At a glance. One lesson of about 60 minutes. Students build y = mx + c from a situation, interpret m and c, find break-even by setting two models equal, price a two-rate plan, and judge the limits of a model beyond its data.
Calculator: no calculator. Nothing here needs one. The numbers are chosen so each break-even is a whole number or a short decimal.
What students will learn
Know
- In a linear model m is the rate and c is the starting amount.
- Break-even: set two models equal and solve.
- A model is reliable only where the pattern was measured; beyond that it is a prediction.
Understand
- The break-even is a boundary: which plan is cheaper changes at it.
- A model's arithmetic always works; whether the answer makes sense is a separate question.
- A piecewise model uses a different straight line for each range.
Do
- Write a linear model from a context and interpret its gradient and intercept.
- Find the break-even point and decide which plan to choose for a given use.
- Comment on the limitations of a model.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving linear equations in context (questions 1 to 9).
- Criterion B. Investigating patterns: the second figure. Ask students to choose a measured range, fit a line, and say where they would stop trusting it, and why.
- Criterion C. Communicating: explaining a recommendation with the break-even and a statement of limits.
- Criterion D. Applying mathematics in real-life contexts: bottled water against a flask, repairs, water charges, a tank and sunflower growth (the whole lesson).
The one thing to do with the figure
Ask for the decision, not the number. Set the flask at 600 rupees and say “a student drinks one bottle a day for 10 days. Which should they choose?” Then move the uses to 20. Ask what changed, and name the crossing as the break-even.
In the second figure, ask at each day whether they would believe it, and what would need to be true for the line to continue.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 8 min | Hook: bottles against a flask; estimates of the break-even. |
| 12 min | First figure: both sliders; the break-even and the boundary. |
| 8 min | Second figure: limits; discuss what makes a model trustworthy. |
| 7 min | Worked example; students finish Plan A against Plan B. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | y = 5x + 3; x = 18; 160. |
|---|---|
| Worked example | 16 uses, 800 rupees; the flask is cheaper by 560 rupees at 30 uses. |
| Faded example | 5 kg, 150 rupees. |
| 1. Building a model | B. A swaps the rate and the fee; C combines them; D subtracts the fee. |
| 2. Using a model | 440. 80 × 3 + 200. |
| 3. Solving for the input | 10. 80h = 800. |
| 4. A falling model | 6. 12.5t = 75. |
| 5. Break-even | 10. Both plans cost 200 rupees. |
| 6. Limit of a model | 12.5. The model only holds for 0 to 12.5 days. |
| 7. A two-rate model | 90. 50 + 40. |
| 8. Predicting from two readings | 500. 30 bags a week. |
| 9. Which plan, by how much | A. B reverses the plans; C and D use the joining fee as the difference. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1; 840 or 680 in question 2 | Swapped the rate and the fixed fee, or combined them. | Ask what is paid once and what is paid every hour, and write the two sentences first. |
| 240 in question 2; 12.5 or 15 in question 3 | Left out the fixed fee, or did not take it off first. | Build the equation, then use inverse operations in order: take off 200, then divide by 80. |
| 8 or 10 in question 4 | Solved for 0% or added 25. | Ask how much charge has been used, from 100 to 25. |
| 6, 15 or 12 in question 5 | Used one rate, or stopped at the difference of the rates. | Write both costs as equations and subtract 8x from both sides. |
| 460, 12 or 540 in question 6 | Gave the amount after one day, rounded down, or added. | Ask what “empty” means for y, and set it to 0. |
| 75, 120 or 105 in question 7 | Used one rate for all units, or swapped the rates. | Draw the units in two groups and label the rate on each. |
| 440 or 350 in question 8 | Used 4 weeks instead of 6, or added one week. | Mark weeks 2, 6 and 12 on a time line with the sales. |
| B, C or D in question 9 | Compared the rates only, or used the joining fee as the difference. | Work out both costs at 12 units, then subtract. |
Questions worth asking
- What does the intercept mean in each model? The fixed amount paid at the start. Listen for “before you use any”.
- Does a lower rate always mean a cheaper plan? No: the fixed fee matters, and so does how much is used. Listen for “depends on how many”.
- Would you trust the sunflower model at day 120? No: it gives 490 cm, a sunflower stops growing at about 2 to 3 m, and we have no measurements after day 10. Listen for “beyond the data” or “the growth will not stay the same”.
- How would you improve the model? More measurements, over a longer range, and a model that allows the growth to slow.
Differentiation
A way in for anyone stuck
- Give a table of costs for several numbers of uses before any equation is written; the crossing can be seen in the table.
- Write the verbal model first: cost = rate × amount + fixed fee.
- Provide a template: write both equations, set equal, solve, check in both.
Stretch
- Question 7, then: write the cost of x units for x greater than 10 as an equation. (C = 50 + 8(x − 10) = 8x − 30.)
- Two plans never break even when their rates are equal. Explain why, and what would have to differ for one always to be cheaper.
Students learning in English as an additional language
“Break-even” means “the point where you neither gain nor lose”. “Model” here means an equation that describes a real situation. Say “fixed fee” and “rate per unit” with a finger on each part of the equation.
| Word | Plain meaning | Example |
|---|---|---|
| model | an equation or graph that describes a real situation | C = 80h + 200 for a repair |
| rate | how much changes for each unit | 5 rupees per refill |
| fixed fee | an amount paid once, whatever the use | a 600 rupee flask |
| break-even | the point where two costs are equal | 15 uses |
| extrapolate | use a model beyond the range of the data | predicting day 120 from 10 days |
| interpolate | use a model inside the range of the data | predicting day 7 from days 0 to 10 |
| piecewise | made of different straight pieces for different ranges | 4 rupees then 7 rupees per unit |
Sentence starters for explaining aloud or in writing:
- The rate is … and the fixed amount is …
- The plans cost the same when …, which is … units.
- I would not trust the model beyond … because …
Two things not to say
“The model says so.” A model describes a pattern; it does not prove the pattern continues.
“Break-even is the answer.” It is the boundary where the best choice changes.
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.
- ModelA taxi charges 50 rupees plus 12 rupees per km. Write the cost for x km and find it for 10 km.
C = 12x + 50; 170 rupees. - SolveFor how many km does the same taxi cost 230 rupees?
15 km.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA gym charges 300 rupees a month plus 20 per visit. Another charges 100 plus 40 per visit. When do they cost the same?
10 visits, 500 rupees. - Earlier work (Lesson 6)Write the line through (2, 3) perpendicular to y = 2x − 1.
y = −x/2 + 4.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA tank is modelled by y = 600 − 25x. For which values of x does the model make sense?
From 0 to 24 days, where y is at least 0. - ReasoningPlan A is 12x + 100 and Plan B is 15x. The company says B is cheaper “for small orders”. Up to how many units?
Up to 33 units (the break-even is 100 ÷ 3 = 33.3), then A is cheaper.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Make a table of costs for 0, 5, 10, 15 and 20 uses for each plan and look for the crossing.