LESSON PLAN

Fractions, Decimals, and Percents

Apothem Team

Grade 7 · Number

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Flash a quiz-show scoreboard: "Contestant A answered $\frac{17}{20}$ correctly, B scored 84%, C scored 0.86. Who won?" Let pairs rank them (thirty seconds) and defend.

The rankings require a common language — and that's the unit: fractions, decimals, and percents are three dialects for one quantity, and fluency means converting without friction. (A: 85%, C: 86%, B: 84% → C wins.)

Explore

Conversion triangle workshop: each pair gets a set of quantities, each given in one dialect only ( $\frac{3}{8}$ , 0.4, 65%, $\frac{7}{4}$ , 1.2, 5%), and must complete all three forms for each, recording the ROUTE they used (fraction→decimal by division; decimal→percent by ×100; percent→fraction by /100 then simplify).

Then the repeating-decimal wrinkle: convert $\frac{1}{3}$ and $\frac{5}{6}$ by division and meet $0.333\ldots$ and $0.8333\ldots$ — some fractions refuse to terminate. Which ones? Pairs test denominators 2, 3, 4, 5, 6, 8, 10, 12 and conjecture: denominators built only from 2s and 5s terminate (they divide powers of ten); anything else repeats.

Formalize

Formalize the conversion routes around the triangle:

$\frac{3}{8} \xrightarrow{3 \div 8} 0.375 \xrightarrow{\times 100} 37.5\% \qquad 45\% = \frac{45}{100} = \frac{9}{20}$

The benchmark family every student should own cold: $\frac{1}{2} = 0.5 = 50\%$ , $\frac{1}{4} = 0.25 = 25\%$ , $\frac{1}{5} = 0.2 = 20\%$ , $\frac{1}{3} = 0.\overline{3} = 33\frac{1}{3}\%$ , $\frac{1}{8} = 0.125 = 12.5\%$ . Unknown quantities get located relative to these before any computation.

Practice

Practice: complete a three-dialect table for eight quantities; order a mixed-dialect set on one number line; two application items (a test score comparison and a "which discount is deeper" judgment); one repeating-decimal conversion with the division shown.

Exit ticket: order $\frac{5}{8}$ , 0.6, 63% from least to greatest, showing the common form used. ( $0.6 < \frac{5}{8} = 0.625 < 0.63$ .)

Exit ticket

Exit ticket: order $\frac{5}{8}$ , 0.6, 63% from least to greatest, showing the common form used. ( $0.6 < \frac{5}{8} = 0.625 < 0.63$ .)

TIP Percent means "per hundred" — restore the phrase whenever confusion appears. "84 per hundred" converts itself; the % symbol alone invites ritual.

WORKED EXAMPLES

Example 1 — Complete the row:

\frac{7}{8}

in all three dialects

Step 1: Fraction → decimal by division: $7 \div 8 = 0.875$ . (Route check: 8 divides 1000 evenly — denominator is all 2s — so termination was guaranteed.)

Step 2: Decimal → percent: $0.875 \times 100 = 87.5\%$ .

Step 3: Locate against benchmarks: above $\frac{3}{4}$ (75%), one eighth (12.5%) short of a whole ✓ internally consistent.

Step 4: The reverse route as verification: $87.5\% = \frac{87.5}{100} = \frac{875}{1000} = \frac{7}{8}$ ✓. Round-tripping a conversion is the self-check that costs ten seconds and catches everything.

Example 2 — The three-store showdown: same jacket, three deals

The jacket lists $80 everywhere. Store A: " $\frac{1}{4}$ off." Store B: "22% off." Store C: "pay 0.77 of the price."

Step 1: Convert all three deals to one dialect — pay-fraction as a percent: A: pay $\frac{3}{4}$ = 75%. B: pay 78%. C: pay 77%.

Step 2: Rank the deals: A (pay 75%) beats C (77%) beats B (78%).

Step 3: Cash it out: A: $80 \times 0.75 = \$60$ ; C: $61.60; B: $62.40.

Step 4: Post-mortem on the marketing: "22% off" SOUNDS bigger than "a quarter off" to many shoppers because 22 > ¼'s visible 4. Dialects can disguise size; conversion is the x-ray. This is the same lesson as 0.6 vs 0.55, now with money attached.

Example 3 — Repeating decimals owned: convert

\frac{5}{12}

and reason about the repeat

Step 1: Divide: $5 \div 12$ : $12$ into $50$ goes 4 ( $48$ ), remainder 2 → bring down: 20, goes 1 ( $12$ ), remainder 8 → 80, goes 6 ( $72$ ), remainder 8 → 80 again — the remainder repeated, so the digits repeat from here: $0.41\overline{6}$ .

Step 2: Say WHY it had to repeat: each step's remainder is one of only 12 possibilities (0–11); with finitely many remainders, one must recur, and recurrence locks the cycle.

Step 3: Predict before dividing (the denominators conjecture): $12 = 2^2 \times 3$ — the 3 dooms termination ✓ matches.

Step 4: Percent form, honestly: $41.\overline{6}\%$ , or "about 41.7%" with the approximation flagged. Repeating notation isn't pedantry — it's the difference between the number itself and its estimate.

MATERIALS

•Conversion triangle mats

•Number lines 0–2

•Benchmark card decks

•Scoreboard scenarios

•Practice set (PDF)

WATCH FOR

!Percents over 100 rejected ("you can't have 120%"). A quantity can exceed its reference: 1.2 = 120% of the original.

!0.6 read as less than 0.55 (or 63% placed below 0.6) — dialect confusion re-invokes place-value comparison; convert to ONE form first.

\frac{1}{3}

written as 0.33 exactly. The bar matters:

0.\overline{3}

; truncation is an approximation, and saying so is the skill.