At a glance. One lesson of about 60 minutes. It assumes laws of indices from Year 9 and the nth term from lesson 1. Students find the ratio, write a times r to the power n minus 1, find terms and positions, and find a ratio or first term from two terms.
Calculator: no calculator. The numbers are chosen so that every power is small enough to calculate by hand. The next lesson uses a scientific calculator for percentage growth.
What students will learn
Know
- A geometric sequence has a constant ratio r.
- The nth term is a × rn − 1.
- A negative ratio alternates the sign; a ratio between 0 and 1 shrinks the terms.
Understand
- The exponent n − 1 counts multiplications, so the first term has exponent 0.
- Growth by multiplying outpaces growth by adding.
- Reading a geometric sequence backwards inverts its ratio.
Do
- Find a ratio by dividing consecutive terms.
- Write and use the nth term.
- Find a ratio or a first term from two non-consecutive terms using a root.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding terms, ratios and positions (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students record the sixth term for several a and r and describe how the bars grow.
- Criterion C. Communicating: the rule with its power, and a sentence explaining why the exponent is n − 1.
- Criterion D. Applying mathematics in real-life contexts: cultures that double, a ratio read as a growth factor (hook and extra questions).
The one thing to do with the figure
Ask for the prediction first. Then keep a = 1 and move r from 2 to 4, asking what happens to the sixth bar. Press the button and ask what the ratio is now.
Ask: “How many times has the ratio been used by the sixth bar?” Five, which is n − 1. This is the idea behind the exponent.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the doubling culture; collect estimates and reasons. |
| 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 B, ask what the difference of two terms tells you in a geometric sequence. |
| 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 | 32; 5n − 2; 33 = 27. |
|---|---|
| Hook | After 12 hours the count is 5 × 212 = 20 480, in the tens of thousands. This is the 13th term of 5, 10, 20, …, because the start is term 1: a good moment for the n − 1 idea. |
| Worked examples | 7 × 2n − 1 and 896; a = 5; −2.5. |
| Faded example | −486. |
| 1. Finding the first term | A. B takes the difference; C uses the difference to go back; D divides by the position. |
| 2. Finding a term | 324. 4 × 34. |
| 3. Negative ratio | −2. −10 ÷ 5. |
| 4. Ratio from two terms | 3. r3 = 27. |
| 5. Which term has a value | 6. 35 = 243. |
| 6. First term from two terms | 4. r = 3, then 36 ÷ 9. |
| 7. Counting terms below a limit | 9. Term 9 is 768; term 10 is 1536. |
| 8. Writing the nth term | C. A is an arithmetic rule through the first two terms; B has the wrong power; D swaps the first term and the ratio. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| B, C or D in question 1 | Took the difference, used it to go back, or divided by the position. | Ask: what do I multiply by to get from the second term to the third? |
| 972 or 60 in question 2 | Used the power n, or multiplied by the position. | List the terms in a row and count the arrows. |
| 2 or −15 in question 3 | Ignored the sign, or subtracted. | Divide, and say the sign out loud. |
| 27 or 9 in question 4 | Gave r cubed, or divided by 3. | Ask: how many multiplications are there between term 1 and term 4? |
| 5 or 243 in question 5 | Gave the power, or the quotient. | Write 243 as 3 to the power something, then add 1. |
| 12 or 27 in question 6 | Divided by r once, or gave r cubed. | Write term 3 = a × r2. |
| 8 or 10 in question 7 | Gave the power, or included the first term above 1000. | Write terms 8, 9, 10 explicitly. |
| A, B or D in question 8 | Fitted an arithmetic rule to two terms, used the wrong power, or swapped the numbers. | Test each option at n = 1, 2 and 3: A fits the first two terms and fails at n = 3 (27, not 75). |
Questions worth asking
- Why is the power n − 1 and not n? Listen for: term 1 has had no multiplications yet.
- How can you tell a geometric sequence from an arithmetic one? Listen for: divide neighbours; they give the same quotient.
- What happens to the terms if the ratio is between 0 and 1? They shrink towards 0.
- Can a geometric sequence have a ratio of 0? Usually not: the definition needs r ≠ 0, because the ratio of the 3rd term to the 2nd would be 0 ÷ 0.
Differentiation
A way in for anyone stuck
- Provide a row of boxes with arrows labelled ×3 between them and ask students to count the arrows.
- Use a table with the columns position, power of the ratio and term.
- Give the frame: “The ratio is … because … ÷ … = …. So the nth term is …”
Stretch
- A geometric sequence has first term 3 and 3rd term 48. Find both possible ratios and the 4th term in each case.
- Show that the terms of a geometric sequence with ratio r are in arithmetic progression only when r = 1 (or the first term is 0).
Students learning in English as an additional language
“Ratio” here is the number you multiply by each time. “Geometric” is the name for a sequence that multiplies. “Power” means a small raised number, such as the 4 in 3 to the power 4. “Double” means multiply by 2. “Alternate” means change from one to the other and back.
| Word | Plain meaning | Example |
|---|---|---|
| geometric sequence | a sequence where each term is the last multiplied by the same number | 9, 18, 36, 72 |
| common ratio | the number each term is multiplied by | the 3 in 7, 21, 63 |
| power | how many times a number is multiplied by itself | 63 = 216 |
| alternate | to change sign each time | +4, −4, +4 |
| nth term | a rule for the term in position n | 8 × 5n − 1 |
Sentence starters for explaining aloud or in writing:
- The ratio is … because …
- Between term … and term … there are … multiplications, so …
- The signs alternate because the ratio is …
Two things not to say
“The power is the position.” Say instead: the power counts the multiplications, which is one fewer.
“Divide to find the ratio from any two terms.” Say instead: divide, then take the root that matches the number of steps.
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 the common ratio of 4, 20, 100, 500.
5. - FindFind the 4th term of 2, 8, 32, …
128.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA viral video in Manila is seen by 10 people in hour 1, and the number of new viewers triples every hour. How many new viewers are there in hour 5?
10 × 34 = 810. - Earlier work (lesson 3)Find the nth term of 3, 8, 15, 24.
n² + 2n.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs 2, 6, 18, 54 the same sequence as 2 × 3n divided by 3? Explain.
Yes: 2 × 3n ÷ 3 = 2 × 3n − 1. - Working backwardsA geometric sequence has first term 1 and its 5th term is 81. Find all possible ratios.
r4 = 81, so r = 3 or r = −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? Fold a sheet of paper in half several times and count the layers.