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Year 8 / Term 1 / Weeks 01 02 / Mathematics

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Ten mathematics lessons, 25 minutes eachYear 8 Maths · T1 W1–2 · Lesson sequence

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Use fictional number cards, graph, sample grid and keys. Every lesson has 2 + 4 + 5 + 7 + 4 + 3 = 25 minutes. Maths may be shown with an equation, table, tactile tokens, graph, calculator/spreadsheet or spoken argument; record the route. Selected partial codes: AC9M8A01, AC9M8A02, AC9M8A03, AC9M8ST01, AC9M8ST02, AC9M8ST03, AC9M8ST04.

Day 1 — Name the variable before calculating

Success: express a fixed-plus-variable cost and state a sensible domain. (AC9M8A01, AC9M8A03)

  1. Read · 2 min. Show Plan A's $18 joining cost and $7 per repair. Ask which cost changes when repairs change.
  2. Model · 4 min. Let n be repairs. Write A(n)=18+7n. The 18 is fixed, the 7 multiplies the number of repairs; n=0,1,2…, not negative or half a repair.
  3. Guided build · 5 min. Learners form B(n)=10n from Plan B's no-joining-cost card. Substitute n=2: A=$32, B=$20.
  4. Independent turn · 7 min. For n=3, learners compute A=$39, B=$30 and write what each term means. Paper, tokens, speech or keyboard accepted.
  5. Error check · 4 min. Repair 18n+7 by testing zero repairs: the incorrect expression yields $7, whereas the joining fee alone would be $18.
  6. Exit · 3 min. “What does 7 represent in A?” Key: $7 for each repair, not $7 fixed joining cost.

Day 2 — A table is a model, not a quotation

Success: calculate a table and identify what the model omits. (AC9M8A03)

  1. Recall · 2 min. Ask for A and B at zero repairs: $18 and $0.
  2. Model · 4 min. Fill n=2 and n=4: A $32/$46, B $20/$40; use substitution so each entry can be checked.
  3. Guided finish · 5 min. Class completes n=6: both $60. Ask why equality at one count does not make all counts equal.
  4. Independent table · 7 min. Learners fill n=8: A $74, B $80, then compare at n=0,2,4,6,8 in a sentence.
  5. Assumption scan · 4 min. Circle omitted parts/quality/travel/eligibility. A correct model answer is not a real cheapest purchase claim.
  6. Exit · 3 min. “At eight repairs, which model total is lower and by how much?” Key: A by $6.

Day 3 — Plot the possible counts

Success: connect a table to a coordinate graph while respecting whole-number domain. (AC9M8A02, AC9M8A03)

  1. Orient · 2 min. On print axes, x is number of repairs and y is total dollars; ask what (2,32) means.
  2. Model · 4 min. Plot A's (0,18), (2,32), (4,46) and B's (0,0), (2,20), (4,40).
  3. Guided extend · 5 min. Plot both at n=6 and n=8. Locate meeting (6,60).
  4. Independent statement · 7 min. Learners write one sentence about slope per repair and one about intercept. A increases $7 per repair from $18; B increases $10 from $0.
  5. Domain check · 4 min. If lines are drawn to show trend, circle integer x-values as actual cases. Ask why n=4.5 is not a possible repair count here.
  6. Exit · 3 min. “What does (6,60) mean?” Key: six repairs cost $60 under either supplied model.

Day 4 — Solve, then substitute

Success: solve the equality and verify it in both formulas. (AC9M8A01, AC9M8A02)

  1. Question · 2 min. Ask which n makes totals equal, without pointing to graph first.
  2. Model · 4 min. Write 18+7n=10n, subtract 7n from both sides to get 18=3n.
  3. Guided finish · 5 min. Divide by 3: n=6. Substitute: A 18+42=60, B 60.
  4. Independent compare · 7 min. Learners choose n=5 and n=7 to show direction changes: at 5, A=$53/B=$50; at 7, A=$67/B=$70.
  5. Method compare · 4 min. Table, graph and algebra agree on six. Ask which one best exposes the exact equality and why.
  6. Exit · 3 min. “Is A lower at five?” Key: no, A=$53, B=$50.

Day 5 — Fresh linear check A

Success: model and compare a new fixed-plus-variable pair independently. Prepare art-studio card. (AC9M8A01, AC9M8A02, AC9M8A03)

  1. Set · 2 min. State that fictional visits are whole, non-negative numbers; no real fee is being quoted.
  2. Clarify · 4 min. Read only the new conditions: C $12 plus $5 per visit; D $8 per visit. Do not model the equation.
  3. Plan · 5 min. Learners choose table, graph or equation, with enough room to show work.
  4. Independent response · 7 min. Find formulas, equality point and which plan is lower at three and six visits.
  5. Self-check · 4 min. Substitute the meeting value into both; check a count on each side. Record any revision separately.
  6. Exit · 3 min. Submit exact work. Key: v=4, $32 each; at 3 D lower, at 6 C lower.

Day 6 — Who is the population?

Success: distinguish a full group from a sample and note practical/ethical limits. (AC9M8ST01, AC9M8ST04)

  1. Frame · 2 min. Read fictional question: “What travel modes occur in the imagined 100-student population?” No real classmates are surveyed.
  2. Model · 4 min. Define population 100, census all 100, sample fewer. A census gives full counts in this model but can be costly or intrusive in a real school.
  3. Guided classify · 5 min. Learners label A/B as constructed simple random samples of 20 and C as convenience sample of 20 by racks.
  4. Independent plan · 7 min. Write a way to select 20 from numbered fictional cards giving each one a known equal chance, without replacing selected cards. State one privacy question a real survey would need to resolve.
  5. Peer audit · 4 min. Partner checks whether selection location favours a mode and whether all 100 could be reached by the proposed method.
  6. Exit · 3 min. “Does 20 responses automatically mean a fair sample?” Key: no; how people were selected matters.

Day 7 — Same size, different results

Success: calculate proportions and describe variation between two same-size model random samples. (AC9M8ST02, AC9M8ST03)

  1. Recall · 2 min. Both A and B have 20 responses. Ask whether identical size requires identical counts.
  2. Model · 4 min. Bicycle share A is 3/20=15%; B is 4/20=20%.
  3. Guided table · 5 min. Compute bus shares A 7/20=35%, B 5/20=25%; ask which category changed more in percentage points.
  4. Independent comparison · 7 min. Learners choose walk/roll or other, compute both shares, and say why two model samples can differ even when each uses a fair chance process.
  5. Visual check · 4 min. Show 20-cell grids or tactile tokens for each sample; prevent a 5-percentage-point difference being mistaken for five people.
  6. Exit · 3 min. “How many percentage points apart are bicycle shares A and B?” Key: 5 points, one person out of 20.

Day 8 — The location changed the answer

Success: critique a convenience sample without calling its arithmetic wrong. (AC9M8ST01, AC9M8ST02, AC9M8ST04)

  1. See · 2 min. Display rack C: 16 bicycles out of 20; ask for its sample percentage.
  2. Model · 4 min. 16/20=80%, which is correct for C. Because selection was beside racks, it may overrepresent cyclists; it cannot estimate all students well.
  3. Guided compare · 5 min. Contrast C's 80%, random model A's 15%, B's 20% and known fictional population's 20%. Ask which source answers each narrower question.
  4. Independent critique · 7 min. Learners write a two-line correction to “80% of all students cycle”: valid sample result plus uncertainty about population.
  5. Fairness talk · 4 min. Ask why a shelter decision based on the rack sample could overlook bus users. Do not ask anyone to disclose their own travel.
  6. Exit · 3 min. “What is wrong: 16/20=80% or calling that 80% of all students?” Key: the population inference.

Day 9 — Report uncertainty, not a fake certainty

Success: make an inference limited by how the sample was chosen. (AC9M8ST03, AC9M8ST04)

  1. Recall · 2 min. Ask why A and B's bicycle proportions differ by one person even though both have 20.
  2. Model · 4 min. Say: “In these two simulated random samples, bicycle estimates were 15% and 20%; another fair sample could differ too.” Show the optional repeated simulation: across 1000 seeds, the average error was 8.95 percentage points for samples of 10 and 3.92 for samples of 40. Do not claim every larger sample must be closer.
  3. Guided claim test · 5 min. Sort “sample A had 3 cyclists” as direct model fact; “all students cycle at 15%” as overreach; “larger fair samples often vary less, but do not guarantee exactness” as a qualified pattern.
  4. Independent brief · 7 min. Learners draft three sentences for a fictional principal: two source figures, one method limitation and one ethical next action. A source label must accompany each number.
  5. Peer check · 4 min. Partner asks how a simple random sample would be drawn and whether a census is necessary or appropriate.
  6. Exit · 3 min. “Can two equal-size random samples give different percentages?” Key: yes; this model has 15% and 20% bicycle.

Day 10 — Fresh sample check B

Success: independently analyse a new sampling claim and propose a fairer response. Prepare arts-club card. (AC9M8ST01, AC9M8ST02, AC9M8ST03, AC9M8ST04)

  1. Set · 2 min. Clarify fictional status and that no learner interest is being asked.
  2. Read · 4 min. Give the 100-person model population and three 20-person samples. Do not name the likely biased sample.
  3. Plan · 5 min. Learners decide how to compare percentages, selection method and the claim's reach.
  4. Independent work · 7 min. Compute shares, identify sampling concern and write a bounded population statement.
  5. Check · 4 min. Multiply or divide to confirm each fraction; label percentages as model sample or known model population.
  6. Exit · 3 min. “Why not use the music-club meeting to speak for all 100?” Key: people there are more likely to be interested. Use rubric/key.

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