LESSON PLAN

Proportional Reasoning — Percents, Rates, and Scale

Apothem Team

Grade 9 · Number

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

The recipe-photo challenge: a pancake recipe photo shows quantities for 4; the caption says "feeds 10 if you scale it." One minute: flour goes from $1\frac{1}{2}$ cups to…? ( $\times 2.5 \to 3\frac{3}{4}$ .) Then the twist question: "the pan is 20 cm across; for 2.5× the batter, how much wider must a single pancake pan be?" — NOT 2.5× (area scales with the square: $\sqrt{2.5} \approx 1.58$ × wider).

Grade 9 proportional reasoning is the capstone: rates, percents, scale — including where naive scaling BREAKS.

Explore

Three-station synthesis: (1) multi-step percent: a $1,250 laptop, 15% student discount, then 5% loyalty coupon on the result, then 12% tax — one multiplier chain ( $\times 0.85 \times 0.95 \times 1.12 = \times 0.9044$ ); (2) rate networks: currency triangulation (CAD→USD→EUR vs CAD→EUR direct — do the rates agree? arbitrage hides in the gap); (3) scale with dimensions: a 1:200 architectural model — lengths ÷200, areas ÷40,000, volumes ÷8,000,000, and the model's 2.5 cm door is a 5 m real door… no: 2.5 × 200 = 500 cm = 5 m ✓.

Station 3 cements the exponent ladder: $k$ , $k^2$ , $k^3$ for length, area, volume — the year's third meeting, now definitive.

Formalize

Formalize the dimensional scaling law and the multiplier chain:

$\text{length} \times k \;\Rightarrow\; \text{area} \times k^2, \;\text{volume} \times k^3 \qquad \text{net multiplier} = \prod (1 + r_i)$

The proportionality audit before any solve: is the relationship direct, inverse, or dimensional ( $k^2$ / $k^3$ )? Pancake-pan problems (area), model-weight problems (volume — weight scales with $k^3$ !), and paint problems (area) each punish the wrong exponent by orders of magnitude.

Practice

Practice: one multiplier chain with reversal (find pre-tax); one currency triangulation audit; two dimensional-scaling problems (model car mass; map area); one "which exponent?" diagnosis set of five one-liners.

Exit ticket: a 1:50 model boat is 40 cm long and weighs 200 g. Real length and (same materials) real weight? (20 m; $200 \times 50^3$ g $= 25{,}000$ kg.)

Exit ticket

Exit ticket: a 1:50 model boat is 40 cm long and weighs 200 g. Real length and (same materials) real weight? (20 m; $200 \times 50^3$ g $= 25{,}000$ kg.)

TIP The

k^3

weight scaling shocks every year (the model weighs grams; the boat weighs tonnes). Let the shock land — it's why ants can't be horse-sized and why giants can't exist, and nobody forgets the lesson afterward.

WORKED EXAMPLES

Example 1 — The multiplier chain with a reversal: the laptop receipt

The deal: sticker \$1,250 → 15% discount → 5% coupon → 12% tax. Final?

Step 1: Chain: $1250 \times 0.85 \times 0.95 \times 1.12$ .

Step 2: Merge: $0.85 \times 0.95 = 0.8075$ ; $\times 1.12 = 0.9044$ . Final: $1250 \times 0.9044 = \$1{,}130.50$ .

Step 3: Read the merged multiplier: after everything, you pay 90.44% of sticker — the two discounts barely outran the tax.

Step 4: Reversal drill: a friend's receipt shows $904.40 final under the same deal — their sticker? $904.40 \div 0.9044 = \$1{,}000$ ✓ division undoes the whole chain at once.

Step 5: Order-independence noted once more: the three multipliers commute — discount-then-tax equals tax-then-discount. What never works: adding the percents ( $-15 - 5 + 12 = -8\%$ → $1,150 ✗ off by $19.50).

Example 2 — The map, the hike, and the water: three exponents in one trip

The map: 1:25,000. The planned loop measures 34 cm on paper. The park on the map covers 80 cm².

Step 1: Trail length (k): $34 \times 25{,}000 = 850{,}000$ cm $= 8.5$ km.

Step 2: Park area ( $k^2$ ): $80 \times 25{,}000^2$ cm² $= 80 \times 6.25 \times 10^8 = 5 \times 10^{10}$ cm² $= 5$ km².

Step 3: Water planning (proportional, not dimensional): at 0.6 L per hour and 5 hours of hiking (8.5 km at ~1.7 km/h with breaks): 3 L.

Step 4: The audit habit: three quantities, three DIFFERENT scaling behaviours — length by $k$ , area by $k^2$ , consumption by time (a rate, not a scale at all). The skill isn't the arithmetic; it's assigning each quantity its law before touching the calculator.

Example 3 — Percent change chains in the news: the "crime up 40%" headline

The headline: "Bike thefts up 40% this year!" The fine print: last year 10 thefts; this year 14.

Step 1: Verify the percent: $\frac{14 - 10}{10} = 40\%$ ✓ — the math is honest.

Step 2: The base-size interrogation: 4 extra thefts in a city of 200,000 — the RATE per person moved from 0.005% to 0.007%. Both headlines ("up 40%!" and "theft touches 0.007% of residents") are true; they manufacture opposite feelings.

Step 3: The compounding follow-up: if thefts rose 40% again next year: $14 \times 1.4 = 19.6 \to$ ~20, and the two-year multiplier is $1.4^2 = 1.96$ — "nearly doubled in two years" would ALSO be true.

Step 4: The literacy takeaway: percent changes amplify small bases and compound multiplicatively — journalists reach for the version with drama, and readers who can recompute the base and the chain are unfoolable. This unit is that armour.

MATERIALS

•Model objects with scales

•Currency rate cards

•Multiplier chain sheets

•Exponent-diagnosis cards

•Practice set (PDF)

WATCH FOR

!Areas and volumes scaled by

k

. The exponent ladder, drilled with the model-boat shock.

!Sequential percents added (the Grade 8 ghost — final exorcism: multipliers or bust).

!Rates chained without unit-tracking, inverting one link silently. Write units; cancel like factors.