Unit 4 Linear Relationships Standards: 8.EE.5 8.EE.6 8.EE.7 8.F.1 8.F.2 8.F.3 8.F.4 8.F.5 Domain and Standards Overview Expressions and Equations Understand the connections between proportional relationships, lines, and linear equations. Analyze and solve linear equations. Functions Define, evaluate, and compare functions. Use functions to model relationships between quantities. Priority and Supporting CCSS 8.EE.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Explanations and Examples 8.EE.5 Students build on their work with unit rates from 6 th grade and proportional relationships in 7 th grade to compare graphs, tables and equations of proportional relationships. Students identify the unit rate (or slope) in graphs, tables and equations to compare two proportional relationships represented in different ways. Example 1: Compare the scenarios to determine which represents a greater speed. Explain your choice including a written description of each scenario. Be sure to include the unit rates in your explanation. Scenario 1: Scenario 2: y = 55x x is time in hours y is distance in miles Solution: Scenario 1 has the greater speed since the unit rate is 60 miles per hour. The graph shows this rate since 60 is the distance traveled in one hour. Scenario 2 has a unit rate of 55 miles per hour shown as the coefficient in the equation. Given an equation of a proportional relationship, students draw a graph of the relationship. Students recognize that the unit rate is the coefficient of x and that this value is also the slope of the line.
Rate of Change = Slope Look for words like per and for each or for every to identify the slope in a word problem. Example 1: The plumber charges a flat rate of $30, plus $40 per hour. Example 2: The graph below shows how much Connor earns each week. The slope is the number in front of the variable. Variables are letters. For y = mx + b, m is the slope of the line and b is the y-intercept. Examples of linear equations: y = 20x + 5 slope = 20 y = 2/3x 8 slope = 2/3 y = -4x + 3 slope = -4 y = 75 + 3.5x slope = 3.5 y = -½x slope = -½x T = 50h 20 slope = 50 y = 4 slope = 0 (same as y = 0x + 4) Slope = delta y over delta x When given a table of values, you can find the slope by finding the change in the y-values and dividing them by the change in the x-values. [note: the x is on top in the table, but goes on bottom in the fraction].
Let s look at these linear equation problems. Rather than solve them, we will only identify the 21 slopes found in the problems to determine which one is the greatest slope. Fill in the table below and circle the largest rate of change (slope). 1. The total cost (c) in dollars of renting a sailboat for n days is given by the equation c =120 + 60n. If the total cost was $360, for how many days was the sailboat rented? 2. Frank wrote the first 5 numbers of the pattern shown below. 11, 15, 19, 23, 27 What is the twentieth number in the pattern? 3. The senior class is ordering T-shirts. The printing company charges a $45 design fee plus $7.50 per shirt. Which equation represents the total cost, C, for ordering s shirts? 4. Melissa has $790 in savings. She wants to purchase a new computer for $1,750. She plans to add $20 each week to her savings. How many weeks will Melissa need to add to her savings before she can purchase the new computer? 5. The values in the table below were generated using a rule. x y 0 23 1 30 2 37 3 44 4 51 5 58 Which rule could have been used to generate the values in the table?
6. Beau charges a $10 base fee plus $5 per hour to mow yards. Jaime charges a $12 base fee plus $4.50 per hour. At what time will Beau and Jaime be charging the same total fee? 7. What equation is the rule for the function illustrated by the table of values? x -2-1 0 1 2 y -4-1 2 5 8 8. The population of a town is 13,000 and is increasing by about 250 people per year. This information can be represented by the following equation, where y represents the number of years and p represents the population. p = 13,000 + 250y According to the equation above, in how many years will the population of the town be 14,500? 9. Pat cleans windows during the summer. He charges $50 per house and an additional $5 per window. Which expression represents how much Pat would charge to clean x windows at one house? 10. Three linear equations are shown below. y 1 y 2 y 3 11. Ada charges a flat rate of $75 for staining a deck plus an additional $6 for each hour she works. Which expression below best describes the total amount of money Ada charges for staining a deck in x hours? 12. A hot air balloon lifted off from an elevation of 425 feet above sea level. The balloon rose at a constant rate of 55 feet per minute. Which expression represents the elevation above sea level of the hot air balloon after t minutes?
Total Dollars Earned 13. The graph below shows how much Connor earns each week. Connor s Earnings 50 45 40 35 30 25 20 15 10 5 0 1 2 3 4 5 Week The pattern continues. How much will Connor have earned after his eleventh week? 14. What is the value of y for the equation shown when x = 8? y = 1.5x 7 15. Amahl does computer repairs in his home. He charges a set fee to analyze the problem, plus an hourly rate for his labor. The table below shows C, the total charge to his customer, based on h, the number of hours of labor required. Number of hours (h) Total Charge (C) 2 $39 4 $63 6 $87 Which equation could Amahl use to determine his customer s total charge?
16. The table below shows the value of the first five terms in a sequence. Which expression can be used to find the value of the n th term? Position Value of Term 1 13 2 8 3 3 4-2 5-7 n? 17. Which equation best describes the relationship shown in the graph? y 5 4 3 2 1 0 0 1 2 3 4 5 x
18. Which could be the table of values that was used to graph the function of x shown? x Y x y 0 4 0-4 A 2 0 C 1-2 4 4 2 0 6 8 3 3 B x Y 0-4 1 0 2 4 3 8 D X y 0-4 2 0 4 4 6 8
Number 1 2 3 4 5 6 Beau 6 Jaime 7 8 9 10 y 1 10 y 2 Slope Fill in the table at left, circle the greatest slope, then complete the standard 8.EE.5 problem below from last year s eighth grade EOG. 21. Limousine Company P and Company R both charge a rental fee plus an additional charge per hour. The equation y = 50 + 30x models the total cost (in dollars), y, of renting a limousine from Company P for x hours. The table below shows the cost to rent a limousine from Company R for different lengths of time. Company R Time (hours) 1 2 3 4 5 Total Cost $100 $125 $150 $175 $200 Which statement accurately compares the per hour charges of the two companies? 10 y 3 11 12 13 14 15 16 17 18 A B C D Company P charges $5 less per hour than Company R. Company P charges $5 more per hour than Company R. Company P charges $25 less per hour than Company R. Company P charges $25 more per hour than Company R. Answer the Schoolnet questions here: 1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16
Cost (dollars) Here are three more questions from last year s North Carolina eighth grade EOG. 20. Two stores sell cherries at different prices per pound. Store P sells 3.5 pounds of cherries for $13.30. The graph below shows the cost to purchase different weights of cherries at Store Q 9 8 7 6 5 4 3 2 1 0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 Weight (pounds) Phillip needs to purchase 10 pounds of cherries. Which statement below is true? A Phillip will spend $8.00 less on cherries at Store P than at Store Q. B Phillip will spend $8.00 more on cherries at Store P than at Store Q. C Phillip will spend $6.00 less on cherries at Store P than at Store Q. D Phillip will spend $6.00 more on cherries at Store P than at Store Q.
Number of Jumping Jacks 33. Alicia and Melissa did jumping jacks. The table below shows the number of jumping jacks that Alicia had done in different amounts of time. Alicia Time (minutes) 1 2 3 4 5 6 7 8 Jumping Jacks 30 60 90 120 150 180 210 240 The graph below shows the number of jumping jacks Melissa had done in different amounts of time. Melissa 160 140 120 100 80 60 40 20 0 1 2 3 4 5 6 7 8 Time (in minutes) Which choice best describes the difference between the rates at which the girls did jumping jacks? A B C D Melissa did 6 more jumping jacks per minute than Alicia. Alicia did 6 more jumping jacks per minute than Melissa. Melissa did 5 more jumping jacks per minute than Alicia. Alicia did 5 more jumping jacks per minute than Melissa.
Gallons 35. Rain is flowing into two containers at different rates. The figure below shows the volume of water in each container at different times. 10 8 6 4 2 0 Container 1 Container 2 Minutes Gallons 5 2 10 4 15 6 20 8 25 10 5 10 15 20 25 Minutes What is the difference in the rate of change between the two containers? A gallon per minute B gallon per minute C gallons per minute D gallons per minute
Student: Class: Date: 1. Alex and Susan are taking a trip. They are both driving at a constant speed. The graph shows the distance Alex has traveled. The table shows the distance Susan has traveled. Time (hours) 2 3 4 Distance (miles) 90 135 180 After 6 hours, which statement is true? A. Alex has driven 5 miles farther than Susan. B. Alex has driven 30 miles farther than Susan. C. Susan has driven 5 miles farther than Alex. D. Susan has driven 30 miles farther than Alex. 8.EE.5 Page 2 of 14
2. Trevor and Jeff mow grass in the summer to earn money. Trevor earns $15 an hour to mow grass. The table below represents how much Jeff earned for different amounts of time spent mowing. Jeff s Earnings from Mowing Hours Worked 2 3 4 Total Earned $26 $39 $52 Which statement below is true? A. Trevor earns $11 more per hour. B. Jeff earns $11 more per hour. C. Trevor earns $2 more per hour. D. Jeff earns $2 more per hour. 3. Thrifty Store sells 3 pounds of potatoes for $4.77 and five pounds of potatoes for $7.95. EcoStore sells the same potatoes for the prices in the table below. EcoStore Weight (lb) Price ($) 4 5.96 6 8.94 8 11.92 Based on the information, which statement is true? A. Thrifty Store charges $0.10 more per pound than EcoStore. B. EcoStore charges $0.10 more per pound than Thrifty Store. C. Thrifty Store charges $0.20 more per pound than EcoStore. D. EcoStore charges $0.20 more per pound than Thrifty Store. 8.EE.5 Page 3 of 14
4. Jamie and Austin are both training for a race. The equation y = 28x represents the number of miles Jamie ran after x weeks. The table below represents the total miles Austin ran over different number of weeks. Austin s Running Total Number of Miles Ran Weeks 3 96 6 192 8 256 After 15 weeks, who ran the most miles and by how many more miles? A. Austin, by 4 miles B. Jamie, by 4 miles C. Austin, by 60 miles D. Jamie, by 60 miles 5. Train E and Train G leave a city s train station at the same time. Train E s average speed can be represented by the equation d = 45t, where d is the distance the train has traveled in miles, and t is the time in hours. The time and distance Train G has traveled are shown in the table below. Time (hours) Train G Distance (miles) 2 70 5 175 7 245 Which train is traveling the fastest and by how much? A. Train E is traveling 25 miles per hour faster than Train G. B. Train G is traveling 25 miles per hour faster than Train E. C. Train E is traveling 10 miles per hour faster than Train G. D. Train G is traveling 10 miles per hour faster than Train E. 8.EE.5 Page 4 of 14
6. Evan and Nathan are electricians. Evan uses the equation y = 30x to calculate the amount he earns for x hours of work. The graph below shows the amount Nathan earns for work. What is the difference in the amounts Evan and Nathan earn per hour? A. $30 B. $20 C. $10 8.EE.5 Page 5 of 14
7. The equation y = 30x represents the amount of money Ben has saved after x weeks. The table below shows the amount of money Lisa has saved over a few weeks. Number of Weeks Lisa s Savings Amount Saved 2 $50 4 $100 6 $150 Which statement is true? A. Lisa saves $20 more per week than Ben. B. Lisa saves $5 more per week than Ben. C. Ben saves $5 more per week than Lisa. 8.EE.5 Page 6 of 14
8. A line is graphed below. What is the slope of the line? A. 0 B. 1 C. 2 D. 4 8.EE.5 Page 7 of 14
9. The graph below shows the total cost, y, of x bananas. What is the slope of the line? A. 0.25 B. 0.5 C. 1 D. 2 8.EE.5 Page 8 of 14
10. A group of students plan to carve a certain number of pumpkins for a contest next week. The graph below represents their plans. At what rate does the group plan to carve the pumpkins? A. 3 pumpkins per hour B. 3.2 pumpkins per hour C. 5 pumpkins per hour D. 6.4 pumpkins per hour 8.EE.5 Page 9 of 14
11. The equation y = 450x represents the number of miles airplane Z traveled, y, for a flight of x hours. The graph below shows the number of miles airplane W traveled for a 5 hour flight. After 2 hours, which airplane had traveled the farthest and by how much? A. Airplane W traveled 50 miles farther. B. Airplane Z traveled 50 miles farther. C. Airplane W traveled 100 miles farther. D. Airplane Z traveled 100 miles farther. 12. Max and Ashley begin traveling on their bicycles in the same direction on the same path. They each maintain their own average speed while traveling. The equation Ashley travels 24 miles on her bicycle in 2 hours. represents the number of miles, d, Max travels on his bicycle in t hours. Which statement best describes how far apart they are after they have traveled hours on their bicycles? A. Ashley will be 7 miles ahead of Max on the path. B. Ashley will be 4 miles ahead of Max on the path. C. Max will be 3 miles behind Ashley on the path. D. Max will be 1 mile behind Ashley on the path. 8.EE.5 Page 10 of 14
13. Susan wants to attend a summer camp. The table below shows the cost, y, to stay at Camp A for x number of nights. Camp A Number of Nights 2 4 7 Total Cost $78 $156 $273 The graph below represents the cost, y, to stay at Camp B for x number of nights. Which statement is true? A. Camp A charges $4 more per day. B. Camp B charges $4 more per day. C. Camp A charges $8 more per day. D. Camp B charges $8 more per day. 8.EE.5 Page 11 of 14
14. Savannah and Trey are playing tennis over the summer. The city park charges $3 per hour to use the tennis courts. The table below shows how much the county park charges to use the tennis courts. Hours 2 3 5 Rental Cost $6.50 $9.75 $16.25 They plan to play tennis for 8 hours this week. How much will Savannah and Trey save by choosing the least expensive park to play at this week? A. $0.25 B. $1.00 C. $2.00 D. $3.50 8.EE.5 Page 12 of 14
15. The graph below shows the relationship between the amount of peanuts and the amount of chocolate in different packages of snack mix. For each ounce of peanuts, how much chocolate is in the mix? A. ounce B. ounce C. ounce D. ounce 8.EE.5 Page 13 of 14
16. The table below shows the cost of tomatoes at Roger s store. Roger s Store Pounds of Cost Tomatoes 3 $7.35 5 $12.25 8 $19.60 The graph below shows the cost of tomatoes at Lisa s store. Which statement below is true? A. Roger s store is cheaper by $0.05 per pound. B. Roger s store is cheaper by $0.15 per pound. C. Lisa s store is cheaper by $0.05 per pound. D. Lisa s store is cheaper by $0.15 per pound. 8.EE.5 Page 14 of 14