Download Chapter 1 Section 8 - Lamar County School District

Chapter 1
Real Numbers and
Introduction to
Algebra
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Bellwork:
1.
7 subtracted from the
quotient of 0 and 5
4.
twice the sum of -3
and -4
2.
-1 added to the
product of -8 and -5
5.
the quotient of -9 and
-30
3.
the quotient of -8 and
-20
6.
the difference of -9
and the product of -4
and -6
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
2
Bellwork:
1.
7 subtracted from the
quotient of 0 and 5
4.
twice the sum of -3
and -4
0/5 – 7 = -7
2.
-1 added to the
product of -8 and -5
5.
the quotient of -9 and
-30
3.
the quotient of -8 and
-20
6.
the difference of -9
and the product of -4
and -6
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
3
Bellwork:
1.
7 subtracted from the
quotient of 0 and 5
4.
twice the sum of -3
and -4
5.
the quotient of -9 and
-30
6.
the difference of -9
and the product of -4
and -6
0/5 – 7 = -7
2.
-1 added to the
product of -8 and -5
(-8)(-5) + (-1) = 40 + (-1) = 39
3.
the quotient of -8 and
-20
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4
Bellwork:
1.
7 subtracted from the
quotient of 0 and 5
4.
twice the sum of -3
and -4
5.
the quotient of -9 and
-30
6.
the difference of -9
and the product of -4
and -6
0/5 – 7 = -7
2.
-1 added to the
product of -8 and -5
(-8)(-5) + (-1) = 40 + (-1) = 39
3.
the quotient of -8 and
-20
-8 / -20 = + 2/5
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5
Bellwork:
1.
7 subtracted from the
quotient of 0 and 5
4.
0/5 – 7 = -7
2.
-1 added to the
product of -8 and -5
twice the sum of -3
and -4
2(-3 + -4) = 2(-7) = -14
5.
the quotient of -9 and
-30
6.
the difference of -9
and the product of -4
and -6
(-8)(-5) + (-1) = 40 + (-1) = 39
3.
the quotient of -8 and
-20
-8 / -20 = + 2/5
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
6
Bellwork:
1.
7 subtracted from the
quotient of 0 and 5
4.
0/5 – 7 = -7
2.
-1 added to the
product of -8 and -5
2(-3 + -4) = 2(-7) = -14
5.
(-8)(-5) + (-1) = 40 + (-1) = 39
3.
the quotient of -8 and
-20
-8 / -20 = + 2/5
twice the sum of -3
and -4
the quotient of -9 and
-30
-9 / -30 = 3/10
6.
the difference of -9
and the product of -4
and -6
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7
Bellwork:
1.
7 subtracted from the
quotient of 0 and 5
4.
0/5 – 7 = -7
2.
2(-3 + -4) = 2(-7) = -14
-1 added to the
product of -8 and -5
5.
(-8)(-5) + (-1) = 40 + (-1) = 39
3.
the quotient of -8 and
-20
-8 / -20 = + 2/5
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
twice the sum of -3
and -4
the quotient of -9 and
-30
-9 / -30 = 3/10
6.
the difference of -9
and the product of -4
and -6
-9 – (-4)(-6) = -9 – (+24) = -33
8
1.7
Properties of Real
Numbers
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Objectives:
Define
and use properties of
real numbers
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10
Commutative Property
•Addition: a + b = b + a
•Multiplication: a · b = b · a
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11
Commutative Property
•Addition: a + b = b + a
•Multiplication: a · b = b · a
“reorder”
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12
Activity
Can
you illustrate the commutative
property using a group of people?
Show what the commutative property
means using a 2 or 4 member group.
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13
Associative Property
•Addition:(a + b) + c = a + (b + c)
•Multiplication: (a · b) · c = a · (b ·
c)
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14
Associative Property
•Addition:(a + b) + c = a + (b + c)
•Multiplication: (a · b) · c = a · (b ·
c)
“regroup”
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15
Activity
Can
you illustrate the associative
property using a group of people?
Show what the associative property
means using a 3 member group.
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16
Example 1
Use the commutative or associative property to complete.
a. x + 8 = ______
8+x
b. 7 · x = ______
x·7
c. 3 + (8 + 1) = _________
(3 + 8) + 1
d. (‒5 ·4) · 2 = _________
‒5(4 · 2)
e. (xy) ·18 = ___________
x · (y ·18)
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17
Example 1
Use the commutative or associative property to complete.
a. x + 8 = ______
8+x
b. 7 · x = ______
x·7
c. 3 + (8 + 1) = _________
(3 + 8) + 1
d. (‒5 ·4) · 2 = _________
‒5(4 · 2)
e. (xy) ·18 = ___________
x · (y ·18)
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18
Example 1
Use the commutative or associative property to complete.
a. x + 8 = ______
8+x
b. 7 · x = ______
x·7
c. 3 + (8 + 1) = _________
(3 + 8) + 1
d. (‒5 ·4) · 2 = _________
‒5(4 · 2)
e. (xy) ·18 = ___________
x · (y ·18)
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19
Example 1
Use the commutative or associative property to complete.
a. x + 8 = ______
8+x
b. 7 · x = ______
x·7
c. 3 + (8 + 1) = _________
(3 + 8) + 1
d. (‒5 ·4) · 2 = _________
‒5(4 · 2)
e. (xy) ·18 = ___________
x · (y ·18)
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20
Example 1
Use the commutative or associative property to complete.
a. x + 8 = ______
8+x
b. 7 · x = ______
x·7
c. 3 + (8 + 1) = _________
(3 + 8) + 1
d. (‒5 ·4) · 2 = _________
‒5(4 · 2)
e. (xy) ·18 = ___________
x · (y ·18)
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
21
Example 1
Use the commutative or associative property to complete.
a. x + 8 = ______
8+x
b. 7 · x = ______
x·7
c. 3 + (8 + 1) = _________
(3 + 8) + 1
d. (‒5 ·4) · 2 = _________
‒5(4 · 2)
e. (xy) ·18 = ___________
x · (y ·18)
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22
Distributive Property
a(b + c) = ab + ac
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23
Distributive Property
a(b + c) = ab + ac
“Multiplication Over Addition”
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24
Identity Properties
 Addition
•
0 is the identity
for addition
a + 0 = a and
0+a=a
 Multiplication
•
1 is the identity
for multiplication
a · 1 = a and
1·a=a
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25
Identity Properties
 Addition
•
0 is the identity
for addition
 Multiplication
•
1 is the identity
for multiplication
“doesn’t change it”
a + 0 = a and
0+a=a
a · 1 = a and
1·a=a
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26
Identity Properties
 Addition
•
0 is the identity
for addition
 Multiplication
•
1 is the identity
for multiplication
“same as what you started with”
a + 0 = a and
0+a=a
a · 1 = a and
1·a=a
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27
Example 2
Use the distributive property to write each expression
without parentheses. Then simplify the result.
a. 7(x + 4y) = 7x + 28y
b. 3(‒5 + 9z) = 3(‒5) + (3)(9z) = ‒15 + 27z
c. ‒(8 + x ‒ w) = (‒1)(8) + (‒1)(x) ‒ (‒1)(w) = ‒8 ‒ x + w
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28
Example 2
Use the distributive property to write each expression
without parentheses. Then simplify the result.
a. 7(x + 4y) = 7x + 28y
b. 3(‒5 + 9z) = 3(‒5) + (3)(9z) = ‒15 + 27z
c. ‒(8 + x ‒ w) = (‒1)(8) + (‒1)(x) ‒ (‒1)(w) = ‒8 ‒ x + w
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
29
Example 2
Use the distributive property to write each expression
without parentheses. Then simplify the result.
a. 7(x + 4y) = 7x + 28y
b. 3(‒5 + 9z) = 3(‒5) + (3)(9z) = ‒15 + 27z
c. ‒(8 + x ‒ w) = (‒1)(8) + (‒1)(x) ‒ (‒1)(w) = ‒8 ‒ x + w
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
30
Example 2
Use the distributive property to write each expression
without parentheses. Then simplify the result.
a. 7(x + 4y) = 7x + 28y
b. 3(‒5 + 9z) = 3(‒5) + (3)(9z) = ‒15 + 27z
c. ‒(8 + x ‒ w) = (‒1)(8) + (‒1)(x) ‒ (‒1)(w) = ‒8 ‒ x + w
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
31
Example 2
Use the distributive property to write each expression
without parentheses. Then simplify the result.
a. 7(x + 4y) = 7x + 28y
b. 3(‒5 + 9z) = 3(‒5) + (3)(9z) = ‒15 + 27z
c. ‒(8 + x ‒ w) = (‒1)(8) + (‒1)(x) ‒ (‒1)(w) = ‒8 ‒ x + w
Copyright © 2012 Pearson Education, Inc. Publishing as Prentice Hall.
32
Example 2
Use the distributive property to write each expression
without parentheses. Then simplify the result.
a. 7(x + 4y) = 7x + 28y
b. 3(‒5 + 9z) = 3(‒5) + (3)(9z) = ‒15 + 27z
c. ‒(8 + x ‒ w) = (‒1)(8) + (‒1)(x) ‒ (‒1)(w) = ‒8 ‒ x + w
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33
Inverse Properties
Additive Inverse
The numbers a and –a are additive inverses or
opposites of each other because their sum is 0;
that is a + (–a) = 0.
Multiplicative Inverse
1
The numbers b and b (for b ≠0) are reciprocals or
multiplicative inverses of each other because their
product is 1;
1
that is b   1
b
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34
Inverse Properties
Additive Inverse
“opposites”
The numbers a and –a are additive inverses or
opposites of each other because their sum is 0;
that is a + (–a) = 0.
Multiplicative Inverse
1
The numbers b and b (for b ≠0) are reciprocals or
multiplicative inverses of each other because their
product is 1;
1
that is b   1
b
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35
Inverse Properties
Additive Inverse
“opposites”
The numbers a and –a are additive inverses or
opposites of each other because their sum is 0;
that is a + (–a) = 0.
Multiplicative Inverse
1
The numbers b and b (for b ≠0) are reciprocals or
multiplicative inverses of each other because their
product is 1;
1
that is b   1
b
“reciprocals”
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36
Inverse Properties
Additive Inverse
“opposites”
The numbers a and –a are additive inverses or
opposites of each other because their sum is 0;
that is a + (–a) = 0.
Multiplicative Inverse
1
The numbers b and b (for b ≠0) are reciprocals or
multiplicative inverses of each other because their
product is 1;
1
that is b   1
b
“flip it”
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37
Example 3
Name the property illustrated by each true statement.
a. 5(x + y) = 5 · x + 5 ·y
Distributive property
b. (n + 0) + 9 = n + 9
Identity element for addition
c. ‒5 · (x · 11) = (‒5 ·11) · x
Commutative and
associative properties of
multiplication
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38
Example 3
Name the property illustrated by each true statement.
a. 5(x + y) = 5 · x + 5 ·y
Distributive property
b. (n + 0) + 9 = n + 9
Identity element for addition
c. ‒5 · (x · 11) = (‒5 ·11) · x
Commutative and
associative properties of
multiplication
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39
Example 3
Name the property illustrated by each true statement.
a. 5(x + y) = 5 · x + 5 ·y
Distributive property
b. (n + 0) + 9 = n + 9
Identity element for addition
c. ‒5 · (x · 11) = (‒5 ·11) · x
Commutative and
associative properties of
multiplication
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40
Example 3
Name the property illustrated by each true statement.
a. 5(x + y) = 5 · x + 5 ·y
Distributive property
b. (n + 0) + 9 = n + 9
Identity element for addition
c. ‒5 · (x · 11) = (‒5 ·11) · x
Commutative and
associative properties of
multiplication
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41
Example 3
Name the property illustrated by each true statement.
d. 5 + (x + y) = (5 + x) + y
Associative property of addition
 1
e. 3      1
 3
Multiplicative inverse property
f. ‒5 + 5 = 0
Additive inverse property
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42
Example 3
Name the property illustrated by each true statement.
d. 5 + (x + y) = (5 + x) + y
Associative property of addition
 1
e. 3      1
 3
Multiplicative inverse property
f. ‒5 + 5 = 0
Additive inverse property
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43
Example 3
Name the property illustrated by each true statement.
d. 5 + (x + y) = (5 + x) + y
Associative property of addition
 1
e. 3      1
 3
Multiplicative inverse property
f. ‒5 + 5 = 0
Additive inverse property
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44
Example 3
Name the property illustrated by each true statement.
d. 5 + (x + y) = (5 + x) + y
Associative property of addition
 1
e. 3      1
 3
Multiplicative inverse property
f. ‒5 + 5 = 0
Additive inverse property
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45
Closure:
I say commutative, you say _______.
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46
Closure:
I say commutative, you say _______.
“reorder”
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47
Closure:
I say associative, you say _______.
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48
Closure:
I say associative, you say _______.
“regroup”
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49
Closure:
I say distributive, you say _______.
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50
Closure:
I say distributive, you say _______.
“multi. over add”
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51
Closure:
I say identity, you say _______.
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52
Closure:
I say identity, you say _______.
“doesn’t change it”
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53
Closure:
I say identity, you say _______.
“same as what you
started with”
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54
Closure:
I say additive
inverse, you say _______.
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55
Closure:
I say additive
inverse, you say _______.
“opposite”
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56
Closure:
I say multiplicative
inverse, you say _______.
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57
Closure:
I say multiplicative
inverse, you say _______.
“reciprocal”
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58
Closure:
I say multiplicative
inverse, you say _______.
“flip it”
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59
Closure:
I
say commutative, you say _______.
I
say associative, you say _______.
I
say distributive, you say _______.
I
say identity, you say _______.
I
say additive
I
say multiplicative
inverse, you say _______.
inverse, you say _______.
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60