HSPA Open-ended Questions O.E. #1 Maurice and Aldo both go to the same library. The last time they saw each other was last Thursday. Maurice goes to the library every seven days. Aldo goes to the library every five days. _ How many days, from their last meeting, will they both be at the library together? Show how you arrived at your answer. 1
Answer 2
O.E. #2 Jason s Discount Store is having a Holiday Sale. During this sale, a pair of pants is reduced by 30%; then an additional 15% is taken off of the reduced price at the cash register. The original cost of the pants is $35.99. How much would a customer pay for these pants? How much would the pants be if they are sold at a one time discount of 45% off the original cost? Why didn t the store sell these pants for 45% off the original cost? 3
Answer: 4
O.E. #3 Angela has applied for a job as a florist. She had to choose between two salary options. Plan 1: $800 a month+ 6% of monthly sales, or Plan 2: 10% of monthly sales. Sales for the prior months were: January - $9,000 February - $15,000 March- $6,000 April- $ 12,000 Based on this information, which salary option should Angela select if she decides to accept the job? Explain your answer. 5
Answer: 6
O.E. #4 Naser owns a rare stamp collection. There are 45 rare stamps in his collection. He wants to divide the stamps among his 3 children. He requested that the oldest child receive 4/5 of the collection, the middle child receive 1/10 of the collection and the youngest child receive 1/15 of the collection. His children met and tried to figure out how to distribute the stamps according to their father s specifications. How many stamps should each child get? Did Naser choose the fractional parts each child was to receive in the best way? Explain why or why not. 7
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O.E. #5 Dorothy is running for president of the student body and wants to create campaign posters to hang throughout the school. She has determined that there are four main hallways that need six posters each. A single poster takes one person 30 minutes to create and costs a total of $1.50. A) What would be the total cost for Dorothy to create all the needed posters? Show your work. B) If two people working together can create a poster in 20 minutes, how much total time would Dorothy save by getting a friend to help her? Show your work. C) If Dorothy works alone for 3 hours, and is then joined by her friend, calculate exactly how much total time it will take to create all the necessary posters. Show your work. D) Omar, Dorothy's opponent, decided to create his posters on a Saturday and get his friends Janice and Beth to help. He knows that he can create 24 posters in 12 hours if he works alone. He also knows that Janice can create 24 posters in 10 hours and Beth can create 24 posters in 9 hours. How long will it take them, if all three of them work together to create the 24 posters? Round all decimals to the nearest hundredths. Show your work. E) When Omar went to purchase his posters, he discovered that the cost of creating a poster had increased by 20%. How many posters will he be able to create if he wants to spend the same amount of money on his posters as Dorothy? Justify your answer. 9
Answer: Sample Response: A) ($1.50) (4 6) = $36 B) (30 24) (20 24) = 240 minutes = 4 hours or 24 10 (time saved per poster) = 240 minutes (3) (4) 2 (4) = 12 8 = 4 hours C) One person can make 6 posters in 3 hours. The remaining 18 posters take 2 people 360 minutes or 6 hours. 3 hours + 6 hours = 9 hours total. h D) 12 h + 10 h + 9 = 1.29h = 1 h = 3.45 hours (note: used rounding process twice) h = 3.40 hours (note: used rounding process only once) OR h h h 180 ( 12 + 10 + 9 ) = 1 (180) 15h + 18h + 20h = 180 53h = 180 180 21 h = 53 hours = 3 53 hours or 3.40 hours E) (1.20) (1.50) = $1.80 per poster $36 1.80 = 20 posters Omar can create 20 posters 10
O.E. #6 A bank offers an interest rate of r compounded n times per year. The formula for the amount of money, A, in an account at the end of t years, is: A = P ( 1 + r ) nt n where P is the amount of money in the account at the beginning of the year (assuming no deposits or withdrawals). A) If at the beginning of the year Joe had $1,000 in an account with 2% interest compounded semiannually, how much money would he have in the account at the end of the year? Show your work or provide an explanation for your answer. B) The effective interest rate, R, is the percent increase in the account over one year. What is the effective interest rate for Joe's account? (Do not round your answer.) Show your work or provide an explanation for your answer. C) Joe had x dollars in his account at the beginning of the year. Describe how to determine the amount of money Joe would have in his account after 1 year using the effective rate you found above. 11
Answer: Sample Response: A) $1,020.10 B) 0.0201 OR 2.01% C) x 1.0201 OR x + x 0.0201 OR 0.02 1,000 1 + 2 20.10 = 1,000 0.0201 2 = 1,020.10 Multiply the amount of money he has at the beginning of the year by 1.0201. 12
O.E. #7 A carpenter positions the bottom of his 45 foot ladder 5 ½ feet from the base of a building to fix a broken window. How high up on the building will the ladder reach to enable the carpenter to fix the window? Explain and show all of your work. 13
Answer: 14
O.E. #8 If a rectangular stick of butter, as shown below, measures 6 in. x 2 in. x 1 in. What is the volume of the stick of butter? (in square inches) If you take ½ (or 50%) off each side of this stick of butter, what us the volume of the new solid? (in square inches) What is the ratio of the volume of the original stick of butter to the volume if the second? Explain and show your work. 15
Answer: 16
O.E. #9 f you line up 30 squares in a row, side by side, as shown below, with each side equal to 6 inches, what would the perimeter of the figure be? 17
Answer: 18
O.E. #10 Quadrilateral ABCD is graphed below with A(- 3, 2), B(0, - 2), C(8, 4), and D(5, 8). A) Calculate the slopes of each side of ABCD and of the two diagonals. Show your work and label your responses. B) Explain mathematically how you know that the quadrilateral is or is not each of the following types of quadrilaterals: 1. parallelogram 2. rectangle 3. rhombus 4. square 19
Answer: Sample Response: 2 ( 2) 4 4 A) slope of AB = 3 0 = 3 = 3 4 8 4 4 slope of DC = 8 5 = 3 = 3 2 4 6 3 slope of BC = 0 8 = 8 = 4 2 8 6 3 slope of AD = 3 5 = 8 = 4 2 4 2 2 slope of AC = 3 8 = 11 = 11 2 8 10 slope of BD = 0 5 = 5 = 2 B) 1) ABCD is a parallelogram. AB and DC have equal slopes so AB is parallel to DC. BC and AD have equal slopes so BC is parallel to AD. ABCD is a parallelogram because if a quadrilateral has both pairs of opposite sides parallel, then it is a parallelogram. 2) ABCD is a rectangle. The slopes of AB and BC are opposite reciprocals, so AB is perpendicular to BC. (note: any pair of consecutive sides can be used here.) ABCD is a rectangle because if a parallelogram has one right angle, then it is a rectangle. 3) ABCD is NOT a rhombus. The slopes of AC and BD are not opposite reciprocals so AC is not perpendicular to BD. Since the diagonals of the parallelogram are not perpendicular, it is NOT a rhombus. 4) ABCD is NOT a square. Since ABCD is not a rhombus it cannot be a square because to be a square a figure must be both a rectangle and a rhombus. 20
O.E. #11 A rectangular board that measures 2 m wide and 10 m long is leaning against a wall, as shown below. The sun is shining directly above the board. A) Sketch the shape of the shadow that the board makes on the ground. B) If the sun's rays are vertical, describe how moving the bottom edge of the board closer to the wall would affect the width and length of the shadow on the ground. C) How close to the wall should the bottom edge of the board be positioned so that the shadow of the board forms a square? Explain. D) Find the height at which the top of the board touches the wall when the shadow on the ground is a square. Explain how you found your answer. 21
Answer: Sample Response: A) Shadow will be a rectangle. B) This would shorten the shadow s length, but this would not affect the shadow s width at all. C) The shadow of the board will be in the shape of a square when the distance between the ramp s base and the wall is the same as the ramp s width of 2 meters. D) 9.79 or 9.8 meters 2 2 2 10 2 = x 100 4 = 96 96 = x x = 9.79 or 9.8 22
O.E. #12 Draw a parallelogram that has the same area as the triangle shown below. Explain and show all of your work. 23
Answer: 24
O.E. #13 A car starts at point A, travels 8 miles east, and then turns and travels 10 miles south to reach point B. Using graph paper, make a scale drawing using vectors to show the cars movement, starting from point A. Draw a vector that would show the direct path from point A to point B. What would be the approximate number of miles the car could have traveled along this path? Approximately how many degrees from north would this path be? Explain how you arrived at this answer. 25
Answer: 26
O.E. #14 A rectangular vacant lot is 100 feet long. Its area (in square feet) is twice its perimeter (in feet). How wide is the lot? Show all of your work and justify your answer. 27
Answer: 28
O.E. #15 In triangle XYZ, angles x and y are congruent. What is the length of YZ? Explain and show all your work. 29
Answer: 30
O.E. #16 Circle A has a radius of 10 inches. How much greater than its area is the area of Circle B whose radius is twice as large? Explain and show how you arrived at your answer. 31
Answer: 32
O.E. #17 Judge Esther Odometer developed a formula to determine the fine for speeding on the parkway in her town. The formula she developed is: F = 12(R 60) + 55 In this equation, F represents the total amount of fine, R represents how fast (rate of speed) the car was going in miles per hour. A car has been stopped for traveling 75 mph, 80 mph, 87 mph, 90 mph, and 100 mph over the course of the past year. Show the domain and range for this relation. Judge Esther charges you a fine of $175 for speeding. Is this a reasonable fine? Why or why not 33
Answer: 34
O.E. #18 Sharifa is offered two jobs: one at Tom s World of Music and one at Rosie s Café. Tom's promises her a salary of $30,000 and a raise of $500 each year. Rosie's offers her a salary of $30,000 and a raise of 4% of her current salary each year. Which job should she accept? Explain your answer. Show all work. 35
Answer: 36
O.E. #19 Mike dropped a basketball ball from his bedroom window a height of 20 feet. Each time the ball bounced, it reached a maximum height of approximately half that of its previous height. Draw a graph to represent the relationship between the number of times the ball bounces and the height reached by the ball. What is the total of the heights the ball reached after the 4th bounce. What do you think this total would have been if the ball had bounced 20 times? 37
Answer: 38
O.E. #20 The length of the rectangular playroom in Sheila's basement is 1½ times its width. The room has no windows and has a height of 8 feet. Sheila has decided to wallpaper the entire room and put a border over the wallpaper at the top around the room. To determine how much wallpaper is needed, she has to add together the area of each wall to get the total surface area of the room. The door to this room measures 2.5 feet wide by 6.5 feet high and will not need wallpaper. A) Let x be the width of the room. Write and simplify an equation in terms of x to determine the total surface area (S) of the playroom's walls. B) The playroom is 12 feet wide. Each roll of wallpaper will cover approximately 56 square feet. The wallpaper has no pattern; therefore, it is not necessary to allow for matching patterns when calculating the amount of wallpaper to purchase. What is the least number of rolls Sheila will need to paper her room? Show your work. C) Sheila plans to spend no more than $350.00 (ignore tax) on the wallpaper and border. The border costs $19.97 per 10-yard roll. The border will be pasted onto the wallpaper at the top of each wall with no overlap. Explain how to determine the most expensive wallpaper Sheila can buy while staying within her budget. 39
Answer: Sample Response: A) S = 8x + 8(1.5x) + 8x + 8(1.5x) (2.5)(6.5) S = 8x + 12x + 8x + 12x 16.25 S = 40 x 16.25 B) S = 40x (2.5)(6.5) S = 40(12) (2.5)(6.5) S = 480 (2.5)(6.5) = 463.75 sq. ft. 463.75 56 8.28 9 rolls 9 rolls Answer C) First, Sheila has to determine how much border she needs. The perimeter of the room is 12 + 18 + 12 + 18 = 60 feet, which is 20 yards. She would need 2 rolls of border which costs $39.94. Subtract $39.94 from $350.00 and that leaves $310.06 to spend on wallpaper. If she 310.06 needs 9 rolls, then 9 $34.45. Sheila can spend no more than $34.45 per roll on wallpaper and still be within her budget. 40
O.E. #21 Belinda wants to determine the number of dots in the 30 th step of the following pattern, but she does not want to actually draw all 30 steps. Explain how Belinda could find the number of dots in Step 30 without actually drawing them. What would be the number of dots in the 30 th step? Write an algebraic expression for the number of dots in the nth step. 41
Answer: 42
O.E. #22 Analyze the pattern below. What letter will be in the 76 th position? Show your work and explain your answer. M A T H M A T H M A T H M A 43
Answer: 44
O.E. #23 Assume that the following table continues forever, to the nth row. (Do not write in the chart below. A duplicate chart has been provided for your use.) Column A Column B Column C Row 1 1 6-2 Row 2 1/2 5 1/2-2 1/2 Row 3 1/3 5 1/3-2 2/3 Row 4? 5 1/4-2 3/4 Row 5 1/5??............ Row 100??? Row n??? A) Complete rows 4 and 5 in the duplicate chart provided. B) What numbers belong in row 100? Write your answers in the chart provided. C) Write expressions to determine the values of the nth row, for columns A, B, and C. D) What number do the values of column A approach? What number do the values of column C approach? Explain your answers or show what the values approach on a number line to support your answer. 45
Answer: A) and B) Row 1 Row 2 Row 3 Column A Column B Column C 1 6 2 1 2 1 3 1 5 2 1 5 3 1 5 4 1 2 2 2 2 3 3 2 4 Row 4 1 4 Row 5 1 1 5 5 5 2 4 5 Row 100 1 100 Row n 1 5 100 99 2 100 C) Column A = Column B = Column C = OR 1 n 1 n + 5 1 n 3 46
New Jersey Department of Education Page 2 of 3 HSPA/SRA Mathematics CLUSTER 4/MACRO A 2 + n 1 n OR 2 + n 1 n or 2n n + 1 n OR 3n + 1 n OR 1 3 + n Note: These expressions may be put into the chart in part A. 1 D) As n gets larger, n approaches zero. Therefore, the values in column A approach zero. The values in column C approach 3.
O.E. #24 Diana had $1200 in her checking account. She withdrew the same amount each month for 5 months to pay for a car loan. At the end of 6 months, she deposited an additional $600 into her account. Her new balance was $800. How much money did she withdraw each month? What was her account balance after the 6 months before her deposit of $600? Explain and show all your work.
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O.E. #25 John and Ellen work in a clothing store after school. John s boss told him to reduce every item in the store for a 2-day sale by 30%. After the sale, John s boss told her to increase every sale item s sale price by 30%. John started marking each item with the original price. Ellen said, That is wrong! If you increase the sale price by 30% you will not get the original price." Who is right? Show your work. You may wish to include a simple, specific example to support your answer.
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O.E. #26 You are the service manager at an auto repair shop. You charge $22 per hour for labor plus the cost of any parts. A car needed $256 of new parts. The final bill for the car was $421. How long did it take to repair the car? Explain your answer. Write an algebraic equation to solve this problem. Show all work.
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O.E. #27 Scientists record data from experiments with hope of eventually finding patterns that will enable them to predict future results. For example, in testing an antibacterial ointment, a chemist might record the number of bacteria present in a tissue culture after using the ointment for different periods of time. The table below shows a record of the number of hours and the number of bacteria. Make a scatter plot of this data. Does there appear to be a relationship between time and the effectiveness of the ointment? Use the scatter plot to support your answer. After how many hours do you expect one-half of the bacteria to be terminated? Do you expect that all the bacteria will be terminated?
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O.E. #28 The data in the table shows the age, t ( in weeks) and the number of hours, h, slept in a day by 24 infants who are less than 1 year old. Draw a scatter plot to represent the data. Using your scatter plot, what type of correlation do t and h have? Explain how you arrived at your answer. Using the scatter plot, predict the number of hours a 65 week old child would sleep.
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O.E. #29 New Jersey s population, p (in millions) from 1900 to 2000 is shown in the table below. Sketch a scatter plot Let t = 0 represent 1900. Label your graph with appropriate values for time and population. Using your scatter plot, what type of correlation do p and t have? Explain how you arrived at your answer. Predict what the population of New Jersey will be in the year 2010. Explain the strategy that you used to make the prediction.
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O.E. #30 A group of friends attending college in Jersey City decide that, during spring vacation, they will visit the following cities by car: Atlantic City, NJ, Philadelphia, PA, and the Delaware Water Gap, NJ. There are many choices as to the order of visiting the cities and returning to Jersey City. The friends want to design a route that minimizes the distance they will be traveling while maximizing the time visiting their friends. Describe the route for the minimum length trip for the friends to take to visit the three cities and to return to Jersey City. Explain why you think this is the minimum distance and the method you used to find it.
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O.E. #31 Study the information in each of the following charts. Based on the different sets of data displayed by the each of the charts, describe the type of graph that should be used to display each set of data. Explain why you selected each type of graph.
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O.E. #32 The chart below shows the gross income (in millions) of Company ABC for the years 1990 to 1999. Find the mean, median, mode and range of the Company s yearly gross income for the years indicated in the table. Predict what the Company s gross income will be for the year 1999-2000.
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O.E. #33 Continental Airlines advertises that its flights are on time. The chart below displays a consecutive 12 day period and is representative of most 12 day periods for the last 3 months. Study the chart, which shows the touchdown times for Flight 288 which had on-time departures at 10:00 A.M. for the 2 hour and 40 minute flight to Florida from Newark. Explain why the advertising claim of on time arrivals is incorrect for this flight. Determine the probability of arriving in Florida late if this flight is taken. If Flight 288 is late, predict how many minutes late the flight will be and explain how you made the prediction.
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O.E. #34 For a probability experiment, Mr. Rivas placed 6 red, 8 blue, 4 white, and 6 yellow marbles, all of which were the same size into a bag. Eighteen students each made 30 random selections from the bag of marbles, making sure to record each pick before replacing the marble. The chart below shows each student s picks. Give the theoretical probability for selecting each color marble. Compare the experimental probability for each color to the theoretical probability for each color. Describe what should happen if the class conducted the same experiment for 5 days and examined the total results for each color for the 3,000 picks.
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O.E. #35 Classroom A Classroom B Media Center Classroom C The technology teacher would like to network 3 different classrooms (A, B, and C) and the media center, so that each classroom is connected to the media center and to each other. One connection runs both ways between classrooms, so once classroom A is connected to classroom B, B is also connected to A. A) Altogether, how many connections will need to be made for the computer network to be completed between the 3 classrooms and the media center? List all the possible connections and draw a diagram showing all the connections. B) After starting with 3 classrooms, more connections are added and by the end of the month there are 15 connections in all. All connections are made for each classroom before adding another. What is the total number of classrooms that are connected to each other and to the media center? List all the possible connections, or draw a diagram showing all the connections. C) Study the table below and complete the duplicate table provided for your use on the following page. Write a formula or rule, in terms of n, that represents the pattern used to determine the number of connections needed for n classrooms to be connected to the media center and each other. Be sure to write your answers on the duplicate chart provided. # of classrooms # of connections 1? 2? 3? 4? 5? 6? 7? n? D) Use your formula from Part C to show the number of connections if 10 classrooms are to be networked.
Answer: Sample Response: A) 6 connections AM, BM, CM, AB, AC, BC OR OR
New Jersey Department of Education Page 2 of 4 HSPA/SRA Mathematics CLUSTER 3/MACRO D B) 5 classrooms AM, BM, CM, DM, EM, AB, AC, AD, AE, BC, BD, BE, CD, CE, DE OR OR
O.E. #35 Before Jason pays 25% tax on the 8% commission he makes on every house he sells at Top Notch Real Estate Agency, he deducts 12% of the commission for expenses to determine the amount of taxes to be paid. Write a set of steps that any Top Notch employee could follow, using the same procedure as Jason. Test out your steps by computing the amount of taxes Jason would pay on a $250,000 home.
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O.E. #36 The picture below shows stages 1 and 2 of a geometric progression that follows this rule: In a triangle, a line is drawn from the middle of each leg to the middle of the hypotenuse. The legs of the new triangles are 1/2 the length of the previous triangle's legs. Stage 1 and Stage 2 of an right triangle are shown below: Find the area of the triangle in Stage 1. Draw stage 3 and Stage 4 Find the total area of the all shaded triangles in Stage 3 and Stage 4. Will the total area of the shaded triangles in any stage ever exceed the area of the triangle in Stage 1? Explain your answer.
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