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 Day

Materials 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 + = 5 has the same solutions as the equation 3x + 14 = 30. Does this mean

that + 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 Vocabulary

Tier 1PropertyOrder of OperationsVariableEqualityEquationCoefficient

Tier 2Associative PropertyCommutative PropertyIdentity PropertyInverse PropertyDistributive PropertyReflexive PropertySymmetric PropertyTransitive Property Properties of Equality

Tier 3JustifyProve

Concepts/Skills to Maintain: Refer to TE Using inverse operations to isolate variables and solve equations Maintaining order of operations Understanding and use properties of exponents

Opening: Opening Activity: Unscrambling Vocabulary Words (Attached)

9th Grade Unit 2 Lesson 1 Day 1

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 + -19

Corresponding Task(s) (if not in work session – there may be several tasks that fit) –

****All Tasks can be found at www.georgiastandards.org ****

Highlight the Mathematical Practices that this lesson incorporates:Make

sense of problems

and persevere in solving

them

Reason abstractly

and quantitatively

Construct viable

arguments and

critique the

reasoning of others

Model with mathematics

Use appropriate

tools strategically

Attend to precision

Look for and make

sure of structure

Look for and

express regularity

in repeated

reasoning

9th Grade Unit 2 Lesson 1 Day 1

Opening 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.

P P

9th Grade Unit 2 Lesson 1 Day 1

Teacher’s NotesThe Properties of Operations

Here 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 addition a + b = b + aAdditive identity property of 0 a + 0 = 0 + a = aExistence of additive inverses For 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 multiplication a × b = b × aMultiplicative identity property of 1 a × 1 = 1 × a = aExistence of multiplicative inverses For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1.Distributive property of multiplication over addition a × (b + c) = a × b + a × c

The Properties of EqualityHere a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.

Reflexive property of equality a = aSymmetric property of equality If a = b, then b = a.Transitive property of equality If a = b and b = c, then a = c.Addition property of equality If a = b, then a + c = b + c.Subtraction property of equality If a = b, then a – c = b – c.Multiplication property of equality If a = b, then a × c = b × c.Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c.Substitution property of equality If a = b, then b may be substituted for a in any expression containing a.

9th Grade Unit 2 Lesson 1 Day 1

9th Grade Unit 2 Lesson 1 Day 1

9th Grade Unit 2 Lesson 1 Day 1

9th Grade Unit 2 Lesson 1 Day 1

9th Grade Unit 2 Lesson 1 Day 1

Name:____________________________ Date:_____________

Guided NotesThe Properties of Operations

Here 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 addition a + b = b + aAdditive identity property of 0 a + 0 = 0 + a = aExistence of additive inverses For 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 multiplication a × b = b × aMultiplicative identity property of 1 a × 1 = 1 × a = aExistence of multiplicative inverses For every a ≠ 0 there exists 1/a so that a × 1/a = 1/a × a = 1.Distributive property of multiplication over addition a × (b + c) = a × b + a × c

The Properties of EqualityHere a, b and c stand for arbitrary numbers in the rational, real, or complex number systems.

Reflexive property of equality a = aSymmetric property of equality If a = b, then b = a.Transitive property of equality If a = b and b = c, then a = c.Addition property of equality If a = b, then a + c = b + c.Subtraction property of equality If a = b, then a – c = b – c.Multiplication property of equality If a = b, then a × c = b × c.Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c.Substitution property of equality If a = b, then b may be substituted for a in any expression containing a.

9th Grade Unit 2 Lesson 1 Day 1

9th Grade Unit 2 Lesson 1 Day 1

9th Grade Unit 2 Lesson 1 Day 1

9th Grade Unit 2 Lesson 1 Day 1

9th Grade Unit 2 Lesson 1 Day 1

Identifying and Applying Properties PracticeName the property shown by each statement.1. 7 ● (-2) = (-2) ● 7

2. -19 + 19 = 0

3. 12 + [(– 3) + 29] = [12 + (-3)] + 29

4. 15 + [8 + (-4)] = [8 + (-4)] + 15

5. (2 ● 3) ● (-9) = 2 ● [3 ● (-9)]

6. 1● (-37) = -37

7. (6 + 0) – 7 = 6 – 7

8. ●7= 1

9. 13 (2 – 6) = 13 (2) – 13(6)

10. (-4 + 3)(5 + 6) = (-4 + 3)(5) + (-4 + 3)(6)

11. 4 + (9 + 6) = (4 + 9) + 6

12. 3(x + 5) = 3 • x + 3 • 5

13. (3 + y) + 0 = 3 + y

14. x • = 1

15. 14xy = 14yx

16. (3 • 9) • 1 = 3 • 9

17. 7 + (-7) = 0

18. 6 • (8 + c) = (8 + c) • 6

19. x + 12 = 12 + x

20. (x + y) • 5 = (y + x) • 5

21. Why is it true that 3(4 + x) = 3(x + 4)?

22. Why is 3(4x) = (3●4)x?

23. Why is 12 – 3x = 3(4 – x)?

Simplify the expression. Justify your steps.24. 3b+ (4b - 6b + 2) –b 25. 2(6x – 5) – 3(5x + 4)