math-2017:8.EE.5
math-2017:8.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.
Learning goal: Graph a proportional relationship, explain why its unit rate is the slope of the graph, and compare two different proportional relationships that are represented in different ways: a graph, a table, an equation or a sentence.
What a proportional relationship looks like on a graph. Two quantities x and y are proportional when y = kx for one fixed number k, the constant of proportionality. The unit rate is the value of y when x is 1: k units of y per one unit of x. When every pair (x, kx) is plotted on a coordinate plane, the points lie on a straight line through the origin, because x = 0 gives y = 0. This is the graph of the proportional relationship, and the condition matters: a straight line that misses the origin is not proportional, and a curve through the origin is not proportional either.
Slope. The slope of a line is the ratio of vertical change to horizontal change between two points on it: slope = rise ÷ run = (change in y) ÷ (change in x). Take the two points (0, 0) and (1, k) on a proportional graph. The rise is k and the run is 1, so the slope is k ÷ 1 = k. That is exactly the unit rate. Any other pair of points gives the same number. Between (2, 2k) and (5, 5k) the rise is 5k − 2k = 3k and the run is 5 − 2 = 3, so the slope is 3k ÷ 3 = k again. Interpreting the unit rate as the slope is therefore not a coincidence of one example; it follows from y = kx for every pair of points.
Reading the unit rate from a graph. On the graph of a proportional relationship the point with x = 1 is not always marked, and it may be hard to read. Use any clearly marked point (x, y) with x not zero and compute y ÷ x; this works only because the graph is a line through the origin, so one point fixes the whole line. If a proportional distance-time graph passes through (4, 30), with time in hours and distance in miles, the unit rate is 30 ÷ 4 = 7.5 miles per hour. Then y = 7.5x, and you can check the equation against another marked point: at x = 2 it predicts 15 miles, which is where the line should be.
A fully worked model: which cyclist is faster? Cyclist P's motion is given as a distance-time graph: a line through (0, 0), (2, 24) and (5, 60), with time in hours on the horizontal axis and distance in miles on the vertical axis. Cyclist Q's motion is given as an equation, d = 10t, with d in miles and t in hours. These are two different proportional relationships represented in different ways, so first put them in the same form. For P, the slope of the graph is (60 − 0) ÷ (5 − 0) = 12 miles per hour; the point (2, 24) confirms it, because 24 ÷ 2 = 12. For Q, the equation already shows the unit rate: the coefficient 10 means 10 miles per hour. Since 12 is greater than 10, P is faster. If Q's line were drawn on the same axes it would be the less steep of the two, passing through (5, 50) rather than (5, 60). After 5 hours P is 10 miles ahead; after 1 hour, 2 miles ahead. Check the claim numerically rather than by eye: at t = 3, P has traveled 12 × 3 = 36 miles and Q has traveled 10 × 3 = 30 miles.
Comparing across representations. A table is consistent with a proportional relationship when y ÷ x is the same number for every row with x not zero, and any row with x = 0 has y = 0 (the quotient y ÷ x has no value at x = 0, so that row is checked separately); that shared quotient is the unit rate. A table shows only its own rows, so it can show consistency; the relationship is known to be proportional only when the situation says the same rate holds for every input. A sentence such as "the pump moves 12 liters every 4 minutes" represents one too: the unit rate is 12 ÷ 4 = 3 liters per minute. An equation y = kx shows k directly. A graph shows k as its slope. To compare two relationships, translate each into its unit rate with the same units, then compare the numbers. A comparison of a rate in meters per second with a rate in meters per minute is meaningless until one of them is converted.
The misconception to avoid. Students often decide that the steeper line is faster, full stop. That is correct only when the two lines are drawn on the same axes with the same scales. If one graph counts minutes on the horizontal axis and the other counts hours, or one vertical axis runs to 100 and the other to 10, the visual steepness tells you nothing. A line rising 1 grid square per square could be 1 mile per minute or 1 mile per hour. Always compute the slope from coordinates, with units, before comparing. A second error is reading the y-value of any marked point as the unit rate. The point (4, 30) does not mean a rate of 30; it means 30 miles in 4 hours, and the rate is the quotient.
Why the origin matters when comparing. Two lines can have the same slope but different starting values, such as y = 3x and y = 3x + 5. Only the first is proportional. In this lesson every relationship passes through the origin, so its slope, its unit rate and its constant of proportionality are all the same number. In the next lesson the intercept is allowed to move away from zero and slope becomes a separate idea from the unit rate.
Worked table to equation to comparison: A tank fills at a constant rate. A table shows 7.5 liters after 3 minutes and 20 liters after 8 minutes. Quotients: 7.5 ÷ 3 = 2.5 and 20 ÷ 8 = 2.5, so the relationship is proportional with unit rate 2.5 liters per minute, equation V = 2.5t, and a graph through the origin with slope 2.5. A second hose is described in words: "12 liters every 4 minutes", a unit rate of 12 ÷ 4 = 3 liters per minute. Since 3 is greater than 2.5, the second hose fills faster. Check the table against the equation: 2.5 × 3 = 7.5 and 2.5 × 8 = 20, both matching.
Worked graph against equation: A car's distance-time graph is a line through (0, 0) and (2.5, 150), hours and miles. Slope = 150 ÷ 2.5 = 60 miles per hour. A truck's motion is d = 55t. The car's unit rate, 60, exceeds the truck's, 55, so the car is faster. After 4 hours the car has gone 60 × 4 = 240 miles and the truck 55 × 4 = 220 miles, a 20-mile gap that grows by 5 miles every hour.
Worked comparison with different units: Swimmer A's distance-time graph is a line through the origin and (120, 180), with seconds on the horizontal axis and meters on the vertical axis: unit rate 180 ÷ 120 = 1.5 meters per second. Swimmer B is described as swimming 50 meters in 40 seconds: 50 ÷ 40 = 1.25 meters per second. Both are now in meters per second, and 1.5 is greater than 1.25, so A is faster. Had B's time been given in minutes, converting first would have been essential.
Worked slope from two non-origin points: A proportional graph passes through (2, 10) and (5, 25). Slope = (25 − 10) ÷ (5 − 2) = 15 ÷ 3 = 5. The unit rate from one point agrees: 10 ÷ 2 = 5 and 25 ÷ 5 = 5. Equation: y = 5x. The point (1, 5) is on the line even if it is not marked.
Representation task: You are given four proportional relationships, each in a different form: a graph through (0, 0) and (6, 45); a table with the pairs (2, 13) and (10, 65); the equation y = 8x; and the sentence "60 units in 12 minutes". For each, compute the unit rate with units, write the equation, and sketch all four on one set of axes with the same scales. Order them from least to greatest rate and explain how the order shows up in the steepness of the lines you drew.
Independent performance: Invent two moving objects. Give one as a distance-time graph (mark at least two points, label both axes with units) and the other as an equation d = kt. Decide which is faster and justify the decision using slopes computed from coordinates, not from appearance. Then redraw the graph with a different vertical scale so that the slower object looks steeper, and write a paragraph explaining why the picture is misleading and the computation is not. Success criteria: unit rates carry units; the equation and graph each pass a substitution check at one point; the explanation names the slope as rise divided by run and names the unit rate as the value of y at x = 1.
An adult may read the prompts aloud. Answer independently with words, objects, drawings or a demonstration before opening the answer guide. Award 1 point for a correct result or claim and 1 point for a matching explanation or demonstration; where a question asks only for a result (a corrected sentence, a spelling, a count), a fully correct result earns both points, and the answer guide shows what a complete response contains. Record prompting separately. The total (two points per question) is local practice feedback, not an Ohio proficiency level.
A proportional graph passes through (0, 0) and (6, 15). What is the unit rate, and why is it also the slope?
The unit rate is 15 ÷ 6 = 2.5 units of y per unit of x. Between (0, 0) and (6, 15) the rise is 15 and the run is 6, so the slope is 15 ÷ 6 = 2.5, the same number. On a proportional graph the slope always equals the unit rate because y = kx makes every rise-over-run ratio equal to k.
Why does the graph of a proportional relationship always pass through the origin?
Because y = kx gives y = 0 when x = 0. Zero of one quantity pairs with zero of the other, so (0, 0) is always on the graph. A line that crosses the vertical axis somewhere else, such as y = 3x + 5, is not proportional.
Object A moves at a steady speed from the start; its distance-time graph is a line through (0, 0) and (3, 21). Object B follows d = 6.5t. Both use hours and miles. Which is faster?
A: its unit rate is 21 ÷ 3 = 7 miles per hour, and B's equation shows a rate of 6.5 miles per hour. Since 7 is greater than 6.5, A is faster. The comparison works because both rates are in the same units.
A table shows (2, 9) and (6, 27). Are these rows consistent with a proportional relationship? If so, give the unit rate and the equation it would have.
Yes: 9 ÷ 2 = 4.5 and 27 ÷ 6 = 4.5, so both quotients agree. The rows fit y = 4.5x, whose unit rate is 4.5 and whose graph is a line through the origin with slope 4.5. A table can only show that every listed pair fits; the relationship is proportional if every pair with a nonzero input, listed or not, gives the same quotient and an input of 0 gives an output of 0.
A hypothetical recipe uses 1.5 cups of flour per batch. Graph the cups of flour against the number of batches for 0 to 4 batches: list the points you plot, describe the graph, and show where the unit rate appears on it.
Batches go on the horizontal axis and cups on the vertical axis, as the prompt asks. The points are (0, 0), (1, 1.5), (2, 3), (3, 4.5) and (4, 6); they lie on a straight line through the origin, so the relationship is proportional, y = 1.5x. The unit rate, 1.5 cups per batch, appears as the point (1, 1.5) and as the slope: every run of 1 batch has a rise of 1.5 cups, for example from (2, 3) to (4, 6) the rise is 3 over a run of 2, and 3 ÷ 2 = 1.5.
Machine A runs at a steady rate from zero; its graph is a line through (0, 0) and (5, 40), parts against minutes. Machine B is described as making 90 parts in 12 minutes. Which machine is faster?
A: 40 ÷ 5 = 8 parts per minute, while B makes 90 ÷ 12 = 7.5 parts per minute. Since 8 is greater than 7.5, A is faster. Both rates are in parts per minute, so they can be compared directly.
Show that the slope between (3, 12) and (7, 28) on a proportional graph equals the unit rate found from either point alone.
Slope = (28 − 12) ÷ (7 − 3) = 16 ÷ 4 = 4. From one point: 12 ÷ 3 = 4 and 28 ÷ 7 = 4. All three agree because the relationship is y = 4x, so any rise divided by its run gives 4.