At a glance. One lesson of about 60 minutes. It assumes lessons 1 and 2. Students halve the second difference, subtract an squared, find the arithmetic rule for the remainder and check the final rule.
Calculator: no calculator. The terms are small and every step is a short subtraction, so no calculator is needed.
What students will learn
Know
- a is half the second difference.
- Subtracting an² from each term leaves an arithmetic sequence bn + c.
- The final rule is checked against two or more original terms.
Understand
- The method reduces a quadratic sequence to an arithmetic one.
- A wrong guess for a leaves a remainder that is not arithmetic, which is how you know.
- A term is in the sequence only if solving for n gives a positive whole number.
Do
- Find an² + bn + c from at least four terms.
- Use the rule to find far terms and positions.
- Set up simultaneous equations from two terms when a is known.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding rules and terms (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students test guesses for a on several sequences and state how the right guess is found.
- Criterion C. Communicating: a three-step layout with a check, and a sentence explaining why the remainder is arithmetic.
- Criterion D. Applying mathematics in real-life contexts: counting stones in growing patterns (hook and extra questions).
The one thing to do with the figure
Ask for the prediction first. Then choose sequence 1 and ask for guesses of a before moving the slider. Move to the right guess and ask what has changed about the bottom row.
Ask: “Which guess is nearly right? How can the bottom row tell you how far off you are?” A guess one too small leaves remainders that still rise in a quadratic way.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the patio stones; a guess for the number in front of n squared. |
| 12 min | Figure, key idea, warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C, ask what subtracting 4 n squared does at n = 1. |
| 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 | 2; 4n + 3; 3n² + n. |
|---|---|
| Hook | The terms 2, 9, 20, 35 give a = 2 and remainders 0, 1, 2, 3, which is n − 1. The nth term is 2n² + n − 1, and the 20th term is 819. The shape confirms it: patio n is (2n − 1) stones by (n + 1) stones, and (2n − 1)(n + 1) = 2n² + n − 1; patio 20 is 39 by 21. |
| Worked examples | 4n² − n + 1; −2n² + 3n + 7; the 7th term, and 60 is not a term. |
| Faded example | 3n² − n + 4. |
| 1. Subtracting the n squared part | D. A adds 2n²; B subtracts 2n; C subtracts 4n². |
| 2. Difference of the remainder | 3. Remainders 4, 7, 10, 13. |
| 3. Building a term from the parts | 63. 48 + 15. |
| 4. The coefficient of n | 2. Remainders 4, 6, 8, 10. |
| 5. A far term from the rule | 302. 3n² + 2. |
| 6. Using two terms to find a constant | −2. b = 5, c = 3 − 5. |
| 7. Solving for the position | 8. n² = 64. |
| 8. Writing the nth term | A. B uses a = 4; C has the wrong sign on the constant; D fits only the first two terms. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 | Added the n² part, subtracted 2n, or used the whole second difference. | Write the guess a = 2 first, then n² for n = 1 to 5, then 2 × each. |
| 9 or 4 in question 2 | Used the first gap of the terms, or the first remainder. | Write the remainders in a row and subtract neighbours. |
| 15 or 111 in question 3 | Gave the remainder only, or used 6n². | Write the full rule 3n² + 4n − 1 before substituting. |
| 4 or 6 in question 4 | Gave the first remainder or the second difference. | Subtract 3n² and look at the gaps of the remainders. |
| 602 or 32 in question 5 | Used 6n², or did not square 10. | Write 3 × 10² with the index visible. |
| 3 or 5 in question 6 | Forgot b, or gave b. | Write both equations and subtract. |
| 128 or 64 in question 7 | Stopped before dividing and rooting, or gave n². | Ask: is this the position or the position squared? |
| B, C or D in question 8 | Used a = 4, wrong constant, or an arithmetic rule through two terms. | Test each option at n = 3. |
Questions worth asking
- Why does subtracting the right an² make the remainder arithmetic? Listen for: the second difference of the remainder is zero, because an² carried all of it.
- How do you check a rule you found? Listen for: substitute n = 1, 2, 3 and compare with the given terms.
- What if the remainder is a constant list? Then b = 0 and the rule is an² + c.
- Why might a term not appear in the sequence? n would not be a positive whole number.
Differentiation
A way in for anyone stuck
- Provide a table with the columns n, term, n², a n² and remainder.
- Do the first two worked examples together and let students try the second step alone.
- Give the frame: “Second difference …, so a = …. The remainder is …, so b = … and c = …”
Stretch
- Prove that the sum of the first n odd numbers is n² by finding the nth term of the totals 1, 4, 9, 16 and justifying it.
- Find the nth term of 2, 10, 24, 44 and use it to predict the 5th term, then check by extending the first differences.
Students learning in English as an additional language
“Subtract” means take away. “Remainder” here means what is left after taking away the n squared part. “Guess” is a first try that you then check. “Halve” means divide by 2. Say “two n squared plus five n minus one” aloud.
| Word | Plain meaning | Example |
|---|---|---|
| remainder | what is left after subtracting the n squared part | after taking away 4, 16, 36 from 5, 17, 37 |
| halve | divide by 2 | half of 18 |
| nth term | a rule that gives the term in position n | 6n² + n |
| coefficient | the number multiplying a letter | the 8 in 8n |
| check | testing a rule on terms you already know | n = 2 in 3n² + 1 |
Sentence starters for explaining aloud or in writing:
- The second difference is …, so a = …
- After subtracting, the remainder is …, which is arithmetic with difference …
- Checking at n = 2 gives …, which matches …
Two things not to say
“Guess and check until it fits.” Say instead: the second difference tells you a straight away.
“The number in front of n squared is the second difference.” Say instead: it is half of 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.
- SubtractSubtract n² from 3, 6, 11, 18. What is left?
2, 2, 2, 2, so the rule is n² + 2. - FindWrite the nth term of 2, 8, 18, 32.
2n².
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextThe number of paving stones in pattern n of a courtyard in Hanoi is n² + 4n. How many stones are in pattern 9?
81 + 36 = 117. - Earlier work (lesson 1)Find the nth term of 20, 17, 14, 11.
23 − 3n.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningShow that the nth term of 4, 9, 16, 25 is (n + 1)².
Second difference 2, so a = 1; remainders 3, 5, 7, 9 are 2n + 1; n² + 2n + 1 = (n + 1)². - Working backwardsA sequence has nth term 3n² + bn + 2 and its 4th term is 62. Find b.
48 + 4b + 2 = 62, so b = 3.
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.
No device? A table on paper with a row for the remainder does the same job.