Download a + b

Lesson 1.6
Properties of Real
Numbers
California Standards
1.0 Students identify and use the arithmetic properties of
subsets of integers and rational, irrational, and real
numbers, including closure properties for the four basic
arithmetic operations where applicable.
24.3 Students use counterexamples to show that an
assertion is false and recognize that a single
counterexample is sufficient to refute an assertion.
Also covered: 25.1
Words to Know
Associate- to group or join together.
Commute- to change locations.
Distribute- to pass out or to give out
shares
Commutative Property…
In
Words
In a sum, you can In a product, you
add terms in any can multiply
order.
factors in any
order.
In Algebra
a+b=b+a
ab=ba
In
Arithmetic
6+9=9+6
47=74
Let’s Try It…
Commutative Property
Associative Property…
In Words
In Algebra
In
Arithmetic
Changing the
grouping of terms
will not change the
sum.
(a + b) + c = a + (b + c)
(9 + 5) + 6 = 9 + (5 + 6)
Changing the
grouping of factors
will not change the
product.
(ab)c = a(bc)
(5  10)  3 = 5  (10  3)
Let’s Try It…
Associative Property
Does It Work With
Subtraction or Division?
7–3=4
NOT EQUAL
8÷4=2
NOT EQUAL
3–7=-4
4 ÷ 8 = 0.5
(9 – 2) – 3 = 4
8 ÷ (4 ÷ 2) = 4
NOT EQUAL
9 – (2 – 3) = 10
NOT EQUAL
(8 ÷ 4) ÷ 2 = 1
.
Name the Property
A. 7(mn) = (7m)n
The grouping is different.
Associative Property of Multiplication
B. (a + 3) + b = a + (3 + b) The grouping is different.
Associative Property of Addition
C. x + (y + z) = x + (z + y)
The order is different.
Commutative Property of Addition
Name the Property
a. n + (–7) = –7 + n
The order is different.
Commutative Property of Addition
b. 1.5 + (g + 2.3) = (1.5 + g) + 2.3 The grouping is
different.
Associative Property of Addition
c. (xy)z = (yx)z
The order is
different.
Commutative Property of Multiplication
Distributive Property
In Algebra
a(b+c) = ab + ac
In Arithmetic
3(4 + 8) = 12 + 24
3(12) = 36
Let’s try it …
Write each product using the
Distributive Property. Then simplify.
A. 5(71)
5(71) = 5(70 + 1)
= 5(70) + 5(1)
= 350 + 5
= 355
B. 4(38)
4(38) = 4(40 – 2)
= 4(40) – 4(2)
= 160 – 8
= 152
Rewrite 71 as 70 + 1.
Use the Distributive Property.
Multiply (mentally).
Add (mentally).
Rewrite 38 as 40 – 2.
Use the Distributive Property.
Multiply (mentally).
Subtract (mentally).
Now You Try
1. 2(3x – 5) =
2(3x) – 2(5)=
=6x – 10
2. -4(y + 9) =
-4(y) + -4(9)
= -4y - 36
Now You Try
(8 – x)3 =
3(8) – 3(x) =
= 24 – 3x
-4(12 + x - y) =
(-4)(12) + (-4)(x) – (-4)(y)
= -48 - 4x + 4y
Use Mental Math to Evaluate the Following
(9 + 14) + 1 =
10 + 14 =
= 24
191 + 12 + 9 =
200 + 12 =
= 212
Compare and Contrast
Commutative
Property
Associative
Property
A set of numbers is said to be closed, or to
have closure, under an operation if the
result of the operation on any two numbers
in the set is also in the set.
Ex: {-1, 0, 1}; the set is closed under
multiplication
(-1)(0) = 0
(-1)(1) = -1
(0)(1) = 0
yes
Finding Counterexamples to Statements About
Closure
Find a counterexample to show that each statement is false.
A. The prime numbers are closed under addition.
Find two prime numbers, a and b, such that their sum
is not a prime number.
Try a = 3 and b = 5.
a+b=3+5=8
Since 8 is not a prime number, this is a
counterexample. The statement is false.
Lesson Quiz: Part I
Name the property that is illustrated in each equation.
1. 6(rs) = (6r)s
Associative Property of Multiplication
2. (3 + n) + p = (n + 3) + p
Commutative Property of Addition
3. (3 + n) + p = 3 + (n + p)
Associative Property of Addition
4. Find a counterexample to disprove the statement “The
Commutative Property is true for division.”
Possible answer: 3 ÷ 6 ≠ 6 ÷ 3
Lesson Quiz: Part II
Write each product using the Distributive Property. Then
simplify.
5. 8(21)
8(20) + 8(1) = 168
6. 5(97)
5(100) – 5(3) = 485
Find a counterexample to show that each statement is false.
7. The natural numbers are closed under subtraction.
Possible answer: 6 and 8 are natural, but 6 – 8 = –2,
which is not natural.
8. The set of even numbers is closed under division.
Possible answer: 12 and 4 are even, but 12 ÷ 4 = 3,
which is not even.