Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Sigma notation and infinite geometric series
Algebra
CCSS HSA.SSE.B.4
Objective and success criteria
Students will: Sigma notation and infinite 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: Arithmetic and geometric sequences and 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 (Arithmetic and geometric sequences and series). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find the sum of the first 36 terms of the arithmetic series −10 + (−8) + (−6) + … → 900Prompt: Sₙ = n/2 × (2a + (n − 1)d).
- A geometric sequence starts 3, −9, 27, … What is the 7th term? → 2187Prompt: Find the common ratio r by dividing consecutive terms.
- Find the sum of the first 5 terms of the geometric series 1 + 3 + 9 + … → 121Prompt: Sₙ = a(rⁿ − 1)/(r − 1).
Hook · 5 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Sigma notation and infinite 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 5 and common ratio 2/5.
- Step 1: S∞ = 5 ÷ 3/5
- Step 2: = 25/3
- Answer: 25/3 (8 1/3)
- 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 1 and common ratio −1/3. → 3/4Prompt: S∞ = a / (1 − r), valid because |r| < 1.
- Evaluate Σ (from k = 1 to 5) of (2k + 5). → 55Prompt: This is an arithmetic series: find the first and last terms.
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.
- Evaluate Σ (from k = 1 to 9) of (2k + 1). → 99Prompt: This is an arithmetic series: find the first and last terms.
- Find the sum to infinity of the geometric series with first term 18 and common ratio −1/4. → 72/5 (14 2/5)Prompt: S∞ = a / (1 − r), valid because |r| < 1.
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: Arithmetic and geometric sequences and 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