LESSON PLAN

Operations with Fractions

Apothem Team

Grade 6 · Number

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Serve the brownie-pan problem: "After the bake sale, one pan is 3/8 full and another is 1/4 full. Can we combine them into one pan?" Hands vote yes/no, then pairs sketch the pans and defend.

The sketch makes the issue visible: eighths and quarters are different-sized pieces, so the counts 3 and 1 can't just add. The whole unit lives inside this warm-up: same-size pieces first, then count.

Explore

Fraction-strip lab for addition and subtraction: pairs build $\frac{3}{8} + \frac{1}{4}$ by re-slicing the quarter into eighths ( $\frac{2}{8}$ ), landing on $\frac{5}{8}$ . Then $\frac{2}{3} - \frac{1}{2}$ via sixths. Each result gets recorded three ways: strips, number line hops, and symbols.

Then multiplication enters as "of": $\frac{1}{2} \times \frac{3}{4}$ means half OF three-quarters — fold a three-quarters strip in half: $\frac{3}{8}$ . The area model on grid paper confirms: a $\frac{1}{2} \times \frac{3}{4}$ rectangle covers 3 of 8 grid cells.

Formalize

Formalize the two operations' different souls. Addition/subtraction requires common denominators — counting needs same-size pieces. Multiplication multiplies across — because "a of b" scales both the slicing and the counting:

$\frac{3}{8} + \frac{1}{4} = \frac{3}{8} + \frac{2}{8} = \frac{5}{8} \qquad \frac{1}{2} \times \frac{3}{4} = \frac{1\times3}{2\times4} = \frac{3}{8}$

The deep contrast to say aloud: adding needs SAME pieces because addition counts; multiplying doesn't because multiplication re-slices. Students who know WHY the rules differ stop hybridizing them (adding tops and bottoms).

Practice

Practice: four add/subtract items with unlike denominators (including one mixed-number), four multiplications (including a fraction of a whole number), and two story problems where students must first choose the operation from the story's structure.

Exit ticket: "Explain to an absent classmate, with one picture, why $\frac{1}{2}+\frac{1}{3}$ is NOT $\frac{2}{5}$ ." (Strips: halves and thirds re-slice into sixths → $\frac{3}{6}+\frac{2}{6} = \frac{5}{6}$ .)

Exit ticket

TIP The error

\frac{1}{2}+\frac{1}{3}=\frac{2}{5}

survives lectures but dies by picture: have every student draw the sixths re-slicing once — the visual becomes the memory.

WORKED EXAMPLES

Example 1 — The pan problem finished:

\frac{3}{8} + \frac{1}{4}

Step 1: Different pieces — eighths and quarters — so re-slice to the common size. Quarters split into eighths: $\frac{1}{4} = \frac{2}{8}$ .

Step 2: Count the same-sized pieces: $\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$ .

Step 3: Answer the story: yes — combining fills one pan to $\frac{5}{8}$ , comfortably under a whole pan.

Step 4: Size-check: $\frac{3}{8}$ is a bit under half, $\frac{1}{4}$ is half of half → together a bit over half ✓ ( $\frac{5}{8}$ is a bit over half). The estimate travels alongside fractions just as it does with decimals.

Example 2 —

2\frac{1}{3} - \frac{3}{4}

: mixed number, unlike pieces, one regroup

Step 1: Common denominator for thirds and quarters: twelfths. Convert: $2\frac{1}{3} = 2\frac{4}{12}$ and $\frac{3}{4} = \frac{9}{12}$ .

Step 2: The fraction part can't give up 9 from 4 — regroup one whole: $2\frac{4}{12} = 1\frac{16}{12}$ .

Step 3: Subtract parts: wholes $1 - 0 = 1$ ; twelfths $16 - 9 = 7$ . Result: $1\frac{7}{12}$ .

Step 4: Check on the number line: start at $2\frac{1}{3}$ , hop back $\frac{3}{4}$ — past 2, past $1\frac{3}{4}$ … landing between $1\frac{1}{2}$ and $1\frac{2}{3}$ ✓ ( $1\frac{7}{12}$ sits right there).

Step 5: Note the family resemblance: the regroup is EXACTLY whole-number borrowing, trading a 1 for twelve twelfths instead of ten ones.

Example 3 — Fraction of a fraction in the wild: the leftover pizza

The story: $\frac{2}{3}$ of a pizza is left. You eat $\frac{3}{4}$ OF the leftover. How much of the whole pizza did you eat?

Step 1: Identify the operation from the word "of": $\frac{3}{4} \times \frac{2}{3}$ .

Step 2: Multiply across: $\frac{3\times2}{4\times3} = \frac{6}{12} = \frac{1}{2}$ .

Step 3: Confirm with the picture: slice the leftover two-thirds into 4 equal strips, take 3 — those 3 strips are 6 of the pizza's 12 equal slices: half the pizza.

Step 4: The follow-up that separates operations: "how much pizza REMAINS now?" That's subtraction: $\frac{2}{3} - \frac{1}{2} = \frac{4}{6}-\frac{3}{6} = \frac{1}{6}$ . One story, both operations, each chosen by what the sentence asks — not by which chapter we're in.

MATERIALS

•Fraction strips

•Grid paper for area models

•Number lines 0–2 in twelfths

•Recipe cards for story problems

•Practice set (PDF)

WATCH FOR

!Adding across:

\frac{1}{2}+\frac{1}{3}=\frac{2}{5}

. Cure with the strips picture and the size-check (

\frac{2}{5} < \frac{1}{2}

— adding something positive can't shrink you).

!Multiplication expected to enlarge;

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

shrinks. "Of" language and the folded strip make the shrink sensible.

!Common denominators dragged into multiplication (workable but wasteful) or skipped in subtraction (fatal). Match the rule to the operation's meaning.