Direct proportion and rates
Two quantities are in direct proportion when doubling one doubles the other. The reliable method is to find the value of one unit first.
Part of your national curriculum
- Ratio, proportion and rates of change: Solve problems involving direct and inverse proportion, including graphical and algebraic representations
- Ratio, proportion and rates of change: Use compound units such as speed, unit pricing and density to solve problems
Lesson overview
What you'll learn in this lesson
Solve problems involving direct proportion
Key learning points
- • The unitary method
- • Spotting when it is not proportional
This lesson at a glance
- 25 minutes
- 12 parts to scroll through
- 3 quick checks
- Marked quiz at the end
- Gentle pace: short sittings with pauses
Words to know
Scroll down — the lesson carries on below
Watch & discover
Part 1 of 12
Watch first
Direct proportion and rates
An English teaching video on this topic from mathantics. Watch it, then scroll down for the lesson.
In a nutshell
Solve problems involving direct proportion
Explore the idea
Part 2 of 12
Learn
The unitary method
Divide to find the value of one, then multiply for the amount you need. It works for costs, distances, feed quantities and speeds alike.
Explore the idea
Part 3 of 12
Learn
Spotting when it is not proportional
If a graph of the relationship does not pass through the origin, it is not direct proportion. A fixed charge plus a rate is a different model.
Explore the idea
Part 4 of 12
Worked example
Worked example
7 kg of feed lasts 4 days. One day needs 7 ÷ 4 = 1.75 kg. Two weeks needs 1.75 × 14 = 24.5 kg.
Reset break
Part 5 of 12
Pause
That's sitting 1 of 2 done
Stretch, get a drink, look out of the window. There is no timer and nothing is counting down — your place is saved, so you can come back in five minutes or tomorrow.
Quick check
Part 6 of 12
Quick check
5 items cost £12.50. What do 8 cost?
Quick check
Part 7 of 12
Quick check
A graph of a proportional relationship must…
Quick check
Part 8 of 12
Quick check
A tracker records 18 km in 3 hours. At the same rate, how far in 7 hours?
Challenge round
Part 9 of 12
Game · Sort it
Which of these are true?
Drag each card into the right column. Tap a card first if dragging is fiddly.
True
Not true
Challenge round
Part 10 of 12
Game · Fill the gaps
Finish the sentences
Choose the word that belongs in each gap.
If a graph of the relationship does not pass through the origin, it is not ____ proportion.
Mastery quiz
Part 11 of 12
Marked quiz
End of lesson quiz: Direct proportion and rates
3 questions, marked with the reasoning shown. No timer.
1. 5 items cost £12.50. What do 8 cost?
2. A graph of a proportional relationship must…
3. A tracker records 18 km in 3 hours. At the same rate, how far in 7 hours?
Lesson round-up
Part 12 of 12
Lesson round-up
Ready when you are
Quiz score
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Games
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Points this lesson
0
Best run
0 in a row
Luna: 0 out of 3 on the practice checks. Only if you feel up to it — one more?
Ask LunaDo the work
Your written work for this lesson
This is the part you'd do in your book at school. Pick the pieces below, write here or on paper, and photograph anything you draw or work out.
The cost of a school trip
A school is planning a trip to a wildlife park. The coach costs £480 however many people go. Entry costs £9.50 for each pupil and each adult goes free. The school says every pupil must pay the same amount, and the trip must not lose money.
If 60 pupils go, the coach works out at £8 each, so the total for each pupil is £17.50. If only 40 pupils go, the coach works out at £12 each, so the price rises to £21.50. The teacher explains that the coach is a fixed cost and the entry is a cost per person — which is why the price per pupil changes when the number changes.
The school decides to charge £20 and to use anything left over for a second trip later in the year.
Written for Good Learning Co.
Reading questions
- Work out the total cost of the trip if 50 pupils go.
- Show why the price per pupil falls when more pupils go, using numbers.
- At £20 each, how many pupils must go before the trip makes money? Show your working.
Working out
Working outDo the practice questions on paper, showing every step — not just the answer. Mark them yourself and circle anything you got wrong.
10–15 questions
Worked example, explained
WritingCopy one question out and write underneath, in words, why each step works.
1 question, 4–5 sentences
Problem in words
Working outWrite your own real-life problem that uses today's method, then solve it.
1 problem + solution
Saved as you type.
Part 1 of 12 · Watch & discover
How this is going
- Not looked at yet
Solve problems involving direct and inverse proportion, including graphical and algebraic representations
- Not looked at yet
Use compound units such as speed, unit pricing and density to solve problems