At a glance. One lesson of about 60 minutes. A school fundraiser opens the lesson. The figure lets students vary price, cost and fixed cost and see break-even move, vanish or become a whole line. The decision card gives a reasoned way to choose among graphing, substitution and elimination.
Calculator: no calculator. Nothing here needs one. Every break-even is a whole number or a short decimal, and the interpretation, not the arithmetic, is the learning.
What students will learn
Know
- Break-even is where cost equals income, the intersection of the two lines.
- Break-even items = fixed cost ÷ (price − cost of each item), if the price is more.
- Substitution, elimination and graphing each suit different forms of a system.
Understand
- A model turns a decision into a calculation that can be tested.
- The solution of a system must be interpreted: it can be a whole number only, or be below zero profit.
- Different methods give the same point; choosing is about efficiency.
Do
- Write cost and income as linear expressions and solve for break-even.
- Interpret a solution, including a non-whole number of items and no solution.
- Choose a method and justify the choice.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: solving and choosing methods (questions 1 to 8).
- Criterion B. Investigating patterns: use the figure to find how the break-even moves when each of price, cost and fixed cost changes, and state a general rule.
- Criterion C. Communicating: present a break-even analysis with a table, a graph and a sentence of interpretation.
- Criterion D. Applying mathematics in real-life contexts: the notebooks, the print shops, the co-op and the plans.
The one thing to do with the figure
Ask for the break-even before sliding. “With a fixed cost of 30, each item costing 2 and a price of 5, how many items do we need?” Let students find it with “each item adds 3” and then test it.
Then change one slider at a time and ask what changes and why: price, cost, fixed cost. Finish with price equal to cost and ask what the lines look like.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook; students predict, then explore the figure with their own numbers. |
| 10 min | Key idea and the notebook worked example, including the profit at 30. |
| 10 min | The print-shop example, the method card, and the faded example in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 8, on paper or on screen. |
The answers
| Warm-up | x = 10; x = 3, y = 6; none. |
|---|---|
| Notebooks | 12 notebooks, 84 cedis; at 30 a profit of 72 cedis. |
| Print shops | Equal at 10 copies (60); P is cheaper for 20. |
| Faded example | 16 notebooks and 96 cedis. |
| 1. Choosing a method | A. B and C are slower; D scales both and then adds terms that do not cancel. |
| 2. Break-even point | 18. 90 + 3x = 8x; 5x = 90. |
| 3. Money at break-even | 180. 20 items at 9 each. |
| 4. First profitable whole number | 13. 90 ÷ 7 = 12.86; at 13 income 156 > cost 155. |
| 5. Elimination in a short form | 6. 2x = 12. |
| 6. A two-unknown problem in context | 8. 25s + 480 − 40s = 360; 15s = 120. |
| 7. Which plan is cheaper after how many | 21. Equal at 20 (240 each); at 21, 246 < 251. |
| 8. Price below cost | B. A uses the fixed cost as items; C reads the loss per item as a break-even; D ignores that each item loses money. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Chose a slower method, or scaled without matching. | Ask what is the same in both equations before choosing. |
| 30, 11.25 or 8.18 in question 2 | Used one of the numbers instead of the difference. | Ask: what does one sale add towards the fixed cost? |
| 20, 100 or 80 in question 3 | Gave the items, the fixed cost or the cost of making. | Underline what is asked: money at break-even. |
| 12 or 12.86 in question 4 | Gave the break-even instead of the first whole number with a profit. | Test the whole numbers either side of the break-even. |
| 12, −6 or 8 in question 5 | Stopped early, subtracted in different orders, or gave y. | Write each subtraction in a column with terms and totals. |
| 4, 120 or −5 in question 6 | Gave baskets, stopped early, or partly expanded. | Label s and b; write the answer line as “sandals =”. |
| 20, 100 or 6 in question 7 | Gave the equal point, stopped early, or added rates. | Work out each plan's cost at 20 and at 21 sessions. |
| A, C or D in question 8 | Ignored that each item loses money. | Work out the profit on one item: price − cost. |
Questions worth asking
- What does the crossing point tell the group? Listen for the number of items and the money, and that profit starts after it.
- Why can a business have no break-even? Each item earns no more than it costs, so the fixed cost is never covered.
- Why can the break-even be a non-whole number? The model is a line; real items are whole, so the first profit is the next whole number.
- How did you choose your method? Listen for matching terms, a y = … equation, or a need for a picture.
Differentiation
A way in for anyone stuck
- Give a table with columns for items, cost, income and profit, and fill in the first rows.
- Start with the price, the cost of each item and the fixed cost in whole numbers with a whole-number answer.
- Offer a method card with the four bullet points of the decision card.
Stretch
- Question 7, then: find the number of sessions at which plan B has cost 100 more than plan A.
- A charity wants a profit of 96 cedis from the notebooks. How many must they sell, and what price would halve that number?
Students learning in English as an additional language
“Break even” means “neither lose nor gain money”. “Fixed cost” is paid once, whatever you sell. “Income” is the money coming in; “profit” is what is left after costs.
| Word | Plain meaning | Example |
|---|---|---|
| break-even | the point where cost equals income | 12 notebooks |
| fixed cost | a cost that does not change with the number sold | hiring equipment for 48 |
| income | money coming in from sales | 7 × 12 = 84 |
| profit | income minus cost | 210 − 138 = 72 |
| model | a mathematical description of a situation | C = 48 + 3x |
| social enterprise | a business that uses its profit for a good cause | notebooks to raise money for a library |
Sentence starters for explaining aloud or in writing:
- The cost is … plus … for each item, so C = …
- The break-even point is (…, …), which means …
- I chose elimination / substitution / a graph because …
Two things not to say
“Break-even is when you make money.” It is when you stop losing; profit comes after.
“Always use elimination.” The aim is the quickest correct method, and a reason for it.
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.
- Break-evenFixed cost 30, each item costs 1, price 4. Items to break even?
10 items. - ProfitIncome 7x, cost 20 + 2x. Profit at 8 items?
56 − 36 = 20.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA tuck shop pays 50 for a licence and 3 for each pie, and sells pies at 5. How many pies to break even?
25 pies. - Earlier workSolve y = 2x − 1 and x + y = 8.
(3, 5).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningA business breaks even at 10 items. If its price rises by 1, will it break even sooner or later? Why?
Sooner: each item adds more towards the fixed cost. - Working backwardsWrite a cost and an income whose break-even is 15 items at 90.
Any correct pair, for example C = 30 + 4x and I = 6x.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
All four 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? Build a table of items, cost and income for 0 to 20 items on paper and look for where income overtakes cost.