(7) Lesson 5.3 - Properties of Operations
- 1. Describe the pattern in each sequence. 1. 2, 22, 42, 62, . . . 2. 2.8, 6, 9.2, 12.4, . . . Write the next three terms of each sequence. 3. 4, 12, 20, 28, . . . 4. 2.1, 2.8, 3.5, 4.2, . . . 5. Every 18 months, National Surveys conducts a population survey of the United States. If they conducted a survey in September 2003, when will they conduct the next four surveys? 6. What is the next term in the sequence 3.2, 12.8, 22.4, 32, …? Course 2, Lesson 5-3
- 2. Answers 1. Add 20 to the previous term. 2. Add 3.2 to the previous term. 3. 36, 44, 52 4. 4.9, 5.6, 6.3 5. March 2005, September 2006, March 2008, September 2009 6. 41.6 Course 2, Lesson 5-3
- 3. HOW can you use numbers and symbols to represent mathematical ideas? Expressions and Equations Course 2, Lesson 5-3
- 4. • 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. • 7.EE.2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. Course 2, Lesson 5-3 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Expressions and Equations
- 5. Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. 5 Use appropriate tools strategically. 7 Look for and make use of structure. Course 2, Lesson 5-3 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. Expressions and Equations
- 6. • To identify properties of operations • To determine if conjectures are true or false and provide counterexamples for false conjectures • To use properties to simplify algebraic expressions Course 2, Lesson 5-3 Expressions and Equations
- 7. Course 2, Lesson 5-3 Expressions and Equations • Commutative Property • Associative Property • property • Additive Identity Property • Multiplicative Identity Property • Multiplicative Property of Zero • counterexample
- 8. Course 2, Lesson 5-3 Expressions and Equations Words The Commutative Property states that the order in which the numbers are added or multiplied does not change the sum or product. Addition Multiplication Symbols a + b = b + a a • b = b • a Examples 6 + 1 = 1 + 6 7 • 3 = 3 • 7
- 9. Course 2, Lesson 5-3 Expressions and Equations Words The Associative Property states that the way in which numbers are grouped when they are added or multiplied does not change the sum or product. Addition Multiplication Symbols a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c Examples 2 + (3 + 8) = (2 + 3) + 8 3 • (4 • 5) = (3 • 4) • 5
- 10. 1 Need Another Example? Step-by-Step Example 1. Name the property shown by the statement 2 • (5 • n) = (2 • 5) • n. The order of the numbers and variable did not change, but their grouping did. This is the Associative Property of Multiplication.
- 11. Answer Need Another Example? Name the property shown by the statement (3 • m) • 2 = 2 • (3 • m). Commutative Property of Multiplication
- 12. 1 Need Another Example? 2 Step-by-Step Example 2. State whether the following conjecture is true or false. If false, provide a counterexample. Write two division expressions using the Commutative Property. Division of whole numbers is commutative. 15 ÷ 3 = 3 ÷ 15 5 ≠ State the conjecture. Divide. ? The conjecture is false. We found a counterexample. That is, 15 ÷ 3 ≠ 3 ÷ 15. So, division is not commutative.
- 13. Answer Need Another Example? State whether the following conjecture is true or false. If false, provide a counterexample. Subtraction of whole numbers is associative. false; Sample answer: (12 – 5) – 3 ≠ 12 – (5 – 3)
- 14. 1 Need Another Example? 2 3 4 Step-by-Step Example 3. Alana wants to buy a sweater that costs $38, sunglasses that costs $14, a pair of jeans that costs $22, and a T-shirt that costs $16. Use mental math to find the total cost before tax. Write an expression for the total cost. You can rearrange the numbers using the properties of math. Look for sums that are multiples of ten. Commutative Property of Addition Associative Property of Addition The total cost of the items is $90. 38 + 14 + 22 + 16 = 38 + 22 + 14 + 16 = (38 + 22) + (14 + 16) Add. Simplify. = 60 + 30 = 90
- 15. Answer Need Another Example? In a garden, a decorative pool in the shape of a box is 2 feet deep, 17 feet long, and 5 feet wide. Use mental math to find the volume of water in the pool. 170 ft3
- 16. 1 Need Another Example? 2 3 Step-by-Step Example 4. Simplify (7 + g) + 5. Justify each step. Commutative Property of Addition Associative Property of Addition (7 + g) + 5 = (g + 7) + 5 Simplify.= g + 12 = g + (7 + 5)
- 17. Answer Need Another Example? Simplify 6 + (d + 8). Justify each step. 6 + (d + 8) = 6 + (8 + d) Commutative (+) = (6 + 8) + d Associative (+) = 14 + d Simplify.
- 18. 1 Need Another Example? 2 3 Step-by-Step Example 5. Simplify (m • 11) • m. Justify each step. Commutative Property of Multiplication Associative Property of Multiplication (m • 11) • m Simplify.= 11m2 = (11 • m) • m = 11 • (m • m)
- 19. Answer Need Another Example? Simplify a • (9 • b). Justify each step. a • (9 • b) = (a • 9) • b Associative (×) = (9 • a) • b Commutative (×) = 9ab Simplify.
- 20. How did what you learned today help you answer the HOW can you use numbers and symbols to represent mathematical ideas? Course 2, Lesson 5-3 Expressions and Equations
- 21. How did what you learned today help you answer the HOW can you use numbers and symbols to represent mathematical ideas? Course 2, Lesson 5-3 Expressions and Equations Sample answers: • Using numbers and symbols to represent mathematical properties • Finding a counterexample to show a conjecture is false • Applying properties and use mental math when solving real-world problems
- 22. Give examples of both the Associative Property and the Commutative Property. Ratios and Proportional RelationshipsExpressions and Equations Course 2, Lesson 5-3