9th Grade Unit 2 Lesson 1 Day 1



9th Grade Math Class; Lesson Number 1 Day 1 Properties of EqualityKey Standards addressed in this Lesson: MCC9‐12.A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.Time allotted for this Lesson: 1 DayMaterials Needed:Key Concepts in Standards: Students should focus on and master linear equations and be able to extend and apply their reasoning to other types of equations in future courses. Students will solve exponential equations with logarithms in future courses.Properties of operations can be used to change expressions on either side of the equation to equivalent expressions. In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant produces an equation with the same solutions. Other operations, such as squaring both sides, may produce equations that have extraneous solutions.Example: Explain why the equation QUOTE + QUOTE = 5 has the same solutions as the equation 3x + 14 = 30. Does this mean that QUOTE + QUOTE is equal to 3x + 14?Essential Question: How do you identify and apply the properties of equality?Vocabulary: Tier 1: already knows Tier 2: needs review Tier 3: New VocabularyTier 1PropertyOrder of OperationsVariableEqualityEquationCoefficient Tier 2Associative PropertyCommutative PropertyIdentity PropertyInverse PropertyDistributive PropertyReflexive PropertySymmetric PropertyTransitive Property Properties of EqualityTier 3JustifyProveConcepts/Skills to Maintain: Refer to TEUsing inverse operations to isolate variables and solve equationsMaintaining order of operationsUnderstanding and use properties of exponentsOpening: Opening Activity: Unscrambling Vocabulary Words (Attached)Work Session:Teacher Notes on Properties of Operations and Equality (attached)Students fill-in the guided notes.Closing:Ticket out the door:Justify each step using the appropriate property:3x -2(3y - 2x + 8) - 33x + - 6y + 4x + -16 - 33x + 4x + - 6y + -16- 3 (3x + 4x) + 6y + (-16 – 3)7x + 6y + -19Corresponding Task(s) (if not in work session – there may be several tasks that fit) – ****All Tasks can be found at ****Highlight the Mathematical Practices that this lesson incorporates:Make sense of problems and persevere in solving themReason abstractly and quantitativelyConstruct viable arguments and critique the reasoning of othersModel with mathematicsUse appropriate tools strategicallyAttend to precisionLook for and make sure of structureLook for and express regularity in repeated reasoningOpening ActivityUnscramble these letters to form mathematical words.C O T M M U A T I V E A C I S A S E T I V O D E I N T I T Y NE I V RS E D U S T I R I V B I T E Now arrange the circled letters to form a mathematical word that is related to the above terms. PPTeacher’s NotesThe Properties of OperationsHere a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.Associative property of addition(a + b) + c = a + (b + c)Commutative property of additiona + b = b + aAdditive identity property of 0a + 0 = 0 + a = aExistence of additive inversesFor every a there exists –a so that a + (–a) = (–a) + a = 0.Associative property of multiplication(a × b) × c = a × (b × c)Commutative property of multiplicationa × b = b × aMultiplicative identity property of 1a × 1 = 1 × a = aExistence of multiplicative inversesFor every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1.Distributive property of multiplication over additiona × (b + c) = a × b + a × cThe Properties of EqualityHere a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.Reflexive property of equalitya = aSymmetric property of equalityIf a = b, then b = a.Transitive property of equalityIf a = b and b = c, then a = c.Addition property of equalityIf a = b, then a + c = b + c.Subtraction property of equalityIf a = b, then a – c = b – c.Multiplication property of equalityIf a = b, then a × c = b × c.Division property of equalityIf a = b and c ≠ 0, then a ÷ c = b ÷ c.Substitution property of equalityIf a = b, then b may be substituted for a in any expression containing a.Name:____________________________Date:_____________Guided NotesThe Properties of OperationsHere a, b and c stand for arbitrary numbers in a given number system. The properties of operations apply to the rational number system, the real number system, and the complex number system.Associative property of addition(a + b) + c = a + (b + c)Commutative property of additiona + b = b + aAdditive identity property of 0a + 0 = 0 + a = aExistence of additive inversesFor every a there exists –a so that a + (–a) = (–a) + a = 0.Associative property of multiplication(a × b) × c = a × (b × c)Commutative property of multiplicationa × b = b × aMultiplicative identity property of 1a × 1 = 1 × a = aExistence of multiplicative inversesFor every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1.Distributive property of multiplication over additiona × (b + c) = a × b + a × cThe Properties of EqualityHere a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.Reflexive property of equalitya = aSymmetric property of equalityIf a = b, then b = a.Transitive property of equalityIf a = b and b = c, then a = c.Addition property of equalityIf a = b, then a + c = b + c.Subtraction property of equalityIf a = b, then a – c = b – c.Multiplication property of equalityIf a = b, then a × c = b × c.Division property of equalityIf a = b and c ≠ 0, then a ÷ c = b ÷ c.Substitution property of equalityIf a = b, then b may be substituted for a in any expression containing a.Identifying and Applying Properties PracticeName the property shown by each statement.7 ● (-2) = (-2) ● 7 -19 + 19 = 0 12 + [(– 3) + 29] = [12 + (-3)] + 29 15 + [8 + (-4)] = [8 + (-4)] + 15 (2 ● 3) ● (-9) = 2 ● [3 ● (-9)] 1● (-37) = -37 (6 + 0) – 7 = 6 – 7 ●7= 113 (2 – 6) = 13 (2) – 13(6) (-4 + 3)(5 + 6) = (-4 + 3)(5) + (-4 + 3)(6)4 + (9 + 6) = (4 + 9) + 63(x + 5) = 3 ? x + 3 ? 5 (3 + y) + 0 = 3 + yx ? = 114xy= 14yx(3 ? 9) ? 1 = 3 ? 97 + (-7) = 06 ? (8 + c) = (8 + c) ? 6x + 12 = 12 + x(x + y) ? 5 = (y + x) ? 5Why is it true that 3(4 + x) = 3(x + 4)?Why is 3(4x) = (3●4)x?Why is 12 – 3x = 3(4 – x)?Simplify the expression. Justify your steps.3b+ (4b - 6b + 2) –b25.2(6x – 5) – 3(5x + 4) ................
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