At a glance. One lesson of about 60 minutes. Students test tables, lines and rules for proportionality, find k, use y = kx, and see why a start amount breaks the proportion.
Calculator: no calculator. All values divide cleanly into halves and tenths. The skill is reading the relationship, not computing.
What students will learn
Know
- A proportional relationship has the rule y = kx.
- k is y ÷ x for every pair with x not 0, the value of one.
- Its graph is a straight line through (0, 0).
Understand
- Table, graph and rule are three views of one relationship.
- A straight line that misses the origin is linear but not proportional.
- Doubling x doubles y exactly when the relationship is proportional.
Do
- Decide whether a table, graph or rule is proportional.
- Find k and write y = kx.
- Use the rule to find any y from any x.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: y = kx, k, the table and the origin test (questions 1 to 6).
- Criterion B. Investigating patterns: set c to 0 and change k, then set k and change c. Ask students to describe what each slider does to the graph and to y ÷ x.
- Criterion C. Communicating: explaining in words and symbols why y = 5x + 1 is not proportional, using a table (questions 6, 8 and 9).
- Criterion D. Applying mathematics in real-life contexts: a tea stall, a tank filling, a phone plan (the hook and questions 5 and 8).
The one thing to do with the figure
Ask for a prediction before the sliders move. “3 dirhams for each cup. Plot cups against cost. What shape? Where does it start?” Take straight line and (0, 0) from the room and ask why 0 cups cost 0 dirhams.
Then set the start amount to 4 and ask what happens to y ÷ x at the two points. Ask whether a line that is nearly through the origin is nearly proportional. Use the two read-outs to settle it.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the figure, with predictions first. Include a start amount. |
| 10 min | Key idea; the three views side by side on the board. |
| 10 min | Both worked examples; students finish the faded table example in pairs. |
| 5 min | Question 1, the check. If many choose A or C, go back to the figure's read-outs. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | (3, 12); 33 rupees; 4n. |
|---|---|
| Worked example 1 | y = 4x. |
| Worked example 2 | k = 2.5 and y = 22.5. |
| Faded example | Yes, y = 3.5x. |
| 1. Proportional from a table | B, y = 4x. A and C go up by a constant amount from a non-zero start; D is y = x − 1. |
| 2. Using y = kx | 40. 5 × 8 = 40. |
| 3. Finding k from a point | 7. 14 ÷ 2 = 7. |
| 4. Using a point to find another | 30. k = 1.5, so y = 1.5 × 20 = 30. |
| 5. A steady rate | 54. 6 × 9 = 54 litres. |
| 6. The rule from a point | B, y = 4x. A is straight but does not go through the origin; C and D use the wrong number. |
| 7. A steeper proportional graph | 35. k = 3.5, doubled is 7, and 7 × 5 = 35. |
| 8. A start charge is not proportional | 5. 12 + 18 = 30, and 30 ÷ 6 = 5. With 2 gigabytes it is 18 ÷ 2 = 9, so it is not constant. |
| 9. What shows a table is proportional | B. A, C and D all hold for lines that are not proportional. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1; A, C or D in question 6 | Chose a table that goes up by a constant amount, or a rule that passes through the point but not the origin. | Ask for y ÷ x in each column and compare, then check where the rule starts. |
| 13 in question 2; 16 in question 3 | Added the numbers. | Ask how many times bigger y is than x. |
| 3 in question 2; 12 in question 3 | Subtracted the numbers. | Ask how many times bigger y is than x, not how much bigger. |
| 1.6 in question 2; 28 in question 3 | Divided or multiplied the wrong way round. | Write k = y ÷ x on the board and say what each letter is. |
| 26 in question 4; 29 in question 5; 19 in question 7 | Added the extra amount to y. | Ask whether doubling x doubles y; then use k. |
| 360 in question 4; 216 in question 5 | Multiplied the two numbers they could see. | Ask: what is k? Write it before any other working. |
| 1.5 in question 4; 6 in question 5 | Stopped at k. | Underline the value of x asked for and finish with it. |
| 17.5 or 28 in question 7 | Used the first graph, or doubled the point instead of k. | Write k for each graph in a small table. |
| A, C or D in question 9 | Took a property that some lines have as the test: rising, a constant step, or y larger than x. | Test each statement on y = 2x + 1: it rises, steps by 2 and has y larger than x, but it is not proportional. |
| 3 or 30 in question 8 | Gave the rate, or the total cost. | Compute the total cost of the month first, then divide. |
Questions worth asking
- Does every straight line show a proportional relationship? No: it must pass through (0, 0). Listen for “it must start at zero”.
- What does k mean in a context? The value of one, the rate, the price per item. Listen for a unit.
- If y ÷ x is 4 at one point, must it be 4 at another? Only if the relationship is proportional. Link to the phone plan in question 8.
- What does the graph of y = kx look like for a bigger k? Steeper. Listen for “it goes up faster”.
Differentiation
A way in for anyone stuck
- Give a table template with a third column headed y ÷ x to be filled for every pair.
- Start with whole-number values of k and graphs plotted on squared paper.
- Say “for every 1 across, k up” as you draw each line.
Stretch
- Question 8, then: find the cost divided by the gigabytes for 2, 6 and 20 gigabytes. What number does it get close to as the gigabytes grow? (9, 5 and 3.6; it gets close to 3.)
- A graph passes through (2, 5) and (6, 15). Is it proportional? What if it passes through (2, 5) and (6, 13)? (Yes, k = 2.5; no, because 5 ÷ 2 = 2.5 but 13 ÷ 6 is about 2.17.)
Students learning in English as an additional language
“Proportional” and “proportion” are used for the same idea. Say “y is always k times x” each time, and point to the straight line through the origin.
| Word | Plain meaning | Example |
|---|---|---|
| proportional | one quantity is always the same multiple of the other | cost and number of cups |
| constant of proportionality | the number k in y = kx | k = 3 for 3 dirhams a cup |
| origin | the point (0, 0) where the axes meet | 0 cups cost 0 dirhams |
| linear | forming a straight line on a graph | y = 2x + 4 is linear |
| slope | how steeply a line rises | k = 5 is steeper than k = 2 |
Sentence starters for explaining aloud or in writing:
- y ÷ x is … for every pair, so …
- The line goes through (0, 0), so the rule is y = …x
- It is not proportional because …
Two things not to say
“It is a straight line, so it is proportional.” The line must pass through the origin.
“k is the slope, learn the formula.” Start with y ÷ x as the value of one.
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.
- Using y = kxy = 6x. Find y when x = 5.
30. - Finding k(3, 15) lies on y = kx. What is k?
5.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA machine makes 12 bolts a minute. Write the rule for the bolts y after x minutes and find y at 7 minutes.
y = 12x, so 84 bolts. - Currency1 dirham is worth 25 rupees. Write a rule for the rupees y for x dirhams and find y for 12 dirhams. Is the graph proportional?
y = 25x, so 300 rupees. Yes: 0 dirhams is 0 rupees, so the line goes through (0, 0). - Earlier work (Lesson 5)8 books cost 44 rupees. Write a rule for the cost y of x books, and find the cost of 11 books.
y = 5.5x, so 60.5 rupees.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningZoya says that y = 2x + 3 is proportional because it goes up by the same amount each step. Is she right?
No. At x = 1, y = 5, and at x = 2, y = 7. It does not double. y ÷ x is 5 and 3.5. - SpottingA table has x = 2, 4, 8 and y = 3, 6, 12. Write the rule, and say what y is when x = 10.
y = 1.5x, so y = 15.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under the figure, “Show me the steps” gives the same states in words.
No device? Plot the points on squared paper and use a ruler to test whether the line passes through the origin.