Recognize and represent proportional relationships between quantities.
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
Two quantities are proportional when one is always the same multiple of the other. If every ticket costs $4.25, the total cost is always 4.25 times the number of tickets, so cost and tickets are in a proportional relationship. That fixed multiplier is the constant of proportionality, and it is the unit rate from 7.RP.A.1 wearing a new name.
Students learn to test for proportionality in several representations. In a table, every y/x ratio is the same. On a graph, the points lie on a straight line that passes through the origin. In an equation, the relationship has the form y = kx with no added constant. They also read the meaning of a point in context: (0, 0) says zero tickets cost nothing, and (1, k) shows the cost of exactly one ticket. A straight line that misses the origin, such as a taxi fare with a starting fee, is linear but not proportional, and telling those two apart is one of the main goals of the grade.
A line for 'a $3 entry fee plus $2 per ride' is straight but starts at (0, 3). Students should check the origin, not just the straightness of the graph.
Seeing y go up by 4 each time x goes up by 1 does not prove proportionality. The table 1, 5 / 2, 9 / 3, 13 rises steadily but its ratios 5, 4.5 and 4.33 are not equal.
If 6 notebooks cost $9, the constant is 9/6 = 1.5 dollars per notebook, not 6/9. Labeling the constant with units shows which quantity goes on top.
Students sometimes say the point (1, 12) means '1 dollar for 12 hours'. Reading the axis labels aloud, x first, fixes the order.
A table shows x = 2, 5, 8 and y = 7, 17.5, 28. Is the relationship proportional? If so, write its equation.
Answer: Yes, it is proportional, with constant 3.5, so y = 3.5x.
Give students the same situation in four forms at once (a story, a table, a graph and an equation) and ask them to find the constant in each. Card sorts that mix proportional and non-proportional relationships work well, especially when the non-proportional cards include straight lines that miss the origin.
Expect test items that ask students to choose the equation for a graph through a given point, identify the constant from a table, or explain the meaning of a labeled point. Asking for the meaning of (1, k) in words is a quick exit ticket for the whole standard.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: A) x: 1, 2, 3 and y: 4, 8, 12
In the first table y/x is 4 every time (4/1, 8/2, 12/3). The other tables change at a steady rate but their y/x ratios are not equal.
Answer: 1.5 (also accepted: 1.50, $1.50)
Divide cost by notebooks: 9 ÷ 6 = 1.5. Each notebook costs $1.50.
Answer: C) y = 2.5x
The constant is y/x = 10/4 = 2.5, so y = 2.5x. The equation y = x + 6 also passes through (4, 10) but it does not go through the origin.
Answer: B) $12 is earned for each hour
The x-value is 1 hour and the y-value is $12, so the worker earns $12 per hour. That is the unit rate, which is also the constant of proportionality.
Answer: 70
The constant is 21 ÷ 3 = 7, so y = 7x. When x = 10, y = 7 × 10 = 70.
Answer: D) A straight line through (0, 0)
A proportional relationship graphs as a straight line through the origin. A line through (0, 3) has a starting value, and a curve does not have a constant ratio.
Answer: 4.25 (also accepted: $4.25)
In t = 4.25n the constant of proportionality is 4.25, which is the cost when n = 1. One ticket costs $4.25.
A full lesson with slides, activities and an exit ticket on recognizing proportional relationships, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.RP.A.2 with an answer key, ready in about a minute.
Make a worksheet →Turn recognizing proportional relationships into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It is the fixed number k in y = kx. It equals y/x for every pair in a proportional relationship and is the same as the unit rate.
No. A linear relationship is proportional only when its graph goes through the origin, so an equation such as y = 2x + 3 is linear but not proportional.