At a glance. One lesson of about 60 minutes. It assumes proportion from Year 8 and rearranging from lessons 6 and 7. Students find k from one pair, write the equation, substitute, handle squares and roots and recognize proportion from a table.
Calculator: GDC optional. The numbers are small, and the point is the structure of the relationship. A GDC graph is offered to check the shape.
What students will learn
Know
- Direct: y = kx, y ÷ x is constant, a line through the origin.
- Inverse: y = k/x, xy is constant, a curve that falls.
- Powers and roots change which combination is constant.
Understand
- The constant k is found from one pair of values.
- A relationship that rises or falls is not necessarily proportional.
- Inverse proportion means doubling one quantity halves the other.
Do
- Find k and write the equation.
- Solve direct and inverse problems, including squares and roots.
- Test a table for proportion.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving proportion problems (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students record the constant combination for each type and describe what doubling x does to y.
- Criterion C. Communicating: writing the equation with k before substituting, and a sentence explaining the type.
- Criterion D. Applying mathematics in real-life contexts: work rates and brightness (hook and extra questions).
The one thing to do with the figure
Ask for the prediction first. Then set the relationship to 1 and double x, asking what happens to y. Switch to relationship 3 and ask the same. Move to 2 and 4 and ask what doubling x does.
Ask: “Which combination of x and y did not change?” Make students say it for each type. The constant is what identifies the relationship.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the banquet; students state their assumption. |
| 12 min | Figure, key idea, warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask what stays constant in inverse proportion. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 64; x = y ÷ 6; 36. |
|---|---|
| Hook | 6 × 8 = 48 cook-hours, so 12 cooks need 4 hours, assuming each cook works at the same rate and does not get in each other's way. |
| Worked examples | 70; 3 hours; 250 g. |
| Faded example | 40 units. |
| 1. Inverse proportion | B. A treats it as direct; C adds the change; D subtracts the change. |
| 2. Direct proportion | 60. k = 5. |
| 3. Inverse proportion | 4. k = 36. |
| 4. Proportional to a square | 288. k = 8. |
| 5. A context for inverse proportion | 15. 120 ÷ 8. |
| 6. Inverse square | 2. k = 72. |
| 7. Proportional to a root | 25. k = 3. |
| 8. Recognizing inverse proportion | D. A has a falling difference; B is direct; C is linear. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Treated it as direct, or added or subtracted the change. | Ask: what is x × y for the first pair? For the new pair? |
| 40 or 2.4 in question 2 | Added the change in x, or found k as x ÷ y. | Find k first and say what it means. |
| 36 or 6 in question 3 | Treated it as direct, or subtracted. | Compare x × y before and after. |
| 96 or 8 in question 4 | Ignored the square, or gave k. | Write y = kx² and substitute the first pair. |
| 35 or 120 in question 5 | Subtracted the extra pumps, or gave the constant. | Ask: how many pump-minutes does the job need? |
| 4 or 72 in question 6 | Treated it as inverse to x, or gave k. | Write y = k/x² and find k. |
| 10 or 5 in question 7 | Treated it as direct to x, or gave the root. | Write y = k√x, find k, then square. |
| A, B or C in question 8 | Chose a falling pattern, a direct one, or a linear one. | Compute x × y for each column. |
Questions worth asking
- How do you know which combination is constant? Listen for: test y ÷ x and y × x on the given pairs.
- Does a graph that falls always show inverse proportion? No: test the product; y = 12 − x falls but is not inverse.
- Why does a direct proportion graph go through the origin? When x = 0, y = k × 0 = 0.
- What does the constant mean in the banquet? The total work in cook-hours.
Differentiation
A way in for anyone stuck
- Provide a table with the columns x, y, y ÷ x and y × x for students to fill in.
- Use the word “cook-hours” or similar to give the constant a meaning.
- Give the frame: “y ÷ x is constant, so y = kx. k = …, so y = …”
Stretch
- y is inversely proportional to x + 1, and y = 6 when x = 3. Find y when x = 7.
- Show that if y is directly proportional to x² then doubling y changes x by a factor of √2.
Students learning in English as an additional language
“Proportional” means that two quantities change together in a fixed way. “Directly” means both go up together; “inversely” means one goes up as the other goes down. “Constant” means it never changes. “Square of” means multiplied by itself.
| Word | Plain meaning | Example |
|---|---|---|
| direct proportion | two quantities whose ratio is always the same | y = 9x |
| inverse proportion | two quantities whose product is always the same | y = 20 ÷ x |
| constant of proportionality | the fixed number k that links the two quantities | the 9 in y = 9x |
| origin | the point (0, 0) where the axes cross | (0, 0) |
| proportional to the square of | equal to a constant times the square | area of a circle and radius |
Sentence starters for explaining aloud or in writing:
- I tested … and it was constant, so …
- k = … because …
- This is not proportional because …
Two things not to say
“It goes up, so it is proportional.” Say instead: test whether y ÷ x is constant.
“Cross-multiply.” Say instead: find k and use it, so that you know why.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Findy is directly proportional to x and y = 12 when x = 3. Find y when x = 5.
20. - Findy = k/x and y = 9 when x = 2. Find k.
18.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextFive workers in Dhaka lay a path in 12 days. How long would 4 workers take, working at the same rate?
5 × 12 = 60 worker-days, so 15 days. - Earlier work (lesson 7)Solve 4x − 3 = 2x + 9.
x = 6.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs y = 2x + 3 directly proportional to x? Explain.
No: when x = 0, y = 3, and y ÷ x is not constant (5, 3.5, …). - Working backwardsy is inversely proportional to x², and y = 2 when x = 6. Find x when y = 18.
k = 72; x² = 4; x = 2.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Tables of values on paper with product and quotient columns do the same job.