LESSON PLAN

Operations with Fractions

Apothem Team

Grade 8 · Computational Fluency

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

The baking emergency: "The recipe serves 12 and needs $2\frac{1}{4}$ cups of flour. You're making it for 8 people. Flour?" Pairs attack freely — the scale factor $\frac{8}{12} = \frac{2}{3}$ emerges, then the multiplication $\frac{2}{3} \times \frac{9}{4}$ .

Grade 8 fractions are all four operations at full strength — mixed numbers, division included — deployed inside real contexts. (Answer: $\frac{3}{2} = 1\frac{1}{2}$ cups.)

Explore

Division-of-fractions summit — the operation that was postponed until it could be understood: (1) Common-denominator route: $\frac{3}{4} \div \frac{1}{8} = \frac{6}{8} \div \frac{1}{8}$ = "how many eighths in six eighths?" = 6. (2) The pattern route to invert-and-multiply: dividing by 2 is halving (× ½); dividing by ½ is doubling (× 2); dividing by $\frac{a}{b}$ is… × $\frac{b}{a}$ — the reciprocal, forced by pattern. (3) Verify both routes agree on $\frac{5}{6} \div \frac{2}{3} = \frac{5}{6} \times \frac{3}{2} = \frac{5}{4}$ .

Then mixed-operation workout with mixed numbers, converting at entry, computing, converting at exit.

Formalize

Formalize all four with the division rule now justified:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \qquad 2\tfrac{1}{4} = \tfrac{9}{4}$

The "why" of invert-and-multiply kept alive in one sentence: division asks "how many of THIS fit in THAT," and fitting $\frac{c}{d}$ -sized pieces goes $\frac{d}{c}$ times as often as fitting whole units. Size-check every quotient: dividing by a number under 1 must ENLARGE.

Practice

Practice: mixed set of 10 spanning all operations (mixed numbers in half); two multi-step recipes; one "how many servings" division story; one error autopsy (invert-the-wrong-fraction).

Exit ticket: $1\frac{2}{3} \div \frac{5}{6}$ , exact, with a one-line size-check. ( $\frac{5}{3} \times \frac{6}{5} = 2$ ; dividing by less than 1 grew it ✓.)

Exit ticket

Practice: mixed set of 10 spanning all operations (mixed numbers in half); two multi-step recipes; one "how many servings" division story; one error autopsy (invert-the-wrong-fraction).

Exit ticket: $1\frac{2}{3} \div \frac{5}{6}$ , exact, with a one-line size-check. ( $\frac{5}{3} \times \frac{6}{5} = 2$ ; dividing by less than 1 grew it ✓.)

TIP Invert-and-multiply without the "how many fit" story degrades into inverting random fractions. Re-tell the story weekly; require the size-check line on every division.

WORKED EXAMPLES

Example 1 — How many servings: division as fitting

The vat: $7\frac{1}{2}$ litres of soup; each serving is $\frac{5}{8}$ L. How many servings?

Step 1: Convert: $7\frac{1}{2} = \frac{15}{2}$ .

Step 2: Divide (fit $\frac{5}{8}$ -sized scoops): $\frac{15}{2} \div \frac{5}{8} = \frac{15}{2} \times \frac{8}{5}$ .

Step 3: Cancel before multiplying (the fluency move): $\frac{15}{5} = 3$ and $\frac{8}{2} = 4$ → $3 \times 4 = 12$ servings.

Step 4: Size-check: dividing by a piece smaller than 1 L must yield MORE servings than litres — 12 > 7.5 ✓.

Step 5: The cancel-early habit isn't cosmetic: it kept every number single-digit. Fraction fluency is mostly factor-spotting.

Example 2 — The paint job: all four operations in one story

The job: a room needs $3\frac{1}{3}$ L of paint. You have $\frac{3}{4}$ L left from before, and paint sells in $1\frac{1}{2}$ L cans. How many cans, and how much surplus?

Step 1: Paint needed beyond stock: $3\frac{1}{3} - \frac{3}{4} = \frac{10}{3} - \frac{3}{4} = \frac{40}{12} - \frac{9}{12} = \frac{31}{12}$ L.

Step 2: Cans: $\frac{31}{12} \div \frac{3}{2} = \frac{31}{12} \times \frac{2}{3} = \frac{31}{18} \approx 1.72$ → buy 2 cans (round UP — the vans problem's logic, in litres).

Step 3: Surplus: bought $2 \times 1\frac{1}{2} = 3$ L; used $\frac{31}{12} \approx 2\frac{7}{12}$ → left over $3 - \frac{31}{12} = \frac{5}{12}$ L.

Step 4: The pipeline audit: subtraction (needed common denominators), division (invert-multiply), a context round-up, and a final subtraction — four decisions, each chosen by the STORY. That choosing is the Grade 8 skill; the mechanics were Grades 6–7.

Example 3 — Error autopsy: the double inversion

The submitted work: $\frac{4}{5} \div \frac{2}{3} = \frac{5}{4} \times \frac{2}{3} = \frac{10}{12} = \frac{5}{6}$ .

Step 1: Spot the move: the student inverted the DIVIDEND ( $\frac{4}{5} \to \frac{5}{4}$ ) and left the divisor alone. Wrong fraction flipped.

Step 2: The size-check would have caught it: dividing $\frac{4}{5}$ by a number less than 1 must produce something BIGGER than $\frac{4}{5}$ — but $\frac{5}{6} < \frac{4}{5}$ ? ( $\frac{25}{30}$ vs $\frac{24}{30}$ — actually $\frac{5}{6}$ is barely bigger… the check is inconclusive here; honesty matters). Use the story check instead: "how many $\frac{2}{3}$ s in $\frac{4}{5}$ ?" — a bit more than one ( $\frac{2}{3} \approx 0.67 < 0.8$ ). The claimed $\frac{5}{6} \approx 0.83$ … also near one. The autopsy needs the actual rule.

Step 3: Correct: $\frac{4}{5} \times \frac{3}{2} = \frac{12}{10} = \frac{6}{5} = 1\frac{1}{5}$ — "a bit more than one" ✓ matching the story estimate.

Step 4: The honest lesson: some errors produce near-plausible answers that slip past loose checks. The rule's ANCHOR (divisor is the piece being fitted; IT gets inverted) is the defence when numerical checks come back fuzzy.

MATERIALS

•Fraction strips

•Measuring cups

•Recipe cards

•Practice set (PDF)

WATCH FOR

!The DIVIDEND inverted instead of the divisor. The story assigns roles: the divisor is the piece-size being fitted.

!Mixed numbers operated on part-by-part in multiplication (

2\frac{1}{2} \times 3\frac{1}{2} = 6\frac{1}{4}

?? — the cross terms vanish). Convert to improper first.

!Division answers expected smaller — killed by the size-check ritual.