Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Find the sum to infinity of a geometric series
Algebra
CCSS HSA.SSE.B.4England: A-level Pure Mathematics
Objective and success criteria
Students will: Find the sum to infinity of a geometric series.
- I can answer questions on today's skill on my own, without help.
- I can explain each step and say why it works.
- I can check my answer and spot a mistake.
Before and after
Needs first: Sum a finite geometric series
Leads to: the next skill in this area.
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Sum a finite geometric series). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find the sum of the first 4 terms of 3 − 6 + 12 + … → -15Prompt: Sₙ = a(rⁿ − 1)/(r − 1).
- Ines saves $1 in week 1 and doubles the amount saved each week. How much has Ines saved in total after 10 weeks? Give the amount as a number. → 1023Prompt: The weekly amounts form a geometric sequence with r = 2.
- Find the sum of the first 7 terms of 384 + 192 + 96 + … → 762Prompt: Here r = 1/2. Use Sₙ = a(1 − rⁿ)/(1 − r).
Hook · 5 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Find the sum to infinity of a geometric series. Read the success criteria together.
I do: model it · 10 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Find the sum to infinity of the geometric series with first term 36 and common ratio 3/4. Give a fraction if needed.
- Step 1: 1 − (3/4) = 1/4
- Step 2: S∞ = 36 ÷ 1/4 = 144
- Answer: 144
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 10 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Find the sum to infinity of the geometric series with first term 16 and common ratio 1/4. Give a fraction if needed. → 64/3 (21 1/3)Prompt: S∞ = a/(1 − r), valid because |r| < 1.
- A ball is dropped from 12 m. Each bounce reaches 2/3 of the previous height. In theory it bounces for ever. What total distance does it travel? → 60Prompt: After the first drop, every bounce goes up and comes down again.
You do: independent practice · 10 min
- Students work on the student sheet. Circulate and check the first two answers of every student early.
- Students who finish: write a question of their own for a partner, with the answer.
The 8 questions are on the student sheet below, with answers in the key.
Exit ticket · 5 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Find the sum to infinity of the geometric series with first term 24 and common ratio 1/4. Give a fraction if needed. → 32Prompt: S∞ = a/(1 − r), valid because |r| < 1.
- A ball is dropped from 8 m. Each bounce reaches 3/4 of the previous height. In theory it bounces for ever. What total distance does it travel? → 56Prompt: After the first drop, every bounce goes up and comes down again.
Watch out for
- Doing an operation to one side of an equation but not the other.
- Reading 3x as 3 + x, or thinking x always stands for one particular number.
Support and stretch
Support
- Review first: Sum a finite geometric series.
- Use concrete materials: balance scales or a balance drawing.
- Give the first step of each question, or a partly worked example to finish.
- Pair with a confident partner for the guided practice.
Stretch
- Explain a common mistake in writing and how to fix it.
- Write a word problem that uses today's skill, then solve a partner's.
Homework and more practice
Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-12.
Free lesson plan from eurekami.com/lesson-plan-builder