At a glance. One lesson of about 60 minutes. It builds on lessons 4 to 6. Students write an average cost as F/n + v, read its floor, solve for the number of items from a target, and find a mean speed from two equal distances.
Calculator: GDC optional. GDC optional: a table and an intersection confirm the answers, but each is solved by hand first.
What students will learn
Know
- Average cost = (F + vn)/n = F/n + v.
- The horizontal asymptote y = v is a floor.
- Mean speed is total distance divided by total time.
Understand
- Dividing a fixed cost among more items makes each share small but never zero.
- A mean of two speeds over equal distances is not their arithmetic mean.
- The domain in a context is whole positive numbers.
Do
- Write and evaluate an average-cost model.
- Solve for the number of items from a target average.
- Find a mean speed.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: rearranging and solving (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe how the floor and the curve depend on F and v.
- Criterion C. Communicating: explaining the floor to a manager in context.
- Criterion D. Applying mathematics in real-life contexts: printing, baking and tickets (hook, questions 2, 7).
The one thing to do with the figure
Ask for the prediction first. Set F = 300, v = 6 and ask students for the average at 20 items before they read it. Then set 50 items and ask what number the average is getting close to.
Ask: “What would the graph look like if F were 0?” The average would be a flat line at v: there is no fixed cost to share.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the poster printing; collect predictions about the floor. |
| 12 min | Figure, key idea and warning card; compute one average by hand. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A or B, ask what 3500/n does as n grows. |
| 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 | 100 rupees; y = 3; x = 4. |
|---|---|
| Hook | It keeps falling but never reaches 12, so it never goes below 12. |
| Worked examples | 11 rupees and more than 900 copies; 24 km/h; 300 copies. |
| Faded example | 6 shillings and 600 items. |
| 1. The floor | C. A divides by the target; B expects enough trips to reach the floor; D puts the whole fixed cost on one trip. |
| 2. Evaluating an average | 38. 1520 ÷ 40. |
| 3. Solving for the number of items | 250. 500 ÷ 2. |
| 4. Mean speed | 36. 180 ÷ 5. |
| 5. A target below a value | 481. n > 480. |
| 6. Finding the fixed cost | 400. F/100 = 4. |
| 7. A ratio target | 600. 3000 ÷ 5. |
| 8. Using the floor and a target | B. A divides the fixed cost by the floor; C thinks the floor can be reached; D stops at 1800/n = 5. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Divided the fixed cost by the target, expected enough trips to reach the floor, or used one trip. | Write 3500/n + 40 = 40 and ask what 3500/n would have to be. |
| 30.2 or 30 in question 2 | Added the per-item cost once, or left it out. | Find the total cost, then divide. |
| 35.7 in question 3 | Divided 500 by the target. | Subtract the per-item cost from the target first. |
| 37.5 in question 4 | Took the arithmetic mean of the speeds. | Find the total time. |
| 480 or 160 in question 5 | Allowed equality, or divided by the wrong number. | Solve the inequality, then say what whole number is needed. |
| 200 or 4 in question 6 | Halved wrongly, or gave the difference. | Express the difference of averages in terms of F. |
| 100 or 120 in question 7 | Forgot to take off the cost per ticket, or divided by 25. | Work out the target first, then subtract. |
| A, C or D in question 8 | Divided the fixed cost by 30, thought the floor can be reached, or gave 1800/n instead of n. | Write 1800/n + 30 = target and solve one line at a time. |
Questions worth asking
- Why does the average never reach v? Listen for: F/n is always above 0.
- What does doubling the fixed cost do to the floor? Listen for: nothing; the floor depends only on v.
- Why is the mean speed not the mean of the speeds? Listen for: more time is spent at the slow speed.
- Why must n be a whole number? Listen for: you cannot make a part of an item.
Differentiation
A way in for anyone stuck
- Give a table of n and average cost, and ask students to extend it and spot the pattern.
- Provide F/n + v already written and ask only for evaluating and solving.
- Give the frame: “The total cost is …. The average is …. The floor is ….”
Stretch
- Show that the mean speed over equal distances at speeds a and b is 2ab/(a + b).
- Find the number of items for which the average cost is within 0.5 rupee of the floor, for F = 900 and v = 5.
Students learning in English as an additional language
“Fixed cost” is a cost that does not change with the number made. “Average” here means the total divided by the number. “Floor” means a lowest limit. “Mean speed” is total distance divided by total time.
| Word | Plain meaning | Example |
|---|---|---|
| fixed cost | a cost that is the same however many are made | the setup of a print job |
| average cost | the total cost divided by the number of items | total ÷ number |
| floor | a level that a value stays above | a river that stays above 2 metres all year has a floor of 2 metres |
| mean speed | total distance divided by total time | 120 km in 5 hours |
| target | the value you want to reach | an average of 9 rupees |
Sentence starters for explaining aloud or in writing:
- The total cost is … and the average is ….
- As n gets larger, the average gets close to … but stays ….
- The total distance is … and the total time is ….
Two things not to say
“The average cost reaches the cost of one item.” Say instead: it gets close to it but never reaches it.
“Add the speeds and halve.” Say instead: divide the total distance by the total time.
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.
- EvaluateA fixed cost of 800 rupees and 4 rupees an item. Average for 50 items?
800/50 + 4 = 20. - FloorState the floor of (1000 + 7n)/n.
7.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Target(750 + 5n)/n = 8. Find n.
n = 250. - Earlier work (lesson 6)Solve 8/x + 2 = 4.
x = 4.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan the average cost per item ever be below the cost of one item? Explain.
No: F/n is positive, so the average is above v. - Working backwardsThe average is 15 for 20 items and 10 for 40 items. Find F and v.
F/20 − F/40 = 5, so F = 200 and v = 5.
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.
A GDC is optional and sits in a closed section; nothing depends on it.