At a glance. One lesson of about 60 minutes. It assumes geometric sequences from Standard Unit 3 and the idea of a series from lesson 1. Students derive the sum formula by cancelling, and use it for a sum, a ratio and a number of terms.
Calculator: no calculator. No calculator: powers are small and the cancelling step is the learning.
What students will learn
Know
- Sn = a(rn − 1) ÷ (r − 1), r not 1.
- rS − S = arn − a.
- When r = 1, Sn = na.
Understand
- Why shifting the sum by one place makes the middle terms match.
- Why the power is n and not n − 1.
- Why the formula needs r not equal to 1.
Do
- Find a geometric sum from a, r and n.
- Find r from a sum, by factorizing.
- Find n by comparing a power of r with a number.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the formula and its parts (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students say which terms are left over and why.
- Criterion C. Communicating: writing the cancelling argument with a few terms and the general term.
- Criterion D. Applying mathematics in real-life contexts: spreading messages and rumours (hook, question 2).
The one thing to do with the figure
Ask for the prediction first. Then set the ratio to 2 and the number of terms to 3, and ask students which terms are left after the subtraction. Add a term and ask what changes.
Ask: “Why is there one box in each row without a partner?” The bottom row has been shifted along by one place.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the rumour; collect predictions as tens, hundreds or thousands. |
| 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 which term has the power 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 | 3; 128; 54. |
|---|---|
| Hook | New people each hour are 2, 4, 8, 16, 32, 64. With Anil that is 1 + 2 + 4 + 8 + 16 + 32 + 64 = 127 people, so about a hundred. |
| Worked examples | 728; 441; 126. |
| Faded example | 2343. |
| 1. Setting up the formula | C. A takes the last power as n; B leaves out the −1; D divides by r. |
| 2. Sum of a geometric sequence | 780. 5 × 156. |
| 3. Ratio less than one | 363. 243 + 81 + 27 + 9 + 3. |
| 4. Finding the number of terms | 8. 2n = 256. |
| 5. Finding the ratio | 3. (r + 4)(r − 3). |
| 6. Ratio from a sum comparison | 2. 1 + 2 + 4 + 8 = 15. |
| 7. First time a sum passes a target | 9. 29 = 512. |
| 8. The cancelling step | B. A uses the last term of S; C leaves out the first term; D subtracts the first term of 3S. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Took the last term's power as n, left out the −1, or divided by r. | Ask: which term is 5 × 30, so which term is 5 × 36? |
| 155 or 156 in question 2 | Stopped after three rounds, or left out the first term. | Write the four rounds as 5, 25, 125, 625. |
| 360 or 351 in question 3 | Added four terms, or three. | Count the terms on the page: the list has five. |
| 255 or 256 in question 4 | Gave a power's value rather than the exponent. | Ask: is the question asking for 2n or for n? |
| 12 or 4 in question 5 | Stopped at r squared plus r, or took the wrong factor. | Substitute r = 3 and r = 4 into 1 + r + r2. |
| 3.75 or 14 in question 6 | Divided by 4, or stopped before 1 was added. | Try r = 2 and add 1 + 2 + 4 + 8. |
| 8 or 335 in question 7 | Gave the last sum below the target, or the value the power must pass. | Tabulate n, 2n and 3(2n − 1). |
| A, C or D in question 8 | Used the last term of S, left out the first term, or subtracted the wrong leftover. | Write the first three and the last two terms of both rows, one place apart. |
Questions worth asking
- Why do we multiply by r and not by something else? Listen for: each term of S times r is the next term of S.
- Which two terms do not cancel? Listen for: the first term of S and the last term of rS.
- What if r = 1? Listen for: there is nothing to cancel against; the sum is na.
- How can you check the formula quickly? Listen for: n = 2 gives a + ar.
Differentiation
A way in for anyone stuck
- Give a table with columns for term number and term, filled with powers of the ratio.
- Provide the picture of S and rS as two printed rows with the matching terms underlined.
- Give the frame: “a = …, r = …, n = …, so S = …(…… − 1) ÷ (… − 1).”
Stretch
- Show that for r < 1 the formula can be written a(1 − rn) ÷ (1 − r), and say why it is easier to use.
- The sum of the first 2 terms of a geometric sequence is 12 and the sum of the first 3 is 28. Find r and a.
Students learning in English as an additional language
“Ratio” is the number you multiply by to get the next term. “Cancel” means two equal amounts that are added and subtracted disappear. “Shifted” means moved along by one place. “Geometric” here means a pattern that multiplies each time.
| Word | Plain meaning | Example |
|---|---|---|
| geometric series | the sum of the terms of a geometric sequence | 1 + 3 + 9 |
| common ratio | the number each term is multiplied by | r = 2 |
| cancel | two equal parts of opposite sign add to nothing | 7 − 7 |
| shifted | written one place along | under the next box |
| formula | a rule written with letters that gives a sum from a, r and n | Sn |
Sentence starters for explaining aloud or in writing:
- The first term is …, the ratio is … and there are … terms.
- When I subtract, everything cancels except … and ….
- The power is … because …
Two things not to say
“The power is n minus 1.” Say instead: n minus 1 is the nth term; the sum has n.
“Just plug in.” Say instead: list a, r and n first.
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 + 4 + 8 + 16 + 32.
63. - FindFind the sum of the first 4 terms of 5, 10, 20, …
75.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA bacteria colony in a lab in Chennai starts with 3 cells and triples every hour. How many cells have there been in total after the first 4 hours' counts, 3 + 9 + 27 + 81?
120. - Earlier work (lesson 3)Totals of an arithmetic sequence are 5, 12, 21. Find the third term.
9.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy can the formula not be used when r = 1, and what is the sum?
It divides by zero. The sum is na. - Working backwardsA geometric sequence has a = 2 and S3 = 62. Find r.
1 + r + r2 = 31, so r = 5 or r = −6 (2 − 12 + 72 = 62 too).
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.