Review for the State Algebra End-of-Course Exam

1. Review for the State Algebra End-of-Course Exam Sign on to www.jamesrahn.com/radical algebra survey.htm and complete the survey before the workshop begins. Jim Rahn Bob Eckert Maureen McNulty LL Teach, Inc. www.jamesrahn.com www.llteach.com james.rahn@verizon.net

2. O: Operations on Numbers and Expressions • O1.a Use properties of number systems within the set of real numbers to verify or refute conjectures or justify reasoning. • O1.b Use rates, ratios and proportions to solve problems, including measurement problems. • O1.B1 Describe and distinguish among the various uses of variables, including: • Symbols for varying quantities (such as 3x) • Symbols for fixed unknown values (such as 3x – 2 = 7) • Symbols for all numbers in properties (such as x + 0 = x) • Symbols for formulas (such as A = l * w) • Symbols for parameters (such as m and b for slope in y = mx + b) • O1.B2 Use matrices to represent and solve problems. • Adding and subtracting matrices. • Multiplying a matrix by a scalar.

3. O: Operations on Numbers and Expressions • O1.c & O2.a Apply the laws of exponents to numerical and algebraic expressions with integral exponents to rewrite them in different but equivalent forms or to solve problems. • O1.d & O2.d Use the properties of radicals to convert numerical or algebraic expressions containing square roots in different but equivalent forms or to solve problems. • O2.b Add, subtract and multiply polynomial expressions. • O2.c Factor simple polynomial expressions.

4. L. Linear Relationships • L1.a Recognize, describe and represent linear relationships using words, tables, numerical patterns, graphs and equations. • L1.b Describe, analyze and use key characteristics of linear functions and their graphs. • L1.c Graph the absolute value of a linear function and determine and analyze its key characteristics. • L1.d Recognize, express and solve problems that can be modeled using linear functions. Interpret solutions in terms of the context of the problem.

5. L. Linear Relationships • L2.a Solve single-variable linear equations and inequalities with rational coefficients. • L2.b Solve equations involving the absolute value of a linear expression. • L2.c Graph and analyze the graph of the solution set of a two-variable linear inequality. • L2.d Solve systems of linear equations in two variables using algebraic and graphic procedure • L2.e Recognize, express and solve problems that can be modeled using single-variable linear equations; one- or two-variable inequalities; or two-variable systems of linear equations.

6. N: Non-linear Relationships • N1.a Recognize, describe, represent and analyze a quadratic function • N1.b Analyze a table, numerical pattern, graph, equation or context to determine whether a linear, quadratic or exponential relationship could be represented. • N1.c Recognize and solve problems that can be modeled using a quadratic function. Interpret the solution in terms of the context of the original problem.

7. N. Non-linear Relationships • N2.a Solve equations involving several variables for one variable in terms of the others. • N2.b Solve single-variable quadratic equations. • N2B1. Provide and describe multiple representations of solutions to simple exponential equations using concrete models, tables, graphs, symbolic expressions, and technology.

8. D: Data, Statistics, and Probability • D1.a. Interpret and compare linear models for data that exhibit a linear trend in the context of a problem. • D1.b Use measures of center and spread to compare and analyze data sets. • D1.c. Evaluate the reliability of reports based on data published in the media. • D2.a Use counting principles to determine the number of ways an event can occur. Interpret and justify solutions. • D2.b Apply probability concepts to determine the likelihood an event will occur in practical situations. • D2.C1 Determine and apply probabilities in complex situations.

9. O: Operations on Numbers and Expressions

11. 7 4X 3X 21X 12X2 5 20X 35

12. W = 3(3x+5) ft W=(9x+15) ft L= 3(4x+7) ft L = (12x+21) ft

13. 12X 21 108X2 189X 9X W = 3(3x+5) ft W=(9x+15) ft 15 180X 280 L= 3(4x+7) ft L = (12x+21) ft

15. x 6 6x x x2 -2 -2x -12

18. Jeopardy Games to review chapters 2-9 of Discovering Algebra Chapter 2: Proportional Reasoning and Variations

19. L. Linear Relationships

20. Let x = the number of skateboards painted by Chris. Let y = the number of skateboards painted by Kim. Total number of skateboards painted is x + y = 100 Relationship between the number painted by Chris and the number painted by Kim is y = 2x + 10.

22. The p simply shifts the shaded area vertically p units. For the origin to be part of the solution set, you must raise the line up vertically or increase p to some positive number.

23. 500 grams/block is a rate of change and represents the change in mass (grams)of a cart every time a metal block is added to the cart. The y-intercept is 5500 grams and represents the weight of the cart with no blocks in it or an empty cart.

25. 167.4 140

27. Original Formula that relates the weights If we add 2 pounds to the earth’s weight the formula will change to The original weight on Neptune was 1.19E. You can see that the new weight has been increased by 2.38 lbs.

28. #6 on page 152 On many packages the weight is given in both pounds and kilograms. The table shows the weights listed on a sample of items. a. Use the information in the table to find an equation that relates weights in pounds and kilograms. Explain what the variables represent in your equation. b. Use your equation to calculate the number of kilograms in 30 pounds. c. Calculate the number of pounds in 25 kilograms. L1 L2 Study L2/L1 to see if they weights are proportional X = weight in kilograms Y= weight in pounds When y=30 pounds, 30=2.2x or x = 13.6 kilograms When y=25 kilograms, y=2.2(25) =55 kilograms

29. # 8 on page 153 Use the table of values to answer each question. a. Are the data in the table related by a direct variation or an inverse variation? Explain. b. Find an equation to fit the data. You may use your calculator graph to see how well the equation fits the data. c. Use your equation to predict the value of y when x is 32. Because the product of the x- and y-values is approximately constant, it is an inverse relationship. Using the average value in L3:

30. #8 on page 208 Suppose a new small-business computer system costs $5,400. Every year its value drops by $525. a. Define variables and write an equation modeling the value of the computer in any given year. b. What is the rate of change, and what does it mean in the context of the problem? c. What is the y-intercept, and what does it mean in the context of the problem? d. What is the x-intercept, and what does it mean in the context of the problem? Y= value in dollars x= number of years The rate of change is 525; in each additional year, the value of the computer system decreases by $525. The y-intercept is 5400; the original value of the computer system is $5,400. The x-intercept represents when the value of the business is zero or at 10.3 years.

31. #5 on page 268 Consider the point-slope equation y=-3.5+2(x+4.5) a. Name the point used to write this equation. b. Write an equivalent equation in intercept form. c. Factor your answer to 5b and name the x-intercept. d. A point on the line has a y-coordinate of 16.5. Find the x-coordinate of this point and use this point to write an equivalent equation in point-slope form. e. Explain how you can verify that all four equations are equivalent. Point: (-4.5, -3.5) Equation: y=-3.5+2(x+4.5) y=-3.5+2x+9 y=2x+5.5 Factored Equation: y=2x+5.5=2(x+2.75) X-intercept: x=-2.75 16.5=2x+5.5 → 2x=11→ x = 5.5 Y=16.5+2(x-5.5) Could graph each equation, use table of values, or put all equations in intercept form.

33. #4 on page 328 Show how to solve this system of equations: Y =16 +4.3(x -5) Y =-7 + 4.2x or Y=16+4.3x-21.5 or Y=-5.5+4.3x 16+4.3(x -5) =-7 + 4.2x 16+4.3x -21.5 =-7 + 4.2x 4.3x -5.5=-7 + 4.2x 0.1x =-1.5 The two lines are nearly parallel. x =-15 Y=-7+4.2(-15)=-70

34. N: Non-linear Relationships

36. First rewrite the inequality in slope intercept form A and C are eliminated Test the origin to see that it satisfies the inequality.

37. Zeros at x = 0 and x = 2 2 0 (1, -2)

38. This equation represents exponential growth or growing value. This equation should show y values that are negative. This equation can be rewritten as y=2-x and contains the point (0,1). This equation can be rewritten as y=2(2-x ) and contains the point (0,2).

39. Think about undoing this equation.

41. The initial height of the ball will take place when t = 0, so we need to find h(0). The student may also enter the equation in their calculator and find the initial height using The home screen a table a graph

42. 37 alt. The height (h) in feet of a ball t seconds after being dropped is given by the function h(t) = 9 – 16t2 . At what time will the ball hit the ground? We’ll select +3/4 since time began at t = 0.

43. #5 on page 384 For the table, find the value of the constants A and r such that or Then use your equations to find the missing values. The table illustrates growth, not decay so choose So 1505.9072

44. #7 on page 430 • Draw graphs that fit these descriptions: • a function that has a domain of -5 ≤x ≤1, a range of -4 ≤y ≤4, and f (-2) =1 • b. a relationship that is not a function and that has inputs on the interval -6 ≤x ≤4 and outputs on the interval 0 ≤y ≤5.

45. # 14 on page 547 Make a rectangle diagram to factor each expression. a. x2 +7x +12

46. # 14 on page 547 Make a rectangle diagram to factor each expression. b. x2 -14x +49

47. # 14 on page 547 Make a rectangle diagram to factor each expression. c. x2 + 3x -28

48. # 14 on page 547 Make a rectangle diagram to factor each expression. d. x2 -81

49. Jeopardy Games to review chapters 2-9 and 11 of Discovering Algebra Chapter 6: Exponents and Exponential Models

50. D. Data Analysis and Statistical Analysis and Probability