Math lesson plan · 11th grade (Year 12 in England) · 45 minutes
Calculate and use z-scores
Statistics and probability
CCSS HSS.ID.A.4England: A-level Statistics
Objective and success criteria
Students will: Calculate and use z-scores.
- 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: Spread: standard deviation
Leads to: Combine regions of a normal curve with the 68–95–99.7 rule; Find normal probabilities with z-scores and technology
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 (Spread: standard deviation). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Find the population standard deviation of: 9, 15 → 3Prompt: Find the mean, then each value's distance from it.
- Set A: 27, 28, 28, 29, 28. Set B: 20, 25, 28, 31, 36. Which set has the larger standard deviation? (A / B / They are equal) → BPrompt: Standard deviation measures how spread out the data are from the mean.
- Find the population standard deviation of: 3, 9 → 3Prompt: Find the mean, then each value's distance from it.
Hook · 5 min
- Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
- Share the objective: Calculate and use z-scores. 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: A test has mean 66 and standard deviation 9. Find the z-score of a mark of 93.
- Step 1: (93 − 66) ÷ 9 = 3
- Answer: 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.
- A test has mean 50 and standard deviation 8. Find the z-score of a mark of 34. → -2Prompt: z = (x − mean) ÷ standard deviation.
- Heights have mean 159 cm and standard deviation 6 cm. What height has a z-score of −2? → 147Prompt: x = mean + z × standard deviation.
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.
- A test has mean 73 and standard deviation 4. Find the z-score of a mark of 83. → 2.5Prompt: z = (x − mean) ÷ standard deviation.
- Leo scored 85 in maths (mean 67, sd 8) and 62 in science (mean 56, sd 13). Relative to the class, which was the better result? (Maths / Science) → MathsPrompt: Raw marks can't be compared directly when the spreads differ.
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: Spread: standard deviation.
- 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
- Move on to: Combine regions of a normal curve with the 68–95–99.7 rule; Find normal probabilities with z-scores and technology.
- 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-11.
Free lesson plan from eurekami.com/lesson-plan-builder