KS3 · Maths

Direct proportion (graphical and algebraic)

Two boxes of popcorn cost £5. Four boxes? You said £10 before you'd finished reading. That instant 'just double it' has a name: direct proportion.

Maths · Direct proportion

More boxes, more cost — in lockstep
£0.00£2.50£5.00£7.50£10.00£12.50£15.00£17.50£20.00each cell is one box — drag to buy more →

Boxes bought: 2/8. Boxes 2. Total cost £5.00. Cost ÷ boxes £2.50 each. Double the boxes 4 boxes → £10.00

Boxes2Total cost£5.00Cost ÷ boxes£2.50 eachDouble the boxes4 boxes → £10.00

Every cell of the bar is one box of popcorn at £2.50. Drag along the bar, then watch the last two readouts: the cost for each box never changes, and doubling the boxes always doubles the cost.

Watch out: Both numbers going up is not what makes this special. What matters is that the cost is ALWAYS the number of boxes × 2.50 — the same multiplier for every pair.

Worked example · a ratio table in the kitchen

Problem

A pancake recipe uses 120 g of flour to make 4 pancakes. (a) How much flour do you need for 10 pancakes? (b) How many pancakes can you make with 450 g of flour?

Maths · Check the units

Where does this answer go wrong?

Petrol costs £1.50 per litre. How much does 800 ml of petrol cost?

A student's ratio table — which line goes wrong?

The same popcorn, drawn as a graph

0246805101520Boxes boughtTotal cost (£)(4, 10)Cost of each extra box (£): 2.50

Boxes bought: 4. Total cost (£): 10. Cost of each extra box (£): 2.50

Slide the point along the line and read both numbers. Try whole numbers of boxes first.

1 box across: The short orange line is one step of 1 box across, from (2, 5) to (3, 5). Straight above its end, the line is at (3, 7.5) — £2.50 higher. Take a 1-box step anywhere on this line and it always climbs £2.50. That rise for each 1 across is the gradient, and it is the same 2.5 that multiplied boxes into cost on the bar. So this line is y = 2.5x.

Watch out: The line joins the points, so it also gives readings like 3.5 boxes for £8.75. The maths is fine, but the shop is not: you cannot buy half a box, so here only whole-number readings make sense. The context always decides.

Maths · Finding k

Your turn: find k from two points

A direct proportion graph passes through (4, 6) and (10, 15). Find the gradient k and write the equation of the line.

  1. Points: (4, 6) and (10, 15). In each pair the first number is x and the second is y.
  2. missing step
Which line is step 2?

Maths · Spot the impostors

Direct proportion — or just pretending?

Is each one direct proportion?

Still to sort

Direct proportion (0)

One multiplier links every pair; the graph is a straight line through (0, 0).

Where the line is: Every pair must pass, not just the first one.

Not direct proportion (0)

At least one pair breaks the multiplier, or the graph misses (0, 0) or bends.

Where the line is: Both quantities increasing is not enough on its own.

7 of 7 still to sort.

Sort each table or graph. Use the tests: the same multiplier for every pair, the doubling check, and a straight line through (0, 0).

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

One multiplier links two quantities — and it is the same number in the table, on the graph and in y = kx.

What you need to know

  • Direct proportion: two variables linked by a constant multiplier — y is always the same number times x.
  • Test it on more than one pair, or use the doubling check: if one doubles, the other must double too.
  • You can show it on a double number line, in a ratio table or on a graph.
  • Its graph is a straight line with a constant positive gradient that passes through the origin (0, 0).
  • The multiplier, the rate per 1, the gradient and the constant of proportionality k are the same number, so y = kx.
  • Ratio tables solve problems in context: known pair first, new value in the right column, then multiply — or use the inverse to go back.

The big picture

Two quantities are in direct proportion when one constant multiplier links every pair of values. That same number is the rate for each 1, the gradient of the graph and the k in y = kx, and the graph is always a straight line through the origin. Ratio tables solve direct proportion problems, as long as the units match.

Key points

1Divide y by x for several pairs: the same answer every time means direct proportion.
2A straight line that misses (0, 0) is NOT direct proportion, and neither is a curve through (0, 0).
3Plot it by completing the table with the multiplier, starting at (0, 0), and joining the points with a ruler.
4Gradient from two points: change in y for a change in x, scaled to a change in x of 1. For y = 3x the multiplier, rate, gradient and k are all 3.
5Make the units match before you fill a ratio table — convert one quantity first.
6Readings between plotted points only count if the context allows them: you can't buy 3.5 boxes.

Worked example

Problem

Are x and y in direct proportion? x = 2, 5, 8 and y = 7, 17.5, 28. If they are, write the equation and find y when x = 11.

⚠ Watch out

Thinking any straight line, or anything where both go up, is direct proportion. y = 2x + 3 is straight, but x = 1 gives 5 and x = 2 gives 7: x doubled, y didn't. You need one multiplier for every pair — and a line through (0, 0).

🧠

Memory hook

One number, four names: multiplier, rate, gradient, k. Find it once and it does every job.

✓

Check yourself

5 pens cost £3.50. What do 8 pens cost — and what does your multiplier mean? (£5.60: 3.50 ÷ 5 = 0.70, the cost of one pen, which is k.)

Flashcards

(14)
What does it mean for two variables to be in direct proportion?
They have a constant multiplicative relationship: one variable is always the same number times the other.
How do you check a table for direct proportion?
Find the multiplier for more than one pair — it must be the same every time. You can work across or down the table.
What is the doubling check?
If one variable doubles, the other must double too. If it doesn't, they are not directly proportional.
Name three ways to represent a direct proportion.
A double number line, a ratio table and a graph.
Give four everyday situations that are direct proportion.
Converting units of measure, scaling a recipe, changing currency with an exchange rate, and buying different amounts of the same item.
What two features does every direct proportion graph have?
It is a straight line with a constant positive gradient, and it passes through the origin (0, 0).
Which two graphs look close but are NOT direct proportion?
A straight line that doesn't start at the origin, and a line through the origin whose gradient changes (a curve).
How do you plot a direct proportion graph?
Complete the table using the multiplier, plot the points starting with (0, 0), then join them with a ruler in a straight line.
What does the gradient of a direct proportion graph tell you?
How much y changes when x goes up by 1 — the constant of proportionality.
What is the general equation of a direct proportion graph?
y = kx, where k is the constant of proportionality.
How do you find the gradient from two points with whole-number coordinates?
Find the change in x and the change in y, then use a ratio table to scale to a change in x of 1.
In a ratio table problem, where does the new value go?
In the same column as the quantity it measures. Then find the multiplier and apply it — or use its inverse when working the opposite way.
Before solving a direct proportion problem in context, what should you check?
That the units are consistent throughout — convert one quantity first if they are not.
You read 3.5 boxes off a cost graph. Why might that not make sense?
The context decides: boxes come in whole numbers, so readings between whole numbers don't mean anything in the shop.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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