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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
A proportional relationship, such as cost growing at a fixed price per pound, graphs as a straight line through the origin. In seventh grade students found the unit rate; in eighth grade they name that unit rate as the slope of the line. A graph through (0, 0) and (1, 4) rises 4 units for every 1 unit across, so its slope is 4 and its equation is y = 4x.
The second half of the standard is comparison across representations. One runner might be described by a distance-time graph, another by the equation d = 5.5t, a third by a table. To decide who is faster, students must pull the unit rate out of each form: read the point where the time is 1 on the graph, read the coefficient in the equation, or divide distance by time in the table. Whichever rate is larger corresponds to the steeper line.
Students divide horizontal change by vertical change and report 1/4 instead of 4. Anchor slope to the unit rate: how much y changes when x goes up by 1.
Two graphs drawn with different axis scales can make the slower relationship look steeper. Compare the computed rates, not the pictures.
A line that does not pass through the origin, like y = 2x + 3, is linear but not proportional. The ratio y/x is not constant.
Taking a point such as (3, 12) and subtracting gives 9 rather than dividing to get 4. For a line through the origin the rate is y ÷ x.
Maya's savings follow the equation s = 12w, where s is dollars and w is weeks. Leo's savings are shown in a table: after 3 weeks he has $39, and after 5 weeks he has $65. Who saves more per week, and by how much?
Answer: Leo saves more, by $1 per week ($13 per week compared with $12 per week).
Give each group a different representation of a relationship (a story, a table, a graph and an equation) and have them find a partner whose representation shows the same unit rate. The discussion of why they match is where the connection between slope and rate becomes solid.
Test items commonly put two relationships side by side in different forms and ask which has the greater rate of change. Model a consistent move: convert everything to 'y per one x' before comparing. This sets up 8.EE.B.6 and the comparison of functions in 8.F.A.2.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 4.5 (also accepted: 9/2)
Slope = change in y ÷ change in x = 18 ÷ 4 = 4.5.
Answer: A) Car B
Car A's rate is 55 mph. Car B's rate is 180 ÷ 3 = 60 mph. Car B is faster.
Answer: B) y = 7x
A proportional relationship has the form y = kx, where k is the unit rate. y = 7x has k = 7.
Answer: 0.8 (also accepted: 0.80, $0.80)
Unit rate = 2.40 ÷ 3 = 0.80 dollars per pound, so the slope is 0.8.
Answer: D) The unit rate r
When x = 1, the y-value equals the unit rate, which is also the slope of the line.
Answer: 3 (also accepted: 3 gallons)
The first pump fills 150 ÷ 6 = 25 gallons per minute. The second fills 28. The difference is 28 - 25 = 3 gallons per minute.
A full lesson with slides, activities and an exit ticket on proportional relationships and slope, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.EE.B.5 with an answer key, ready in about a minute.
Make a worksheet →Turn proportional relationships and slope into a quiz students answer online that marks itself, with a class summary for you.
Build a test →For a proportional relationship, yes. The unit rate (y per one x) is exactly the slope of the line through the origin.
Students might get one relationship as a graph and another as an equation or table, and must find and compare the unit rate of each.