The shape of the lesson
By the end of the lesson, learners can
- Direct proportion: In the lab, pick a type of proportion (\(y \propto x\), \(y \propto x^2\),...) and set \(k\).
- Inverse proportion: Inverse proportion means one quantity goes UP as the other goes DOWN, so that their product (or the product with a power) stays the same.
- Proportion in context: Science and real-life questions use proportion all the time.
- Equation: replace \(\propto\) by \(= k \times\) (direct) or \(= \dfrac{k}{\ldots}\) (inverse).
- Power: use the exact one in the question: square, cube, square root, cube root.
- Constant: substitute the given pair of values and solve for \(k\).
- Formula: rewrite the equation with the number for \(k\).
- Answer: substitute; if finding \(x\), undo the power at the end.
One picture every idea returns to
Turn \(\propto\) into \(= k \times\). Direct: \(y = kx^n\). Inverse: \(y = \dfrac{k}{x^n}\). Use the given pair of values to find \(k\), write the formula, then use it.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, One constant, k.. Extended. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 13 min | Teach and check: Direct proportion | In the lab, pick a type of proportion (\(y \propto x\), \(y \propto x^2\),...) and set \(k\). Project tab 5 of the lab (Sequences and proportion lab, proportion) and let a student drive it while the class predicts. Model the worked example Linear on the board, one line at a time. | Whole class, then pairs |
| 13 to 21 min | Teach and check: Inverse proportion | Inverse proportion means one quantity goes UP as the other goes DOWN, so that their product (or the product with a power) stays the same. Model the worked example Reciprocal on the board, one line at a time. Your turn: \(y\) is inversely proportional to \(x^2\). When \(x = 2\), \(y = 5\). Find \(y\) when \(x = 4\). | Pairs |
| 21 to 30 min | Teach and check: Proportion in context | Science and real-life questions use proportion all the time. Model the worked example Falling object on the board, one line at a time. Your turn: \(y\) is proportional to the cube root of \(x\). When \(x = 8\), \(y = 6\). Find \(y\) when \(x = 125\). | Rally Coach |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 19 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Proportion in context, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Rally Coach
Pairs share one worksheet. One partner solves the first question aloud while the other coaches, checking each line and offering a prompt rather than the answer; they swap roles for the next question.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Always find \(k\) and write the formula first. | Scaling by eye ('x went from 5 to 3, so...') goes wrong as soon as there is a square or a root involved. |
| In \(y = \dfrac{k}{\sqrt{x}}\), substitute \(\sqrt{9} = 3\), not 9. | Read the question for the exact power: 'the square of', 'the square root of' and 'the cube of' all change the equation. |
| If \(y \propto x^2\) and \(x\) doubles, \(y\) is multiplied by \(2^2 = 4\), not 2. | The multiplier for \(y\) is the multiplier for \(x\) raised to the same power. |
Direct and inverse proportion: worksheet
6 questions, 19 marks. Written in the style of the papers; no past paper question is reproduced.
Write each statement as an equation using a constant \(k\). (a) \(p\) is proportional to the cube of \(q\). (b) \(m\) is inversely proportional to the square root of \(n\).
\(y\) is directly proportional to \(x\). When \(x = 4\), \(y = 18\). (a) Find \(y\) when \(x = 10\). (b) Find \(x\) when \(y = 63\).
\(y \propto x^2\). When \(x = 5\), \(y = 75\). (a) Find \(y\) when \(x = 8\). (b) Find the positive value of \(x\) when \(y = 27\).
\(y\) is inversely proportional to \(x\). When \(x = 6\), \(y = 8\). Find \(y\) when \(x = 16\).
The frequency \(f\) hertz of a pendulum is inversely proportional to the square root of its length \(l\) metres. When \(l = 1\), \(f = 0.5\).
(a) Find \(f\) when \(l = 0.25\). (b) Find the length of a pendulum with frequency 0.2 hertz.
The mass \(m\) grams of a steel ball is proportional to the cube of its radius \(r\) cm. A ball of radius 1.5 cm has mass 54 g.
(a) Find a formula for \(m\) in terms of \(r\). (b) Find the mass of a ball of radius 2 cm. (c) Find the radius of a ball of mass 250 g. (d) One ball has twice the radius of another. How many times heavier is it?
Answer key and marking notes
- (a) \(p = kq^3\)
- (b) \(m = \dfrac{k}{\sqrt{n}}\)
- \(18 = 4k\), so \(k = 4.5\) and \(y = 4.5x\).
- (a) \(y = 4.5 \times 10 = 45\).
- (b) \(63 = 4.5x\), so \(x = 14\).
- \(75 = 25k\), so \(k = 3\) and \(y = 3x^2\).
- (a) \(y = 3 \times 64 = 192\).
- (b) \(27 = 3x^2\), so \(x^2 = 9\) and \(x = 3\).
- \(8 = \dfrac{k}{6}\), so \(k = 48\).
- \(y = \dfrac{48}{16} = 3\).
- \(f = \dfrac{k}{\sqrt{l}}\): \(0.5 = \dfrac{k}{1}\), so \(k = 0.5\).
- (a) \(f = \dfrac{0.5}{\sqrt{0.25}} = \dfrac{0.5}{0.5} = 1\) hertz.
- (b) \(0.2 = \dfrac{0.5}{\sqrt{l}}\), so \(\sqrt{l} = 2.5\).
- \(l = 2.5^2 = 6.25\) m.
- (a) \(m = kr^3\): \(54 = k \times 1.5^3 = 3.375k\), so \(k = 16\) and \(m = 16r^3\).
- (b) \(m = 16 \times 2^3 = 128\) g.
- (c) \(250 = 16r^3\), so \(r^3 = 15.625\) and \(r = \sqrt[3]{15.625} = 2.5\) cm.
- (d) \(16(2r)^3 = 8 \times 16r^3\), so it is 8 times heavier.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
\(y\) is directly proportional to the cube of \(x\). Which equation is correct?
- A\(y = kx^3\)
- B\(y = \dfrac{k}{x^3}\)
- C\(y = k\sqrt[3]{x}\)
- D\(y = 3kx\)
\(y\) is proportional to \(x^2\). When \(x = 3\), \(y = 36\). Find \(y\) when \(x = 5\).
- A100
- B60
- C12.96
- D400
\(y\) is inversely proportional to the square root of \(x\). When \(x = 16\), \(y = 6\). Find \(y\) when \(x = 4\).
- A12
- B24
- C3
- D1.5
The frequency \(f\) Hz of a vibrating string is inversely proportional to its length \(L\) cm. When \(L = 60\), \(f = 440\). Find \(f\) when \(L = 48\).
- A550
- B352
- C687.5
- D428
\(y\) is inversely proportional to \(x^2\). \(x\) is multiplied by 3. What happens to \(y\)?
- A\(y\) is divided by 9
- B\(y\) is divided by 3
- C\(y\) is multiplied by 9
- D\(y\) is divided by 6
\(y\) is proportional to the cube of \(x\). When \(x = 3\), \(y = 54\). Find \(x\) when \(y = 250\).
- A5
- B125
- C13.9
- D6.45
Quiz key
\(y \propto x^3\) becomes \(y = kx^3\). \(y = \dfrac{k}{x^3}\) is inverse proportion to the cube; \(y = k\sqrt[3]{x}\) is the cube ROOT; \(3kx\) confuses cubing with multiplying by 3.
\(36 = k \times 9\), so \(k = 4\) and \(y = 4x^2\). When \(x = 5\), \(y = 4 \times 25 = 100\). 60 treats it as \(y \propto x\); 12.96 uses inverse proportion; 400 squares \(kx\) instead of just \(x\).
\(6 = \dfrac{k}{\sqrt{16}} = \dfrac{k}{4}\), so \(k = 24\). When \(x = 4\), \(y = \dfrac{24}{\sqrt{4}} = \dfrac{24}{2} = 12\). 24 comes from \(y = \dfrac{k}{x}\) (no square root); 3 uses direct proportion to \(\sqrt{x}\).
\(440 = \dfrac{k}{60}\), so \(k = 26\,400\). When \(L = 48\), \(f = \dfrac{26\,400}{48} = 550\) Hz. 352 uses direct proportion (\(440 \times \dfrac{48}{60}\)), which wrongly lowers the frequency; 687.5 uses inverse proportion to \(L^2\); 428 just subtracts the 12 cm change.
\(\dfrac{k}{(3x)^2} = \dfrac{k}{9x^2} = \dfrac{1}{9} \times \dfrac{k}{x^2}\), so \(y\) is divided by \(3^2 = 9\). Dividing by 3 forgets the square; multiplying by 9 is the DIRECT proportion answer; 6 doubles the 3 instead of squaring it.
\(54 = k \times 27\), so \(k = 2\) and \(y = 2x^3\). Then \(250 = 2x^3\), \(x^3 = 125\), \(x = \sqrt[3]{125} = 5\). 125 forgets to take the cube root; 13.9 uses \(y \propto x\); 6.45 uses \(y \propto x^2\).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.