130i - Consolidation and Course Review

Lesson 130 — Consolidation: Chance Experiments and Simulations; Course Review

Strand: Probability | Descriptor: AC9M7P02 | Duration: 45 minutes

Final lesson of the Year 7 course. The first half consolidates the probability strand; the second reviews the year.

Learning Intentions

• To consolidate chance experiments, simulations and the comparison of predictions with results.
• To review the connections across the year’s mathematics.

Success Criteria

I can:

1. Conduct, simulate and interpret a chance experiment end to end.
2. Explain differences between predicted and observed results correctly.
3. Identify connections between topics studied this year.
4. Name the mathematical habits that apply across every strand.

Warmup

(5 minutes — probability synthesis, individually then check)

Answer from memory:

1. What are the two ends of the probability scale?
2. Does two outcomes mean a fifty-fifty chance?
3. Expected frequency ?
4. Relative frequency ?
5. After five heads, what is ?
6. What does a bigger number of trials improve?

Answers: and ; no — only if the outcomes are equally likely; trials; occurrences trials; — trials are independent; the reliability of the estimate, never the probability itself.

Activities

Activity 1 — Probability Consolidation (16 min)

Pairs. One task, running the whole P02 cycle.

The three-outcome spinner.

A spinner has equal sectors: red, blue, gold.

1. State the theoretical probability of each outcome and check they total .
2. Predict the results of spins.
3. A class ran spins and recorded: red , blue , gold . Find each relative frequency (3 d.p.).
4. Compare each with prediction, giving absolute and proportional differences.
5. Judge: is there evidence the spinner is not as described? Explain.
6. Describe how you would simulate this spinner in a spreadsheet.
7. A student says “gold came up times instead of , so gold is due next spin.” Refute this.

Socratic scaffolding:

Prompt	Purpose
Q1: probabilities?	, , ; total ✓
Q2: predicted counts?	, , .
Q4: the differences?	Red (); blue (); gold ().
Q5: judgement?	All three are small deviations over trials — entirely ordinary variation. No evidence of a problem.
Q6: the simulation?	=RANDBETWEEN(1,8), counting – as red, – as blue, as gold.
Q7: the refutation?	The gambler’s fallacy — the spinner has no memory; is on every spin regardless of history.

(Answers: 1. , , ; 2. , , ; 3. , , ; 4. as scaffolded; 5. no evidence; 6. as scaffolded; 7. as scaffolded.)

Quick extension for early finishers: the gold sector pays 10$1$2240$48030 \times 10 + 90 \times 1 = $390$90$.)*

Activity 2 — The Year in Connections (16 min)

Pairs, then whole class. The course review.

Below are eight pairs of topics from this year. For each, write one sentence explaining how they connect.

	Topic A	Topic B
1	Prime factorisation (L5)	Simplifying fractions (L55)
2	Solving equations (L32)	Finding a missing angle (L47)
3	Ratios (L70)	Probability (L122)
4	Tables of values (L88)	Straight lines on a plane (L89)
5	The mean (L108)	Multi-variable formulas (L91)
6	Reading graphs (L83)	Misleading data displays (L112)
7	Algebraic proof (L30)	Testing algorithms (L105)
8	Modelling assumptions (L74)	Statistical limitations (L118)

Expected connections:

1. Prime factorisation finds the HCF, which simplifies a fraction in one step.
2. Both find an unknown by writing what is known as an equation and undoing operations.
3. A ratio gives probabilities and — part-to-part becomes part-to-whole.
4. A rule’s table of values plots as a straight line; the constant difference is the steepness.
5. Both require recovering a total before finding a missing value: mean count total.
6. Both depend on checking the axes before trusting the visual impression.
7. Both distinguish testing examples from proving: examples can only ever disprove.
8. Both require stating what your answer depends on, so a reader can judge how far to trust it.

Then, the bigger question — whole class:

Across every strand this year, what habits kept appearing?

Collect on the board. The expected list:

Habit	Where it appeared
Estimate before calculating	L60 onwards — every calculation lesson
Check your answer	Substitution (L34), forward checks (L25), verification everywhere
State your assumptions	Modelling (L74–77), financial models (L81), statistics (L118)
Examples disprove; they don’t prove	Algebra (L30), shapes (L44), angles (L46), algorithms (L105)
Read the axes / read the question	Graphs (L83–86), displays (L112), rounding contexts (L61)
More data fixes chance, not bias	Statistics (L120), probability (L123–128)
Say which measure, and give the spread	Central tendency (L109), reporting (L118)

The closing point for the year: the calculations get harder in Year 8, but these habits do not change. They are what makes an answer trustworthy rather than merely produced.

Activity 3 — Course Reflection (6 min)

Individually, collected.

Answer briefly:

1. Which topic did you find most interesting this year, and why?
2. Which idea surprised you most?
3. Which topic would you most like more practice with?
4. Name one thing you can do now that you could not do in February.
5. Where outside school have you noticed mathematics from this course?

Candidates for Q2, worth mentioning if nobody raises them: the rope around the Earth (L95); that reflecting twice gives a rotation (L99); that a 25% rise then a 25% fall does not return the price (L79); that median and range cannot pin down a distribution’s shape (L113); that the gambler’s fallacy is wrong (L129).

Checks for Understanding

(2 minutes — one question, collected)

Reasoning. A friend in another school says: “I rolled a die times and got four sixes, so my die must be loaded.” Write a reply of two or three sentences using what you have learned about chance.

Expected answer: Four sixes in twelve rolls gives a relative frequency of against the expected — noticeably high, but twelve rolls is far too few to judge. Chance produces runs like this regularly with perfectly fair dice. To find out, roll it several hundred times and compare the relative frequency then; a persistent gap over many trials would be convincing, but a short run proves nothing.

Common Misconceptions

Misconception	How to pre-empt it
The gambler’s fallacy.	Activity 1’s Q7 and the exit question.
Judging bias from a short run.	The exit question makes it the final thing assessed.
Believing topics are unconnected.	The eight connection pairs.
Thinking mathematics is only calculation.	The habits table — every entry is a judgement, not an operation.
Treating an unremarkable result as a failure.	Activity 1’s Q5: “no evidence of a problem” is a complete answer.
Expecting simulation to verify a probability.	Carried from L127; Q6 asks only how it would be set up.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A spinner has sectors: red, blue, gold. In spins, how many golds would you expect?

Answer

.

E2 (AMC Junior style). A die rolled times gives sixes. Find both differences and judge the result.

Answer

Predicted ; absolute ; proportional . Extremely close over a large number of trials — exactly what a fair die produces.

E3 (Challenge). A game costs 312$201$52360$ plays, find the operator’s expected profit, and the price that would make the game exactly fair.

Answer

Income 108030 \times 20 + 60 \times 5 = 600 + 300 = $900$180$900360$2.50$.

E4 (Challenge). Explain, using ideas from at least three different topics this year, why the claim “our new product is better” is difficult to assess.

Answer

Percentages (L79): ” better” gives no base — better than what, measured how? Statistics (L118): no sample size, no spread, and possibly a biased sample. Graphs (L112): any accompanying chart may use a truncated axis to exaggerate a small difference. The claim is unassessable until the measure, the comparison and the evidence are stated.

E5 (Challenge). Choose any two topics from this year that seem unrelated, and find a genuine connection between them.

Answer

Student’s own. Strong examples: prime factorisation (L5) and square roots (L9), since halving even exponents gives the root; transformations (L98) and congruence (L102), since all six preserve lengths and angles; ratios (L71) and probability (L122), since both are part-whole reasoning; algorithms (L104) and shape classification (L42), since a decision tree encodes the nested hierarchy.

Homework

(Final homework of the course.)

1. A spinner has sectors: red, blue, green. (a) State each probability. (b) Predict the results of spins. (c) Observed: red , blue , green . Find each relative frequency and both differences. (d) Judge whether the spinner matches its description.
2. Describe how you would simulate the Q1 spinner in a spreadsheet, giving the formula and the counting rules.
3. A coin tossed times gives heads. (a) Both differences. (b) Your judgement, with a reason.
4. A student claims a die is loaded after getting five sixes in rolls. Write a two-sentence reply.
5. Explain, in your own words, why a coin has no memory.
6. Name the three possible explanations for a result differing from a prediction, with an example of each.
7. Write one sentence connecting each pair: (a) prime factorisation and simplifying fractions (b) solving equations and finding unknown angles (c) ratios and probability.
8. Reasoning. Name three mathematical habits from this year that apply across more than one strand, and give an example of each.
9. Reasoning. Explain why “more trials” improves an estimate but cannot correct a biased method.
10. Challenge. Design a fair carnival game: state the equipment, the price to play, the prizes and their probabilities, and show that the expected payout equals the expected income.

Answers: Q1 — (a) , , (b) , , (c) red , , ; blue , , ; green , , (d) all three sit very close to prediction — the spinner matches its description. Q2 — =RANDBETWEEN(1,12): – red, – blue, – green. Q3 — (a) predicted ; ; (b) a small proportional gap over many trials — consistent with a fair coin, though a persistent gap of this size across several such experiments would be worth watching. Q4 — five sixes in rolls is a relative frequency of , double the expected, but rolls is far too few to judge; several hundred more rolls would be needed before drawing any conclusion. Q5 — the coin is an object with no mechanism for storing or acting on its history; each toss faces identical physical conditions and identical probabilities. Q6 — ordinary variation (a coin giving heads in ); bias (a die giving sixes over rolls); a wrong model (assuming a spinner’s sectors are equal when they are not). Q10 — student’s own; the check is that price plays sum of (probability prize plays).

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Course Complete

This is the final lesson of the Year 7 sequence. Across lessons the course has covered all content descriptors of the Australian Curriculum Version 9 for Year 7 Mathematics:

Strand	Descriptors	Lessons
Number	N01–N09	1–14, 55–73, 78–82
Algebra	A01–A06	23–36, 83–94
Measurement	M01–M06	15–22, 46–54, 74–77, 95–97
Space	SP01–SP04	37–45, 98–106
Statistics	ST01–ST03	107–120
Probability	P01–P02	121–130