Direct and inverse proportion When x doubles, what happens to y?

The lesson, step by step

45 minutes in total. The stretch is optional.

1. 1 · Think5 min

   Double x

   Table A: x = 1, 2, 4 and y = 3, 6, 12

   Table B: x = 1, 2, 4 and y = 12, 6, 3

   When x doubles, what happens to y?

   Write a rule for each table.

   Write here

   5 min 1 · Hook · students think

   Double x

   Students find a rule for two tables before anyone explains.

   Say “Two tables. In each one, look at what happens to y when x doubles. Write a rule for each. Don't show anyone yet.”

   Ask

   • What do you multiply x by to get y in table A?
   • What stays the same in table B?

   Watch for Seeing only 'y goes down' in table B: ask what x × y is each time.

   For the teacher

   Give two minutes of silent work, then gather rules. Table A: y is always 3 times x. Table B: x × y is always 12.

   Don't name them yet. The question is the point: what would each table look like as a graph?

   Gather rules with Ask the class. Keep y = 3x and x × y = 12 for the next step.

   Shorter: Show both tables with their rules; skip the guessing round (−2 min).

2. 2 · Watch12 min

   Direct: y = kx

   x doubles, y doubles.

   Through (0, 0), always.

   k = y/x

   Predict, then move k.

   2 · Watch

   Inverse: y = k/x

   x doubles, y halves.

   x × y = k, every time.

   It never meets an axis.

   At x = 2, then 4: what is y?

   2 · Watch

   Find k, then use it

   y ∝ x means y is proportional to x.

   y ∝ x, and y = 12 when x = 3

   12 = k × 3, so k = 4: y = 4x. At x = 5, y = 20

   y ∝ 1/x, and y = 6 when x = 2

   6 = k ÷ 2, so k = 12: y = 12/x. At x = 3, y = 4

   First k, then the formula, then the answer.

   12 min 2 · Explain · you explain

   Two graphs, then finding k

   Move k on y = kx and on y = k/x; then find k from one pair and use it.

   Say “Both kinds have a k. Find k first; then the formula answers every question.”

   Ask

   • Before I move k: where will the line be when x is 0?
   • On the curve, x goes from 2 to 4. What happens to y?

   Watch for Doubling y in inverse proportion: on the curve, doubling x halves y.

   Direct: find k, then y

   y is proportional to x; y = 12 when x = 3

   12 = k × 3, so k = 4: y = 4x

   When x = 5, y = 20

   Inverse: find k, then y

   y is inversely proportional to x; y = 6 when x = 2

   6 = k ÷ 2, so k = 12: y = 12/x

   When x = 3, y = 4

   For the teacher

   First slide: y = kx. Move k: the line gets steeper, but always goes through (0, 0), because 0 × k = 0. Doubling x doubles y; k = y/x.

   Second slide: y = k/x. Move k: the curve moves out, but never meets an axis. Doubling x halves y, because x × y is always k.

   Third slide: one pair of values gives k. Write the formula, then use it. ∝ means 'is proportional to': y ∝ x, and y ∝ 1/x for inverse.

   Vote before each move (Ask the class): k to 4 on the line, then 8 on the curve.

   Shorter: Move k on one graph only; one worked example (−4 min).

3. 3 · Your turn8 min

   Worked, then yours

   I do: y ∝ x, y = 12 when x = 3. y when x = 5?

   k = 12 ÷ 3 = 4, so y = 4 × 5 = 20

   You do: y ∝ x, y = 10 when x = 2. y when x = 7?

   Find k first.

   Write here

   3 · Your turn

   Two more pairs

   I do: y ∝ 1/x, y = 6 when x = 2. y at x = 3?

   k = 6 × 2 = 12, so y = 12 ÷ 3 = 4

   You do: y ∝ 1/x, y = 8 when x = 3. y at x = 6?

   I do: 3 painters take 8 hours; 6 take 4 hours.

   You do: 4 taps fill a tank in 6 minutes. 3 taps?

   Write here

   Its answers' slide comes with your copy.

   8 min 3 · Your turn · students write

   Worked, then yours

   Each worked example beside a twin that students solve themselves.

   Say “Watch me do one. Then you do its twin, which is almost the same.”

   Ask

   • Direct or inverse? How do you know before any numbers?

   Watch for Using the old pair's numbers again: find k first.

   Direct

   y ∝ x, y = 12 when x = 3: y when x = 5?

   k = 12 ÷ 3 = 4

   y = 4 × 5 = 20

   Then theirs

   y ∝ x, y = 10 when x = 2: y when x = 7?

   Inverse

   y ∝ 1/x, y = 6 when x = 2: y when x = 3?

   k = 6 × 2 = 12

   y = 12 ÷ 3 = 4

   Then theirs

   y ∝ 1/x, y = 8 when x = 3: y when x = 6?

   In context

   3 painters take 8 hours. How long do 6 take?

   Twice the painters, half the time

   8 ÷ 2 = 4 hours

   Then theirs

   4 taps fill a tank in 6 minutes. How long do 3 take?

   For the teacher

   Work the first example aloud, writing each line. Then students solve its twin while you watch who finds k first.

   Do the same for each pair. The taps catch students who treat every context as direct: fewer taps take longer.

   Open the answers slide once most have finished their twins.

   Shorter: Do the first two pairs only (−3 min).

4. 4 · Vote and practise15 min

   Vote first

   y ∝ 1/x, and y = 10 when x = 2.

   What is y when x = 4?

   A: 5

   B: 20

   C: 12

   D: 8

   Choose on your own, then say why.

   4 · Vote and practise

   Practise: find k first

   1. y ∝ x, y = 15 when x = 3. y when x = 7?

   2. y = kx, and y = 18 when x = 6. k = ?

   3. y ∝ 1/x, y = 4 when x = 6. y when x = 8?

   4. Which table is inverse proportion?

   (a) x: 1, 2, 3 and y: 5, 10, 15

   (b) x: 1, 2, 3 and y: 12, 6, 4

   (c) x: 1, 2, 3 and y: 10, 9, 8

   Stuck on 3? k = x × y.

   Write here

   4 · Vote and practise

   Reason it out

   5. Which graph shows direct proportion?

   (a) a line through (0, 0), (b) a line through (0, 3)

   (c) a curve that gets closer to both axes

   6. y = 12/x, and x is multiplied by 3. y?

   7. 3 hours at 40 mph. How long at 60 mph?

   Stretch: y ∝ x², y = 12 when x = 2. y at x = 3?

   15 min 4 · Who's stuck? · students vote

   Vote, then practise

   One question shows who's stuck; then practice in levels, and a stretch.

   Say “Choose an answer on your own. Nobody talks until everyone has chosen.”

   Ask

   • What is x × y before, and after? Is it the same?

   Watch for Treating inverse proportion as direct: doubling y when x doubles.

   Show your working on paper: find k first. The stretch is optional.

   The question that shows who's stuck

   1. y is inversely proportional to x. When x = 2, y = 10. What is y when x = 4?
      1. 5
      2. 20
      3. 12
      4. 8

   • A10
   • B9
   • C4
   • D2

   Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.

   1. y is directly proportional to x. y = 15 when x = 3. Find y when x = 7.
   2. y = kx, and y = 18 when x = 6. What is k?
   3. y is inversely proportional to x. y = 4 when x = 6. Find y when x = 8.
   4. Which table shows inverse proportion?
      1. x: 1, 2, 3 and y: 5, 10, 15
      2. x: 1, 2, 3 and y: 12, 6, 4
      3. x: 1, 2, 3 and y: 10, 9, 8
   5. Which graph shows y directly proportional to x?
      1. A straight line through (0, 0)
      2. A straight line crossing the y-axis at 3
      3. A curve that gets closer and closer to both axes
   6. y = 12/x. x is multiplied by 3. What happens to y?
      1. It is divided by 3
      2. It is multiplied by 3
      3. It goes down by 3
   7. A journey takes 3 hours at 40 mph. How many hours does the same journey take at 60 mph?
   8. Stretch: y is proportional to x². y = 12 when x = 2. Find y when x = 3.

   Support · For question 3: work out k = x × y from the pair you are given. Then y = k/x for the new x. Check: x × y is the same as before.

   For the teacher

   Everyone chooses before anyone explains. If they doubled y, ask whether x × y is the same for both pairs. If they added or took away 2, ask whether proportion adds and takes away, or multiplies and divides.

   Questions 1–2 build fluency; 3–4 change the look; 5–6 ask for reasons; 7 is a journey. The stretch is proportional to x².

   Students show their working on paper or the board; the quiz keeps their final answers.

   If many treated it as direct, put both pairs into x × y together before they practise.

   Shorter: After the vote, questions 1–4 only (−6 min).

   Longer: The stretch, then: what happens to y when x doubles if y ∝ x²? (+10 min).

5. 5 · Show what you know5 min

   Line or curve?

   Sketch y = 3x and y = 12/x on one pair of axes.

   Mark where they meet.

   Then: a real example of inverse proportion.

   Write here

   5 min 5 · Exit · students show

   Line or curve?

   Two sketches and one example show who tells the two kinds apart.

   Say “Sketch y = 3x and y = 12/x on one pair of axes. Then answer the exit question.”

   Watch for A curve drawn through (0, 0), or a line that misses it.

   One or two sentences are enough.

   1. Give a real example of two things in inverse proportion, and say why doubling one halves the other.

   For the teacher

   Sketch y = 3x and y = 12/x for x from 0 to 6 on one pair of axes, and mark where they meet.

   Then answer the exit question in one or two sentences.

   The examples show who can tell direct from inverse in context: start the next lesson with two of them. Set the homework.

   Longer: Where do the line and the curve meet, and why there? (+5 min).

The answers, their slides and the mistake behind each wrong answer come with your copy. The worked examples above are open to everyone.

One-to-one: ask the student to say whether each situation is direct or inverse before any numbers, and to check every answer by working out k again from it.

Homework: find k and use it

Say first whether each is direct or inverse. Find k, write the formula, then answer. Check each answer by working out k again from it.

1. y is directly proportional to x. y = 20 when x = 4. Find y when x = 9.
2. y is directly proportional to x. y = 14 when x = 2. Find x when y = 35.
3. y is inversely proportional to x. y = 9 when x = 4. Find y when x = 6.
4. 5 workers build a wall in 12 days. How long would 6 workers take, working at the same rate?
5. Sketch y = 2x and y = 18/x for x from 0 to 6 on one pair of axes. Where do they meet?
6. x: 2, 4, 8 and y: 16, 8, 4. Direct, inverse or neither? Write the formula, and find y when x = 16.

Print the worksheet with working space and homework. Teacher answers come with your Library copy.