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Learn to extract k from ratio tables and proportional lines, write unit-rate statements, and check work with a printable worksheet plan.
The constant of proportionality k links two proportional quantities: y = kx. In a ratio table, k is the quotient y/x for every pair. On a graph, a proportional line passes through the origin, and k is the vertical change divided by the horizontal change between any two points on the line. This lesson shows how to compute k, state it as a unit rate, and verify consistency. A printable worksheet with six items is planned but not yet built; the resource plan below describes its intended contents.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
A ratio table lists paired values of two proportional quantities. Because the ratio y/x is constant, choose any row, divide the y-value by the x-value, and simplify. The result is k. Check a second row to confirm the same quotient.
Write the unit-rate statement in words: for each 1 unit of x, y changes by k units. Keep units consistent and label them explicitly.
A proportional graph is a straight line through the origin. Choose two points on the line, preferably with integer coordinates, and compute the vertical change divided by the horizontal change. That quotient equals k.
If the line passes through (0,0) and (a,b), then k = b/a. The point (1,k) is always on the line, which gives a quick visual check.
A unit-rate statement connects k to the context. For example, if k = 3 and x is hours while y is dollars, write: for each 1 hour, the cost is 3 dollars. Avoid swapping the quantities; the statement must match the table or graph labels.
To check, substitute a known pair into y = kx. If the equation holds for every pair, the value of k is consistent.
A table shows x = 2, y = 7; x = 4, y = 14; x = 6, y = 21. Find k and write a unit-rate statement if x is hours and y is pages.
k = 3.5 pages per hour. For each 1 hour, 3.5 pages are produced.
Why: 7 divided by 2 equals 3.5; 14 divided by 4 equals 3.5; 21 divided by 6 equals 3.5. The quotient is constant, so k = 3.5.
Check: Substitute (2,7) into y = 3.5x: 7 = 3.5 times 2, true. Substitute (6,21): 21 = 3.5 times 6, true.
Table: x = 3, y = 12; x = 5, y = 20; x = 8, y = 32. Find k and state the unit rate if x is bags and y is apples.
k = 4 apples per bag.
Why: 12 divided by 3 equals 4; 20 divided by 5 equals 4; 32 divided by 8 equals 4. The ratio is constant.
Check: Substitute (3,12) into y = 4x: 12 = 4 times 3, true. Substitute (8,32): 32 = 4 times 8, true.
A proportional line passes through the origin and the point (4, 10). Find k and write a unit-rate statement if x is minutes and y is liters.
k = 2.5 liters per minute.
Why: Vertical change is 10, horizontal change is 4, so k = 10/4 = 2.5. The line passes through the origin, so the ratio is constant.
Check: The point (1, 2.5) should lie on the line. Substitute (4,10) into y = 2.5x: 10 = 2.5 times 4, true.
Table: x = 6, y = 9; x = 10, y = 15; x = 14, y = 21. Find k and state the unit rate if x is trays and y is muffins.
k = 1.5 muffins per tray.
Why: 9 divided by 6 equals 1.5; 15 divided by 10 equals 1.5; 21 divided by 14 equals 1.5. The quotient is constant.
Check: Substitute (10,15) into y = 1.5x: 15 = 1.5 times 10, true. Substitute (14,21): 21 = 1.5 times 14, true.
A proportional line passes through the origin and the point (5, 2). Find k and write a unit-rate statement if x is hours and y is kilometers.
k = 0.4 kilometers per hour.
Why: Vertical change is 2, horizontal change is 5, so k = 2/5 = 0.4. The line passes through the origin.
Check: The point (1, 0.4) should lie on the line. Substitute (5,2) into y = 0.4x: 2 = 0.4 times 5, true.
Yes. Any nonzero constant is valid. Simplify fractions and keep decimals consistent with the table values.
Skip that row for computing k because division by zero is undefined. Use another row with nonzero x.
The graph must be a straight line through the origin. If it does not pass through (0,0), it is not proportional.
A single pair gives a candidate k. Checking a second pair confirms the ratio is constant and catches arithmetic errors.