At a glance. One lesson of about 60 minutes. It assumes second differences from Year 10 and the sum formula from lesson 1. Students find the second difference of running totals, write the total as a quadratic, and recover terms from totals.
Calculator: no calculator. No calculator: the algebra of S(n) minus S(n minus 1) is the learning, and the numbers are small.
What students will learn
Know
- The second difference of the totals of an arithmetic sequence is d.
- Sn = (d ÷ 2)n2 + (a − d ÷ 2)n.
- un = Sn − Sn−1, and u1 = S1.
Understand
- Why the gaps between totals are the terms.
- Why a linear sequence summed is quadratic, and what the coefficient means.
- The difference between a total and a term.
Do
- Find the second difference of running totals.
- Write S(n) from the first term and difference.
- Recover the nth term or a term position from S(n).
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: totals, terms and the sum rule (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students predict the second difference before reading it.
- Criterion C. Communicating: explaining why the totals are quadratic using the gaps.
- Criterion D. Applying mathematics in real-life contexts: weekly savings (hook) and a training plan (question 3).
The one thing to do with the figure
Ask for the prediction first. Then set the difference to 0 and ask what the totals do, and then to 1. Ask students to say the second difference before the figure shows it.
Ask: “What would you have to change in the sequence to make the second difference 6?” The difference of the terms would be 6.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: running savings; collect predictions about the gaps. |
| 12 min | Figure, key idea and warning card; derive the quadratic from the second difference. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose B, ask which totals are being compared. |
| 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; n2 − 2n + 1; 21. |
|---|---|
| Hook | The terms are 20, 25, 30, …, the totals are 20, 45, 75, 110, … with gaps 25, 30, 35, so the second difference is 5. |
| Worked examples | 2n2 + n; 6n − 1; 4, 7, 10 with d = 3. |
| Faded example | 23. |
| 1. Term from the totals | A. B is the total of three terms; C is the total of two; D is the first term. |
| 2. Term from the totals | 17. 44 − 27. |
| 3. Using the sum rule | 315. 225 + 90. |
| 4. Totals to difference | 4. Gaps 13, 17, 21. |
| 5. Difference from a total rule | 12. 2 × 6. |
| 6. Unknown in a total rule | 10. 6n − 3 + p. |
| 7. Which term | 18. 2n + 5 = 41. |
| 8. Terms from running totals | C. A is the totals; B leaves out u1 = S1; D drops the −5 in 5(n − 1). |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Gave the total of three terms, of two, or the first term. | Ask which two totals have the 3rd term as the gap. |
| 44 or 27 in question 2 | Gave the total of four terms, or of three. | Draw the stack: the 4th term is the top piece. |
| 225 or 330 in question 3 | Left out the second part, or used a as the coefficient. | Check at n = 1: the rule must give the first term, 7. |
| 2 or 9 in question 4 | Gave half the second difference, or the first term. | Write the gaps and ask how they change. |
| 6 or 5 in question 5 | Gave the n squared coefficient, or the first term. | Find S1 and S2 and subtract. |
| 7 or 19 in question 6 | Dropped the constant from the expansion, or used the wrong linear part. | Expand 3(n − 1)2 carefully, line by line. |
| 20.5 or 36 in question 7 | Left out the constant in the term rule, or stopped at 2n. | Test the rule: 2n + 5 should give 7 at n = 1, which is S1. |
| A, B or D in question 8 | Gave the totals, left out the first term, or dropped a term when expanding 5(n − 1). | Ask: what is S1, and which term of the sequence is it? |
Questions worth asking
- Why are the gaps between totals the terms? Listen for: each total is the one before plus the new term.
- Why is the coefficient of n squared half of d? Listen for: the second difference of a n squared is 2a.
- What does S1 equal? Listen for: the first term.
- What would the totals of 4, 4, 4, 4 look like? Listen for: 4, 8, 12, 16, linear, second difference 0.
Differentiation
A way in for anyone stuck
- Give a table with columns for term, running total and gap, filled one row at a time.
- Provide the expansion of (n − 1)2 and 3(n − 1)2 on a card.
- Give the frame: “The terms are the gaps …, which rise by …, so d = ….”
Stretch
- Show from Sn = (d ÷ 2)n2 + (a − d ÷ 2)n that un = a + (n − 1)d.
- The totals of a sequence are 3, 10, 21, 36, … Find Sn and the nth term.
Students learning in English as an additional language
“Total” is the sum of the terms so far. “Running total” means a total that is updated after each new term. “Gap” is the difference between two neighbouring numbers in a list. “Recover” means get back.
| Word | Plain meaning | Example |
|---|---|---|
| running total | the sum of the terms so far, updated at each term | S5 |
| partial sum | another name for the sum of the first n terms | Sn |
| second difference | the difference between neighbouring gaps | the gaps rise by 2 |
| coefficient | the number multiplying a letter | the 3 in 3n |
| quadratic | a rule with an n squared term and no higher power | n2 + 4 |
Sentence starters for explaining aloud or in writing:
- The gaps between the totals are …, which are the ….
- The totals rise by … more each time, so d = ….
- To get the nth term I subtract … from ….
Two things not to say
“Sn is the nth term.” Say instead: Sn is the total of the first n terms.
“The coefficient of n squared is d.” Say instead: it is d divided by 2.
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.
- FindTerms 4, 6, 8, 10, … Write the first four totals.
4, 10, 18, 28. - FindSn = n2. Find the 5th term.
25 − 16 = 9.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA shop in Accra sells 12 phones on Monday, and 3 more each day than the day before. Write the total sold after n days.
Sn = 1.5n2 + 10.5n. - Earlier work (lesson 2)Evaluate the sum from k = 1 to 4 of 5k.
50.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan Sn = 2n2 + 5 be the total of an arithmetic sequence? Explain.
S1 = 7, S2 = 13, S3 = 23: gaps 6, 10, so terms 7, 6, 10 are not arithmetic. - Working backwardsAn arithmetic sequence has S2 = 8 and S4 = 24. Find a and d.
2a + d = 8 and 2a + 3d = 12 give d = 2 and a = 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 calculator is needed.