Grade 8, Unit 5 Practice Problems - Open Up Resources

Lesson 21Lesson 1Problem 1Given the rule:Complete the table for the function rule for the following input values:input0246810outputSolution2, 2.5, 3, 3.5, 4, 4.5Problem 2Here is an input-output rule:Complete the table for the input-output rule:input-3outputSolution-2-10123

1, 0, 1, 0, 1, 0, 1, respectivelyProblem 3(from Unit 4, Lesson 15)Andre’s school orders some new supplies for the chemistry lab. The online store shows a pack of 10 test tubes costs 4 less than a set of nestedbeakers. In order to fully equip the lab, the school orders 12 sets of beakers and 8 packs of test tubes.1. Write an equation that shows the cost of a pack of test tubes, , in terms of the cost of a set of beakers, .2. The school office receives a bill for the supplies in the amount of 348. Write an equation with andthat describes this situation.3. Since is in terms of from the first equation, this expression can be substituted into the second equation where appears. Write an equationthat shows this substitution.4. Solve the equation for .5. How much did the school pay for a set of beakers? For a pack of test tubes?Solution1.2.3.4.5. 19 and 15Problem 4(from Unit 4, Lesson 14)Solve:Solution

. Substituting, we getforinto the second equation, we get. Solving this equation gives.Problem 5(from Unit 4, Lesson 9)For what value of do the expressionsandhave the same value?SolutionLesson 2Problem 1Here are several function rules. Calculate the output for each rule when you use -6 as the input.SolutionRule 1: -13Rule 2: 36Rule 3: -2Rule 4:. Substitutinginto

Rule 5:Rule 6: -6 is not a valid input for this rule since it doesn't make sense to express a side length with a negative number.Problem 2A group of students is timed while sprinting 100 meters. Each student’s speed can be found by dividing 100 m by their time. Is each statement trueor false? Explain your reasoning.1. Speed is a function of time.2. Time is a function of distance.3. Speed is a function of number of students racing.4. Time is a function of speed.Solution1. True. For each time, one speed is generated.2. False. For each distance (100 m), many times are generated.3. False. The number of students racing does not affect any student’s speed, and the same speed may be reached for more than one student ina group of the same size.4. True. For each speed calculated, there is only one possible time.Problem 3(from Unit 4, Lesson 15)Diego’s history teacher writes a test for the class with 26 questions. The test is worth 123 points and has two types of questions: multiple choiceworth 3 points each, and essays worth 8 points each. How many essay questions are on the test? Explain or show your reasoning.Solution9 essay questions. Explanations vary. Sample response: Use to represent multiple choice questions andasand, and solve it by substitutinginto the second equation.for essay questions. Write the system

Problem 4These tables correspond to inputs and outputs. Which of these input and output tables could represent a function rule, and which ones could not?Explain or show your reasoning.Table A:inputoutput-24-11001124Table B:inputoutput4-21-1001142Table C:inputoutput102030Table D:

inputoutput010203SolutionTable A and Table C represent functions, but Table B and Table D do not. Explanations vary. Sample response: Tables B and D have multipleoutputs for the same input, but functions take each input to only one output. On the other hand, it is okay for a function rule to take different inputs tothe same output.Lesson 3Problem 1Here is an equation that represents a function:.Select all the different equations that describe the same function:A.B.C.D.E.F.G.SolutionA, B, C, EProblem 2(from Unit 4, Lesson 13)1. Graph a system of linear equations with no solutions.2. Write an equation for each line you graph.

SolutionAnswers vary. The graph could be any two lines that are parallel.Problem 3Brown rice costs 2 per pound, and beans cost 1.60 per pound. Lin has 10 to spend on these items to make a large meal of beans and rice for apotluck dinner. Let be the number of pounds of beans Lin buys and be the number of pounds of rice she buys when she spends all her moneyon this meal.1. Write an equation relating the two variables.2. Rearrange the equation sois the independent variable.3. Rearrange the equation sois the independent variable.Solution1.2.3.

Problem 4(from Unit 4, Lesson 6)Solve each equation and check your answer.1.2.3.Solution1. x 1.2. z \frac{\text-13}{7}3. q 2Lesson 4Problem 1The graph and the table show the high temperatures in a city over a 10-day period.

day12345678910temperature(degrees F)606163616261606567631. What was the high temperature on Day 7?2. On which days was the high temperature 61 degrees?3. Is the high temperature a function of the day? Explain how you know.4. Is the day a function of the high temperature? Explain how you know.Solution1. 60 degrees F2. Days 2, 4, 63. The high temperature is a function of the day. There are no different outputs for the same input. That is, there is no day with two different hightemperatures.4. Day could not be a function of temperature as there are multiple days that have the same high temperature. There would be the differentoutputs for the same input.Problem 2The amount Lin’s sister earns at her part-time job is proportional to the number of hours she works. She earns 9.60 per hour.1. Write an equation in the form y kx to describe this situation, where x represents the hours she works and y represents the dollars she earns.2. Is y a function of x? Explain how you know.3. Write an equation describing x as a function of y.Solution1. y 9.6x, where x is number of hours worked and y is amount earned in dollars2. y is a function of x because there is only one output for each input.

3. x \frac{1}{9.6}yProblem 3Use the equation 2m 4s 16 to complete the table, then graph the line using s as the dependent variable.m0s-230Solutionm02-28s4350

Problem 4(from Unit 4, Lesson 13)Solve the system of equations: \begin{cases} y 7x 10 \\ y \text-4x-23 \\ \end{cases}Solution(\text-3, \text-11)Lesson 5Problem 1Match each diagram to the function described, then label the axes appropriately

1. The function inputs the age of an oak tree a and outputs a prediction of the height of the tree h.2. The function inputs the edge length e of a cube and outputs the volume v.3. The function inputs the distance traveled d and predicts the amount of fuel left in the tank f.4. The function inputs the height h of a triangle with base 12 and outputs the area a.5. The function inputs the time of day t and predicts the temperature T.6. The function inputs the time of day t and predicts the number of cars washed at a student car wash c.Solution1. F2. A3. E4. C5. B6. D

Problem 2(from Unit 4, Lesson 13)The solution to a system of equations is (6,\text-3). Choose two equations that might make up the system.A. y \text-3x 6B. y 2x-9C. y \text-5x 27D. y 2x-15E. y \text-4x 27SolutionC and DProblem 3(from Unit 5, Lesson 3)A car is traveling on a small highway and is either going 55 miles per hour or 35 miles per hour, depending on the speed limits, until it reaches itsdestination 200 miles away. Letting x represent the amount of time in hours that the car is going 55 miles per hour, and y being the time in hoursthat the car is going 35 miles per hour, an equation describing the relationship is: 55x 35y 2001. If the car spends 2.5 hours going 35 miles per hour on the trip, how long does it spend going 55 miles per hour?2. If the car spends 3 hours going 55 miles per hour on the trip, how long does it spend going 35 miles per hour?3. If the car spends no time going 35 miles per hour, how long would the trip take? Explain your reasoning.Solution1. About 2.05 hours2. 1 hour3. About 3.64 hours. If the car spent the entire trip going 55 mph, the trip would be completed in about 3.64 hours.Problem 4

The graph represents an object that is shot upwards from a tower and then falls to the ground. The independent variable is time in seconds and thedependent variable is the object’s height above the ground in meters.1. How tall is the tower from which the object was shot?2. When did the object hit the ground?3. Estimate the greatest height the object reached and the time it took to reach that height. Indicate this situation on the graph.Solution1. 10 meters2. 6 seconds after it was shot

3. Approximately 93 meters high at 2.9 seconds. A point should be plotted at (2.9, 93).Lesson 6Problem 1Match the graph to the following situations (you can use a graph multiple times). For each match, name possible independent and dependentvariables and how you would label the axes.1. Tyler pours the same amount of milk from a bottle every morning.2. A plant grows the same amount every week.3. The day started very warm but then it got colder.4. A carnival has an entry fee of 5 and tickets for rides cost 1 each.Solution1. B2. A3. C4. AProblem 2Jada fills her aquarium with water.The graph shows the height of the water, in cm, in the aquarium as a function of time in minutes. Invent a story of how Jada fills the aquarium thatfits the graph.

SolutionAnswers vary. One possible story: Jada turns on the water faucet, and the water in the aquarium is increasing at a constant rate for the first twominutes to a height of 10 cm. Then Jada’s mom calls her to take out the trash, so she turns off the faucet for the minute it takes her to take out thetrash. After she comes back, she turns on the water higher than before, and the water increases to a height of 30 cm in the next two minutes. This ishigh enough, and Jada turns off the water. Unfortunately, there is a slow leak, and the water height decreases to 25 cm. After two minutes, Jadanotices the leak. She stops it, and the water stays constant after that.Problem 3(from Unit 5, Lesson 4)Recall the formula for area of a circle.1. Write an equation relating a circle’s radius, r, and area, A.2. Is area a function of the radius? Is radius a function of the area?3. Fill in the missing parts of the table.rA3\frac1216\pi100\pi

Solution1. A \pi r 22. Yes for both (Using, for example, 2 and -2 for r would produce the same output, but -2 is not a valid input in this situation, since a radius cannotbe Problem 4(from Unit 3, Lesson 11)The points with coordinates (4,8), (2,10), and (5,7) all lie on the line 2x 2y 24.1. Create a graph, plot the points, and sketch the line.2. What is the slope of the line you graphed?3. What does this slope tell you about the relationship between lengths and widths of rectangles with perimeter 24?

Solution1.2. -13. A slope of -1 means that for rectangles of perimeter 24, every extra unit of length put into the width is one less unit of length that can be putinto the length.Lesson 7Problem 1The equation and the tables represent two different functions. Use the equation b 4a-5 and the table to answer the questions. This table representsc as a function of a.

a-30251012c-20732119451. When a is -3, is b or c greater?2. When c is 21, what is the value of a? What is the value of b that goes with this value of a?3. When a is 6, is b or c greater?4. For what values of a do we know that c is greater than b?Solution1. b2. a 5, b 153. There is not enough information to answer this question, since 6 is not in the table for a.4. 0 and 12Problem 2(from Unit 5, Lesson 2)Match each function rule with the value that could not be a possible input for that function.A. 3 divided by the inputB. Add 4 to the input, then divide this value into 3C. Subtract 3 from the input, then divide this value into 11. 32. 43. -4

4. 05. 1SolutionA. 4B. 3C. 1Problem 3Elena and Lin are training for a race. Elena runs her mile a constant speed of 7.5 miles per hour.Lin’s times are recorded every 210.320.410.530.620.730.8511. Who finished their mile first?2. This is a graph of Lin’s progress. Draw a graph to represent Elena’s mile on the same axes.