Download Multiplying and Dividing Integers

3 Multiplying and Dividing
Integers
Unit 1: Number System
Common Core Standards
 CCSS.Math.Content.7.NS.A.2 Apply and extend previous
understandings of multiplication and division and of fractions to
multiply and divide rational numbers.
 CCSS.Math.Content.7.NS.A.2a Understand that multiplication is
extended from fractions to rational numbers by requiring that
operations continue to satisfy the properties of operations,
particularly the distributive property, leading to products such as (–
1)(–1) = 1 and the rules for multiplying signed numbers. Interpret
products of rational numbers by describing real-world contexts.
 CCSS.Math.Content.7.NS.A.2b Understand that integers can be
divided, provided that the divisor is not zero, and every quotient of
integers (with non-zero divisor) is a rational number. If p and q are
integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational
numbers by describing real-world contexts.
 CCSS.Math.Content.7.NS.A.2c Apply properties of operations as
strategies to multiply and divide rational numbers.
Vocabulary/Properties
Multiplication Property of Zero
The product of a number and 0 is 0. (-4 x 0 = 0)
Identity Property of Multiplication
The product of a number and the multiplicative
identity, 1, is the number. (4 x 1 = 4)
For your notebook:
Why does a negative number times a
negative number equal a positive product?
Here’s one explanation: Follow the Pattern:
Notice as one factor is multiplied by a
decreasing factor, the product decreases:
3X3=9
3X2=6
3X1=3
3X0=0
3X-1=-3
3X-2=-6
3X-3=-9
So following the pattern so far it makes sense that
a positive times a negative results in a negative.
Now let’s decrease the first factor but leave
the second; notice how the product changes:
3X3=9
3X2=6
3X1=3
3X0=0
3X-1=-3
3X-2=-6
3X-3=-9
3X-3=-9
2X-3=-6
1X-3=-3
0X-3=0
-1X-3=3
-2X-3=6
-3X-3=9
So following the pattern it makes sense that a
negative times a negative results in a positive.
Example 1: Multiplying Integers
Multiply.
A. –6(4)
–24
B. –8(–5)(2)
–8(–5)(2)
40(2)
80
Signs are different.
Answer is negative.
Multiply two integers.
Signs are the same.
Answer is positive.
Check It Out! Example 1
Multiply.
A. 5(–2)
Signs are different.
Answer is negative.
–10
B. –3(–2)(4)
–3(–2)(4)
Signs are the same.
6(4)
Answer is positive.
6(4)
Signs are the same.
24
Answer is positive.
Example 2: Finding Powers of Integers
Simplify.
A. (–3)4
(-3) (-3) (-3) (-3)
(9)(9)
81
B. -34
Write as a repeated product.
Multiply in groups
Negative sign is not in parenthesis.
-(3)4 = -[ (3)(3)(3)(3)] Use order of operations
-81
Evaluate power. Multiply by -1
Example 3: Using Multiplication Properties
Find the product.
A. 6(1)
6
Identity Property of Multiplication
B. –15(0)
0
Multiplication Property of Zero
Evaluating an Expression Involving
Multiplication
MOVIES
A stuntman working on a movie set falls from a
building’s roof 90 feet above an air cushion. The
expression –16t 2 + 90 gives the stuntman’s height (in
feet) above the air cushion after t seconds. What is
the height of the stuntman after 2 seconds?
SOLUTION
Evaluate the expression for the height when t = 2.
–16t 2 + 90 = –16( 2 ) 2 + 90
= –16( 4 ) + 90
= - 64 + 90 = 26
Substitute 2 for t.
Evaluate the power.
26 feet
Dividing Integers
FOR YOUR NOTEBOOK:
Zero
The quotient of 0 and any nonzero integer is 0.
0 =0
12
0 =0
-12
But you can not divide by zero.
12 = undefined
0
Example 4: Dividing Integers
Divide.
C.
D.
–18
2
Signs are different.
–9
Answer is negative.
–25
–5
Signs are the same.
5
Answer is positive.
Check It Out! Example 4
Divide.
C.
D.
–24
3
Signs are different.
–8
Answer is negative.
–12
–2
Signs are the same.
6
Answer is positive.
Mean of a Data Set
FOR YOUR NOTEBOOK:
Mean: Find the sum of all values in a data
set and divide by the number of values.
Mean = Sum of values
# of values
Example 5: Finding Mean (Multiple Choice
Practice)
One of the coldest places on Earth is a Russian town
located near the Arctic Circle. To the nearest degree,
what is the mean of the average high temperatures
shown in the table for winter in the Russian town?
Winter Temperatures
Month
Dec
Jan
Feb
Average high – 41°F – 40°F – 48°F
– 147°F
– 48°F
– 43°F
Mar
– 18°F
– 37°F
SOLUTION
STEP 1 Find the sum of the temperatures.
– 41 + ( – 40) + ( – 48) + ( – 18) = – 147
STEP 2 Divide the sum by the number of temperatures.
– 147
4
= – 36.75
To the nearest degree, the mean of the temperatures
is – 37°F.
ANSWER
The correct answer is D.
Evaluating an Expression, Class Example
ab
Evaluate the expression
when a = – 24, b = 8 , and
c
c = – 4.
ab – 24 ( 8 )
=
c
–4
– 192
=
–4
= 48
Substitute values.
Multiply.
Divide. Same sign, so quotient
is positive.
Evaluating an Expression, Example 2
a2+b
Evaluate the expression
when a = 6, b = -6, and
c
c = -2 .
62+(-6)
a2+b
=
c
Substitute values.
-2
=
36+(-6)
–2
= 30 = - 15
-2
Evaluate exponent.
Divide. Different sign, so quotient
is negative.