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Year 7 / Mathematics / Term 1 / Weeks 03 04

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Mathematics · Days 11–20Year 7 Maths · T1 W3–4 · Lesson sequence

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Teacher copy · prompts and answer keys

Teacher copy: This page may include teaching prompts or answer keys. Answer keys in this public library can be viewed by anyone. Give learners a clean prompt, use checks as formative evidence, and change a case locally when prior access matters.

Each day is 25 minutes: launch 2 + model 5 + guide 6 + one practice choice 6 + exit 4 + note 2. On Days 15/20, the ten minutes for practice/exit become a fresh check; those days' three routes are later practice. Use clean learner prompts, original print/text aids, and a private staff key. Record exact mathematics, representation, access route and any content hint separately. A read-aloud can preserve mathematical reasoning without establishing independent reading.

Week 3 · Choose and check operations with positive rational numbers

The first fortnight introduced equivalent rational forms, rounding, fraction addition/subtraction, decimal money arithmetic and simple percentages. This week revisits and extends these, adding multiplication/division. The physical whole or unit must be named. Every numeric situation below is invented for teaching; no purchase or personal finance disclosure is required.

Day 11 · Add fractions without changing the whole

Codes: AC9M7N04, AC9M7N06. Goal: explain 1/2+3/8 and 5/6−1/3 with common units. Prepare: operation and check mat, matching fraction strips.

  1. Launch · 2 min. Ask if half a fictional page plus three eighths of the same page is above or below one whole.
  2. Model · 5 min. Rename 1/2=4/8; then 4/8+3/8=7/8, below one. Show the same whole strip, not two differently sized pages.
  3. Guide · 6 min. Rename 1/3=2/6; 5/6−2/6=3/6=1/2. Learners explain why adding/subtracting denominators would change unit size. Compare result with zero and one.
  4. Choice · 6 min. D11-A/B/C: original mosaic, time-segment or error-repair route. Every route needs an equivalent-name step and plausibility check.
  5. Exit · 4 min. Solve 3/4−1/8=6/8−1/8=5/8, then say why it is less than 3/4. This revisits a familiar operation to separate method from new numbers.
  6. Note · 2 min. Record whether the learner preserved one whole and justified denominator choice; recheck a fresh pair if not.

Optional home/extension: Draw a same-length strip and solve 1/2+1/4=3/4. Extension: two strategies for 5/6−1/3.

Day 12 · Decimal operations and a reasonable estimate

Codes: AC9M7N05, AC9M7N06. Goal: add and subtract decimals by place value, then check scale. Prepare: operation mat, decimal place cards.

  1. Launch · 2 min. Two fictional design lengths are 4.75 m and 2.60 m. Expect a total near 7–8 m, not 70 m.
  2. Model · 5 min. Align decimal places: 4.75+2.60=7.35 m. Round each input to the nearest whole metre (4.75→5, 2.60→3) for a loose estimate 5+3=8 m; this detects a decimal-shift error but does not replace the exact sum.
  3. Guide · 6 min. From a fictional 10.00 m roll, subtract 7.35 m: 10.00−7.35=2.65 m. Check 7.35+2.65=10.00. Ask why metres cannot be silently changed to centimetres.
  4. Choice · 6 min. D12-A/B/C uses a fictional art strip, data-storage readout or sports path; each shows written/inverse check and unit.
  5. Exit · 4 min. Repair 4.75+2.60=6.135: show correct 7.35 and one magnitude reason. Do not credit a close estimate as the exact answer.
  6. Note · 2 min. Save original decimal alignment and inverse. If error is in place value, use a different tenths/hundredths example next.

Optional home/extension: Use paper decimal columns; extension: express 2.65 m as 265 cm with the unit conversion stated.

Day 13 · Multiply fractions as part of a part

Codes: AC9M7N04, AC9M7N06. Goal: explain positive fraction multiplication without a memorised rule alone. Prepare: same-whole rectangle/grid or raised partitions.

  1. Launch · 2 min. In a fictional graphic, what should 3/4 of 2/3 be compared with 2/3: larger or smaller? Smaller.
  2. Model · 5 min. Partition one whole into thirds one way and quarters the other. The overlap for 3/4×2/3 is 6/12=1/2. Explain the one-whole area model and then numerator/denominator product.
  3. Guide · 6 min. Use 2/5×3/4=6/20=3/10. Confirm 3/10<3/4 and 3/10<2/5 because both factors are below 1.
  4. Choice · 6 min. D13-A/B/C: page panel, signal interval or rectangular grid. Learners represent an overlap and exact simplified result.
  5. Exit · 4 min. Is 1/2×3/4 equal to 3/8 or 4/6? 3/8; explain with an eight-cell area or a half of three quarters.
  6. Note · 2 min. Record whether learner understood “of” as a second partition of the same whole; recheck a new area model if only a rule was repeated.

Optional home/extension: Draw a 4-by-3 blank rectangle. Extension: explain why 3/4×2/3=2/3×3/4 by rotated partitions.

Day 14 · Division and percentage are operation choices

Codes: AC9M7N05, AC9M7N06. Goal: identify how many 0.6-unit lengths fit in 2.4 units and find a familiar percentage. Prepare: number-line/decimal cards, operation mat.

  1. Launch · 2 min. A fictional path is 2.4 units long; markers are 0.6 units apart. Predict roughly four intervals.
  2. Model · 5 min. 2.4÷0.6=4; multiply both quantities by 10 to use 24÷6, then check 4×0.6=2.4. Keep the quotient unit as intervals, not 4 units of length.
  3. Guide · 6 min. Another fictional 1.5-unit strip in quarter-unit pieces: 1.5÷0.25=6 pieces; check six quarters =1.5. Then 12.5%=1/8, so 12.5% of 48 equal panels is 6. These are two operation types, with the whole named in each.
  4. Choice · 6 min. D14-A/B/C gives a repeated-decimal grouping, percentage partition or operation-choice explanation; require the unit and an inverse/benchmark check.
  5. Exit · 4 min. “Is 2.4÷0.6 equal to 0.4?” No; four groups of 0.6 fill 2.4. A quotient smaller than one cannot count four visible groups.
  6. Note · 2 min. Separate grouping-division reasoning from percentage-of-whole reasoning; plan the missing strand as a next move.

Optional home/extension: Draw six equal quarter lengths to cover 1.5; extension: compare 1.5÷0.25 with 1.5×0.25 and explain why they differ.

Day 15 · Fresh rational-operation check

Codes: AC9M7N05, AC9M7N06. Goal: transfer to unseen 2/3+1/4, 7.2÷0.3 and 15% of 80. Prepare: closed learner check, separate staff key. Do not show check figures in routes before this point.

  1. Launch · 2 min. Explain that a new problem shows which strategy to teach next, not a rank of students.
  2. Model · 5 min. Rehearse different figures: 1/2+1/4=3/4 and 2.4÷0.6=4; put them away.
  3. Guide · 6 min. Find 25% of 40 =10 through one quarter. Ask what 100% names; do not show the held-out percent.
  4. Independent check · 6 min. Give unseen Items 1–2. Read wording neutrally or supply a blank tactile representation without choosing steps.
  5. Check exit · 4 min. Item 3 asks for a percentage of a named whole and a check. Preserve first responses; D15 routes are later practice.
  6. Note · 2 min. Record equivalence, place value/grouping, percent whole and estimation separately. If access/time prevented a response, mark not observed.

Optional home/extension: Later practice only; do not coach the held-out page as homework.

Week 4 · Prime products, powers and square roots

A prime factorisation uses only primes. Exponent notation compresses repeated multiplication: 2³=2×2×2=8, not 2×3. Distinct factor trees for one number must yield the same prime product. A positive perfect square has an integer square root; √144=12 because 12²=144, with the principal square root positive in this pack. The code links below are partial encounters, not broad mastery claims.

Day 16 · Break a number into prime leaves

Code: AC9M7N02. Goal: factorise 60 and 84 with prime leaves and exponents. Prepare: factor-tree mat or text/tactile card route.

  1. Launch · 2 min. Ask why 6×10=60 is not yet a prime factorisation: 6 and 10 are composite.
  2. Model · 5 min. Split 60 as 6×10=(2×3)×(2×5)=2×2×3×5=2²×3×5. Check 4×3×5=60.
  3. Guide · 6 min. Factor 84 via 4×21=(2×2)×(3×7)=2²×3×7; verify 4×3×7=84. Other valid tree shapes are welcome.
  4. Choice · 6 min. D16-A/B/C: tree, prime-leaf cards or repair of an unfinished product. Each ends with primes only and a multiplication check.
  5. Exit · 4 min. Is 2²×3×5 equal to 60? Yes. Is 4×3×5 already a prime product? No, because 4 is composite.
  6. Note · 2 min. Record where factorisation stopped and whether exponent notation matches leaf count.

Optional home/extension: Draw a 60 tree on paper. Extension: start 60 with 3×20 and compare final leaves.

Day 17 · Read powers as repeated prime factors

Code: AC9M7N02. Goal: represent 72 accurately as 2³×3² and distinguish base/exponent. Prepare: factor cards and tree.

  1. Launch · 2 min. Ask what the small 3 in 2³ counts: three factors of 2.
  2. Model · 5 min. Factor 72=8×9=(2×2×2)×(3×3)=2³×3². Check 8×9=72; 2³ is 8, not 6.
  3. Guide · 6 min. Start another tree 72=12×6=(2×2×3)×(2×3); regroup to 2³×3². A different branch order changes no prime counts.
  4. Choice · 6 min. D17-A/B/C: tree comparison, spoken exponent description or error diagnosis with a new composite.
  5. Exit · 4 min. Evaluate 2³×3² as 8×9=72, and expand 3² as 3×3.
  6. Note · 2 min. If learner reads 2³ as 2×3, place three 2-cards and revisit with 3² as a contrasting card.

Optional home/extension: Make base/exponent cards; extension: compare 2²×3²=36 with 2³×3²=72 and explain the extra factor of 2.

Day 18 · Prime factors can simplify a comparison

Codes: AC9M7N02, AC9M7N04. Goal: factorise 90 and 150, then explain a shared factor and equivalent fraction. Prepare: tree mat, blank ratio/fraction line.

  1. Launch · 2 min. Two invented collections have 90 and 150 paper cards. Ask what common equal group size might simplify the comparison.
  2. Model · 5 min. 90=2×3²×5; 150=2×3×5². Both include 2×3×5=30, so 90/150=(90÷30)/(150÷30)=3/5. Verify 3/5=0.6; do not infer that the collections have the same total.
  3. Guide · 6 min. Draw two factor trees, then circle one matching 2, 3 and 5 in each. Check 90=2×9×5; 150=2×3×25. Discuss why a leftover 3 versus 5 matters.
  4. Choice · 6 min. D18-A/B/C: compare original cards, factor-leaf match or critique a false reduction. Each keeps both totals and explains the division by 30.
  5. Exit · 4 min. Complete 90/150=__/__ in simplest form: 3/5. Name 30 as a shared factor and check 150×3/5=90.
  6. Note · 2 min. Record whether cancellation paired equal prime factors, not arbitrary digits.

Optional home/extension: Factor 90 and 150 on paper. Extension: find another common factor and show it reaches the same 3/5 after further reduction.

Day 19 · Square products and their positive roots

Codes: AC9M7N01, AC9M7N02. Goal: connect 144=12², √144=12 and prime powers. Prepare: square-root link mat or tactile grid.

  1. Launch · 2 min. A fictional 12-by-12 pixel panel has how many cells? 144.
  2. Model · 5 min. 12²=12×12=144, so √144=12. Factor 12=2²×3; squaring gives 144=2⁴×3². Check 16×9=144. The root asks for the nonnegative side length, not ±12 in this setting.
  3. Guide · 6 min. 100=10²; 10=2×5, so 100=2²×5², and √100=10. Compare 81=9², 9=3², so 81=3⁴ and root9.
  4. Choice · 6 min. D19-A/B/C: square grid, exponent pairs or square-root error repair. Require both multiplication and root explanation.
  5. Exit · 4 min. “What is √81, and why?” 9 because 9×9=81; 3⁴=81 does not mean the root is 3.
  6. Note · 2 min. Check side/area units and distinguish evaluating a power from taking a square root; recheck 64 after instruction if needed.

Optional home/extension: Sketch 10×10 and 12×12 arrays. Extension: explain why 2⁴×3² can be grouped into two equal 2²×3 products.

Day 20 · Fresh prime-power and square-root check

Codes: AC9M7N01, AC9M7N02. Goal: transfer factorisation and square-root reasoning to unseen 108, 196 and 75. Prepare: closed learner check, staff key, blank factor tree/root mat.

  1. Launch · 2 min. Remind learners to multiply prime leaves back and test a root by squaring it.
  2. Model · 5 min. Revisit 60=2²×3×5 and 100=10²; put those models away. Keep check values unseen.
  3. Guide · 6 min. Factor 36 as 2²×3², check 6²=36, then put away. Do not display fresh numbers.
  4. Independent check · 6 min. Give unseen Items 1–2. A blank tree or large/tactile cards may be supplied; do not choose factor leaves.
  5. Check exit · 4 min. Item 3 asks for error detection in a proposed prime product. Preserve first work; D20 routes are later.
  6. Note · 2 min. Record prime leaves, exponent counts, inverse square check and error explanation separately. A missing access format means not yet observed.

Optional home/extension: D20 extras after the check use different values; do not coach this held-out sheet at home.

The optional extra cards add short practice, not an entire mathematics day. Original SubjectNest lessons © NeuroForgeIO Pty Ltd 2026, CC BY 4.0. Exact ACARA rows and taught limits appear in the crosswalk.

Source: https://neuroforge.io/subjectnest/library/resources/year-7/mathematics/term-1/weeks-03-04/LESSONS · Original material © NeuroForgeIO Pty Ltd 2026 · CC BY 4.0: https://creativecommons.org/licenses/by/4.0/ · Credit the source and note changes. External sources keep their own terms.