Summer Review Packet for Students Entering Honors Algebra (9-4) in September

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1 Page 1 of 14 Summer Review Packet for Students Entering Honors Algebra (9-4) in September Introduction The learning objectives and sample problems that follow were adapted from the Honors 8th grade math curriculum for the South Orange Maplewood School District. If your student is from another district or from a private school there is a possibility he or she may not have been taught these topics. The packet will be collected on the first full day of school at the start of class. The packet will be graded based on completion and will count as a quiz. Late submissions will lose 10% per day late. There will be a test on this material in the middle of the second week of school. The following statement is taken from the 8 th grade math at a glance document: Students will solve problems with scientific notation, rational, and irrational numbers. They use linear equations to represent real world situations, and they strategically select efficient strategies to solve them. Students use functions to describe mathematical relationships. They use a constant rate of change, slope, and proportionality to find missing values. Students learning will emphasize function notation, and students will describe functions as continuous or non-continuous, increasing or decreasing, and linear or nonlinear. Grade 8 Honors uses quadratic functions to solve equations and represent realworld data; this includes factoring quadratics and writing quadratic equations. Learning Objective 1: Represent equivalent forms of exponential expressions. Represent situations and solve problems using scientific notation. Know and apply the properties of integer exponents to generate equivalent numerical and/or variable expressions. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

2 Page 2 of 14 Sample problems 1. An algebra class had this problem on a quiz: Find the value of students reasoned differently. Student 1 Two times three is six. Six squared is thirty-six. Student 2 Three squared is nine. Two times nine is eighteen. 2 2x when x = 3. Two Who was correct? Explain why. 2. Simplify each expression without a calculator by using properties of exponents x 2x 2 5 ( 3 ) x y 3 xy 3. 12x 2 3x x ( 2x) ( 2x) (2 3) Use the distributive property and the properties of exponents to write an equivalent expression without parentheses ( 2 x )( x 3 x )

3 Page 3 of Write three different expressions that are equivalent to 12x. Try to use at least three different exponent rules In 2002, about pounds of cotton were produced in California. The cotton was 5 planted on acres of land. What was the average number of pounds of cotton produced per acre? Round your answer to the nearest whole number. 11. In 1954, 50 swarms of locusts were observed in Kenya. The largest swarm covered an area of 200 square kilometers. The average number of locusts in a swarm is about locusts per square kilometer. a. About how many locusts were in Kenya s largest swarm? Write your answer in scientific notation. b. The average mass of a desert locust is 2 grams. What was the total mass (in kilograms)of Kenya s largest swarm? Write your answer in scientific notation. Learning Objective #2: Solve linear equations in one variable. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

4 Page 4 of 14 Sample problems Solve each equation, if possible. If the equation has no solution, say so x + 6 = 8x = -3(x + 4) 13. 2y 8 = (c + 1) = 9 +5c 14. 3(x + 4) x = 2(x 6) 19. 4(y 1) = 2y (1 + n) = 2(7 + n) 20. 2(m 3) + 5 = 2(m + 1) (y + 2) = (x + 3) = 3x x Learning Objective #3: Recognize that the primary characteristic of linearity is a constant rate of change. Generate a table given a recursive routine and a starting point Find the rate of change given an equation, graph, table, or real-world problem

5 Page 5 of 14 Develop fluency in connecting the rate of change represented in equations and tables, tables and graphs, and graphs and equations Find the equation of a line given a table of values Sample problems Manny has a part-time job as a waiter. He makes $45 per day plus tips. He has calculated that his average tip is 12% of the total amount his customers spend on food and beverages. 22. Define variables and write an equation in intercept form to represent Manny s daily income in terms of the amount his customers spend on food and beverages. 23. Graph this relationship for food and beverage amounts between $0 and $ Write and evaluate an expression to find how much Manny makes in one day if his customers spend $312 for food and beverages. 25. What amounts spent on food and beverages will give him a daily income between $105 and $120? A car is moving at a speed of 68 miles per hour from Dallas to San Antonio. Dallas is 272 miles from San Antonio.

6 Page 6 of Create a table of values relating time to distance from San Antonio for 0 to 5 hours in 1 hour intervals 27. Graph the information in your table. 28. Draw a line through the points of your plot. What is the real-world meaning of this line? What does the line represent that the points alone do not? 29. When is the car within 100 miles of San Antonio? Explain how you got your answer. 30. How long does it take the car to reach San Antonio? Explain how you got your answer Learning Objective #4: Define and calculate slope. Graph a line given various attributes. Determine the equation given a table, graph, or situation, with or without context. Write equations in point-slope form. Graph a line using two points, a point and the slope, or points from a table.

7 Page 7 of 14 Sample problems 31. Find the slope of the line through the points (2, 4) and (4, 7). Then name another point on the same line. Explain how you got your point. Use the table below to complete the problems. X Y Find the rate of change in the table. Explain how you found this value. 33. Find the output value that corresponds to an input value of 0. What is this value called? 34. Use your results to write an equation in intercept form. Each table gives the coordinates of four points on a different line. Without calculating, can you tell whether the slope of the line through each set of points is positive, negative, zero, or undefined? Explain how you can tell. Then write an equation for each table of values. 35. Positive/negative/zero? Equation? X Y

8 Page 8 of Positive/negative/zero? Equation? X Y Positive/negative/zero? Equation? X Y Positive/negative/zero? Equation? X Y A line has slope and goes through the point (3, 5). Graph the line, and then name 4 two other points on this same line. Use the slope formula to check that the slope 3 between each of the two new points and the point (8, 12) is equivalent to. 4

9 Page 9 of Give the equation for each graphed line in the most convenient form. a. b. equation equation c. Pick a line from problem 2. Write the equation of a line parallel to the line you chose. Then, draw the graph of that line. (on the same coordinate plane) 41. Write the equation of the line described in either intercept form or point-slope form. a. The line passes through the point (6,11) and has a slope of 3 2. b. The line passes through the point (-4,10) and has a slope of 1. 4

10 Page 10 of 14 Learning Objective #5: Model and solve systems of linear equations. Use the standard form of a linear equation to represent real-world problems. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. Solve systems of linear equations by graphing and estimating solutions. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. Solve systems of two linear equations in two variables algebraically, using substitution or elimination. Solve mathematical problems related to systems, for example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Sample problems Solve each system using either elimination or substitution. If a system has no solution or infinite solutions, say so. Otherwise, express the solution as an (x, y) pair. 3x+ 2y = 17 3x 4y = x 2y = x 2y = 4 4x 2y = 6 y = 50 8x 44. 8x 4y = x+ 7 y = 44

11 Page 11 of y = 7x 5 4x+ 3y = x 8y = y = x 4 2 Kendra is painting her dining room white and her living room blue. She spends a total of $132 on 5 cans of paint. The white paint costs $24 per can, and the blue paint costs $28 per can. 48. Write a linear equation in standard form that expresses the relationship between the number of cans of blue paint and the number of cans of white paint Kendra bought. Identify your variables. 49. Write a linear equation in standard form that expresses the relationship between the cost of each color of paint and the total cost of the paint. 50. Solve this system to find the number of cans of each color of paint that Kendra bought. 51. How much would Kendra have saved by switching the colors of the dining room and living room? Explain.

12 Page 12 of Solve the following systems by graphing: a. 3 y = x y = x 3 2 b. y = 3x+ 1 2x+ y = 1 DeShawn and Shayna are selling flower bulbs for a school fundraiser. Customers can buy bags of windflower bulbs and bags of daffodil bulbs. DeShawn sold 10 bags of windflower bulbs and 12 bags of daffodil bulbs for a total of $380. Shayna sold 6 bags of windflower bulbs and 8 bags of daffodil bulbs for a total of $ Write an equation expressing the relationship between the amount of bags of each type Deshawn sold and the total money he raised. Identify your variables. 54. Write an equation expressing the relationship between the amount of bags of each type Shayna sold and the total money she raised.

13 Page 13 of Solve the system. 56. Interpret your solution in the context of the problem. Chris pays $0.39 for each digital photo he has printed. Debbie buys a photo printer for $48. It costs $0.24 per photo for ink and paper to print a photo using the printer. 57. Write an equation that relates the total cost Chris pays as a function of the number of prints he has printed. 58. Write an equation that relates the total cost Debbie pays as a function of the number of prints she has printed. 59. Solve the system of equations using substitution. 60. What is the real-world meaning of the point of intersection?

14 Page 14 of 14 Learning Objective #6: Develop, represent, explain, and apply the concept of function. Evaluate functions that model relationships between quantities to answer real world questions. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Jo mows lawns after school. She finds that she can use the equation P= N to calculate her profit. 61. Give some possible real-world meanings for the numbers -300 and 15 and the variable, N. 62. Invent two questions related to this situation and then answer them. 63. Make a graph showing the relationship between your variable N and your variable P. Label the axes and provide a scale.