Math lesson plan · 12th grade (Year 13 in England) · 45 minutes
Expected value
Statistics and probability
CCSS HSS.MD.A.2CCSS HSS.MD.B.5
Objective and success criteria
Students will: Expected value.
- 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: Experimental probability and expected results; Permutations and combinations
Leads to: Use expected value to judge whether a game is fair
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 (Experimental probability and expected results; Permutations and combinations). Students answer on mini whiteboards.
- Show answers together and note who needs support in the main task.
- Mei flips a drawing pin 60 times. It lands point up 13 times. What is the relative frequency of landing point up? Give a fraction in simplest form. → 13/60Prompt: Relative frequency = number of times it happened ÷ number of trials.
- In how many ways can 1st, 2nd place be awarded among 7 runners? → 42Prompt: Order matters: use permutations nPr.
- The probability of an event is 1/4. The experiment is done 36 times. How many times would you expect the event to happen? → 9Prompt: Expected number = probability × number of trials.
Hook · 5 min
- Take a quick class survey (favorite fruit, travel to school) and ask what the results tell us.
- Share the objective: Expected value. 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 game pays 9 points with probability 0.5, 8 points with probability 0.3, 5 points with probability 0.2. What is the expected number of points?
- Step 1: 9 × 0.5 + 8 × 0.3 + 5 × 0.2 = 7.9
- Answer: 7.9
- 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 game pays 8 points with probability 0.1, 1 points with probability 0.4, 10 points with probability 0.5. What is the expected number of points? → 6.2Prompt: E(X) = Σ x × P(x).
- A raffle ticket costs 5 points. There is a 0.25 chance of winning 10 points. What is the expected gain per ticket (prize value minus cost)? It may be negative. → -2.5Prompt: Expected winnings = probability × prize.
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 raffle ticket costs 1 points. There is a 0.05 chance of winning 11 points. What is the expected gain per ticket (prize value minus cost)? It may be negative. → -0.45Prompt: Expected winnings = probability × prize.
- A game pays 5 points with probability 0.1, 7 points with probability 0.4, 3 points with probability 0.5. What is the expected number of points? → 4.8Prompt: E(X) = Σ x × P(x).
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: Experimental probability and expected results.
- 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: Use expected value to judge whether a game is fair.
- 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