Download Properties of Numbers - Alliance Gertz

Lesson 1
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TAKS
CAGrade
Standard
# Objective
Alg 1 1.0
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(#.#)(X)
Properties
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of Numbers
Lesson Title >>
Three properties of equality govern the way that variables and quantities can
be moved around within an expression. These properties are the commutative
property, the associative property, and the distributive property.
New Vocabulary
• <<Vocab
commutative
New
Word >>
property
• associative property
• distributive
property
Using the Commutative, Associative, and Distributive
Properties
The Commutative Property of Addition states
that the order in which numbers are added
does not change their sum:
abba
The Commutative Property of Multiplication states
that the order in which numbers are multiplied
does not change their product:
a?bb?a
In the equation
a(b ⴙ c) ⴝ ab ⴙ ac,
the factor a is
distributed to
each term of the
sum (b ⴙ c).
The Associative Property of Addition states that
the order in which numbers are grouped does not
change their sum:
(a + b) + c a + (b + c)
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
The Associative Property of Multiplication states
that the order in which numbers are grouped
does not change their product:
(a ? b) ? c a ? (b ? c)
The Distributive Property states that multiplying
a number by a sum or difference gives the same
result as the sum or difference of the products of
the number and each term:
a(b + c) ab + ac
EXAMPLE 1
Name the property that each equation illustrates.
352325
The equation involves addition. The order of the numbers is
changed. The equation illustrates the Commutative Property
of Addition.
5 ? (3 ? 2) (5 ? 3) ? 2
The equation involves multiplication. The grouping of the
numbers is changed. The equation illustrates the Associative
Property of Multiplication.
2(3 5) 2 ? 3 2 ? 5
The equation involves multiplication and subtraction. The
multiplication is applied after taking the difference on one
side, and applied before taking the difference on the other.
The equation illustrates the Distributive Property.
CA Standards Check 1
Name the property that each equation illustrates.
1a. 7 (2 3) (7 2) 3
CA Standards Review
1b. 6(3) 6(5) 6(3 5)
LESSON 1
■
Properties of Numbers
1
CA Standard Alg 1 1.0
LESSON 1
Closure Properties
The real number system is the union of two sets of numbers, rational
numbers and irrational numbers. Rational numbers are traditionally
classified into three subsets.
A rational number is any number that can be written as a fraction where the
numerator and denominator are integers and the denominator does not
equal zero. This includes all terminating and repeating decimals.
The set of counting numbers (also called natural numbers) is a subset of the
rational numbers. The set of counting numbers is {1, 2, 3, 4, 5, …}.
The set of whole numbers is a subset of the rational numbers. The set of
whole numbers is the set of counting numbers and zero.
The set of integers is a subset of the rational number. Integers are all whole
numbers and their opposites.
An irrational number is any number that cannot be written as a fraction
where the numerator and denominator are integers. This includes
nonterminating and nonrepeating decimals.
The Closure Property of Addition for the set of real numbers states that
when you add two real numbers, their sum is also a real number. The Closure
Property of Multiplication for the set of real numbers states that when you
multiply two real numbers, their product is also a real number.
Write a valid argument to show that the set of whole numbers is closed for subtraction or
use a counterexample to show that it is not.
Choose two whole numbers to subtract that have a difference that is a whole number.
Subtract 8 from 3.
3 is a whole number. 8 is a whole number.
5 is not a whole number; therefore, the set
of whole number is not closed for subtraction.
3 8 5
CA Standards Check 2
2a. Write a valid argument to show that the set of integers is closed for division or use a
counterexample to show that it is not.
2b. Write a valid argument to show that the set of counting numbers is closed for multiplication or
use a counterexample to show that it is not.
2
LESSON 1
■
Properties of Numbers
CA Standards Review
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
EXAMPLE 2
Name__________________________Class____________Date________
1 Name the property illustrated by the
equation below:
(a b) c a (b c)
A Commutative Property of Addition
B Distributive Property
C Associative Property of Addition
D Commutative Property of Multiplication
2 Which property does the equation below
illustrate?
a(b c) ab ac
5 Which expression is an example of
commutative property for the expression
16 ? 2?
A 4?4?2
B 2(8 1)
C 2(8 1)
D 2 ? 16
6 Which expression is an example of the
associative property for the expression
2 (5 13)
A 13 2 5
A Distributive Property
B (2 5) 13
B Commutative Property of Multiplication
C 20
C Associative Property of Addition
D 2 ( 13 5)
Copyright © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved.
D Associative Property of Multiplication
3 Which is not a counterexample that shows
that the set of whole numbers is not closed
for division?
7 Which is a counterexample that shows that
the set of counting numbers is not closed for
subtraction?
A 12 2 10
A 0 12 0
B 1 2 1
B 5 0 is undefined.
C 0 1 1
C 3 2 1.5
D 413
D 4 12 13
4 Closure Property of Addition for the set of
real numbers states that the sum of two real
numbers must be
8 For which operation is the set of integers not
closed?
A addition
B subtraction
A a positive number.
C multiplication
B a rational number.
D division
C an integer
D a real number
CA Standards Review
LESSON 1
■
Properties of Numbers
3