7.RP.A.2Common CoreMathRatios and Proportional RelationshipsGrade 7
7.RP.A.2: Recognizing and Representing Proportional Relationships
In plain English: 7.RP.A.2 is the Common Core grade 7 math standard that asks students to recognize and represent proportional relationships. Students test tables for equivalent ratios and check whether a graph is a straight line through the origin, find the constant of proportionality in tables, graphs, equations, diagrams and words, write equations such as t = pn, and explain what points like (0, 0) and (1, r) mean.
Recognize and represent proportional relationships between quantities.
a.Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
b.Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
c.Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
d.Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Common Core State Standards for Mathematics · Domain: Ratios and Proportional Relationships (RP) · Cluster: Analyze proportional relationships and use them to solve real-world and mathematical problems. Also written as 7.RP.2 · Official standard
Two quantities are in a proportional relationship when one is always the same number times the other. For example, if each notebook costs $3, the total cost is always 3 times the number of notebooks. That fixed number is the constant of proportionality, usually called k. It is the same as the unit rate: the amount of y for exactly 1 unit of x. Every proportional relationship can be written as an equation y = kx.
The standard has four parts, and the lesson covers each one. Part a: decide whether a relationship is proportional, by checking a table for equivalent ratios (ratios that name the same comparison, like 2 to 3 and 4 to 6) or by graphing it on a coordinate plane (a grid with a horizontal x-axis and a vertical y-axis) and seeing whether the graph is a straight line through the origin, the point (0, 0). Part b: find k in tables, graphs, equations, diagrams and word descriptions. Part c: write the equation. Part d: explain what a point (x, y) on the graph means in the situation, especially (0, 0) and (1, r), where r is the unit rate.
Learning Objectives
By the end of this lesson, students will be able to:
Decide whether two quantities are proportional by testing a table for equivalent ratios or by graphing and looking for a straight line through (0, 0)
Find the constant of proportionality in a table, a graph, an equation, a diagram and a word description
Write an equation of the form y = kx for a proportional relationship, such as t = pn for a total cost
Explain what a point (x, y) on the graph means in the situation, including (0, 0) and (1, r)
Prior Knowledge Required
Students should already be comfortable with:
Unit rates, including unit rates of fractions 7.RP.A.1
Tables of equivalent ratios and plotting their pairs on a coordinate plane 6.RP.A.3
Writing an equation for one quantity in terms of another, such as d = 50t 6.EE.C.9
Plotting points (x, y) in the first quadrant, the part of the grid where both numbers are positive 5.G.A.2
Show two price lists for stickers and ask students which store keeps the same price per sticker.
Warm-up sticker prices
Stickers
Store A cost
Store B cost
4
$1
$2
8
$2
$3
12
$3
$4
Warm-Up Prompt
"At which store does every sticker cost the same amount? How do you know? What would 20 stickers cost at that store?"
At Store A, every row gives 1/4 dollar per sticker: 1 ÷ 4, 2 ÷ 8 and 3 ÷ 12 are all $0.25. So 20 stickers cost $5. At Store B, the price per sticker changes (2 ÷ 4 = $0.50 but 4 ÷ 12 is about $0.33), because Store B adds a $1 charge to every order. Tell students that Store A shows a proportional relationship and Store B does not, and that today they will learn four ways to tell.
Direct Instruction20 minutes
Part a: Is it proportional? Show both tests.
Table test: divide y by x in every row. If every quotient (the answer to a division) is the same number, the ratios are equivalent and the relationship is proportional.
Graph test: plot the pairs. If the points lie on a straight line that goes through the origin (0, 0), the relationship is proportional. A curve, or a straight line that misses (0, 0), is not proportional.
Find k: k = y ÷ x for any pair (not (0, 0)). On a graph, k is the y-value where x = 1.
Write the equation: y = kx, using letters that fit the story, such as t for total cost and n for number of items.
Table, proportional (parts a, b and c)
A part-time worker's pay: 2 hours, $27; 3 hours, $40.50; 5 hours, $67.50; 8 hours, $108. Is pay proportional to hours?
Equation: 27 ÷ 2 = 13.50, 40.50 ÷ 3 = 13.50, 67.50 ÷ 5 = 13.50 and 108 ÷ 8 = 13.50. Every ratio is the same, so it is proportional with k = 13.50 dollars per hour. Equation: p = 13.5h.
Table, not proportional (part a)
Bowling: 1 game costs $7, 2 games cost $11 and 3 games cost $15, because shoe rental adds $3 to every visit.
Equation: 7 ÷ 1 = 7, 11 ÷ 2 = 5.50 and 15 ÷ 3 = 5. The ratios are not equal, so the cost is not proportional to the number of games. The graph is a straight line, but it crosses the y-axis at (0, 3), not at the origin.
Verbal description to equation: the official example (part c)
Total cost t is proportional to the number n of items bought at a constant price p. Pens cost $1.25 each.
Equation: t = pn, so t = 1.25n. The constant of proportionality is the price, p = 1.25. For 6 pens, t = 1.25 × 6 = $7.50.
Graph, and what its points mean (parts b and d)
Diagram 1 shows the water in Tub A, which starts empty and fills from a hose. The line passes through (0, 0), (1, 4) and (3, 12).
Equation: (0, 0): at 0 minutes, the tub holds 0 gallons. (1, 4): after 1 minute it holds 4 gallons, so r = 4 gallons per minute. (3, 12): after 3 minutes it holds 12 gallons. Equation: g = 4m.
Verbal description and a diagram (part b)
A printer prints 45 pages every 3 minutes at a steady speed. Diagram 2 shows this on a double number line (two number lines lined up so that matching amounts sit one above the other).
Equation: 45 ÷ 3 = 15, so k = 15 pages per minute and p = 15m. On the double number line, the 1-minute mark lines up with 15 pages.
A double number line is a diagram of a proportional relationship: to find k, follow the lines to 1 on the bottom and read the top. In an equation y = kx, k is the number multiplied by x. Equations such as y = 4x + 6 or y = x + 5 are not proportional, because y is not 0 when x is 0.
Part d: What do the points mean? A point (x, y) on the graph is one pair of matching amounts, such as "after 3 minutes, 12 gallons." Two points need special attention. The origin (0, 0) means "none of x goes with none of y," which every proportional graph must include. The point (1, r) shows the unit rate: the amount of y for 1 unit of x. In Diagram 1, Tub B starts with 6 gallons, so its line begins at (0, 6). It rises at the same speed as Tub A, but it is not proportional. Ask: "Where would Tub B's line need to start for the gallons to be proportional to the minutes?" (At the origin.)
Guided Practice15 minutes
Pairs work through four problems. For each, one partner decides whether it is proportional and the other finds k and the equation, then they switch for the next problem. Ask pairs to say which test they used.
Guided practice problems with answers
Problem
Answer
Museum tickets: 3 tickets cost $22.50, 5 cost $37.50 and 8 cost $60. Proportional? If so, find k and the equation.
Yes. Every ratio is 7.50, so k = 7.50 dollars per ticket and c = 7.5t
A sunflower's height: week 1, 4 cm; week 2, 7 cm; week 3, 10 cm. Proportional?
No. 4 ÷ 1 = 4, 7 ÷ 2 = 3.5 and 10 ÷ 3 is about 3.33. The ratios are not equal
Bananas: y = 0.6x, where x is pounds and y is the cost in dollars. What is k, and what does the point (5, 3) mean?
k = 0.60 dollars per pound. The point (5, 3) means 5 pounds of bananas cost $3
A car goes 186 miles on 6 gallons of gas, at the same rate throughout. Find k and write an equation.
k = 186 ÷ 6 = 31 miles per gallon, so d = 31g
Listen for the error of checking only that y goes up by the same amount each row. The sunflower grows 3 cm each week, but its ratios are not equal, so it is not proportional. Ask: "Would a graph of the sunflower pass through (0, 0)?" (No. At week 0 it would already be 1 cm tall.)
Independent Practice15 minutes
Students decide whether each relationship is proportional. For the proportional ones, they give k and an equation. For the graph, they sketch it on grid paper first.
Independent practice problems with answers
Problem
Answer
Table: x = 4, 6, 10; y = 10, 15, 25
Proportional, k = 2.5, y = 2.5x
Table: x = 2, 4, 6; y = 5, 9, 13
Not proportional (5 ÷ 2 = 2.5, but 9 ÷ 4 = 2.25)
Equation: y = (2/3)x
Proportional, k = 2/3; the graph goes through (1, 2/3)
Equation: y = x + 4
Not proportional: when x = 0, y = 4
Graph: a straight line through (0, 0) and (5, 40)
Proportional, k = 40 ÷ 5 = 8, y = 8x
Words: Kiara earns $36 for 4 hours of tutoring, at the same hourly rate
Proportional, k = 9 dollars per hour, e = 9h
Closure5 minutes
Exit ticket: (1) Is the table x = 3, 6, 9; y = 12, 24, 39 proportional? (No: 12 ÷ 3 = 4 and 24 ÷ 6 = 4, but 39 ÷ 9 is about 4.33.) (2) A graph of a walker's distance in miles against time in hours goes through (0, 0) and (1, 2.5). What does (1, 2.5) mean, and what is the equation? (The walker goes 2.5 miles in the first hour, the unit rate; d = 2.5t.) (3) In one sentence: why must every proportional graph pass through (0, 0)?
Differentiation Strategies
For Struggling Students
Add a third column to every table for y ÷ x, so the table test becomes "are all the numbers in this column the same?"
Give a sentence frame for points: "The point (__, __) means __ [x units] go with __ [y units]."
Use whole-number constants first, then move to decimals such as 13.50 and fractions such as 3/4
For Advanced Students
Give two proportional relationships, one as a table and one as an equation, and ask which has the greater unit rate and how the graphs would differ
Ask students to write a story for y = 4x + 6 and explain in words why it is not proportional even though the graph is a straight line
Ask students to show that if (a, b) is on a proportional graph, then (2a, 2b) and (a/2, b/2) are on it too
Assessment Guidance
What to Look For
Check that students divide y by x in every row of a table, rather than looking only at how much y goes up. On graphs, look for both conditions: a straight line and a line through (0, 0). When finding k, students should name its units, such as dollars per ticket. In equations, check that k multiplies the input letter (t = 1.25n, not n = 1.25t). For part d, students should explain a point in a full sentence with both quantities and their units, and connect (1, r) to the unit rate.
02
Classroom Activities
3 Activities
1
Proportional or Not? Card Sort
15 minPairs
Pairs sort 8 cards into two piles, proportional and not proportional. Each card shows a relationship in a different form: a table, an equation, words or a graph described in words. For every proportional card, pairs write k and an equation.
The 8 Cards
C1 (table): x = 2, 3, 7; y = 9, 13.5, 31.5
C2 (table): x = 1, 2, 4; y = 5, 8, 14
C3 (equation): y = 12x
C4 (equation): y = 5 + 2x
C5 (words): Each bag of apples costs $4.25.
C6 (words): A taxi charges $3 to start plus $2 per mile.
C7 (graph): a straight line through (0, 0) and (5, 8)
C8 (graph): a straight line through (0, 6) and (2, 10)
Answer Key
Proportional: C1 (k = 4.5, y = 4.5x), C3 (k = 12), C5 (k = 4.25, c = 4.25b) and C7 (k = 8/5 = 1.6, y = 1.6x)
Not proportional: C2 (5 ÷ 1 = 5 but 8 ÷ 2 = 4), C4 and C6 (both have a starting amount) and C8 (the line misses the origin)
Discussion Questions
Exactly four cards are proportional. What do the four "not" cards have in common?
C8 is a straight line. Why is it still not proportional?
C4 and C6 describe the same kind of rule. Write C6 as an equation (c = 2m + 3) and compare it with C4.
Modification for Distance Learning
Put the 8 cards on a shared slide with two boxes labeled "proportional" and "not proportional." Pairs drag each card into a box and type k and the equation next to each proportional card.
2
Penny Stacks
20 minGroups of 3-4
Groups measure the height of stacks of pennies and graph the results on grid paper. A US penny is about 1.5 mm thick (1.52 mm), so the height of a stack is proportional to the number of pennies. Measured heights will be close to 7.6, 15.2, 22.8 and 30.4 mm for 5, 10, 15 and 20 pennies.
Procedure
Build stacks of 5, 10, 15 and 20 pennies and measure each height with a millimeter ruler
Record the pairs in a table and divide height by number of pennies for each row
Plot the points on grid paper, with pennies on the x-axis and height in mm on the y-axis, and draw a line through them
Now put each stack on a block 10 mm tall and measure from the table to the top of the stack. Record and graph these heights in a second color
Discussion Questions
Does your first line go through (0, 0)? What does the point (0, 0) mean for a stack of pennies?
What does the point (1, 1.52) mean? How is it related to the numbers in your y ÷ x column?
Is the height with the block proportional to the number of pennies? Use your table and your graph to explain. (No: it starts at (0, 10), and 17.6 ÷ 5 is not equal to 25.2 ÷ 10.)
Challenge Variation
Groups predict the height of a stack of 50 pennies from their equation, then check by building it. They explain why a small measuring error in one stack does not change k much.
3
Point Stories
15 minPairs
Pairs get two proportional graphs and write a sentence for what each labeled point means, then decide whether two new points belong on each graph. This practices part d of the standard.
Graph Cards
Graph 1 (babysitting pay): a line through (0, 0), (1, 12), (3, 36) and (5, 60), with hours on the x-axis and dollars on the y-axis
Graph 2 (buying gas): a line through (0, 0), (1, 3.80) and (5, 19), with gallons on the x-axis and dollars on the y-axis
Tasks
Write one sentence for each labeled point, with units
Circle the point that shows the unit rate on each graph, and name the rate
Decide whether (2, 30) belongs on Graph 1 and whether (10, 38) belongs on Graph 2 (No: 2 hours pay $24. Yes: 10 gallons cost $38.)
Write the equation of each graph (p = 12h and c = 3.8g)
Discussion Questions
Why does (0, 0) make sense for babysitting pay but not for a job that pays a $10 sign-up bonus?
On Graph 2, what point shows the cost of half a gallon? Is it on the line?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: A Proportional and a Non-Proportional Graph
Two tubs fill at 4 gallons per minute. Tub A starts empty, so its graph is a straight line through the origin: the gallons are proportional to the minutes, with y = 4x. The point (0, 0) means 0 gallons at 0 minutes, and (1, 4) shows the unit rate. Tub B starts with 6 gallons, so its straight line starts at (0, 6) and is not proportional. Drawn to scale.
Diagram 2: Finding k on a Double Number Line
The top line shows pages and the bottom line shows minutes, lined up so matching amounts sit one above the other. The given pair is 45 pages in 3 minutes. Dividing both by 3 gives 15 pages in 1 minute, so the constant of proportionality is 15 pages per minute.
04
Homework Assignment
~30 min
7.RP.A.2 Homework: Proportional Relationships
Directions: Show your work. For each table, write the ratio y ÷ x for every row. Use grid paper for the graphs. Give every constant of proportionality with its units, and answer questions about points in full sentences.
Part 1: Is It Proportional? (Problems 1-2)
Decide whether each table shows a proportional relationship, and explain. (a) A faucet: 3 minutes, 4.5 liters; 6 minutes, 9 liters; 10 minutes, 15 liters. (b) A bamboo shoot: day 1, 8 cm; day 2, 10 cm; day 3, 12 cm.
Graph both tables on the same grid. Table A: x = 1, 2, 3, 4 and y = 3, 6, 9, 12. Table B: x = 1, 2, 3, 4 and y = 3, 5, 7, 9. Use the graphs to explain which one is proportional.
Part 2: The Constant and the Equation (Problems 3-4)
Find the constant of proportionality in each form, with units, and write an equation. (a) Words: a machine fills 150 bottles every 4 minutes. (b) Equation: y = (2/5)x, where y is cups of sugar and x is cups of flour. (c) Diagram: a double number line shows 0, 7, 14 and 21 kilometers lined up with 0, 1/2, 1 and 1 1/2 hours of cycling.
Movie tickets cost $11.50 each. The total cost t is proportional to the number of tickets n. Write the equation, make a table for n = 0, 1, 2 and 4, and find the cost of 6 tickets.
Part 3: What the Points Mean (Problems 5-6)
The equation y = 18x gives a worker's pay y in dollars for x hours. Explain what the points (0, 0), (1, 18) and (5, 90) mean. Is (4, 70) on the graph? Explain.
A graph of a drip hose's water use passes through (0, 0) and (8, 6), with hours on the x-axis and gallons on the y-axis. Find the point (1, r) and explain what it means. Write the equation, and find how much water the hose uses in 12 hours.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Deciding
Ratio or graph test used correctly and explained
Correct decision with a weak reason
Incorrect or no reason
Constant of Proportionality
Correct k with units in every form
Correct k but units missing or one form wrong
Most constants incorrect
Equations
Equations of the form y = kx with fitting letters
Right idea, k in the wrong place once
No equations or incorrect
Meaning of Points
Full sentences with both quantities and units, including (0, 0) and (1, r)
Sentences missing units or one point
Missing or incorrect
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
Which table shows a proportional relationship between x and y?
Answer: B
In choice B, 3 ÷ 2, 6 ÷ 4 and 9 ÷ 6 all equal 1.5, so the ratios are equivalent. Choice A goes up by 3 each time, but its ratios are 4, 3.5 and about 3.33. Choice C adds 1 to x (y = x + 1), so its ratios are 2, about 1.33 and 1.2. Choice D squares x, and its ratios are 2, 3 and 4.
Question 2 of 20 · Multiple Choice
A store's apple prices: 2 pounds cost $3.50, 4 pounds cost $7.00 and 6 pounds cost $10.50. What is the constant of proportionality?
Answer: A
Divide cost by pounds in any row: 3.50 ÷ 2 = 1.75, and the other rows agree. Choice B uses the cost of 2 pounds as if it were the cost of 1 pound. Choice C divides pounds by cost (2 ÷ 3.50 is about 0.57), which is pounds per dollar. Choice D uses the cost of 6 pounds as if it were the cost of 1 pound.
Question 3 of 20 · Multiple Choice
On a highway trip, d = 65h gives the distance d in miles after h hours. What is the constant of proportionality, and what does it mean?
Answer: B
In d = 65h, the number multiplied by h is 65, so the car goes 65 miles per hour. Choice A is a true point on the graph, (2, 130), but it is not the constant. Choice C flips the rate. Choice D uses the right number but reverses the units.
Question 4 of 20 · Multiple Choice
Which graph shows a proportional relationship?
Answer: D
A proportional graph must be a straight line through the origin, and only choice D is both. Choice A is straight but starts at (0, 3). Choice B goes through the origin but is a curve. Choice C is straight, but it crosses the y-axis at (0, -1), so it misses the origin.
Question 5 of 20 · Multiple Choice
A proportional graph passes through (0, 0) and (4, 22). What is the constant of proportionality?
Answer: C
k = y ÷ x = 22 ÷ 4 = 5 1/2. Choice A divides x by y (4 ÷ 22). Choice B uses the y-value alone. Choice D subtracts 4 from 22.
Question 6 of 20 · Multiple Choice
A graph shows the cost y in dollars of x pounds of cherries. It passes through (1, 4.50). What does this point mean?
Answer: C
In (x, y), x is pounds and y is dollars, so (1, 4.50) means 1 pound costs $4.50, the unit rate. Choice A swaps the two quantities. Choice B ignores the x-value, as if the cost never changed. Choice D describes the point (0, 4.50), which is not on a proportional graph.
Question 7 of 20 · Multiple Choice
A graph shows the calories y a person burns after x minutes of jumping rope. It is a straight line through the origin. What does the point (0, 0) mean?
Answer: B
The origin pairs 0 minutes with 0 calories: before the person starts, no calories have been burned. Choice A confuses the starting point with the rate. Choice C reverses the rule: proportional graphs must start at (0, 0). Choice D describes a point (1, 1), which the question does not give.
Question 8 of 20 · Multiple Choice
A table shows a server's pay: 2 hours, $31; 5 hours, $77.50; 9 hours, $139.50. Which equation shows the relationship between hours x and pay y?
Answer: D
31 ÷ 2 = 15.50, 77.50 ÷ 5 = 15.50 and 139.50 ÷ 9 = 15.50, so k = 15.50 and y = 15.5x. Choice A uses the pay for 2 hours as the pay for 1 hour. Choice B puts k in front of the wrong letter. Choice C uses the difference in the first row (31 - 2 = 29) and fails for the other rows.
Question 9 of 20 · Multiple Choice
Mangoes cost $1.20 each. Which equation gives the total cost t in dollars for n mangoes?
Answer: B
The total cost is the price times the number of items, t = pn, so t = 1.20n. Choice A adds the price once instead of once per mango. Choice C switches the letters. Choice D divides the price by the number of mangoes, so more mangoes would cost less.
Question 10 of 20 · Multiple Choice
A double number line for a drink mix shows 0, 9, 18 and 27 cups of water lined up with 0, 2, 4 and 6 scoops of powder. What is the constant of proportionality in cups of water per scoop?
Answer: C
9 cups go with 2 scoops, so 9 ÷ 2 = 4 1/2 cups of water per scoop, and 18 ÷ 4 and 27 ÷ 6 agree. Choice A divides scoops by cups. Choice B reads the first water mark without dividing by 2. Choice D reads the first scoop mark.
Question 11 of 20 · Multiple Choice
A worker packs 84 boxes in 6 hours at a steady rate. What is the constant of proportionality in boxes per hour?
Answer: A
84 ÷ 6 = 14 boxes per hour. Choice B divides hours by boxes, which is hours per box. Choice C subtracts 6 from 84. Choice D multiplies 84 × 6.
Question 12 of 20 · Multiple Choice
Which equation represents a proportional relationship?
Answer: D
Only y = 7x has the form y = kx, with k = 7. The other three all give y ≠ 0 when x = 0: choice A gives 2, and choices B and C give 7. Choice A still grows by the same amount each time, but its ratios change (5, 4, about 3.67).
Question 13 of 20 · Multiple Choice
Photocopies cost the same amount each. The graph of cost y in dollars against number of copies x passes through (40, 6). What does this point mean?
Answer: A
In (40, 6), x = 40 copies and y = $6, so 40 copies cost $6 (and each copy costs 6 ÷ 40 = $0.15). Choice B switches x and y. Choice C reads the y-value as the price of one copy. Choice D pairs the number of copies with the unit price instead of the total cost.
Question 14 of 20 · Multiple Choice
A graph of a proportional relationship passes through (6, 8). Which other point is also on the graph?
Answer: C
k = 8 ÷ 6 = 4/3, and 4 ÷ 3 = 4/3, so (3, 4) is on the line. Choice A adds 2 to each number, and choice B adds 4, which does not keep the ratio. Choice D subtracts 3 from each number, and 5 ÷ 3 is not 4/3.
Question 15 of 20 · Short Answer
A car's trip log: 3 gallons, 84 miles; 5 gallons, 140 miles; 8 gallons, 224 miles. Is distance proportional to gas used? If so, find k and write an equation.
84 ÷ 3 = 28, 140 ÷ 5 = 28 and 224 ÷ 8 = 28, so yes, it is proportional, with k = 28 miles per gallon and m = 28g.
Question 16 of 20 · Short Answer
Is the relationship in this table proportional? x = 2, 4, 6 and y = 9, 16, 23. Explain.
No. 9 ÷ 2 = 4.5, 16 ÷ 4 = 4 and 23 ÷ 6 is about 3.83, so the ratios are not equivalent. (The rule is y = 3.5x + 2, which does not pass through the origin.)
Question 17 of 20 · Short Answer
A graph shows the distance d in miles a cyclist rides in t hours. It is a straight line through (0, 0) and (2, 23). Explain what (0, 0) means, find the point (1, r), and explain what it means.
(0, 0): at 0 hours, the cyclist has ridden 0 miles. r = 23 ÷ 2 = 11.5, so the point is (1, 11.5): the cyclist rides 11.5 miles in 1 hour, the unit rate. The equation is d = 11.5t.
Question 18 of 20 · Short Answer
A school club buys T-shirts for $8.75 each. Write an equation for the total cost t of n shirts, and find the cost of 24 shirts.
t = 8.75n. For 24 shirts, t = 8.75 × 24 = $210.
Question 19 of 20 · Short Answer
A double number line shows 0, 14, 28 and 42 dollars lined up with 0, 4, 8 and 12 pounds of birdseed. Find the constant of proportionality, write an equation, and find the cost of 10 pounds.
k = 14 ÷ 4 = 3.50 dollars per pound, so c = 3.5p. For 10 pounds, c = 3.5 × 10 = $35.
Question 20 of 20 · Short Answer
Smoothies cost $6 each, so y = 6x gives the cost y in dollars of x smoothies. Ella says the point (3, 20) is on the graph. Is she right? What is the correct point for 3 smoothies, and what does it mean?
No. When x = 3, y = 6 × 3 = 18, so the point is (3, 18): 3 smoothies cost $18. The point (3, 20) would mean $20 for 3 smoothies, which does not match the price.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.RP.A.2 mean?
7.RP.A.2 means students can tell whether two quantities are proportional and can show the relationship in several ways. They test tables and graphs (part a), find the constant of proportionality in any form (part b), write equations such as y = kx (part c) and explain what points on the graph mean (part d). For example, if every ticket costs $9, the total cost is proportional to the number of tickets, with c = 9n.
What is a proportional relationship in simple words?
It is a relationship where one quantity is always the same number times the other. Double one, and the other doubles; take none of one, and you have none of the other. Buying gas at one price per gallon is proportional. A phone plan with a monthly fee plus a charge per minute is not, because you pay the fee even for 0 minutes.
Is the constant of proportionality the same as the unit rate?
Yes, for a proportional relationship they are the same number. The unit rate is the amount of y for 1 unit of x, and the constant of proportionality k is the number in y = kx. The standard itself writes "constant of proportionality (unit rate)." On a graph, it is the y-value of the point where x = 1.
How do you know if a graph is proportional?
A graph is proportional when it is a straight line that passes through the origin, (0, 0). Both conditions matter. A straight line that starts above the origin, such as the graph of a taxi fare with a starting charge, is not proportional. A curve through the origin is not proportional either.
Why isn't y = 2x + 3 proportional if its graph is a straight line?
It is not proportional because y is 3, not 0, when x is 0, so the ratios y ÷ x change. At x = 1 the ratio is 5, and at x = 3 it is 3. In grade 8, students learn that y = mx + b describes all straight lines and that only the lines with b = 0 are proportional (8.EE.B.6).
What do the points (0, 0) and (1, r) mean on a proportional graph?
The point (0, 0) means that zero of the first quantity goes with zero of the second, such as 0 hours worked and $0 earned. The point (1, r) shows the unit rate: for 1 unit of x, you get r units of y, such as $16 for 1 hour. Part d of the standard asks students to explain these points in words that fit the situation.
What grade is 7.RP.A.2, and what comes next?
It is a grade 7 standard in Ratios and Proportional Relationships. It builds on unit rates (7.RP.A.1) and leads to percent and multistep ratio problems in grade 7 (7.RP.A.3). In grade 8, students graph proportional relationships and call the unit rate the slope (8.EE.B.5). In Algebra I, proportional relationships become a special case of linear functions.
What mistakes should teachers watch for?
A common mistake is deciding a table is proportional because y goes up by the same amount each row, which only shows it is a straight line. Other frequent errors are forgetting to check the origin on a graph, dividing x by y instead of y by x, putting k in front of the wrong letter, and describing a point without its units.
Does 7.RP.A.2 include the word slope?
No. Slope is introduced in grade 8 (8.EE.B.5 and 8.EE.B.6). In grade 7, students talk about the constant of proportionality or the unit rate. Teachers can mention that the steepness of a proportional graph shows how big k is, but slope formulas belong to the next grade.
How can parents help with proportional relationships at home?
Parents can ask "Is it proportional?" about everyday prices. For example, at $2.50 per pound, 2 pounds of grapes cost $5 and 4 pounds cost $10, which is proportional. A gym with a joining fee is not. Ask your child to find the price for 1 unit, to write an equation, and to say what "0 pounds" would cost.
07
Related Standards
6 standards
These standards connect to 7.RP.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.RP.A.3Prerequisite
Use ratio and rate reasoning with tables, tape diagrams and double number lines
