1 4 Properties of Real Numbers

1-4 Properties of Real NumbersI CAN:Identify and use properties of real numbers

Get Ready!• Tell whether each pair of expressions is equal by completing

each statement with = or ≠. Explain your answers.1. 34 + 12 ___ 12 + 342. 100 – 1 ____ 1 – 100 3. 0 + 180 ____ 1804. 18 ÷ ____ 15. 45 – 1 ____ 456. 6 · ____ 1

Focus Question• Why are the properties of real numbers, such as commutative

and associative properties, useful?

BIG Ideas• Relationships that are always true for real numbers are called

properties, which are rules used to rewrite and compare expressions.

• Important properties include commutative, associative and identity properties, the zero property of multiplication and the multiplication property of -1.

Vocabulary to Know • Equivalent Expressions• Algebraic expressions that have the same value for all values of

the variables(s).

Vocabulary to Know• Property• Rules used to rewrite and compare expressions

Properties of Real Numbers• Commutative Properties of Addition and Multiplication• Changing the order of the addends does not change the sum.• Changing the order of the factors does not change the product.

Properties of Real Numbers• Commutative Properties of Addition and Multiplication• Changing the order of the addends does not change the sum.• Changing the order of the factors does not change the product.

Algebra Example

Addition a + b = b + a 18 + 54 = 54 + 18

Multiplication a · b = b · a 12 · = · 12

Properties of Real Numbers• Associative Properties of Addition and Multiplication• Changing the grouping of the addends does not change the sum.• Changing the grouping of the factors does not change the

product.

Properties of Real Numbers• Associative Properties of Addition and Multiplication• Changing the grouping of the addends does not change the sum.• Changing the grouping of the factors does not change the

product.

Algebra Example

Addition (a + b) + c = a + (b + c) (23 + 9) + 4 = 23 + (9 + 4)

Multiplication (a · b) · c = a · (b · c) (7 · 9) · 10 = 7 · (9 · 10)

Properties of Real Numbers• Identity Properties of Addition and Multiplication• The sum of any real number and 0 is the original number.• The product of any real number and 1 is the original number.

Properties of Real Numbers• Identity Properties of Addition and Multiplication• The sum of any real number and 0 is the original number.• The product of any real number and 1 is the original number.

Algebra Example

Addition a + 0 = 1 + 0 =

Multiplication a · 1 = a 67 · 1 = 67

Properties of Real Numbers• Zero Property of Multiplication• The product of a and 0 is 0. a · 0 = 0 18 · 0 = 18

Properties of Real Numbers• Zero Property of Multiplication• The product of a and 0 is 0. a · 0 = 0 18 · 0 = 18

• Multiplication Property of -1• The product of -1 and a is –a. -1 ·a = - a -1 · 9 = -9

Identifying Properties• What property is illustrated by each statement?• 42 · 0 = 42• (y + 2.5) + 28 = y + (2.5 + 28)• 10x + 0 = 10x• 4x · 1 = 4x• x + ( + z) = x + (z + ()

Identifying Properties• What property is illustrated by each statement?• 42 · 0 = 42 Zero Property of Multiplication• (y + 2.5) + 28 = y + (2.5 + 28) Associative Property of Addition• 10x + 0 = 10x Identity Property of Addition• 4x · 1 = 4x Identity Property of Multiplication• x + ( + z) = x + (z + () Commutative Property of Addition

Properties & Mental Math• You can use properties to help you solve some problems using

mental math.• A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs

$1.25. What is the total cost for a ticket, a drink, and popcorn? Use mental math.




• Use the Commutative Property of Addition• (7.75 + 2.40) + 1.25 = (2.40 + 7.75) + 1.25





• Use the Associative Property of Addition• 2.40 + (7.75 + 1.25)






• Simplify inside parentheses• 2.40 + 9






• Simplify inside parentheses• 2.40 + 9

• Add• 11.40

• State your solution: The total cost is $11.40.

Properties & Mental Meth• A can holds 3 tennis balls. A box holds 4 cans. A can holds 6

boxes. How many tennis balls are in 10 cases? Use mental math!

Writing Equivalent Expressions

• 5(3n)


• 5(3n)• Put numbers together…• (5 · 3)n


• 5(3n)• Put numbers together…• (5 · 3)n• Simplify 15n


• (4 + 7b) + 8


• (4 + 7b) + 8• Use the Commutative Property of Addition• (7b + 4) + 8


• (4 + 7b) + 8• Use the Commutative Property of Addition• (7b + 4) + 8• Use the Associative Property of Addition• 7b + (4 + 8)


• (4 + 7b) + 8• Use the Commutative Property of Addition• (7b + 4) + 8• Use the Associative Property of Addition• 7b + (4 + 8)• Simplify• 7b + 12


6 𝑥𝑦𝑦


• Rewrite the denominator using the Identity Property of Multiplication



• Use the rule for multiplying fractions.




• x ÷ 1 = x and y ÷ y = 1• 6x · 1

Writing Equivalent Expressions• Rewrite the denominator using the Identity Property of

Multiplication


• x ÷ 1 = x and y ÷ y = 1• 6x · 1• Use the Identity Property of Multiplication• 6x

Simplify Each Expression• 2.1(4.5x)• 6 + (4h + 3)

Vocabulary to Know• Deductive Reasoning• the process of reasoning logically from given facts to a

conclusion.

Vocabulary to Know• Counterexample• To show that a statement is not true, find an example for which

the statement is not true. A example showing that a statement is false is a counterexample. You need only one counterexample to prove that a statement is false.

Using Deductive Reasoning and Counterexamples

• Is the statement true or false? If false, give a counterexample.• a · b = b + a


• Is the statement true or false? If false, give a counterexample.• a · b = b + a• False• 5 · 3 ≠ 3 + 5


• (a + b) + c = b + (a + c)


• (a + b) + c = b + (a + c)• True. You can use the properties of real numbers to prove this

is true.



is true.• Use the Associative Property of Addition• (a + b) + c = (b + a) + c



is true.• Use the Associative Property of Addition• (a + b) + c = (b + a) + c

• Use the Commutative Property of Addition• (a + b) + c = (a + b) + c

Got It?• Is each statement true or false? If it is false, give a

counterexample. If it is true, use the properties of real numbers that the expressions are equivalent.

• a. For all real numbers j and k, j · k = (k + 0) · j• b. For all real numbers m and n, m(n + 1) = mn + 1

Got It?• Is each statement true or false? If it is false, give a

counterexample. If it is true, use the properties of real numbers that the expressions are equivalent.

• a. For all real numbers j and k, j · k = (k + 0) · j• b. For all real numbers m and n, m(n + 1) = mn + 1

• c. Is the statement in part (A) false for every pair of real numbers a and b?

Focus Question Answer• Why are the properties of real numbers, such as the

commutative and associative properties, useful?

BIG Ideas• Relationships that are always true for real numbers are called

properties, which are rules used to rewrite and compare expressions.

• Important properties include commutative, associative and identity properties, the zero property of multiplication and the multiplication property of -1.

Assignment• Pages 29-31• 1-4• 7-19 odd• 20-34 even• 35-45• 47-56

1 4 Properties of Real Numbers

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