1. Algebra 2 Algebraic Expressions Lesson 1-3

2. Goals Goal To evaluate algebraic expressions. To simplify algebraic expressions. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

3. Essential Question Big Idea: Variables, Properties How can you use the properties of real numbers to simplify algebraic expressions? –Some mathematical phrases and real-world quantities can be represented using algebraic expressions. –Variables can represent variable quantities in real world situations and in patterns. –The properties that apply to real numbers also apply to variables that represent them.

4. Vocabulary Evaluate Term Coefficient Constant Term Like Terms

5. Modeling Words with Algebraic Expressions To represent real-world situations containing variable quantities with mathematical phrases you must translate word phrases into algebraic expressions, looking for words that describe mathematical operations (addition, subtraction, multiplication, division).

6. What words indicate a particular operation? Addition Sum Plus More than Increase(d) by Perimeter Deposit Gain Greater (than) Total Subtraction Minus Take away Difference Reduce(d) by Decrease(d) by Withdrawal Less than Fewer (than) Loss of

7. Words for Operations - Examples

9. What words indicate a particular operation? Multiply Times Product Multiplied by Of Twice (×2), double (×2), triple (×3), etc. Half (×½), Third (×⅓), Quarter (×¼) Percent (of) Divide Quotient Divided by Half (÷2), Third (÷3), Quarter (÷4) Into Per Percent (out of 100) Split into __ parts

10. Words for Operations - Examples

12. Write an algebraic expression for each word phrase. A. 9 less than a number w 9 less than a number w w – 9 B. 3 increased by the difference of p and 5 3 increased by the difference of p and 5 3 + (p – 5) Example:

13. Write an algebraic expression for each word phrase. A. 88 times the difference of h and 4 88 times the difference of h and 4 88 (h – 4) B. the quotient of a number f and 6 quotient of f and 6 f  6 f 6 Your Turn:

14. Your turn: 1) m increased by 5. 2)7 times the product of x and t. 3)11 less than 4 times a number. 4)two more than 6 times a number. 5)the quotient of a number and 12. 1) m + 5 2)7xt 3)4n - 11 4)6n + 2 5)

15. Your Turn: 1.Twice the sum of x and three D 2.Two less than the product of 3 and x E 3.Three times the difference of x and two B 4.Three less than twice a number x A 5.Two more than three times a number x C A.2x – 3 B.3(x – 2) C.3x + 2 D.2(x + 3) E.3x – 2 Match the verbal phrase and the expression

16. Write a word phrase for the algebraic expression 9 – 3c. 9 – 3c 9 minus the product of 3 and c Example:

17. Write a word phrase for the algebraic expression 7 + 19b. 7 + 19b 7 plus the product of 19 and b Your Turn:

18. Modeling a Situation To model a situation with an algebraic expression, do the following: 1.Identify the actions that suggest operations. 2.Define one or more variables to represent the unknowns. 3.Represent the actions using the variables and the operations.

19. Example: You start with $20 and save $6 each week. What algebraic expression models the total amount you save? Identify action Operations starting amount plus amount saved times number of weeks Define Variables Letw= the number of weeks. Write Expression 20+6 w The expression 20 + 6w models the situation.

20. Write an algebraic expression to represent each situation. A. the number of apples in a basket of 12 after n more are added B. the number of days it will take to walk 100 miles if you walk M miles per day Add n to 12. Divide 100 by M. 12 + n Your Turn:

21. You had $150, but you are spending $2 each day. What algebraic expression models this situation? Answer: Let d = the number of days 150 – 2d

22. Definition Evaluate – To evaluate an expression is to find its value. To evaluate an algebraic expression, substitute numbers for the variables in the expression and then simplify the expression.

23. Order of Operations Rules for arithmetic and algebra expressions that describe what sequence to follow to evaluate an expression involving more than one operation.

24. Remember the phrase “Please Excuse My Dear Aunt Sally” or PEMDAS. ORDER OF OPERATIONS 1. Parentheses - ( ) or [ ] 2. Exponents or Powers 3. Multiply and Divide (from left to right) 4. Add and Subtract (from left to right)

25. The Rules Step 1: First perform operations that are within grouping symbols such as parenthesis (), brackets [], and braces {}, and as indicated by fraction bars. Parenthesis within parenthesis are called nested parenthesis (( )). If an expression contains more than one set of grouping symbols, evaluate the expression from the innermost set first. Step 2: Evaluate Powers (exponents) or roots. Step 3: Perform multiplication or division operations in order from left to right. Step 4: Perform addition or subtraction operations in order by from left to right.

26. Evaluate the expression for the given values of the variables. Substitute 5 for x and 2 for y. Multiply from left to right. Add and subtract from left to right. 2(5) – (5)(2) + 4(2) 10 – 10 + 8 0 + 8 8 2x – xy + 4y for x = 5 and y = 2 Example:

27. Evaluate the expression for the given values of the variables. q 2 + 4qr – r 2 for r = 3 and q = 7 Substitute 3 for r and 7 for q. Multiply from left to right. Add and subtract from left to right. 49 + 4(7)(3) – 9 49 + 84 – 9 124 Evaluate exponential expressions. (7) 2 + 4(7)(3) – (3) 2 Your Turn:

28. Substitute 2 for x and 5 for y. Multiply from left to right. Add and subtract from left to right. 4(5) – 2(25) + 3(5) 20 – 50 + 15 –15 Evaluate exponential expressions. Evaluate x 2 y – xy 2 + 3y for x = 2 and y = 5. (2) 2 (5) – (2)(5) 2 + 3(5) Your Turn:

29. Definition Term – an expression that is a number, a variable, or the product of a number and one or more variables. Example: –The expression 6x + yz – 7 has 3 terms, 6x, yz, and 7. 6x and yz are variable terms; their values vary as x, y and z vary. 7 is a constant term; 7 is always 7.

30. Definition Coefficient – The numerical factor of a term. Example: –The coefficient of 3x 2 is 3.

31. Definition Like Terms – terms in which the variables and the exponents of the variables are identical. –The coefficients of like terms may be different. Example: –3x 2 and 6x 2 are like terms. –ab and 3ab are like terms. –2x and 2x 3 are not like terms.

32. Definition Constant Term– is a term with no variable. –Constants terms are like terms. –Example: in the expression - 4x + 3y 2 – 5, the constant term is – 5.

33. Recall that the terms of an algebraic expression are separated by addition or subtraction symbols. Like terms have the same variables raised to the same exponents. Constant terms are like terms that always have the same value. Example:

34. Terms that are written without a coefficient have an understood coefficient of 1. x 2 = 1x 2 Remember! To simplify an algebraic expression, combine like terms by adding or subtracting their coefficients. Algebraic expressions are equivalent if they contain exactly the same terms when simplified. Simplifying Algebraic Expressions

36. Simplify the expression. Identify like terms. Combine like terms. 3x 2 + 4x 2 = 7x 2 3x 2 + 2x – 3y + 4x 2 7x 2 + 2x – 3y Example:

37. Simplify the expression. Distribute, and identify like terms. Combine like terms. 7jk – 7jk = 0 j(6k 2 + 7k) + 9jk 2 – 7jk 6jk 2 + 7jk + 9jk 2 – 7jk 15jk 2 Example:

38. –6x + 3xy – 9y – 11xy Simplify the expression –3(2x – xy + 3y) – 11xy. Distribute, and identify like terms. Combine like terms. 3xy – 11xy = –8xy –6x – 8xy – 9y Your Turn:

39. Apples cost $2 per pound, and grapes cost $3 per pound. Write and simplify an expression for the total cost if you buy 10 lb of apples and grapes combined. 2A + 3(10 – A) Combine like terms. Let A be the number of pounds of apples. Distribute 3. = 30 – A Then 10 – A is the number of pounds of grapes. = 2A + 30 – 3A Example: Application

40. Apples cost $2 per pound, and grapes cost $3 per pound. What is the total cost if 2 lb of the 10 lb are apples? Evaluate 30 – A for A = 2. 30 – (2)= 28 The total cost is $28 if 2 lb are apples. Example: Continued

41. Distribute 80. A travel agent is selling 100 discount packages. He makes $50 for each Hawaii package and $80 for each Cancún package. Write an expression to represent the total the agent will make selling a combination of the two packages. Let h be the number of Hawaii packages. 50h + 80(100 –h) = 8000 – 30h Combine like terms. Then 100 – h is the remaining Cancun packages. = 50h + 8000 –80h Your Turn:

42. How much will he make if he sells 28 Hawaii packages? 8000–30(28) = 8000–840 Evaluate 8000 –30h for h = 28. He will make $7160. = 7160 Continued

43. Assignment Section 1-3, Pg 22 – 24: #1 – 9 all, 10 – 54 even.