The unit starts from an idea students already own: a sequence that adds the same amount each time. Writing the sum forwards and backwards shows that every column totals the same, which gives the arithmetic sum formula without learning it as a rule. Sigma notation then gives a short way to write and split such sums, and the running totals of an arithmetic sequence turn out to form a quadratic sequence, joining the two kinds of sequence from earlier in the year.
The geometric sum is found by multiplying by the ratio and subtracting, so that all the middle terms cancel. When the ratio is between −1 and 1 the powers shrink and the sums settle under a ceiling, the sum to infinity. That one idea explains recurring decimals, a bouncing ball, regular savings, loan repayments and the level of a medicine that is taken repeatedly.
The last lesson turns back to letters: the sum formulae are rearranged for a different unknown, and the power in a formula is read for how its answer scales when its inputs are multiplied.
The lessons, in order
- Lesson 1 · Adding an arithmetic sequence The sum of the first n terms of an arithmetic sequence by pairing, the two forms of the formula, and using it to find a sum, a number of terms or a difference. Set a first term, a difference and a number of terms, and see why the reversed copy makes every column the same height. No calculator
- Lesson 2 · Sigma notation Reading and writing sigma notation, the number of terms from the limits, the sum of a constant, of a multiple and of a sum of rules, and the sums of k and of the odd numbers. Pick a rule and the first and last value of k, and see which terms are added. No calculator
- Lesson 3 · Partial sums and quadratic sequences The partial sums of an arithmetic sequence as a quadratic sequence with second difference d, the form (d divided by 2)n squared plus (a minus d divided by 2)n, and recovering the nth term as S(n) minus S(n minus 1). Set a first term and a difference and see the running totals stack up, with their first and second differences. No calculator
- Lesson 4 · Adding a geometric sequence The sum of the first n terms of a geometric sequence by multiplying by the ratio and subtracting, the formula, the case of ratio 1, and using it to find a sum, a ratio or a number of terms. Set a first term, a ratio and a number of terms, and see which terms of S and rS line up and cancel. No calculator
- Lesson 5 · The sum to infinity Partial sums of a geometric series as the ratio changes, the sum to infinity a divided by (1 minus r) for a ratio between minus one and one, why it fails otherwise, recurring decimals and a bouncing ball. Set a ratio and a number of terms, and see whether the partial sums settle under a ceiling. No calculator
- Lesson 6 · Loans, savings and repeated doses Regular deposits, loan balances and repeated doses as geometric series, the formulae for each, the long-run ceiling of a repeated dose, and when a target is first passed. Set the fraction of a drug still in the body and the number of doses, and see the level just before and just after each dose. Scientific calculator
- Lesson 7 · Rearranging and scaling formulae Rearranging the sum formulae and a formula with a squared term, direct, inverse and joint variation with powers, and finding the multiplier of an output from the multipliers of its inputs. Pick a power and how many times bigger x becomes, and see what y is multiplied by. No calculator
Assessing this unit
Your teacher chooses which of these tasks to set, and when: nobody does all of them. Each sheet says where it fits in the unit, how long it takes and how your work is judged. To print a sheet, open it first and then print the page.
| Task | Criteria | When | Time |
|---|---|---|---|
| A1: Quick check: arithmetic series and sigma | A | After lesson 3 | 25 minutes |
| A2: End-of-unit test | A | End of unit, after lesson 7 | 50 minutes |
| B1: Running totals of cubes | B | Mid-unit, after lesson 3 | 50 minutes |
| C1: Explain it to someone who missed it | C | After lesson 5 | 35 minutes |
| D1: The loom fund | D and C | After lesson 6 | 50 minutes |
| D2: The clinic dosage plan | D and C | End of unit, after lesson 7 | 50 minutes |
D1 and D2 are alternatives. Both assess criteria D and C, in two different real-life contexts, a village savings group and a clinic dosage plan: one is set, not both.
Task A1: Quick check: arithmetic series and sigma (criterion A)
When: After lesson 3. Time: 25 minutes. Calculator: no calculator. Criterion: A, knowing and understanding.
Answer every question you can and show your working: a correct method earns credit even if the final answer slips. Questions 5 and 6 are meant to make you think, so try something rather than leaving them blank.
Part 1. Adding an arithmetic sequence
- Find the sum of the first 20 terms of 7, 11, 15, 19, … (2 marks)
- Find the sum of the series 5 + 9 + 13 + … + 101. Say how many terms it has. (3 marks)
Part 2. Sigma notation
- Write out the first three terms and the last term of Σk=112 (3k − 2), then find its value. (3 marks)
Part 3. Running totals
- The sum of the first n terms of an arithmetic sequence is Sn = n2 + 4n. Find the first term, the common difference and the 10th term. (3 marks)
Part 4. Think it through
- An arithmetic series has first term 4 and 20 terms, and its sum is 650. Find the common difference. (2 marks)
- Find a formula, in terms of n, for the sum of the first n even numbers 2 + 4 + 6 + … + 2n. Use it to find 2 + 4 + 6 + … + 100. (3 marks)
How your work is judged
| Level | Criterion A in task A1: what it looks like |
|---|---|
| 1–2 | You use a formula or a method correctly in some of questions 1 to 3, even if the number of terms or the arithmetic slips. |
| 3–4 | You find the number of terms and the sums in questions 1 to 3 correctly and make a start on question 4. |
| 5–6 | You answer questions 1 to 4 correctly with working that shows your method, and make a good start on question 5 or 6. |
| 7–8 | You answer questions 1 to 4 correctly and solve question 5 or 6 with reasoning that shows your answer works. |
Task A2: End-of-unit test (criterion A)
When: End of unit, after lesson 7. Time: 50 minutes. Calculator: scientific calculator. Criterion: A, knowing and understanding.
Three sections, from familiar questions to ones you have not met in this form. Show your working everywhere: it earns credit even when an answer slips. A scientific calculator is allowed.
Section 1. Familiar questions
- Find the sum of the first 8 terms of the geometric sequence 3, 6, 12, 24, … (2 marks)
- Find the sum to infinity of 18 + 6 + 2 + … (2 marks)
- Write the recurring decimal 0.727272… as a fraction in its simplest form, using a geometric series. (3 marks)
- Make d the subject of S = n/2 (2a + (n − 1)d). (2 marks)
- y varies inversely as the square of x. When x = 2, y = 12. Find y when x = 6. (2 marks)
Section 2. More complex questions
- A geometric series has first term 5 and common ratio 1.2. Find the smallest number of terms for which the sum is more than 100. Show the sums that decide it. (3 marks)
- An arithmetic sequence has third term 11 and eighth term 31. Find the sum of its first 20 terms. (3 marks)
- Show that Σk=1n (4k + 1) = 2n2 + 3n. Hence find the value of n for which the sum is 324. (4 marks)
Section 3. Unfamiliar questions
- The sum to infinity of a geometric series is 40 and its first term is 10. Find the common ratio and the sum of the first three terms. (3 marks)
- Eli says: “If you double the number of terms of an arithmetic series, the sum doubles.” Is Eli right? Give a test with numbers, and say exactly when his claim holds. (3 marks)
- The power P of a wind turbine varies as the cube of the wind speed v and as the square of the blade length b. The wind speed rises by 20% and the blades are made 10% shorter. By what factor does P change? Say whether it has gone up or down, and by about how much. (3 marks)
How your work is judged
| Level | Criterion A in task A2: what it looks like |
|---|---|
| 1–2 | You answer some Section 1 questions correctly, for example a sum or a rearrangement. |
| 3–4 | You answer most of Section 1 correctly and make progress in Section 2. |
| 5–6 | You answer Section 1 and most of Section 2 correctly, with clear methods, and make a start in Section 3. |
| 7–8 | You answer Sections 1 and 2 correctly and solve most of Section 3, with reasoning that shows why your answers hold. |
Task B1: Running totals of cubes (criterion B)
When: Mid-unit, after lesson 3. Time: 50 minutes. Calculator: scientific calculator. Criterion: B, investigating patterns.
Anyone can write down 13 + 23 + 33 + … for a few terms, but nobody wants to add 20 cubes one by one. This task is about finding a short way, and about knowing how far you can trust it.
Part 1. Collect results
- Make a table with a row for each n from 1 to 8. Record n, the cube n3 and the running total 13 + 23 + … + n3. Design the table yourself.
Part 2. Describe what you find
- Look at your running totals. Describe what you notice, in words, as a rule a classmate could use to find the running total for any n.
- Use your rule to predict the running totals for n = 9 and n = 10. Then add 93 and 103 to your table to check.
Part 3. Test your rule
- Without adding every cube, find the running total for n = 20. Show how your rule gives it.
- Choose a value of n that you have not used. Work the total out the long way and test your rule against it. Say how confident you now are, and why.
Part 4. Explain
- Explain why your rule works, or show that if your rule is right for one value of n then it must be right for the next.
- Rosa says: “Your rule shows 13 + 23 + … for 100 terms will be a perfect match, because it worked eight times.” Comment on Rosa’s reasoning.
How your work is judged
| Level | Criterion B in task B1: what it looks like |
|---|---|
| 1–2 | With help, you work out some cubes and running totals and say one thing you notice about the totals. |
| 3–4 | You complete your table, describe a rule that fits your totals, and use it to predict and check n = 9 and n = 10. |
| 5–6 | You organize your results clearly, describe a rule that fits every total, use it for n = 20, and test it on a total you did not use to find it, saying how far you trust it. |
| 7–8 | You describe a general rule, use it for large n, test it on new cases and justify it, for example by showing how the step from one total to the next fits it; you say what your evidence does and does not prove. |
Task C1: Explain it to someone who missed it (criterion C)
When: After lesson 5. Time: 35 minutes. Calculator: no calculator. Criterion: C, communicating.
Nia was away for the lessons on arithmetic and geometric series. She tried two questions from a classmate’s notes and her working has mistakes in it.
Nia’s working
(a) Find the sum of 5 + 8 + 11 + … + 62.
Number of terms = (62 − 5) ÷ 3 = 19. Sum = 19/2 × (5 + 62) = 636.5.
(b) Find Σk=16 3 × 2k.
First term 3, ratio 2, so the sum = 3(26 − 1) ÷ (2 − 1) = 189.
Write a one-page explanation for Nia. She should be able to use it on her own, without asking anyone.
Part 1. Put it right
- Find every mistake in Nia’s working. For each one, say what she did, why it is wrong, and write the correct working.
Part 2. Teach the idea
- Explain to Nia how to count the terms of an arithmetic series from its first term, last term and common difference, and why the answer is one more than the number of steps. Use a small example of your own.
- Explain how to read the first term and the number of terms from a sum written with Σ, using Nia’s question (b).
Part 3. Check your explanation
- Give Nia a quick test she can use on any answer she finds, to see whether it is sensible. Show it on both of her results.
- Read your page as if you were Nia. Check that each step is on its own line, that every symbol is introduced, and that the equals sign is only between equal things.
How your work is judged
| Level | Criterion C in task C1: what it looks like |
|---|---|
| 1–2 | Parts of your explanation can be followed, but steps or symbols are missing and the order is hard to follow. |
| 3–4 | You explain what went wrong in both questions with some correct notation and words; your steps are complete and in a sensible order, so Nia could follow most of it. |
| 5–6 | You correct both questions and explain why each was wrong; your steps are laid out one per line, you usually use correct notation and terms, and your example shows the idea. |
| 7–8 | Every step is explained and ordered so Nia could learn from it alone; your notation (sigma limits, equals signs, powers) and key words are correct throughout, and your quick test is clear and shown on both results. |
Task D1: The loom fund (criteria D and C)
When: After lesson 6. Time: 50 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A village weaving group wants to save 50 000 coins for a new loom. Each member’s share is paid into the group’s account as 2 000 coins at the end of every month. The account adds 0.5% interest at the end of every month, on everything in it.
Part 1. Building a model
- Work out the total in the account at the end of 3 months. Show how many months of interest each payment has earned.
- Show that the total after n months can be written as 400 000(1.005n − 1), explaining each step.
Part 2. The target
- Find the smallest whole number of months after which the account holds at least 50 000 coins. Show the two totals that decide it.
- Without any interest, how many months would it take? How much of the 50 000 coins is interest in your answer to question 3?
Part 3. What if things change?
- The bank cuts the interest to 0.3% a month. How many months does it take now? Is it better than no interest at all?
- A member says: “Saving for 24 months gives exactly twice as much as saving for 12.” Use your model to test the claim.
Part 4. Does it make sense?
- State what you assumed to make the model work, and say which assumption is most likely to be wrong in real life.
- Write a short note to the group treasurer with your recommendation, so that someone who has not seen your working can understand it.
How your work is judged
| Level | Criterion D in task D1: what it looks like |
|---|---|
| 1–2 | You work out some of the right numbers, for example the total after 3 months, but without a model or a decision. |
| 3–4 | You find the total after 3 months, a formula, and the number of months for the target, and say whether it makes sense. |
| 5–6 | You build and use the model correctly, test the claim and the lower rate with it, state your assumptions and say how they affect your answer. |
| 7–8 | You choose and use suitable methods throughout, reach correct answers, explain why the model and the target are reasonable and what could make them wrong, and justify your recommendation. |
| Level | Criterion C in task D1: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see how you reached the number of months. |
| 3–4 | Your working can be followed, with some correct notation and units in coins. |
| 5–6 | Your working is clear, usually in order with units and notation, and your note to the treasurer is mostly well organized. |
| 7–8 | Your working is clear, complete and well ordered, units and notation are always correct, and your note to the treasurer makes sense on its own. |
Task D2: The clinic dosage plan (criteria D and C)
When: End of unit, after lesson 7. Time: 50 minutes. Calculator: scientific calculator. Criteria: D, applying mathematics in real-life contexts; C, communicating.
A clinic in Phuket gives a medicine to a patient every 8 hours. A standard dose for an adult of 60 kg is 150 mg. By the time of the next dose, 60% of the previous amount is still in the patient’s body. The clinic’s safe limit for the level of the medicine in the body is 350 mg.
Part 1. Doses add up
- Find the level in the body just after the 1st, 2nd and 3rd doses.
- Find the long-run level the patient approaches, just after a dose. Explain why the level never quite reaches it.
- Which dose is the first to take the level above the safe limit of 350 mg? Show the two levels that decide it.
Part 2. Adjusting the dose
- The long-run level L is related to the dose D and the fraction f that remains by L = D ÷ (1 − f). Make D the subject. What is the largest dose that keeps the long-run level from going above 350 mg?
- A dose varies directly with body mass. What dose should a 40 kg child receive if the same fraction remains? What long-run level results?
Part 3. A patient who clears the medicine slowly
- A 40 kg child clears the medicine more slowly: 80% remains at the next dose. Is the child’s dose from question 5 still safe in the long run? What dose would keep the long-run level at 350 mg?
- Show that halving the dose and halving the amount that leaves (1 − f) leaves the long-run level exactly the same. Why does this matter for a patient with slow clearance?
Part 4. Does it make sense?
- State what the model assumes, and say what could make the real level different from your answers.
- Write a short reply to the head nurse with your advice, so that someone who has not seen your working can understand it.
How your work is judged
| Level | Criterion D in task D2: what it looks like |
|---|---|
| 1–2 | You find some of the levels correctly, for example after the first doses, without the long-run level or a decision. |
| 3–4 | You find the levels, the long-run level and the first dose over the limit, and say whether the plan is safe. |
| 5–6 | You use suitable methods correctly for the levels, the rearranged formula, the proportional dose and the slow-clearance case, and say how reliable the model is. |
| 7–8 | You choose and use suitable methods throughout, reach correct answers, explain what the model leaves out and how that affects safety, and justify your advice. |
| Level | Criterion C in task D2: what it looks like |
|---|---|
| 1–2 | Some working is shown, but it is hard to see how the levels or the doses were reached. |
| 3–4 | Your working can be followed, with some correct notation and units in mg. |
| 5–6 | Your working is clear, usually in order with units and notation, and your reply to the head nurse is mostly well organized. |
| 7–8 | Your working is clear, complete and well ordered, units and notation are always correct, and your reply makes sense on its own. |
Teachers: the assessment teacher page has the markschemes, exemplar answers, the MYP descriptors for this year and support for each task.
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.