LESSON PLAN

Percentages — Discount, Tax, and Tip

Apothem Team

Grade 8 · 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 restaurant bill lands on the doc camera: subtotal \$54.80, and three tasks race — 15% tip mentally, the 12% tax exactly, and the grand total. Fastest table explains their routes (10% + 5% for the tip: 5.48 + 2.74 = 8.22).

Grade 8 percents go professional: multi-step compositions, percent CHANGE, and reverse problems, with the multiplier method carrying everything.

Explore

Percent-change lab: prices that moved — a game $60 → $75 (increase $\frac{15}{60} = 25\%$ ), a phone $800 → $600 (decrease 25%). The base rule discovered by contradiction: is +25% then −25% a wash? $60 \to 75 \to 56.25$ — NO, because the bases differ. Percent change always measures against the ORIGINAL.

Then reverse-engineering stations: "a jacket costs $68 AFTER 15% off — original?" ( $68 \div 0.85 = 80$ ); "a town grew 8% to 27,000 — before?" ( $27{,}000 \div 1.08 = 25{,}000$ ). Both run on divide-by-the-multiplier — never subtract-the-percent.

Formalize

Formalize percent change and the reversal principle:

$\%\Delta = \frac{\text{new} - \text{original}}{\text{original}} \times 100 \qquad \text{original} = \frac{\text{final}}{\text{multiplier}}$

The asymmetry theorem of daily life: a p% rise followed by a p% fall lands BELOW start (multipliers: $(1+p)(1-p) = 1 - p^2 < 1$ ). Stocks, discounts, populations — the down-leg acts on a bigger base. Grade 8 students who own this out-reason most adults.

Practice

Practice: two percent-change computations (one increase, one decrease); two reversals; one rise-then-fall audit; one full receipt with discount, tax, and tip composed as multipliers.

Exit ticket: "A stock rose 20% Monday and fell 20% Tuesday. Net percent change?" ( $1.2 \times 0.8 = 0.96$ → down 4%.)

Exit ticket

Practice: two percent-change computations (one increase, one decrease); two reversals; one rise-then-fall audit; one full receipt with discount, tax, and tip composed as multipliers.

Exit ticket: "A stock rose 20% Monday and fell 20% Tuesday. Net percent change?" ( $1.2 \times 0.8 = 0.96$ → down 4%.)

TIP Percent change errors are base errors 95% of the time. The reflex question — "percent OF WHAT?" — is still the whole game, now with before/after wrinkles.

WORKED EXAMPLES

Example 1 — Percent change with a verdict: the sneaker market

The data: sneakers released at \$140, resold at \$189.

Step 1: Change: $189 - 140 = 49$ .

Step 2: Against the ORIGINAL: $\frac{49}{140} = 0.35 \to 35\%$ increase.

Step 3: The wrong-base version, computed to bury it: $\frac{49}{189} \approx 26\%$ — a different, wrong number; note it UNDERSTATES the markup, which is why sellers might prefer quoting it.

Step 4: Reverse check: $140 \times 1.35 = 189$ ✓.

Step 5: The reporting habit: percent changes are meaningless without their base named. "Up 35% (from release price)" is a fact; "up 35%" is a vibe.

Example 2 — The reverse problem: what was the pre-sale price?

The tag: "NOW \$102 — 40% OFF!"

Step 1: Model forward: original × 0.60 = 102.

Step 2: Reverse by division: original $= 102 \div 0.60 = \$170$ .

Step 3: Verify: $170 \times 0.6 = 102$ ✓.

Step 4: The tempting wrong route, dismantled: "add 40% back": $102 \times 1.40 = 142.80$ — off by $27.20, because the 40% was taken from 170, not from 102. Undo a multiplication with a DIVISION; percents never subtract-to-undo.

Step 5: The consumer angle: reverse computations expose whether "was \$170" claims are honest. Some stores inflate the "original" — and now the class can audit them from the discount math alone.

Example 3 — Compound change: the two-year enrolment story

The school: enrolment rose 15% one year, then fell 10% the next. Started at 800.

Step 1: Chain multipliers: $800 \times 1.15 \times 0.90$ .

Step 2: Compute: $800 \times 1.035 = 828$ . (The merged multiplier $1.15 \times 0.90 = 1.035$ — net +3.5%.)

Step 3: The trap answer: $+15 - 10 = +5\%$ → 840. Wrong by 12 students — the −10% acted on the LARGER 920, clawing back more than a tenth of the original.

Step 4: The general law, stated for keeps: sequential percent changes MULTIPLY as $(1 + r_1)(1 + r_2)\cdots$ ; they never add. Adding is only approximately right when the percents are tiny — and knowing when an approximation is safe is itself a Grade 12 topic being seeded.

Step 5: One more turn: what single-year change matches the two years? +3.5% — the merged multiplier IS the answer to that question.

MATERIALS

•Receipt sets

•Price-change cards

•Multiplier chain sheets

•Practice set (PDF)

WATCH FOR

!Percent change computed against the NEW value. The original is the base — always.

!+p% then −p% believed to cancel. The

(1-p^2)

identity, demonstrated with $100.

!Reversals attempted by subtracting the percent from the final. Divide by the multiplier.