At a glance. One lesson of about 60 minutes. It assumes the geometric sum from lesson 4 and the sum to infinity from lesson 5. Students model regular saving, a loan balance and a repeated dose as geometric series, find a long-run level, and find when a target is first passed. A scientific calculator is used for powers.
Calculator: scientific calculator. Powers such as 1.05 cubed are quick on a calculator and slow by hand. Students write the factor and the structure first, so the calculator confirms.
What students will learn
Know
- Savings: P((1 + i)n − 1) ÷ i.
- Loan: Bn = L(1 + i)n − P((1 + i)n − 1) ÷ i.
- Dose: D(1 − fn) ÷ (1 − f), ceiling D ÷ (1 − f).
Understand
- Each model is a recurrence that unrolls to a geometric series.
- A ceiling exists only when the factor is less than 1.
- Payments earlier in the sequence earn or save more interest.
Do
- Find a savings total, a loan balance or a dose level.
- Find a long-run level.
- Find a repayment or when a target is first passed.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the formulae and the factor (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students describe how the ceiling depends on the fraction left.
- Criterion C. Communicating: setting out a recurrence and its unrolled sum.
- Criterion D. Applying mathematics in real-life contexts: savings, loans and medicine (all questions).
The one thing to do with the figure
Ask for the prediction first. Then set the fraction to 50% and ask students to say the level after dose 2, then check it. Move to 90% and ask why the dashed line is so much higher.
Ask: “If the drug is removed more slowly, what happens to the ceiling?” More is left each time, so the ceiling rises.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tablet; collect predictions about the build-up. |
| 12 min | Figure, key idea, warning card, calculator note. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C, ask how much of the first dose is left. |
| 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 | 840; 40; 1.21. |
|---|---|
| Hook | With f = 0.5 the ceiling is 500 ÷ 0.5 = 1000 mg, and the level just after a tablet approaches 1000 mg but does not pass it. |
| Worked examples | 3310; 7551.68; 262.5 mg and 266.67 mg. |
| Faded example | 150. |
| 1. Level after two doses | D. A is one dose; B is what is left of the first dose; C keeps all of it. |
| 2. Regular savings | 12 610. 4000 × 3.1525. |
| 3. Level after several doses | 556. 400 × 1.39. |
| 4. Long-run ceiling | 7.5. 3 ÷ 0.4. |
| 5. Loan balance | 4095.5. 4550 × 1.01 − 500. |
| 6. Repayment that clears a loan | 2420. 5082 ÷ 2.1. |
| 7. Doses to pass a level | 5. 108.8 then 115.28. |
| 8. Setting up the formula | A. B treats the deposits as made at the end of each year; C is one deposit; D adds simple interest once. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Gave one dose, what is left of the first, or kept all of the first. | Ask: how much of the first dose is still there when the second is taken? |
| 12000 or 13240.5 in question 2 | Added no interest, or gave each deposit a full extra year. | Draw a timeline: the last deposit earns nothing. |
| 520 or 1200 in question 3 | Stopped after two doses, or kept every dose whole. | Write the bracket 1 + f + f squared and count the terms. |
| 5 or 1.8 in question 4 | Took the fraction lost as f (or divided by the fraction left), or gave one remainder. | Say aloud: 40% lost, 60% left, so f is what? |
| 4000 or 4100 in question 5 | Ignored interest, or took interest on the original loan every time. | Write the balance at the end of each month in a table. |
| 2310 or 2520 in question 6 | Shared the balance equally, or double-charged interest. | Set the balance after 2 years to zero and list the terms. |
| 4 or 2.2 in question 7 | Stopped one dose early, or ignored the loss. | Tabulate levels after each dose. |
| B, C or D in question 8 | Used the end-of-year pattern for start-of-year deposits, used one deposit, or added simple interest once. | Draw a timeline from the start of year 1 to the end of year 4 and count the years each deposit earns. |
Questions worth asking
- Why does the earliest deposit earn the most interest? Listen for: it stays in the account longest.
- How does a loan differ from savings in the formula? Listen for: the payment is taken off, so the sum is subtracted.
- When does a repeated dose not settle? Listen for: when nothing leaves the body, f = 1, or more is added each time.
- What does the ceiling mean for the patient? Listen for: the level never exceeds it however many doses.
Differentiation
A way in for anyone stuck
- Give a table with a row for each period and columns for starting amount, factor and ending amount.
- Provide the three formulae on a card with a worked number beside each.
- Give the frame: “The factor is … because …. After … periods I have ….”
Stretch
- Find the monthly payment that clears a loan of 6000 in 3 equal monthly payments at 1% a month.
- Find the dose D so that the long-run level just after a dose is 300 mg when 70% is still there at the next dose.
Students learning in English as an additional language
“Deposit” is money paid into an account. “Repayment” is money paid back to the lender. “Balance” is the amount still owed or held. “Dose” is one measured amount of medicine. “Remains” means is still there.
| Word | Plain meaning | Example |
|---|---|---|
| deposit | money paid into an account | paid at year end |
| balance | the amount owed or held at a time | after month 2 |
| repayment | money paid towards a loan | equal each month |
| dose | one amount of medicine taken at one time | 400 mg |
| ceiling | the level the amount approaches and does not pass | the dashed line |
Sentence starters for explaining aloud or in writing:
- The factor each period is … because …
- The first deposit earns … periods of interest.
- The long-run level is … ÷ (1 − …).
Two things not to say
“Use the percentage lost as the factor.” Say instead: use the fraction that remains.
“Add the repayments to the interest.” Say instead: the balance is multiplied, then the payment is subtracted.
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.
- FindFind 500 × (1.042 + 1.04 + 1).
1560.8. - FindA dose of 60 mg with half left at the next dose: ceiling?
120 mg.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA trader in Lagos saves 10 000 naira at the end of each year at 10% a year. How much is there after 3 years?
10 000 × 3.31 = 33 100. - Earlier work (lesson 5)Find the sum to infinity of 24 + 12 + 6 + …
48.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy does a repeated dose with f = 1 never settle?
Nothing leaves the body, so the level rises by the dose every time. - Working backwardsA long-run level of 240 mg arises from a dose of 60 mg. Find the fraction left.
1 − f = 60 ÷ 240, so f = 0.75.
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 scientific calculator is needed for the powers; a phone calculator in landscape also works.