Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Find normal probabilities with z-scores and technology
Statistics and probability
CCSS HSS.ID.A.4England: A-level Statistics
Objective and success criteria
Students will: Find normal probabilities with z-scores and technology.
- 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: Combine regions of a normal curve with the 68–95–99.7 rule; Calculate and use z-scores
Leads to: the next skill in this area.
Materials: Squared paper for charts, Class survey data, Sticky notes for a live chart, Mini whiteboards and pens, Printed student sheet (below).
Warm-up · 5 min
- Quick retrieval from earlier learning (Combine regions of a normal curve with the 68–95–99.7 rule; Calculate and use z-scores). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- A test has mean 41 and standard deviation 8. Find the z-score of a mark of 25. → -2Prompt: z = (x − mean) ÷ standard deviation.
- A quantity is normally distributed with mean 55 and standard deviation 7. What percentage of values lie between 55 and 69? → 47.5Prompt: Convert each end to 'number of standard deviations from the mean'.
- A factory fills 200 bottles. Volumes are normally distributed with mean 251 ml and standard deviation 14 ml. About how many bottles hold less than 223 ml? → 5Prompt: That value is 2 standard deviations from the mean.
Hook · 5 min
- Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
- Share the objective: Find normal probabilities with z-scores and technology. 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: X is normally distributed with mean 153 and standard deviation 20. Find P(X < 134.8) to 3 decimal places.
- Step 1: z = (134.8 − 153)/20 ≈ −0.91
- Step 2: P ≈ 0.181
- Answer: 0.181
- 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.
- X is normally distributed with mean 56 and standard deviation 10. Find P(X < 74.2) to 3 decimal places. → 0.966Prompt: Standardise: z = (x − μ)/σ.
- Battery life is normally distributed with mean 125 minutes and standard deviation 29 minutes. Find the probability that a battery lasts between 69 and 145 minutes, to 3 decimal places. → 0.728Prompt: P(a < X < b) = P(X < b) − P(X < a).
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.
- Battery life is normally distributed with mean 367 minutes and standard deviation 18 minutes. Find the probability that a battery lasts between 354 and 373 minutes, to 3 decimal places. → 0.395Prompt: P(a < X < b) = P(X < b) − P(X < a).
- X is normally distributed with mean 80 and standard deviation 12. Find P(X < 100.9) to 3 decimal places. → 0.959Prompt: Standardise: z = (x − μ)/σ.
Watch out for
- Forgetting to put data in order before finding the median.
- Reading a bar's height wrongly when the scale goes up in 2s or 5s.
Support and stretch
Support
- Review first: Combine regions of a normal curve with the 68–95–99.7 rule.
- Use concrete materials: squared paper for charts.
- 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