Predict and test

Classroom resources

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Students will be able to

• I can predict how often an outcome should happen in a number of trials.
• I can run an experiment, record counts and work out relative frequencies.
• I can explain why results differ from predictions in a short experiment.

Timing

Predict and Run 10 min · Investigate 12 min · Modify 12 min · Practise and Make 11 min

Before class

Do the Run board first. Students need Sample spaces and probability.

The one idea: expected count = probability × trials; observed counts vary around it. The common mistake is expecting the predicted count exactly, and calling a fair device unfair after a short experiment; it is planted in Noah's reasoning.

If you have real dice or coins, run the 60-roll experiment physically first and pool the class's counts; the board then lets students run far more trials than a lesson allows.

Running the lesson

1. Predict: take a show of hands; ask students who chose 'exactly 6' what 6 is.
2. Run: collect everyone's count of sixes on the board as a dot plot; ask how many got exactly 10.
3. Investigate: link relative frequency to Fractions, decimals and percentages.
4. Modify: before the two-students question, ask what range of heads in 10 flips is normal.
5. Make: ask for the general rule: trials = expected count ÷ probability.

When students get stuck

Answers to the checks

1. how many sixes?: You roll a fair dice 60 times. About how many sixes do you expect?
   About 10: 60 × 1/6 = 10. The real count will usually be near 10, but not always exactly 10.
2. roll 60 times: Roll 60 times (press Run 10 six times). How many sixes did you get? Was your prediction right?
   
3. Expected, then observed: You flip a fair coin 50 times. How many heads do you expect?
   25
   50 × 1/2 = 25.
4. Find the mistake: Noah tests a dice. Click the line where his reasoning goes wrong.
   line 4
   Line 4: in 60 rolls the count of sixes often lands a few away from 10; 7 is common for a fair dice. The expected count is what happens on average, not every time. To test a dice, roll it many more times.
5. Relative frequency: In an experiment, a coin landed heads 13 times in 20 flips. What is the relative frequency of heads? (A decimal.)
   0.65
   13 ÷ 20 = 0.65, more than the probability 0.5: chance varies in 20 flips.
6. Flip 100: Flip the coin 100 times. Compare the number of heads with the expected 50.
   
   Usually between about 40 and 60: close to 50, rarely exactly.
7. An unequal spinner: A spinner is $\tfrac{3}{8}$ red. How many reds do you expect in 80 spins?
   30
   80 × 3/8 = 30.
8. Test the spinner: Spin this spinner (3 red, 5 blue sectors) at least 80 times. Compare the reds with the expected count.
   
   Expected 30 reds in 80 spins; most runs land within about 8 of 30.
9. Two results: Two students flip a coin 10 times each. Ana gets 7 heads; Ben gets 5. Is Ana's coin unfair?
   Not necessarily: 10 flips is too few to tell
   With 10 flips, results vary a lot. To judge a coin, flip it many more times.
10. Not a whole number: Edge case: you roll a dice 50 times. What is the expected number of sixes?
    About 8.3 (50 × 1/6): you can expect a value you can never actually get
    50 × 1/6 = 8.33…: an average over many experiments, not a count from one.
11. Expected count: A bag has 2 red and 8 blue counters. You take one and put it back, 40 times. How many reds do you expect?
    8
    P(red) = 2/10 = 1/5, and 40 × 1/5 = 8.
12. Relative frequency: A spinner landed on green 18 times in 60 spins. What is the relative frequency of green? (A decimal.)
    0.3
    18 ÷ 60 = 0.3.
13. Suspicious: Which result would make you most suspicious of a coin?
    900 heads in 1000 flips
    Over 1000 flips you expect about 500; 900 is far too many for chance.
14. Review: probability: Review: roll a fair dice. What is $P(\text{a number less than } 3)$? Simplest form.
    1/3
    {1, 2}: 2/6 = 1/3.
15. plan an experiment: You want to expect 25 sixes from a fair dice. How many times should you roll it?
    150
    Expected sixes = rolls × 1/6, so rolls = 25 × 6 = 150.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Flip 100

Open task: any program that does what the step asks.

one change at a time › An unequal spinner

Open task: any program that does what the step asks.

one change at a time › Test the spinner

Open task: any program that does what the step asks.

one change at a time › Two results

Open task: any program that does what the step asks.

one change at a time › Not a whole number

Open task: any program that does what the step asks.

Stretch and support

Stretch: Each of 30 students rolls a dice 60 times. Predict roughly how many of them will get exactly 10 sixes. Then simulate it with the board (about 1 in 7).

Support: Use the board's Expected count column to check each prediction.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 7 student in Australia. Help me understand the ideas in ONE lesson only, from the lesson "Predict and test".

The ideas:
- Predicting relative frequencies: I can Students use theoretical probability to predict how often an outcome will occur in a given number of trials.
- Running experiments and recording results: I can Students conduct repeated chance experiments and record observed frequencies systematically.

What understanding looks like:
- Correctly predicts the relative frequency for a given number of trials.
- Understands that predictions are expected values, not certainties.
- Can adjust predictions for non-fair devices.
- Records observed frequencies in an organised frequency table.
- Calculates relative frequency from observed results.
- Compares observed relative frequency to theoretical probability.

Mistakes students often make here:
- A student expects the predicted count exactly (Noah's mistake).
- A student gives the probability as the expected count.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- If I seem to be missing the idea that comes before this one (Assigning probabilities to outcomes), check it quickly, then come back.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Predict and test".

Exit ticket

A spinner has P(blue) = 0.4. How many blues do you expect in 50 spins? You get 23. Is the spinner unfair?

20 expected. 23 is close to 20, so there is no reason to think it is unfair.

What each step is for

• how many sixes?: Committing to the number of sixes in 60 rolls brings out the expected count, and whether they expect it to happen exactly.
• roll 60 times: Running 60 rolls and comparing the count of sixes with 10 shows that the expected count is a good guide, not a promise.
• expected and observed: Expected count = probability × trials; the observed count varies around it, and the relative frequency is count ÷ trials. Noah's mistake treats a small difference in a short experiment as proof the dice is unfair.
• one change at a time: One change at a time: a coin on the board, an unequal spinner, more trials, comparing two results, then the edge case of an expected count that is not a whole number.
• expected counts: Expected counts and relative frequencies in new contexts, mixed; the expect-exactly slip planted again; the review keeps sample spaces fresh.
• plan an experiment: Choosing the number of trials for a target expected count reverses the lesson: trials = expected count ÷ probability.

Where it leads

Next: Many trials (AC9M7P02 node 2): what happens to the relative frequency over hundreds and thousands of trials.

Where it leads: simulating compound events (Year 8) and statistical significance much later.