3.1.1: Properties of Equality/Operations (8/29/2018) When we solve an equation, we are rewriting it into a simpler, equivalent equation that helps us find the unknown value. Properties of Equality, and Properties of Operations, justify our reasoning, and also help us to understand our own thinking. When operations are performed on one side of the equation, the Properties of Operations are generally followed. When an operation is performed on both sides of the equation, the Properties of Equality are generally followed.(Balance the equation) 3.1.1: Properties of Equality

Key Concepts, continued Properties of Equality Property Symbolically In words Reflexive property of equality(Same) a = a A number is equal to itself. Symmetric property of equality(Switch) If a = b, then b = a. If numbers are equal, they will still be equal if the order is changed. Transitive property of equality(Three) If a = b and b = c, then a = c. If numbers are equal to the same number, then they are equal to each other. Addition property of equality If a = b, then a + c = b + c. Adding the same number to both sides of an equation does not change the equality of the equation. 3.1.1: Properties of Equality

Key Concepts, continued Properties of Equality, continued Property Symbolically In words Subtraction property of equality If a = b, then a – c = b – c. Subtracting the same number from both sides of an equation does not change the equality of the equation. Multiplication If a = b and c ≠ 0, then a • c = b • c. Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation. Division property of equality a ÷ c = b ÷c. Dividing both sides of the equation by the same number, other than 0, does not change the equality of the equation. 3.1.1: Properties of Equality

Key Concepts, continued Properties of Equality, continued Property Symbolically In words Substitution If a = b, then b may be substituted for a in any expression containing a. If two numbers are equal, then substituting one in for another does not change the equality of the equation. 3.1.1: Properties of Equality

Key Concepts, continued Properties of Operations: explain the effect that the operations of addition, subtraction, multiplication, and division have on equations. Property General rule Specific example Commutative property of addition a + b = b + a 3 + 8 = 8 + 3 Associative property of addition (a + b) + c = a + (b + c) (3 + 8) + 2 = 3 + (8 + 2) Commutative property of multiplication a • b = b • a 3 • 8 = 8 • 3 Associative property of (a • b) • c = a • (b • c) (3 • 8) • 2 = 3 • (8 • 2) Distributive property of multiplication over addition a • (b + c) = a • b + a • c 3 • (8 + 2) = 3 • 8 + 3 • 2 3.1.1: Properties of Equality

Common Errors/Misconceptions incorrectly identifying operations incorrectly identifying properties performing a step that is not justifiable or does not follow the properties of equality and/or the properties of operations 3.1.1: Properties of Equality

Guided Practice Example #1, P.9: Which property of equality is missing in the steps to solve the equation –7x + 22 = 50? Equation Steps 1) –7x + 22 = 50 1) Original equation(Given) 2) –7x = 28 3) x = –4 3) Division property of equality 2) Subtraction Property of Equality 3.1.1: Properties of Equality

Guided Practice Example #2, P.10 Which property of equality is missing in the steps to solve the equation ? Equation Steps 1) Original equation(Given) 2) Addition property of equality 3) –x = 42 4) x = –42 4) Division property of equality 3) Multiplication Property of Equality 3.1.1: Properties of Equality

Guided Practice Example #3, P.10: Which property of equality is missing in the steps to solve the equation 76 = 5x – 15 + 2x ? Equation Steps 1) 76 = 5x – 15 + 2x 1) Original Equation(Given) 2) 76 = 5x + 2x - 15 2) Commutative Property of Addition 3) 76 = 7x - 15 3) Combine Like Terms(Simplify) 4) 91 = 7x 4) Addition Property of Equality 5) 13 = x 5) Division Property of Equality 6) x = 13 6) Symmetric Property of Equality 3.1.1: Properties of Equality

Guided Practice Example #4, P.10: Which property of equality is missing in the steps to solve the equation 5x + 3(x + 4) = 28 ? Equation Steps 1) 5x + 3(x+4) = 28 1) Original Equation(Given) 2) 5x + 3x + 12 = 28 2) Distributive Property 3) 8x + 12 = 28 3) Combine Like Terms(Simplify) 4) 8x = 16 4) Subtraction Property of Equality 5) x = 2 5) Division Property of Equality 3.1.1: Properties of Equality

8/29/2018: Classwork Workbook(3.1.1): P.13,14 #1-8 5 minutes!!!!! http://www.walch.com/ei/00004

3.1.1, P.13/14 #1-8 Equation Steps #1- 2) Add P of E #2- 2) Div P of E #5- 3) 2x = 2.6 #5- 4) Div P of E #6- 4) x = - 4 #6- 3) Subt P of E #7- 2) Div P of E #7- 3) Add P of E Common Core State Standard: A–REI.3 #8- 2) CLT (Simplify) #8- 3) Subt P of E #8- 4) Div P of E 3.1.2: Solving Linear Equations

8/29/2018: 2) 3.1.2 Notes(U3-11) Homework Finish Algebra Proof worksheet 2) 3.1.2 Notes(U3-11) a) Intro: read only b) Key Concepts: copy the 1 Chart(U3-11) c) Workbook: P.19, Example #1 http://www.walch.com/ei/00004

3.1.1 Handout, #5(On back of handout) Equation Steps 1) 3x - 4 = 7x -11 1) Given 2) -4x - 4 = -11 2) Subtraction Property of Equality(P of E) 3) -4x = -7 3) Addition P of E   4) Division P of E OR Equation Steps 1) 3x - 4 = 7x -11 1) Given 2) -4x - 4 = -11 2) Subtraction Property of Equality(P of E) 3) -4x = -7 3) Addition P of E   4) Division P of E 3.1.1: Properties of Equality