8.EE.B.5Common CoreMathExpressions and EquationsGrade 8
8.EE.B.5: Graphing Proportional Relationships and Unit Rate as Slope
In plain English: 8.EE.B.5 is the Common Core grade 8 math standard that asks students to graph proportional relationships and to read the unit rate as the slope of the line. Students also compare two proportional relationships shown in different ways, such as a graph and an equation, to decide which one has the greater rate. It links grade 7 ratio work to the linear equations of Grade 8 Math.
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.
Common Core State Standards for Mathematics · Domain: Expressions and Equations (EE) · Cluster: Understand the connections between proportional relationships, lines, and linear equations. Also written as 8.EE.5 · Official standard
In grade 7, students learned that a proportional relationship (one quantity is always the same number times the other, y = kx) has a straight-line graph through the origin, the point (0, 0). In this lesson they add one new idea: the slope of a line, which measures how steep it is. Slope is the rise (the vertical change between two points) divided by the run (the horizontal change). For a proportional relationship, the slope equals the unit rate, the amount of y for 1 unit of x.
Students graph proportional relationships from words, tables and equations, find the slope with small right triangles drawn under the line, and say what the slope means with units, such as "7.5 megabytes per second." Then they compare two proportional relationships given in different forms, including the official example: a distance-time graph against a distance-time equation, to decide which object moves faster. Lines that do not pass through the origin (y = mx + b) belong to 8.EE.B.6 and 8.F.A.3 and appear here only as a contrast.
Learning Objectives
By the end of this lesson, students will be able to:
Graph a proportional relationship from a description, a table or an equation y = kx
Find the slope of a proportional graph as rise divided by run, and explain that it equals the unit rate
State the meaning of the slope in context, with units
Compare two proportional relationships given in different forms (graph, table, equation, words) by comparing their unit rates
Decide which of two moving objects is faster from a distance-time graph and a distance-time equation
Prior Knowledge Required
Students should already be comfortable with:
Finding a unit rate, such as dollars per pound or miles per hour 6.RP.A.2
Recognizing proportional relationships and the constant of proportionality k in y = kx 7.RP.A.2
Plotting points in the first quadrant of the coordinate plane 5.G.A.1
Sketch two lines on the board. Both start at (0, 0). The horizontal axis is time in minutes and the vertical axis is distance in meters. Line A is steeper than line B. Do not write any numbers yet.
Warm-Up Prompt
"Two friends walk home from school. The lines show how far each one has walked. Who walks faster? Can you tell without any numbers? Now suppose line A passes through (10, 800) and line B passes through (10, 700). How many meters does each friend walk in 1 minute?"
Students usually say that the steeper line is the faster walker, because it covers more distance in the same time. With the numbers, friend A walks 800 ÷ 10 = 80 meters per minute and friend B walks 700 ÷ 10 = 70 meters per minute. Tell students that "how steep" has a name, slope, and that today they will see that the slope of a proportional graph is exactly the unit rate.
Direct Instruction20 minutes
Write these words on an anchor chart, one at a time, with a small sketch for each:
Proportional relationship (from grade 7): y = kx, where k is the constant of proportionality, the number you multiply x by. Its graph is a straight line through the origin (0, 0).
Unit rate: the amount of y for 1 unit of x, such as 80 meters per minute. In y = kx, the unit rate is k, and the point (1, k) is on the graph.
Rise and run: between two points on a line, the run is the horizontal change (how far right) and the rise is the vertical change (how far up).
Slope: the steepness of a line, found as rise ÷ run. A slope triangle is a right triangle drawn under the line, with the run as its horizontal side and the rise as its vertical side.
The key fact: for a proportional relationship, slope = unit rate = k. When two graphs share the same axes, the steeper line has the greater unit rate.
Work through the four examples. For each one, ask students to name the units of x and y before they find the slope, because the slope's unit is "y-units per x-unit."
Graphing from words
A car wash fundraiser charges $6 per car. Graph the money raised y against the number of cars x, for 0 to 4 cars, and find the slope.
Equation: Table: (0, 0), (1, 6), (2, 12), (3, 18), (4, 24). From (1, 6) to (3, 18), the rise is 12 and the run is 2, so the slope is 12 ÷ 2 = 6: $6 per car, the unit rate. Equation: y = 6x. Since nobody washes half a car, the graph is five dots on a line through the origin, not a solid line.
Unit rate from a graph (Diagram 1)
A video game update downloads at a steady rate. The graph passes through (0, 0), (2, 15) and (4, 30), with time in seconds and data in megabytes (MB).
Equation: From (0, 0) to (2, 15): slope = 15 ÷ 2 = 7.5. From (2, 15) to (4, 30): 15 ÷ 2 = 7.5 again. The unit rate is 7.5 MB per second, so (1, 7.5) is on the line and y = 7.5x.
The official example: distance-time graph against an equation (Diagram 2)
Maria's distance-time graph passes through (0, 0), (4, 22) and (8, 44), with time in seconds and distance in meters. Theo's distance is given by d = 6t. Who runs faster?
Equation: Maria's slope is 22 ÷ 4 = 5.5 meters per second. Theo's unit rate is the 6 in d = 6t: 6 meters per second. Since 6 > 5.5, Theo is faster, and his line is the steeper one. After 10 seconds Theo is 60 - 55 = 5 meters ahead.
A table against a graph
Ana's reading log shows 16 pages in 20 minutes and 36 pages in 45 minutes. Ben's graph of pages against minutes is a line through (0, 0) and (30, 21). Who reads more pages per minute?
Equation: Ana: 16 ÷ 20 = 0.8 and 36 ÷ 45 = 0.8 pages per minute. Ben: slope = 21 ÷ 30 = 0.7 pages per minute. Ana reads 0.1 page per minute more, which is 6 more pages in an hour.
Use Diagram 1 to show that a slope triangle can be any size: the large triangle and the small one give the same slope, and the triangle with a run of 1 shows the unit rate directly. Then use Diagram 2 for the official example. Point out that you do not need to graph Theo's equation to compare, because the equation already shows his unit rate; graphing it simply confirms that the greater unit rate gives the steeper line. Warn students that "steeper means greater" works only when both graphs use the same quantities on the same axes.
Guided Practice15 minutes
Pairs work through four problems. One partner finds each unit rate, the other checks it on a graph or with a slope triangle, and they switch roles for the next problem.
Guided practice problems with answers
Problem
Answer
An ice rink charges $8 per hour of skating. Make a table for 0 to 3 hours, graph it and give the slope.
(0, 0), (1, 8), (2, 16), (3, 24). Slope 8, or $8 per hour; y = 8x
An elevator rises at a steady speed. Its graph of height against time passes through (0, 0) and (6, 9), with seconds and meters. Find the slope and say what it means.
9 ÷ 6 = 1.5: the elevator rises 1.5 meters each second, so (1, 1.5) is on the line
Bus A: d = 45t, with d in miles and t in hours. Bus B's table: 2 hours, 104 miles; 3 hours, 156 miles. Which bus is faster, and by how much?
Bus B: 104 ÷ 2 = 52 and 156 ÷ 3 = 52 miles per hour. Bus B is faster by 7 miles per hour
Without graphing, which line is steeper: y = 0.8x or y = 1.25x? Check with the points where x = 4.
y = 1.25x, because 1.25 > 0.8. At x = 4 it reaches 5, while y = 0.8x reaches 3.2
Listen for students who divide run by rise (6 ÷ 9 in the elevator problem), and for students who compare the numbers they see first (104 against 45) instead of the unit rates.
Independent Practice15 minutes
Students work alone, then compare answers with a partner. Graphs go on grid paper.
Independent practice problems with answers
Problem
Answer
Graph y = 3x for x = 0 to 4. What is the slope?
(0, 0), (1, 3), (2, 6), (3, 9), (4, 12); slope 3
A graph of the cost of renting a kayak passes through (0, 0) and (4, 72), with hours and dollars. Find the slope and say what it means.
72 ÷ 4 = 18: renting costs $18 per hour
Trail mix: 12 ounces of Brand A cost $7.20. Brand B costs c = 0.55w dollars for w ounces. Which costs less per ounce?
Brand A: 7.20 ÷ 12 = $0.60 per ounce. Brand B: $0.55 per ounce, so Brand B costs less
Swimmer A's graph passes through (0, 0) and (40, 50), with seconds and meters. Swimmer B swims d = 1.1t. Who is faster?
Swimmer A: 50 ÷ 40 = 1.25 meters per second, which is more than 1.1, so Swimmer A is faster
In one sentence, explain why the steeper of two proportional graphs on the same axes has the greater unit rate.
For each 1 unit to the right, the steeper line goes up more, so it has more y for each unit of x
Closure5-10 minutes
Exit ticket: (1) A graph of the pages a printer prints passes through (0, 0) and (5, 60), with minutes and pages. What is the unit rate? (60 ÷ 5 = 12 pages per minute.) (2) Car A's distance is d = 58t, in miles and hours. Car B's distance-time graph is a line through (0, 0) and (2, 118). Which car is faster? (Car B: 118 ÷ 2 = 59 miles per hour, 1 more than Car A.) (3) Finish the sentence: "On the graph of a proportional relationship, the unit rate is ..."
Differentiation Strategies
For Struggling Students
Give a slope-triangle template: a right triangle with the labels "run (across)" and "rise (up)" and the reminder "slope = rise ÷ run"
Always find the point (1, k) first: go 1 unit to the right from the origin and read how far up the line is
Have students write the units next to every number, so that "7.5" becomes "7.5 MB per second"
For Advanced Students
Ask: if a distance-time graph has distance on the x-axis and time on the y-axis, what does its slope mean, and does steeper still mean faster?
Give three relationships in three forms (a graph, a table and words) and have students order them from greatest to least unit rate, then write a fourth, in equation form, that fits between two of them
Challenge (beyond this standard): graph y = 2x and y = 2x + 3 on the same grid and describe what is the same and what is different. This previews 8.EE.B.6
Assessment Guidance
What to Look For
A strong answer gives the unit rate with units and connects it to the graph: "the slope is 1.5, so the elevator rises 1.5 meters each second." Watch for three errors: dividing run by rise, reading only the y-value of one point as the slope, and comparing two relationships by the first numbers they show instead of by their unit rates. When students compare, check that both unit rates use the same units.
02
Classroom Activities
3 Activities
1
Match the Rates: Eight Travelers
20 minPairs
Each pair gets 8 cards. Every card describes a traveler moving at a steady speed, in miles and hours, as a graph, a table, an equation or words. Pairs find each unit rate, match the cards that show the same speed in two different forms, and then order the speeds.
Clip matching cards together; each pair of matched cards shows two different forms
Graph all four speeds on one grid, with hours from 0 to 3 on the x-axis and miles from 0 to 36 on the y-axis, in four colors
Label each line with its slope
Discussion Questions
Card 2 gives the time in minutes. What did you have to do before comparing it with the others?
On your grid, the steepest line belongs to Cards 3 and 7. Why does that match the fastest speed?
Which form made the unit rate easiest to see, and which made it hardest?
Modification for Distance Learning
Put the cards on a shared slide. Pairs drag matching cards next to each other and type the unit rate in a text box, then graph the four lines in an online graphing tool.
2
Walk the Line: Measured Speeds
25 minGroups of 3 (walker, timer, recorder)
Groups time a normal walk and a brisk walk over a marked 20-meter course, find each unit rate in meters per second, graph both walks as distance-time lines, and compare them with an equation the teacher gives.
Sample Data (invented)
An invented group recorded a normal walk of 20 meters in 16 seconds and a brisk walk of 20 meters in 12.5 seconds.
Procedure
The walker starts walking before the start line so the speed is steady, and the timer starts the stopwatch as the walker crosses it
Find each unit rate: 20 ÷ 16 = 1.25 and 20 ÷ 12.5 = 1.6 meters per second in the sample
Graph both walks on one grid, with seconds from 0 to 20 on the x-axis and meters from 0 to 32 on the y-axis. Each line passes through the origin
The teacher writes d = 1.4t on the board as a typical adult walking speed. Add it to the grid as a dashed line
Discussion Questions
In the sample, the dashed line d = 1.4t lies between the two walks. What does that tell you about the three speeds?
How long would each sample walk take to cover 100 meters?
Why must the timer start after the walker is already moving?
3
Better Buy Stations
20 minGroups of 3
Groups rotate through three stations. Each station compares the prices at two stores, shown in two different forms. Groups find both unit prices (the cost of 1 unit, such as 1 pound or 1 gallon), decide which is the better buy, and explain their choice in one sentence.
The Three Stations
Station 1 (graph against equation): dog food at Store X has a cost graph through (0, 0) and (10, 18), in pounds and dollars. At Store Y, c = 1.65p, where p is pounds
Station 2 (table against words): gas at Station P costs $19.75 for 5 gallons and $31.60 for 8 gallons. At Station Q, 12 gallons cost $46.20
Station 3 (words against graph): Print Shop M prints 150 flyers for $12. Print Shop N has a cost graph through (0, 0) and (200, 14)
Answer Key
Station 1: Store X, $1.80 per pound; Store Y, $1.65 per pound. Store Y is cheaper by 15 cents per pound
Station 2: Station P, $3.95 per gallon; Station Q, $3.85 per gallon. Station Q is cheaper by 10 cents per gallon
Station 3: Shop M, $0.08 per flyer; Shop N, $0.07 per flyer. Shop N is cheaper by 1 cent per flyer
Discussion Questions
If you graphed both stores at a station on one grid, which line would be steeper: the better buy or the other store?
At Station 3, is a difference of 1 cent per flyer worth caring about for an order of 2,000 flyers? How much is saved?
Challenge Variation
Each group writes its own station: two stores, two different forms, and unit prices that differ by less than 10 cents. Groups trade stations and solve each other's.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Slope Triangles Show the Unit Rate
The graph of a steady download passes through (0, 0), (2, 15) and (4, 30), drawn to scale. The large slope triangle has a rise of 15 and a run of 2, and the small one has a rise of 7.5 and a run of 1. Both give a slope of 7.5, the unit rate of 7.5 MB per second.
Diagram 2: The Official Example, a Graph Against an Equation
Maria's distance-time graph (solid) passes through (4, 22) and (8, 44), so her speed is 5.5 meters per second. Theo's speed comes from his equation, d = 6t: 6 meters per second. His line (dashed), graphed from the equation, is steeper.
04
Homework Assignment
~30 min
8.EE.B.5 Homework: Unit Rate as Slope
Directions: Show your work. Write every unit rate with its units. Use grid paper for the graphs, and label both axes.
Part 1: Graph and Find the Slope (Problems 1-3)
A fitness tracker counts 110 steps per minute during a brisk walk. Write an equation for the steps s after m minutes, make a table for m = 0, 5, 10 and 15, and graph it. What is the slope, and what does it mean?
A graph of the cost of floor tile passes through (0, 0) and (6, 27), with square feet on the x-axis and dollars on the y-axis. (a) Find the slope. (b) Name the point on the line where x = 1 and explain what it means. (c) How much do 10 square feet cost?
During a steady snowfall, a graph of snow depth against time passes through (0, 0) and (8, 2), with hours on the x-axis and inches on the y-axis. Jae says the slope is 4. (a) What did Jae do wrong? (b) Find the correct slope and say what it means. (c) What does Jae's number 4 actually measure?
Part 2: Compare Two Relationships (Problems 4-6)
Kayaker A paddles at a steady speed given by d = 4.5t, where d is kilometers and t is hours. Kayaker B's distance-time graph is a line through (0, 0) and (2, 10). (a) Who paddles faster? Explain with unit rates. (b) How far apart are they after 3 hours if they start together?
Detergent A: a table shows 32 ounces for $6.40 and 50 ounces for $10.00. Detergent B: a 64-ounce bottle costs $11.52. (a) Find both unit prices. (b) Which is the better buy, and how much do you save on 100 ounces?
Tia saves money at a steady rate: y = 12x, where y is dollars and x is weeks. Dev's savings table shows $30 after 3 weeks and $50 after 5 weeks. (a) Graph both on one grid for 0 to 8 weeks. (b) Whose line is steeper, and what does that mean? (c) How much more has the faster saver saved after 8 weeks?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Graphs
Axes labeled with units; points plotted to scale on a line through the origin
Points mostly right, or labels missing
Graph missing or incorrect
Slope and Unit Rate
Every slope is rise ÷ run and matches the unit rate
One slope reversed or miscalculated
Slopes missing or reversed throughout
Meaning in Context
Each rate is explained in a sentence with units
Units or meaning missing once or twice
No interpretation
Comparisons
Both unit rates found and the correct choice explained
Correct choice with a weak or missing reason
Wrong choice or no comparison
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
On the graph of a proportional relationship, what does the slope tell you?
Answer: A
The slope of a proportional graph equals its unit rate: the change in y for each 1-unit increase in x. Choice B describes the starting value, which is always 0 for a proportional relationship, so it tells you nothing about the rate. Choice C depends only on how much of the graph is drawn.
Question 2 of 20 · Multiple Choice
What is the slope of the line through (0, 0) and (4, 14)?
Answer: C
The rise is 14 and the run is 4, so the slope is 14 ÷ 4 = 3.5. Choice A divides run by rise (4 ÷ 14 = 2/7). Choice B subtracts (14 - 4). Choice D uses the rise alone.
Question 3 of 20 · Multiple Choice
A graph shows the cost of rice. It is a straight line through (0, 0) and (5, 8.75), with pounds on the x-axis and dollars on the y-axis. What is the unit rate?
Answer: B
The slope is 8.75 ÷ 5 = 1.75, so rice costs $1.75 per pound. Choice A divides 5 by 8.75, which gives pounds per dollar, not dollars per pound. Choice C subtracts 8.75 - 5. Choice D is the cost of all 5 pounds.
Question 4 of 20 · Multiple Choice
A proportional graph passes through (0, 0) and (3, 2). Which equation does it show?
Answer: D
The slope is rise ÷ run = 2 ÷ 3, so y = (2/3)x. Check: when x = 3, y = (2/3)(3) = 2. Choice A divides run by rise. Choices B and C use only one coordinate of the point.
Question 5 of 20 · Multiple Choice
Car A travels according to d = 62t, where d is miles and t is hours. Car B's distance-time graph is a line through (0, 0) and (3, 195). Which car is faster?
Answer: A
Car B's slope is 195 ÷ 3 = 65 miles per hour, and Car A's unit rate is 62 miles per hour, so Car B is faster by 65 - 62 = 3 miles per hour. Choice C subtracts 62 from 195, the distance at 3 hours, instead of comparing unit rates. Choice B reads the difference correctly but picks the slower car.
Question 6 of 20 · Multiple Choice
Hose A fills a pool according to g = 6m, where g is gallons and m is minutes. A table for Hose B shows 35 gallons after 5 minutes and 56 gallons after 8 minutes. Which hose fills faster?
Answer: C
Hose B: 35 ÷ 5 = 7 and 56 ÷ 8 = 7 gallons per minute. Hose A: 6 gallons per minute. So Hose B is faster by 1 gallon per minute. Choices B and D subtract 6 from 35, a table value, instead of comparing unit rates.
Question 7 of 20 · Multiple Choice
A graph shows the distance a high-speed train travels, in miles, against time, in hours. The line passes through the origin, and its slope is 150. What does the slope mean?
Answer: B
The slope is the unit rate, miles per hour: 150 miles for each hour. Choice A treats the slope as a total. Choice D reverses the units, as if the slope were hours per mile.
Question 8 of 20 · Multiple Choice
A graph of cups of flour y against batches of muffins x is a line through (0, 0) and (1, 2.25). What is the slope, and what does it mean?
Answer: D
The point (1, 2.25) means 1 batch uses 2.25 cups, so the slope and the unit rate are 2.25 cups of flour per batch. Choice A reads the x-coordinate. Choice B adds the coordinates. Choice C divides 1 by 2.25, which gives batches per cup, the reverse rate.
Question 9 of 20 · Multiple Choice
Which table, when graphed, gives a line through the origin with a slope of 4?
Answer: C
In choice C, y is always 4 times x, so the points (0, 0), (1, 4) and (2, 8) lie on y = 4x. Choice A rises 4 for each 1 across, but it starts at (0, 4), not at the origin. Choice B swaps x and y, which gives a slope of 1/4. Choice D rises only 1 for each 1 across.
Question 10 of 20 · Multiple Choice
Liam types 210 words in 5 minutes. A graph of Ana's typing is a line through (0, 0) and (3, 135), with minutes and words. Who types faster?
Answer: B
Liam: 210 ÷ 5 = 42 words per minute. Ana: 135 ÷ 3 = 45 words per minute. Ana is faster by 3. Choice D subtracts 135 from 210, which compares totals for different times. Choice A picks the slower typist.
Question 11 of 20 · Multiple Choice
The lines y = 3x and y = 5x are drawn on the same grid. Which statement is true?
Answer: D
Both are proportional, so both pass through the origin. For each 1 unit to the right, y = 5x rises 5 and y = 3x rises 3, so y = 5x is steeper. Choice C adds the two slopes; the lines meet only at the origin, because at x = 1 they reach 3 and 5.
Question 12 of 20 · Multiple Choice
A hiker walks at a steady pace. A graph of distance against time is a line through (0, 0) and (2.5, 10), with hours on the x-axis and kilometers on the y-axis. What is the hiker's speed?
Answer: C
The slope is 10 ÷ 2.5 = 4, so the hiker walks 4 kilometers per hour. Choice A divides 2.5 by 10, which is hours per kilometer. Choice B subtracts 10 - 2.5, and choice D adds the two coordinates.
Question 13 of 20 · Multiple Choice
Which proportional relationship has a steeper graph than y = 2x, when both are drawn on the same axes?
Answer: B
Compare slopes with 2. Choice B: 10 ÷ 4 = 2.5, which is greater than 2. Choice A has slope 1.5. Choice C has slope 5 ÷ 3, about 1.67. Choice D has slope 8 ÷ 5 = 1.6. A student who sees the y-value 10 in choice C and does not divide it by its x-value, 6, may pick C by mistake.
Question 14 of 20 · Multiple Choice
Nora graphs a walk with distance in miles on the x-axis and time in minutes on the y-axis. Her line passes through (0, 0) and (2, 36). What does the slope mean?
Answer: A
The slope is 36 ÷ 2 = 18, and its units are y-units per x-unit: minutes per mile. So Nora takes 18 minutes for each mile, about 3.3 miles per hour. Choice B uses the units in the wrong order. Choice D reads the point backward. Here a steeper line would mean a slower walker.
Question 15 of 20 · Short Answer
Graph y = 2.5x for x = 0, 2, 4 and 6. List the points, give the slope, and explain how the graph shows a proportional relationship.
The points are (0, 0), (2, 5), (4, 10) and (6, 15). Between any two of them, the rise is 2.5 times the run, so the slope is 2.5. The graph is a straight line through the origin, which shows a proportional relationship.
Question 16 of 20 · Short Answer
Robot A's distance-time graph is a line through (0, 0) and (5, 3.5), with seconds and meters. Robot B moves according to d = 0.8t. Which robot is faster? Explain with unit rates.
Robot A: slope = 3.5 ÷ 5 = 0.7 meters per second. Robot B: 0.8 meters per second, read from the equation. Robot B is faster, by 0.1 meters per second. Its line would be the steeper one.
Question 17 of 20 · Short Answer
Jada's pay table shows $62 for 4 hours and $108.50 for 7 hours. Marco earns $15 per hour. Who earns more per hour, and how much more does that person earn in 20 hours?
Jada: 62 ÷ 4 = 15.50 and 108.50 ÷ 7 = 15.50 dollars per hour. Jada earns more, $0.50 more per hour, so in 20 hours she earns 20 × 0.50 = $10 more ($310 against $300).
Question 18 of 20 · Short Answer
A 3D printer builds a model at a steady rate. A graph of the model's height against time is a line through (0, 0) and (12, 9), with hours on the x-axis and centimeters on the y-axis. Find the slope, explain what it means, and name the point on the line where x = 1.
Slope = 9 ÷ 12 = 0.75: the model grows 0.75 centimeter taller each hour. The point is (1, 0.75).
Question 19 of 20 · Short Answer
A resting heart beats 75 times per minute. Make a table of total beats for 0, 1, 2 and 4 minutes, write an equation, and give the slope of its graph.
Table: (0, 0), (1, 75), (2, 150), (4, 300). Equation: b = 75m, where b is beats and m is minutes. The slope is 75 beats per minute, the unit rate.
Question 20 of 20 · Short Answer
Two volunteer groups plant trees. Group A's graph of trees planted against hours is a line through (0, 0) and (4, 18). Group B plants 40 trees in 8 hours. Which group plants faster, and how many more trees does it plant in 6 hours?
Group A: 18 ÷ 4 = 4.5 trees per hour. Group B: 40 ÷ 8 = 5 trees per hour. Group B is faster. In 6 hours Group B plants 30 trees and Group A plants 27, so Group B plants 3 more.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 8.EE.B.5 mean?
8.EE.B.5 means students graph proportional relationships and understand that the slope of the graph is the unit rate. They also compare two proportional relationships shown in different ways, such as a graph and an equation, by comparing their unit rates. The official example compares a distance-time graph with a distance-time equation to find the faster object.
Is 8.EE.B.5 taught in grade 8 or in Algebra I?
8.EE.B.5 is a grade 8 standard, usually taught at the start of the unit on lines and linear equations. Schools that teach Algebra I in grade 8 cover it in the first weeks of that course. High school builds on it when students interpret slope as a rate of change in HSF.IF.B.6.
Why is the unit rate the same as the slope?
Because both measure how much y changes for each 1 unit of x. The slope is rise ÷ run between two points. If you choose a run of 1, the rise is the amount of y for 1 unit of x, which is the unit rate. For y = kx, both equal k.
How is 8.EE.B.5 different from 7.RP.A.2?
In 7.RP.A.2, students decide whether a relationship is proportional and find the constant of proportionality k. In 8.EE.B.5, they give k a new geometric meaning, the slope of the line, and use it to compare two relationships given in different forms. It is the bridge from ratio reasoning to the equations of lines.
How do you compare two proportional relationships shown in different ways?
Find the unit rate of each one, in the same units, and compare the two numbers. From a graph, divide rise by run; from a table, divide y by x in any row; from an equation y = kx, read k; from words, divide the two amounts. The greater unit rate has the steeper graph when both use the same axes.
What mistakes do students make when finding slope from a graph?
A common mistake is dividing run by rise, which gives the reverse rate (for example, hours per mile instead of miles per hour). Others read only the y-value of one point, or count grid squares without checking the scale on each axis. Asking for units with every slope catches many of these errors.
Does a steeper line always mean a greater rate?
Only when both graphs have the same quantities on the same axes. If one graph puts time on the x-axis and another puts time on the y-axis, the slopes measure different things: miles per hour against hours per mile. Then a steeper line can mean a slower object.
Why does a proportional graph have to pass through the origin?
Because y = kx gives y = 0 when x = 0: zero hours of work earn zero dollars, and zero seconds of running cover zero meters. A line that crosses the y-axis somewhere else, such as y = 2x + 3, has a slope but is not proportional. Students study those lines in 8.EE.B.6 and 8.F.A.3.
Do students need to graph the equation to compare it with a graph?
No. The equation y = kx already shows the unit rate k, so students can compare it directly with the slope of the graph. Graphing the equation on the same grid is still a good check: the greater unit rate should give the steeper line.
How can parents help with 8.EE.B.5 at home?
Use everyday rates. Compare the price per ounce of two package sizes at the store, or the speeds of two ways to get to school. Ask your child to say each rate as "so much per one," and to sketch which line would be steeper on a graph.
07
Related Standards
6 standards
These standards connect to 8.EE.B.5: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.RP.A.2Prerequisite
Understand a unit rate a/b associated with a ratio a:b, and use rate language
