Year 8 · Unit 1 Ratio and proportion · Lesson 6 of 6

Proportional relationships: table, graph, rule

A table, a straight line through the origin and the rule y = kx are three views of one relationship. Move the line and test each view, then see what a start amount does to all three.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. (Year 7, Unit 6, lesson 6) On a line graph, a point is 3 across and 12 up. Write it as a coordinate pair.
    Answer(3, 12). The number across comes first, then the number up.
  2. (Lesson 5) 4 pens cost 22 rupees. How much do 6 pens cost?
    Answer33 rupees, because one pen costs 22 ÷ 4 = 5.5 and 6 × 5.5 = 33.
  3. (Year 7, Unit 3, lesson 3) What is the position rule for the sequence 4, 8, 12, 16, …?
    Answer4n, because each term is 4 times its position.
Words used in this lesson
proportional
one quantity is always the same multiple of the other. For example: cost and number of cups.
constant of proportionality
the number k in y = kx. For example: k = 3 for 3 dirhams a cup.
origin
the point (0, 0) where the axes meet. For example: 0 cups cost 0 dirhams.
linear
forming a straight line on a graph. For example: y = 2x + 4 is linear.
slope
how steeply a line rises. For example: k = 5 is steeper than k = 2.

Three views of one rule

A tea stall charges 3 dirhams for each cup. Make a table of cups against cost: 1 cup, 2 cups, 3 cups, and so on. Then plot the points, with cups along the bottom and cost up the side.

Predict first: what shape do the points make, and where does the line start? Then use the figure to test lines of your own, with and without a start amount.

2
3
3.5y ÷ x at x = 2
2.75y ÷ x at x = 4
noproportional?

y = 2x + 3: at x = 2, y is 7 and y ÷ x is 3.5; at x = 4, y is 11 and y ÷ x is 2.75. Different values, and the line starts at (0, 3): not proportional.

Show me the steps
  1. y = 2x: the points (2, 4) and (4, 8) both have y ÷ x = 2, and the line goes through (0, 0). Proportional.
  2. y = 4x: (2, 8) and (4, 16) both have y ÷ x = 4. Steeper, and still through (0, 0).
  3. y = 2x + 4: (2, 8) gives y ÷ x = 4 but (4, 12) gives 3. The line starts at (0, 4), so it is not proportional.
  4. y = 3x + 6: (2, 12) gives 6 and (4, 18) gives 4.5. Different values, and the line misses (0, 0): not proportional.

A proportional relationship is y = kx. The number k is the constant of proportionality: y ÷ x is the same for every pair with x not 0, and it is the value of one.

Three views, one relationship: in a table, y ÷ x is the same each time; in a graph, the points lie on a straight line through (0, 0); in an equation, y = kx. Doubling x doubles y.

A straight line that misses (0, 0) is a linear relationship, y = kx + c, but it is not proportional.

Back to the tea stall: y = 3x. One cup costs 3, two cost 6, three cost 9, so the points lie on a straight line, and it starts at (0, 0) because 0 cups cost 0 dirhams.

Cups, x1234
Cost, y36912
y ÷ x3333

A common mistake: “The points make a straight line, so it is proportional.” The line y = 5x + 1 is straight, but x = 1 gives 6 and x = 2 gives 11, not 12. Doubling x does not double y. A proportional graph must also pass through (0, 0).

Worked example: from a table to a rule

A table has x = 3, 6, 10 and y = 12, 24, 40. Is it proportional? If so, write the rule.

  1. Divide each y by its x: 12 ÷ 3 = 4, 24 ÷ 6 = 4, 40 ÷ 10 = 4.
  2. The value is the same each time, so the relationship is proportional with k = 4.
  3. The rule is y = 4x, and its graph is a line through (0, 0).

A proportional graph passes through (4, 10). Find k, then find y when x = 9.

  1. k = y ÷ x = 10 ÷ 4 = 2.5.
  2. The rule is y = 2.5x, so at x = 9, y = 2.5 × 9 = 22.5.
  3. Check: 9 is 2.25 times 4, and 10 × 2.25 = 22.5. k = 2.5 and y = 22.5.

Your go. The first step is given. A table has x = 4, 6, 10 and y = 14, 21, 35. Is it proportional? If so, write the rule.

  1. Divide each y by its x. The first is 14 ÷ 4 = 3.5.
  2. Now divide the other two, decide, and write the rule, then check.
Check the answer

21 ÷ 6 = 3.5 and 35 ÷ 10 = 3.5, so the value is the same each time. It is proportional, with y = 3.5x.

Check before you go on

1. Which table shows a proportional relationship?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A proportional relationship has the rule y = 5x. What is y when x = 8?

3. The point (2, 14) lies on the graph of a proportional relationship y = kx. What is k?

Level 2 · Building

4. A proportional graph passes through (12, 18). What is y when x = 20?

5. A tap fills an empty tank at a steady rate: 24 litres are in it after 4 minutes. How many litres are in the tank after 9 minutes?

6. A proportional graph passes through (3, 12). Which is its rule?

Level 3 · Stretch

7. A proportional graph passes through (4, 14). A second proportional graph is twice as steep. What is y on the second graph when x = 5?

8. A phone plan charges a fixed 12 pesos a month plus 3 pesos for each gigabyte. In a month with 6 gigabytes, what is the cost divided by the gigabytes, in pesos per gigabyte?

Something to think about

Draw y = 2x and y = 2x + 3 on the same axes. What is the same about the two lines, and what is different? Which one doubles y when x doubles?

One way to think about it

They are parallel, because both go up 2 for every 1 across. They differ in where they start: y = 2x passes through (0, 0) and y = 2x + 3 through (0, 3). Only y = 2x doubles: x = 1 gives 2 and x = 2 gives 4. For y = 2x + 3, x = 1 gives 5 and x = 2 gives 7, which is not 10.

Exit ticket

9. Which single fact shows that a table of values is proportional?