Math lesson plan · 12th grade (Year 13 in England) · 60 minutes
Solve equations with e and natural logarithms
Algebra
CCSS HSF.LE.A.4England: A-level Pure Mathematics
Objective and success criteria
Students will: Solve equations with e and natural logarithms.
- 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: Continuous growth and the number e; Solve exponential equations using logarithms
Leads to: Compare compounding periods and continuous interest
Materials: Balance scales or a balance drawing, Algebra tiles, Function machine cards, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 6 min
- Quick retrieval from earlier learning (Continuous growth and the number e; Solve exponential equations using logarithms). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- 500 is invested at 3% interest compounded continuously, A = Pe^(rt). Find A after 6 years, to 2 decimal places. → 598.61Prompt: r = 0.03 as a decimal.
- Solve 3^x = 32. Give x to 3 decimal places. → 3.155Prompt: Take logs of both sides.
- 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).
Hook · 6 min
- Show a "think of a number" puzzle and ask the class to work out the starting number.
- Share the objective: Solve equations with e and natural logarithms. Read the success criteria together.
I do: model it · 12 min
- Work through the example out loud, saying why you do each step, not just what you do.
- Example: Solve eˣ = 16. Give x to 3 decimal places.
- Step 1: x = ln 16
- Step 2: x ≈ 2.773
- Answer: 2.773
- Pause after each step and ask a student to say what comes next.
We do: guided practice · 12 min
- Do these together. Students try each step on whiteboards before you reveal it.
- Use the hints as prompts rather than giving the answer.
- Solve eˣ = 24. Give x to 3 decimal places. → 3.178Prompt: ln is the inverse of eˣ.
- Solve ln x = 2.2. Give x to 2 decimal places. → 9.03Prompt: Rewrite in exponential form.
You do: independent practice · 17 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 · 7 min
- Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
- Solve eˣ = 33. Give x to 3 decimal places. → 3.497Prompt: ln is the inverse of eˣ.
- A bacteria population is modelled by P = 1100e^(0.08t), with t in hours. How long until it reaches 1650? Give 1 decimal place. → 5.1Prompt: Divide by 1100 first.
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: Continuous growth and the number e.
- 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: Compare compounding periods and continuous interest.
- 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