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1. Algebraic Expressions Lesson 5-1

2. Evaluate an Algebraic Expression The branch of mathematics that involve expressions with variables is called algebra. In algebra, the multiplication sign is often omitted. The numerical factor of a multiplication expression that contains a variable is called a coefficient. So, 6 is the coefficient of 6d.

3. Example 1 Evaluate 2(n + 3) if n = -4. 2(n + 3) = 2(-4 + 3) = 2(-1) = -2

4. Example 2 Evaluate 8w – 2v if w = 5 and v = 3. 8w – 2v = 8(5) – 2(3) = 40 – 6 = 34

5. Example 3 Evaluate 4y3 + 2 if y = 3. 4y3 + 2 = 4(3)3 + 2 = 4(27) + 2 = 110

6. Got it? 1, 2 & 3 Evaluate each expression if c = 8 and d = -5. a. c – 3 b. 15 – c c. 3(c + d) d. 2c – 4d e. d2 – c2 f. 2d2 + 5d 5 7 9 36 25 17

7. Example 4 Athletic trainers use the formula , where a is a person’s age, to find the minimum training heart rate. Find Latrina’s minimum training heart if she is 15 years old. = = = = 123 Latrina’s minimum training heart rate is 123 beats per minute.

8. Got it? 4 To find the area of a triangle, use the formula , where b is the base and h is the height. What is the area in square inches of a triangle with a height of 6 inches and a base of 8 inches?

9. Example 5 To translate a verbal phrase to an algebraic expression, the first step is to define a variable. When you define a variable, you choose a variable to represent the unknown. Marisa wants to buy a DVD player that costs \$150. She already saved \$25 and plans to save an additional \$10 each week. Write an expression that represents the total amount of money Marissa has saved after any number of weeks. 25 + 10w represents the total saved after any number of weeks.

10. Example 6 Refer to Example 5. Will Marisa have saved enough money to buy the \$150 DVD player in 11 weeks? Use the expression 25 + 10w. 25 + 10w = 25 + 10(11) = 25 + 110 = 135 Marisa will only have saved \$135. She needs \$150, so she does not have enough.

11. Got it? 5 & 6 An iPod costs \$70 and song downloads cost \$0.85 each. Write an expression that represents the cost of the iPod and x number of downloaded songs. Then find the cost if 20 songs are downloaded. 70 + 0.85x \$87

12. Sequences Lesson 5-2

13. Vocabulary • Sequence – an ordered list of numbers • 1, 3, 5, 7, 9, 11, 13… • Term – one of the numbers in the sequence • 7 is a term in the sequence above • Arithmetic Sequence – when the difference is consistent between to consecutive terms. • the difference between any two consecutive numbers is the same • Common Difference – the difference between two terms • The common difference is 2

14. Example 1 In an arithmetic sequence, the terms can be whole numbers, fractions, or decimals. Describe the relationship between the terms in the arithmetic sequence 8, 13, 18, 23,…. Then write the next three terms In the sequence. 23 + 5 = 28 28 + 5 = 33 33 + 5 = 38 The next three terms are 28, 33, and 38.

15. Example 2 Describe the relationship between the terms in the arithmetic sequence 0.4, 0.6, 0.8, 1.0,…. Then write the next three terms In the sequence. 1.0 + 0.2 = 1.2 1.2 + 0.2 = 1.4 1.4 + 0.2 = 1.6 The next three terms are 1.2, 1.4, and 1.6

16. Got it? 1 & 2 Describe the relationship between the terms in each arithmetic sequence. Then write the next terms in the sequence. a. 0, 13, 26, 39, … b. 4, 7, 10, 13… c. 1.0, 1.3, 1.6, 1.9… d. 2.5, 3.0, 3.5, 4.0…

17. Write an Algebraic Expression In a sequence, each term has a specific position within the sequence. Consider the sequence 2, 4, 6, 8… Notice that the position number increases by 1, the value of the term increases by 2.

18. Write an Algebraic Expression You can also write an algebraic expression to represent the relationship between any term in a sequence and its position in sequence. In this case, if n represents the position in the sequence, the value of the term is 2n.

19. Example 3 The greeting cards that Meredith makes are sold in boxes at a gift store. The first week, the store sold 5 boxes. Each week, the store sells five more boxes. This patterns continues. What algebraic expression can be used to find the total number of boxes the end of the 100thweek? What is the total? Each term is 5 times its position. So, the expression is 5n. 5n = 5(100) = 500 In 100 week, 500 boxes will be sold.

20. Got it? 3 If the pattern continues, what algebraic expression can be used to find the number of circles used in any figure. How many circles will be in the 50th figure?

21. Properties Lesson 5-3

22. Commutative Properties Words: the order in which numbers are added or multiplied does not change the sum or product. Symbols: a + b = b + a a ∙ b = b ∙ a Examples: 6 + 8 = 8 + 6 4 ∙ 7 = 7 ∙ 4

23. Associative Properties Words: the order in which numbers are grouped when added or multiplied does not change the sum or product. Symbols: (a + b) + c = a + (b + c) (a ∙ b) ∙ c = a ∙ (b ∙ c) Examples: (3 + 6) + 8 = 3 + (6 + 8) (5 ∙ 2) ∙ 7 = 5 ∙ (2 ∙ 7)

24. Number Properties A property is a statement that is true for any number. The following properties are also true for any numbers.

25. Example 1 Name the property shown by the statement. 2  (5  n) = (2  5)  n The order of the numbers and variable does not change, but their grouping did. This is the Associative property of Multiplication.

26. Got it? 1 Name the property shown by the statement. a. 42 + x + y = 42 + y + x Communicative (+) b. 3x + 0 = 3x Identity (+)

27. Counterexample A counterexample is an example that shows a statement is not true. Statement: All songs are only 3 minutes. Counterexample: The song “We Are The Champions” by Queen is 4 minutes 21 seconds.

28. Example 2 State whether the following conjecture is true or false. If false, provide a counterexample. Division of whole numbers is commutative. Write two division expressions that are commutative. Let’s pick some nice numbers… like 27, 9 and 3 27 ÷ 9 = 9 ÷ 27 3 ≠ We found a counterexample, so division is not commutative.

29. Got it? 2 State whether the following conjecture is true or false. If false, provide a counterexample. The difference of two different whole numbers is always less than both of the two numbers. false; 8 – 2  2 – 8

30. Example 3 Alana wants to buy a sweater that cost \$28, sunglasses that cost \$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. 38 + 14 + 16 + 22 = (38 + 22) + (14 + 16) =60 + 30 =90 The total cost is \$90.

31. Got it? 3 Lance made four phone calls from his cell phone today. The calls lasted 4.7, 9.4, 2.3, and 10.6 minutes. Use mental math to find the total cost amount of time he spent on his phone. 27 minutes

32. Example 4 Simplify the expression. Justify each step. (3 + e) + 7 (3 + e) + 7 = (e + 3) + 7 Commutative Property of Addition = e + (3 + 7) Associative Property of Addition = e + 10

33. Example 5 Simplify the expression. Justify each step. x ∙ (8 ∙ x) x ∙ (8 ∙ x) = x  (x  8) Commutative Property of Multiplication = (x ∙ x) ∙ 8 Associate Property of Multiplication = 8x2

34. Got it? 4 & 5 Simplify the expression. Justify each step. 4 ∙ (3c ∙ 2) 4 ∙ (3c ∙ 2) = 4  (2  3c) Commutative Property of Multiplication = (4 ∙ 2) ∙ 3c Associate Property of Multiplication = (4 ∙ 2 ∙ 3) c Associate Property of Multiplication = 24c

35. Ticket Out The Door Which of the following is an example of the Community Property of Addition? A. (3 ∙ 4) + 5 = 5 + (3 ∙ 4) B. (7 + 8) + 2 = 7 + (8 + 2) C. 8 ∙ 9 = 9 ∙ 8 D. 1 + 0 = 1

36. The Distributive Property Lesson 5-4

37. Distributive Property Words: To multiply a sum or different by a number, multiply each term inside the parentheses by the number outside the parentheses. Symbols: a(b + c) = ab + ac a(b – c) = ab – ac Example: 5(6 + 7) = 5(6) + 5(7) 4(2 – 8) = 4(2) – 4(8)

38. Distributive Property You can model the Distributive Property with algebraic expressions using algebra tiles. The expression 2(x + 2) is modeled. Model x + 2 using algebra tiles. Double the amount of tiles to represent 2(x + 2). 2(x + 2) = 2(x) + 2(2) = 2x + 4 No matter what x is, 2(x + 2) will always equal 2x + 4. Rearrange the tiles by grouping together the ones with the same shapes.

39. Example 1 Use the Distributive Property to write the expression as an equivalent expression. The evaluate the expression. • 5(12 + 4) 5(12 + 4) = 5(12) + 5(4) 5(16) = 60 + 20 80 = 80 • (20 – 3)8 (20 – 3)8 = 8(20) – 8(3) 17 ∙ 8 = 160 – 24 136 = 136

40. Got it? 1 Use the Distributive Property to write the expression as an equivalent expression. The evaluate the expression. a. 5(-9 + 11) b. 7(10 – 5) c. (12 – 8)9 35 10 36

41. Example 2 & 3 Use the Distributive Property to write each expression as an equivalent algebraic expression. • 4(x + 5) 4(x + 5) = 4x + 4(5) = 4x + 20 • 6(y – 10) 6(y - 10) = 6y – 6(10) = 6y – 60

42. Example 4 & 5 Use the Distributive Property to write each expression as an equivalent algebraic expression. • -3(m – 4) -3(m – 4) = -3m – (-3)(4) = -3m + 12 • 9(-3n – 7y) 9(-3n – 7y) = 9(-3n) – (9 ∙ 7y) = -27n – 63y

43. Example 6 Use the Distributive Property to write the expression as an equivalent algebraic expression: (x – 6) (x – 6) = (x) – (6) = x - 2

44. Got it? 2-6 a. 6(a + 4) b. (m + 3n)8 c. -3(y – 10) d. (w – 4) 8m + 24n 6a + 24 w – 2 -3y + 30

45. Example 7 – how to solve story problems with the Distributive Property On a school visit to Washington D.C., Daniel and his class visited the Smithsonian Air and Space Museum. Tickets to the IMAX movie cost \$8.99. Find the total cost of 20 students to see the movie. =20(9 – 0.01) = 20(9) – 20(0.01) 180 – 0.20 \$179.80

46. Got it? 7 A sports club rents dirt bikes for \$37.50 each. Find the total cost for the club to rent 20 bikes. Justify your answer by using the Distributive Property. \$750 =20(37 + 0.50) = 20(37) + 20(0.50)

47. Simplifying Algebraic Expressions Lesson 5-5

48. Vocabulary Term: the expression 5x + 8y – 9 has 3 terms. Coefficient: 5 in 5x is the coefficient. Constant: 9 in the expression 5x + 8y – 9 is the constant. Like terms: 5x, 6x, and 7x are like terms since they all have an x.

49. Identify Parts of an Expression Like terms contain the same variables to the same powers. For example, 3x2 and -7x2 are like terms. So are 8xy2 and 12xy2. But 10x2z and 22xz2 are not like terms.

50. Example 1 Identify the terms, like terms, coefficients, and constant in the expression 6n – 7n – 4 + n 6n – 7n – 4 + n = 6n + (-7n) + (-4) + n Terms: 6n, -7n, -4, n Like terms: 6n, -7n, n (all of these terms have the same variable) Coefficients: 6, -7, 1 Constant: -4 (This is the only term without a variable)