At a glance. One lesson of about 60 minutes. It assumes the nth term of an arithmetic sequence from Standard Unit 3 or Year 10. Students derive the sum formula by pairing, use it for a sum, a term count and an unknown difference, and solve a quadratic that arises from a sum.
Calculator: no calculator. No calculator: the numbers are chosen so that the arithmetic is mental, and the thinking is in setting up the formula.
What students will learn
Know
- Sn = n ÷ 2 × (2a + (n − 1)d) and Sn = n ÷ 2 × (first + last).
- The last of n terms is a + (n − 1)d.
- The number of terms from first to last is (last − first) ÷ d + 1.
Understand
- Why every column of the forward-and-reverse picture has the same total.
- Why the pairing proof halves, and why it works for an odd number of terms.
- A sum is a new sequence: the number of terms is the position in it.
Do
- Find a sum from first term, difference and n.
- Find the number of terms or the difference from a given sum.
- Form and solve an equation in n from a sum.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using the formula in a sum, a count and an unknown (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students state what stays the same in every column and why.
- Criterion C. Communicating: writing the forward-and-reverse argument in a few lines.
- Criterion D. Applying mathematics in real-life contexts: rows of seats and trees (hook and question 3) and a swimmer's training (question 8).
The one thing to do with the figure
Ask for the prediction first. Then set the difference to 0 and ask what the columns look like and why the total is still n times the first term. Move the difference up one step at a time and ask what happens to the left column and the right column.
Ask: “What is the same about every column, and what changes as you go along?” One number rises by d and its partner falls by d.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the hall of seats; collect predictions about the pairs. |
| 12 min | Figure, key idea and warning card; write the argument as a class. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose A, ask what the 8 columns add up to, and how many copies of the sum that is. |
| 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 | 31; 8; 150. |
|---|---|
| Hook | The last row has 12 + 14 × 4 = 68 seats, each pair adds to 80, and 15 ÷ 2 × 80 = 600 seats. |
| Worked examples | 860; 35 350; 10. |
| Faded example | 282. |
| 1. Using the sum formula | B. A forgets to halve; C uses only the last term; D is one column. |
| 2. Sum of the first terms | 270. 10th term 45; 5 × 54. |
| 3. Sum in context | 357. Last row 45; 7 × 51. |
| 4. Finding the difference | 4. 8 + 9d = 44. |
| 5. Counting terms then adding | 1650. 33 terms; 33 ÷ 2 × 100. |
| 6. Two terms give the sequence | 820. d = 4, a = 3. |
| 7. Two sums give the first term | 5. d = 3. |
| 8. Working backwards from a sum | D. A is the mean day; B is first + last; C treats the mean as first + last. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Gave twice the sum, used only the last term, or gave one column. | Ask: how many columns are there, and how many copies of the sum do they hold? |
| 540 or 290 in question 2 | Forgot to halve, or used a + nd as the last term. | Write out the first four terms and count the steps from the first to the 10th. |
| 378 or 45 in question 3 | Used a + nd as the last row, or gave the last row only. | Ask for the 2nd row: how many steps from the first row? Then ask what the question wants the total of. |
| 3.6 or 36 in question 4 | Divided by n instead of n − 1, or stopped at 9d. | Ask: in 10 terms, how many differences are there? |
| 1600 or 1617 in question 5 | Counted 32 terms, or used only the last term. | Test the rule on 2 + 5 + 8: (8 − 2) ÷ 3 + 1 = 3 terms. |
| 980 or 740 in question 6 | Took the 3rd term as the first, or went back three steps from it. | Ask: how many steps are there from the 1st term to the 3rd? |
| 10 or 3.5 in question 7 | Gave 2a, or used n differences instead of n − 1. | Write out S2 as 2 ÷ 2 × (2a + d) to see the single difference. |
| A, B or C in question 8 | Gave the mean day, gave first + last, or treated the mean as first + last. | Ask: why is 930 ÷ 30 the mean, and which two days is it halfway between? |
Questions worth asking
- Why must the total be halved? Listen for: the columns hold the sum twice, once forwards and once backwards.
- Would the formula work for a decreasing sequence? Listen for: yes, d is negative and the pairs still match.
- Which form of the formula is quicker when you know the last term? Listen for: n ÷ 2 times (first + last).
- How do you find n from a sum? Listen for: an equation in n, often a quadratic, with the negative root rejected.
Differentiation
A way in for anyone stuck
- Give the five terms 5, 9, 13, 17, 21 written twice, forwards and backwards, with the column totals shown.
- Provide a template: first term, difference, n, last term, then n ÷ 2 × (first + last).
- Give the frame: “The last term is … because there are … steps of …. Each column adds to ….”
Stretch
- Show that the sum of the first n odd numbers is n2, and say what shape this suggests.
- A sequence has first term 3 and its sum of 12 terms is 168. Find d, then the sum of the next 12 terms.
Students learning in English as an additional language
“Series” means the sum of the terms of a sequence. “Common difference” is the amount added each time. “Column” here means a pair of numbers one above the other. “Reversed” means written in the opposite order.
| Word | Plain meaning | Example |
|---|---|---|
| series | the sum of the terms of a sequence | 4 + 8 + 12 |
| partial sum | the sum of only the first n terms | S3 |
| first term | the term in position 1, written a | a = 6 |
| common difference | the amount added to get from each term to the next, written d | d = −2 |
| last term | the term in position n, equal to a + (n − 1)d | the 5th of 5 |
Sentence starters for explaining aloud or in writing:
- Every column adds to … because …
- There are … terms, so the sum is … ÷ 2 × …
- I counted the terms by …
Two things not to say
“Add the first and last and multiply by n.” Say instead: that is twice the sum, so halve it.
“The nth term is a + nd.” Say instead: the first term has no differences added.
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 1 + 2 + 3 + … + 50.
1275. - FindFind the sum of the first 6 terms of 10, 7, 4, …
15.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cyclist in Nairobi rides 8 km on the first day and 2 km further each day. How far has she ridden after 9 days?
9 ÷ 2 × (8 + 24) = 144 km. - Earlier work (Year 10)Find the 15th term of 5, 8, 11, …
47.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningThe sum of the first n terms of 5, 9, 13, … is 230. Find n.
n(2n + 3) = 230, n = 10. - Working backwardsAn arithmetic sequence has S1 = 7 and S2 = 19. Find the first three terms.
7, 12, 17.
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.