At a glance. One lesson of about 60 minutes. Expected frequency as probability times trials, simulating an event with random digits, and comparing a simulation with theory.
Calculator: no calculator. The numbers are chosen so that each expected count is a whole number by simple fractions or percentages, so the work is in interpretation.
What students will learn
Know
- Expected frequency = probability × number of trials.
- A simulation uses random numbers in place of a real event.
- Digits for “yes” must be in the same proportion as the probability.
Understand
- Expected frequency is an average, not a promise.
- A simulation is an estimate, and longer runs are more reliable.
- Theory and experiment can be compared, and neither is exact in every sense.
Do
- Find expected frequencies.
- Predict from sample data to a larger group.
- Describe a simulation using random digits.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: expected frequency calculations (questions 2 to 5).
- Criterion B. Investigating patterns: with the figure, compare relative frequency at 10, 50, 100 and 200 families and describe how it moves relative to the theory.
- Criterion C. Communicating: describing how digits stand for outcomes, and the reasoning in the exit ticket.
- Criterion D. Applying mathematics in real-life contexts: surveys, seeds and games, and deciding what a simulation can and cannot tell you.
The one thing to do with the figure
Have students write a fraction for the hook before they open anything, then run the slider to 200 without pressing the button. Ask what they now think the probability is, and how sure they are.
Then press “Show the theory” and ask them to explain 7/8 from a list of the 8 outcomes. This pulls together Lessons 2 to 5: the sample space, the complement and the experiment.
Then pick up the gap left in Lesson 5: there the run ended at 0.15 against 1/6 ≈ 0.167, and here 0.845 against 0.875. Both are the same story, a long run close to the theory and not equal to it.
A possible order
| 5 min | Warm-up, then answers. |
|---|---|
| 10 min | Hook: write a fraction; share and compare. |
| 15 min | The figure; the theory reveal; expected frequency. |
| 10 min | Worked examples; the bakery in pairs. |
| 5 min | Question 1, the check. |
| 10 min | Your turn, Levels 1 to 3. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 0.4; 30; 7/20. |
|---|---|
| Worked examples | Die: 50 sixes. Free throws: digits 0 to 6 (7 digits). |
| Faded example | 30 numbers. |
| 1. Expected frequency | B, 20. D multiplies by 6 instead of dividing. |
| 2. Expected count | 180 seeds. |
| 3. Spinner | 60 blues. |
| 4. From a survey | 360 people. |
| 5. Above expectation | 9 more heads. |
| 6. Why simulate | B. A treats an estimate as exact. |
| 7. Two faces win | 30 wins. |
| 8. Working back to trials | 200 trials. |
| 9. Choosing digits | A. B is 50%; D leaves 0 out, so yes becomes 4/9. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| 20 in question 2 | Gave the number expected not to happen. | Ask which group the question wants and underline it. |
| 75 or 30 in question 3 | Divided by the wrong number. | Draw the spinner and shade the blue sections. |
| 1140 in question 4 | Gave the complement. | Read the question again, ending at “cycle”. |
| 109 or 91 in question 5 | Gave the actual count, or the number of tails. | Ask for the expected number first and write it down. |
| 15 or 45 in question 7 | Counted only one winning face, or treated 2 of 6 as 1/2. | Write the winning faces: 5 and 6. |
Questions worth asking
- If you expect 50, will you get 50? Listen for “probably near it, but not exactly”.
- Why use ten digits for a 30% event? Listen for “30% of ten is three”.
- How could we make the simulation more reliable? More trials; the figure shows it settling.
Differentiation
A way in for anyone stuck
- Give a random-digit table and a half-built tally for the families.
- Use percentages that give whole numbers: 10%, 20%, 25%, 50%.
- Let students roll a die in pairs and compare with 1/6 of their number of rolls.
Stretch
- Simulate a family of 3 using a coin: how would you decide a child's sex, and how many coin tosses are needed per family? (One toss per child, so 3 per family.)
- A bag has 3 red and 2 blue counters and one is drawn 100 times with replacement. How many reds do you expect, and how would you simulate one draw with the digits 0 to 9? (60; digits 0 to 5 for red, 6 to 9 for blue.)
Students learning in English as an additional language
Pair each word with an action: a rolling hand for “trial”, a held-up hand for “expect”.
| Word | Plain meaning | Example |
|---|---|---|
| expected frequency | the number of times to expect an event, probability times trials | 30 sixes in 180 rolls |
| simulation | using random numbers to act out a real event many times | digits standing for children |
| trial | one go at the experiment or simulation | one simulated family |
| random | each outcome equally likely, with no pattern | a digit chosen from 0 to 9 |
| estimate | a value found from an experiment, close to but not exactly the truth | 0.845 for 7/8 |
Sentence starters for explaining aloud or in writing:
- The expected number is … × … = …
- I let the digits … stand for … because …
- The simulation is close to / far from the theory, so I would …
Two things not to say
“You should get exactly that many.” Expected frequencies are averages, and some are not whole numbers.
“Simulation is just guessing.” With a sound method and enough trials it is a structured estimate.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ExpectedA spinner has a probability of 1/4 of landing on red. It is spun 80 times. How many reds would you expect?
20. - PercentA shop expects 5% of items to be faulty. In 300 items, how many?
15.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- SurveyIn a sample of 50 students, 12 walk to school. About how many of 400 walk?
96. - Earlier work (experiments)A coin lands heads 33 times in 60 tosses. What is the relative frequency?
0.55.
3Challenge
Reasoning, generalizing and working backwards.
- DigitsA player wins 35% of her games. How many of the two-digit numbers 00 to 99 should stand for a win?
35. - Working backA machine is expected to produce 12 faulty items, a 3% rate. How many items is that out of?
400.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The slider and the button work with the keyboard and by touch; “Show me the steps” gives four cases in words.
On paper: a printed table of random digits does the same job as the figure.