Math lesson plan · 12th grade (Year 13 in England) · 30 minutes
Continuous growth and the number e
Algebra
CCSS HSF.LE.A.4CCSS HSF.LE.B.5
Objective and success criteria
Students will: Continuous growth and the number e.
- 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: Exponential models; Logarithms
Leads to: Solve equations with e and natural logarithms
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 4 min
- Quick retrieval from earlier learning (Exponential models; Logarithms). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- A population of 1400 grows by 8% each year. Write the model P(t) = 1400(b)ᵗ and find P(3), to the nearest whole number. → 1764Prompt: The growth factor b is 1 + r = 1.08.
- Evaluate log_5(1/25). → -2Prompt: Ask: 5 to what power gives 1/25?
- An exponential function y = abˣ passes through (1, 12) and (2, 36). What is b? → 3Prompt: Divide the two y-values: the a cancels.
Hook · 3 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Continuous growth and the number e. Read the success criteria together.
I do: model it · 7 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: 4500 is invested at 4% interest compounded continuously, A = Pe^(rt). Find A after 13 years, to 2 decimal places.
- Step 1: rt = 0.52
- Step 2: A = 4500e^0.52 ≈ 7569.12
- Answer: 7569.12
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 6 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- 2000 is invested at 2% interest compounded continuously, A = Pe^(rt). Find A after 7 years, to 2 decimal places. → 2300.55Prompt: r = 0.02 as a decimal.
- How long does it take money to double at 5% compounded continuously? Give the answer in years to 1 decimal place. → 13.9Prompt: Solve 2 = e^(rt).
You do: independent practice · 7 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 6 questions are on the student sheet below, with answers in the key.
Exit ticket · 3 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- How long does it take money to double at 3% compounded continuously? Give the answer in years to 1 decimal place. → 23.1Prompt: Solve 2 = e^(rt).
- 1100 is invested at 2% interest compounded continuously, A = Pe^(rt). Find A after 12 years, to 2 decimal places. → 1398.37Prompt: r = 0.02 as a decimal.
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: Exponential models.
- 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
- Move on to: Solve equations with e and natural logarithms.
- 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