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Focus Question• Why are the properties of real numbers, such as commutative and associative properties, useful?
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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.
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Vocabulary to Know• Equivalent Expressions • Algebraic expressions that have the same value for all values of the variables(s).
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Vocabulary to Know• Property • Rules used to rewrite and compare expressions
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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.
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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 + 18Multiplication a·b=b·a
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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.
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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 ExampleAddition (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)
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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.
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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=1Multiplication a·1=a 67 · 1 = 67
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Properties of Real Numbers• Zero Property of Multiplication • The product of a and 0 is 0. a · 0 = 0 18 · 0 = 18
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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
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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.
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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
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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)
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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
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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• Add • 11.40• State your solution: The total cost is $11.40.
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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!
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Vocabulary to Know• Deductive Reasoning • the process of reasoning logically from given facts to a conclusion.
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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.
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Using Deductive Reasoning and Counterexamples• Is the statement true or false? If false, give a counterexample.• a·b=b+a
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Using Deductive Reasoning and Counterexamples• Is the statement true or false? If false, give a counterexample.• a·b=b+a• False• 5·3≠3+5
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Using Deductive Reasoning and Counterexamples• (a + b) + c = b + (a + c)
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Using Deductive Reasoning and Counterexamples• (a + b) + c = b + (a + c)• True. You can use the properties of real numbers to prove this is true.
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Using Deductive Reasoning and Counterexamples• (a + b) + c = b + (a + c)• True. You can use the properties of real numbers to prove this is true.• Use the Associative Property of Addition • (a + b) + c = (b + a) + c
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Using Deductive Reasoning and Counterexamples• (a + b) + c = b + (a + c)• True. You can use the properties of real numbers to prove this 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
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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
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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?
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Focus Question Answer• Why are the properties of real numbers, such as the commutative and associative properties, useful?
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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.
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