1-3 Notes Algebra 1 Properties of Numbers

Equivalent Expressions 2 expressions that represent the same number

Properties of Equality Reflexive Property–Any quantity is equal to itself. a = a, 3 = 3 Symmetric Property—If the left side equals the right side of an equation, then the right side equals the left side of an equation. If 3+x=7, then 7=3+x Transitive Property—If the 1st quantity equals the 2nd quantity and the 2nd quantity equals a 3rd quantity, then the 1st quantity equals the 3rd quantity. If 2 + a = 8 and 8 = 3 + 5, then 2 + a = 3 + 5 Substitution Property—A quantity may be substituted for its equal in any expression or equation. If n = 4, then 7n = 7(4)

Addition Properties Additive identity—For any number a, the sum of a and 0 is a. a + 0 = 0 + a = a 5 + 0 = 5 0 + 5 = 5 Additive inverse—A number and its opposite are additive inverses of each other. 𝑎+ −𝑎 =0 3+ −3 =0 3−3=0

Multiplication Properties Multiplicative Inverse—For any number a the product of a and 1 is a. 𝑎∙1=𝑎 1∙9=9 9∙1=9 Multiplicative properties of zero—For any number a, the product of a and 0 is 0. 𝑎∙0=𝑎 9∙0=0 0∙9=0 Multiplicative Inverse (Reciprocal)—for every number 𝑎 𝑏 , where 𝑎,𝑏≠0 there is exactly one number 𝑏 𝑎 such that the product of 𝑎 𝑏 and 𝑏 𝑎 is 1. 𝑎 𝑏 ∙ 𝑏 𝑎 =1 2 7 ∙ 7 2 = 14 14 =1

Example 1 pg. 17 Evaluate using Properties Evaluate and name the property used in each step. 1 4 12−8 +3 15÷5−2

Example 1 pg. 17 Evaluate using Properties Evaluate and name the property used in each step. 1 4 12−8 +3 15÷5−2 1 4 4 +3 15÷5−2 Substitution 12−8 =4 1 4 4 +3 3−2 Substitution (15÷5)=3 1 4 4 +3 1 Substitution (3−2)=1 1+3 1 Multiplicative inverse 1 4 4 =1 1+3 Multiplicative identity 3 1 =3 4 Substitution 1+3=4

Commutative Property The order in which you add or multiply numbers does not change their sum or product. 4+12=12+4 4∙12=12∙4

Associative Property The way you group three or more numbers when adding or multiplying does not change their sum or product. 2+ 4+7 = 2+4 +7 2∙ 4∙7 = 2∙4 ∙7

Example 2 pg. 18 Real World Example HORSEBACK RIDING Gina made a list of trail lengths to find the total miles she rode. Find the total miles Gina rode her horse. Trails Name Miles Bent Tree 4.25 Knob Hill 6.50 Meadowrun 9.00 Pinehurst 7.75

Example 2 pg. 18 Real World Example HORSEBACK RIDING Gina made a list of trail lengths to find the total miles she rode. Find the total miles Gina rode her horse. 27.5 miles Trails Name Miles Bent Tree 4.25 Knob Hill 6.50 Meadowrun 9.00 Pinehurst 7.75

Example 3 pg. 19 Use Multiplication Properties Evaluate using the properties of numbers. Name the property used in each step. 2∙8∙5∙7

Example 3 pg. 19 Use Multiplication Properties Evaluate using the properties of numbers. Name the property used in each step. 2∙8∙5∙7 2∙5∙8∙7 Commutative Property of Multiplication 2∙5 ∙ 8∙7 Associative Property of Multiplication 10∙56 Substitution 560 Substitution

1-3 pg. 19 9-29o, 30, 31-47o, 63-78(x3)