Title: Warm Up
1Preview
Warm Up
California Standards
Lesson Presentation
2Warm Up Add. 1. 6 (4) 2. 17 (5) 3. (9)
7 Subtract. 4. 12 (4) 5. 3 (5) 6. 7 15
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3 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
4Vocabulary
counterexample closure
5The Commutative and Associative Properties of
Addition and Multiplication allow you to
rearrange an expression.
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7Additional Example 1 Identifying Properties
Name the property that is illustrated in each
equation.
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
8Check It Out! Example 1
Name the property that is illustrated in each
equation.
The order is different.
a. n (7) 7 n
Commutative Property of Addition
b. 1.5 (g 2.3) (1.5 g) 2.3
The grouping is different.
Associative Property of Addition
The order is different.
c. (xy)z (yx)z
Commutative Property of Multiplication
9The Commutative and Associative Properties are
true for addition and multiplication. They may
not be true for other operations. A
counterexample is an example that disproves a
statement, or shows that it is false. One
counterexample is enough to disprove a statement.
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11 Counterexamples
Statement Counterexample
February has fewer than 30 days, so the statement
is false.
No month has fewer than 30 days.
Every integer that is divisible by 2 is also
divisible by 4.
The integer 18 is divisible by 2 but is not by 4,
so the statement is false.
12Additional Example 2 Finding Counterexamples to
Statements About Properties
Find a counterexample to disprove the statement
The Commutative Property is true for raising to
a power.
Find four real numbers a, b, c, and d such that
a³ b and c² d, so a³ ? c².
Try a³ 2³, and c² 3².
a³ b
c² d
2³ 8
3² 9
Since 2³ ? 3², this is a counterexample. The
statement is false.
13Check It Out! Example 2
Find a counterexample to disprove the statement
The Commutative Property is true for division.
Try a 4 and b 8.
The statement is false.
14The Distributive Property also works with
subtraction because subtraction is the same as
adding the opposite.
15Additional Example 3 Using the Distributive
Property with Mental Math
Write each product using the Distributive
Property. Then simplify.
A. 5(71)
5(71) 5(70 1)
Rewrite 71 as 70 1.
5(70) 5(1)
Use the Distributive Property.
350 5
Multiply (mentally).
355
Add (mentally).
B. 4(38)
4(38) 4(40 2)
Rewrite 38 as 40 2.
4(40) 4(2)
Use the Distributive Property.
Multiply (mentally).
160 8
152
Subtract (mentally).
16Check It Out! Example 3
Write each product using the Distributive
Property. Then simplify.
a. 9(52)
9(52) 9(50 2)
Rewrite 52 as 50 2.
9(50) 9(2)
Use the Distributive Property.
450 18
Multiply (mentally).
468
Add (mentally).
b. 12(98)
12(98) 12(100 2)
Rewrite 98 as 100 2.
12(100) 12(2)
Use the Distributive Property.
Multiply (mentally).
1200 24
1176
Subtract (mentally).
17Check It Out! Example 3
Write each product using the Distributive
Property. Then simplify.
c. 7(34)
7(34) 7(30 4)
Rewrite 34 as 30 4.
7(30) 7(4)
Use the Distributive Property.
Multiply (mentally).
210 28
238
Subtract (mentally).
18A 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.
19Closure Property of Real Numbers
20Additional Example 4 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.
21Additional Example 4 Finding Counterexamples to
Statements About Closure
Find a counterexample to show that each statement
is false.
B. The set of odd numbers is closed under
subtraction.
Find two odd numbers, a and b, such that the
difference a b is not an odd number.
Try a 11 and b 9.
a b 11 9 2
11 and 9 are odd numbers, but 11 9 2, which
is not an odd number. The statement is false.
22Check It Out! Example 4
Find a counterexample to show that each statement
is false.
a. The set of negative integers is closed under
multiplication.
Find two negative integers, a and b, such that
the product a ? b is not a negative integer.
Try a 2 and b 1.
a ? b 2(1) 2
Since 2 is not a negative integer, this is a
counterexample. The statement is false.
23Check It Out! Example 4
Find a counterexample to show that each statement
is false.
b. The whole numbers are closed under the
operation of taking a square root.
Try a 15.
24Lesson 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
25Lesson 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.