = 5
Divided by
Divided by
5
4 20
Being divided
20
5
4
Being divided
Divided by
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 9
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.
3
7 21
21
3
7
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
quotient
divisor dividend
dividend
 quotient
divisor
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 11
Parallel
Example 2
Identifying the Parts of a Division
Problem
Identify the dividend, divisor, and quotient.
a. 36 ÷ 9 = 4
36 ÷ 9 = 4
dividend
quotient
divisor
b. 75  3
dividend
25
75
3
25
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
Parallel
Example 3
Dividing 0 by a Number
Divide.
a. 0 ÷ 21 = 0
b. 0 ÷ 1290 = 0
c.
0
0
275
d.
0
130 0
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 14
Parallel
Example 4
Changing Division Problems to
Multiplication
Change each division problem to a multiplication
problem.
27
a. 3  9
becomes 3  9 = 27 or 9  3 = 27
7
b. 8 56
becomes 8  7 = 56 or 7  8 = 56
c. 90 ÷ 9 = 10 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
Parallel
Example 5
Dividing 0 by a Number
All the following are undefined.
8
a.
0
is undefined
b. 0 12 is undefined
c. 25 ÷ 0 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
Parallel
Example 6
Dividing a Nonzero Number by
Itself
Divide.
a. 25 ÷ 25 = 1
1
b. 49 49
c.
82
 1
82
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 19
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 20
Parallel
Example 7
Dividing Numbers by 1
Divide.
a. 15 ÷ 1 = 15
72
b. 1 72
34
c. 1  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
Parallel
Example 11
Checking Division by Using
Multiplication
Check each answer.
135 R2
a. 4 542
(divisor  quotient) + remainder = dividend
(4

135)
540
+ 2
+ 2
= 542
Matches
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 23
Parallel
Example 11
Checking Division by Using
Multiplication
Check each answer.
154 R3
b. 8 1236
(divisor  quotient) + remainder = dividend
(8

154)
1232
+ 3
+ 3
= 1235
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
Parallel
Example 12
Testing for Divisibility by 2
Are the following numbers divisible by 2?
a.
128
Because the number ends in 8, which is an even
number, the number is divisible by 2.
Ends in 8
b.
4329
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
Parallel
Example 13
Testing for Divisibility by 3
Are the following numbers divisible by 3?
a.
2128
Add the digits.
2 + 1 + 2 + 8 = 13
Because 13 is not divisible by 3, the number
is not divisible by 3.
b. 27,306
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
Parallel
Example 14
Testing for Divisibility by 5
Are the following numbers divisible by 5?
a. 17,900
The number ends in 0 and is divisible by 5.
b. 5525
The number ends in 5 and is divisible by 5.
c. 657
The number ends in 7 and is not divisible by 5.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 31
Parallel
Example 15
Testing for Divisibility by 10
Are the following numbers divisible by 10?
a. 18,240
The number ends in 0 and is divisible by 10.
b. 3225
The number ends in 5 and is not divisible by 10.
c. 248
The number ends in 8 and is not divisible by 10.
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 1.5- 32
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
Parallel
Using the Order of Operations
Example 1
a.
18 + 20 ÷ 4
18 + 5
23
b.
3 − 24 ÷ 4 + 9
3−
6
+9
3 + (−6)
−3
+9
+9
6
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 9.4- 35
Evaluate:
2  4  6  1   3  5
A.
B.
C.
D.
-14
-30
20
-8
Evaluate:
2
7(3)  2  5  8 
A.
B.
C.
D.
315
-105
3
-76
Parallel
Parentheses and the Order of
Example 2
Operations
a.
−5(8 – 4) – 3
−5(4) – 3
−20 – 3
−20 + (– 3)
−23
b.
3 + 4(4 – 9)(20 ÷ 4)
3 + 4(–5)(20 ÷ 4)
3 + 4(–5) (5)
3 + (–20) (5)
3 + (–100)
–97
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 9.4- 38
Parallel
Exponents and the Order of
Example 3
Operations
a.
52 − (−2)2
b.
(−7)2 − (5 − 8)2 (−4)
(−7)2 − (− 3)2 (−4)
25 − 4
21
49 − 9(−4)
49 − (−36)
49 + (+36)
85
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 9.4- 39
Parallel
Fraction Bars and the Order of
Example 4
Operations
14  3  5  7 
Simplify.
6  42  8
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
Numerator
Denominator
20
 5
4
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
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
Translating Word Phrases into
Expressions
Write each word phrase as an expression
Words
A 4 plus nine
7 more than 3
−12 added to 4
Expression
4 + 9 or 9 + 4
3 + 7 or 7 + 3
−12 + 4 or 4 + (−12)
3 less than 8
8–3
7 decreased by 1
7–1
14 minus -8
14 – (-8)
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 9.8- 43
Translating Word Phrases into
Expressions
Write each word phrase as an expression.
Words
Algebraic Expression
3 times 4
3(4)
Twice the number 5
2(5)
The quotient of 8 and -2
30 divided by 15
The result is
8
2
30
15
=
Copyright © 2010 Pearson Education, Inc. All rights reserved.
Slide 9.8- 44