Chapter 1 The Tools of Algebra

1-2 Expressions & Word Problems A company rents a house boat for $200 plus and extra $30 per day. Write an expression that can be used to find the total cost to rent a house boat. Words: two hundred dollar rental fee plus thirty dollars per day Variable: let d represent the number of days Expression: 200 + 30d

Suppose… Suppose the Gregoran family wants to rent a house boat for six days. What will be the total cost? Expression: 200 + 30d 200 + 30d = 200 + 6(30) = 200 + 180 or 380 The total cost will be $380

SALES At a garage sale, Georgia found some used DVDs and CDs that she wanted to buy. Each DVD was marked at $5 and each CD was marked at $3. Write an expression to find the total cost to buy some DVDs and CDs. Words: Variable: Expression:

Suppose… Suppose she wanted to buy 4 DVDs and 7 CDs. Expression: 5d + 3c The total cost will be $

EXIT SLIP! A studio charges a sitting fee of $25 plus $7 for each portrait sheet ordered. Write an expression that can be used to find the total cost to have photographs taken. Then find the cost of purchasing twelve portrait sheets.

1-3 Properties When you make a peanut butter and jelly sandwich, do you spread the peanut butter first or the jelly? The order does not matter because in the end you have a tasty sandwich. Name two other activities where order does not matter. Name two activities where order does matter. Name a mathematical operation in which you can switch the numbers and still have the same value.

Properties of Addition and Multiplication Properties- statements that are true for any numbers Example: 30 + 10 and 10 + 30 have the same value. This illustrates the Commutative Property of Addition.

Commutative Properties Definition: Commutative Property- the order in which numbers are added or multiplied does not change the sum Symbols: For any numbers a and b, a + b = b + a For any numbers a and b, a x b = b x a Examples: 6 + 9 = 9 + 6 4 x 7 = 7 x 4

Associative Properties Definition: Associative Property- the order in which numbers are grouped when they are added or multiplied does not change the sum or product. Symbols: For any numbers a, b, and c, (a + b) + c = a + (b + c) For any numbers a, b, and c, (a x b) x c = a x (b x c) Examples: (3+6) + 1 = 3 + (6 +1) (5 x 9) x 2 = 5 x (9 x 2)

Additive Identity Definition: Additive Identity- when 0 is added to any number, the sum is the number Symbols: For any numbers a, a + 0 = 0 + a = a Examples: 5 + 0 = 5 0 + 5 = 5

Multiplicative Identity Definition: Multiplicative Identity- when any number is multiplied by 1, the product is the number. Symbols: For any numbers a,, a x 1 = 1 x a = a Examples: 8 x 1 = 8 1 x 8 = 8

Multiplicative Property of Zero Definition: Multiplicative Property of Zero- when any number is multiplied by 0, the product is 0. Symbols: For any number a, a x 0 = 0 x a = 0 Examples: 3 x 0 = 0 0 x 3 = 0

Identify Properties 4 + (a + 3) = (a + 3) + 4 The order of the numbers and variables changed. This is the Commutative Property of Addition. 1 x (3c) = 3c The expression was multiplied by 1 and remained the same. This is the Multiplicative Identity Property.

Let’s Practice d + 0 = d 8 x 1 = 8 5 x 7 x 2 = 7 x 2 x 5 14 + (9 + 10) = (14 + 9) + 10 8 x 1 = 8 5 x 7 x 2 = 7 x 2 x 5

Do these properties apply to subtraction or division? One way to find out is to look for a counterexample. Counterexample- an example that shows a conjecture is not true

Find a Counterexample Is division of whole numbers associative? If not, give a counterexample. The Associative Property of Multiplication states (a x b) x c = a x (b x c). Check: (a ÷ b) ÷ c = a ÷ (b ÷ c).

Check: (a ÷ b) ÷ c = a ÷ (b ÷ c). Pick values for a, b, and c. a= 27 b=9 c=3 (27 ÷ 9) ÷ 3 = 27 ÷ (9 ÷ 3) Simplify. (3) ÷ 3 = 27 ÷ (3) 1 = 9?

Your turn! Is subtraction of decimals associative? If not, give a counterexample. Hint: (a + b) + c = a + (b + c)

Simplify Algebraic Expressions (3 + e) + 7 = (e + 3) +7 = e + (3+ 7) = e + 10 8 ● (x ● 5) 8 ● (x ● 5) = 8 ● (5 ● x) = (8 ● 5) ● x = 40x

Your turn! 12 ● (10 ● z) = 10 + (p + 18) =