063i - Multiplying and Dividing Fractions

Lesson 63 — Multiplying and Dividing Fractions

Strand: Number | Descriptor: AC9M7N06 | Duration: 45 minutes

Learning Intentions

• To multiply and divide fractions, understanding why the procedures work.
• To recognise that multiplying by a proper fraction makes a quantity smaller.

Success Criteria

I can:

1. Multiply fractions, cancelling common factors before multiplying.
2. Explain why “of” means multiply.
3. Divide by a fraction by multiplying by its reciprocal, and explain why.
4. Work with mixed numbers by converting to improper fractions first.

Warmup

(6 minutes — “of” and halving, mini whiteboards)

1. What is of ?
2. What is of ?
3. What is of ? (Think of three-quarters of a pizza shared among 3.)
4. How many quarters are in ?
5. How many halves are in ?

Answers: 1. ; 2. ; 3. ; 4. ; 5. .

Notice: Q1–3 are multiplication (“of”), Q4–5 are division (“how many … in …”). Both meanings drive today’s methods.

Activities

Activity 1 — Explicit Instruction: Multiplying Fractions (12 min)

The rule, with its reason.

The area model — why it works. Draw a unit square. Shade of its width, then of its height. The doubly shaded region is a grid of pieces out of :

Cancel before multiplying — model this as the default:

The key conceptual point. Multiplying by a fraction less than makes the answer smaller. “Multiplication makes bigger” is a primary-school habit that fails here:

Whole numbers and mixed numbers: write whole numbers over ; convert mixed numbers to improper fractions first.

We do: ; ; of ; .

Activity 2 — Explicit Instruction: Dividing Fractions (14 min)

Build the idea from “how many fit?” — do not start with the rule.

Observe the pattern: dividing by is the same as multiplying by . Dividing by multiplies by .

The rule. Dividing by a fraction is multiplying by its reciprocal (the fraction flipped):

Why flipping works — one clean argument. Division asks “how many groups?” A group of size fits into exactly times (e.g. fits into exactly times). So it fits into exactly times.

I do:

Sense-check the middle example: is half of , so it should fit exactly twice ✓

Another surprise to name: dividing by a fraction less than makes the answer bigger. “Division makes smaller” also fails.

You do:

(Answers: ; ; ; ; ; .)

Activity 3 — Inquiry: Fraction of a Fraction of a Fraction (8 min)

Pairs.

A chocolate bar is shared in stages: Ana takes of the bar. Ben takes of what remains. Cara takes of what then remains.

1. What fraction of the original bar does each person get?
2. What fraction is left at the end?
3. What do you notice about the three shares and the remainder?

Socratic scaffolding:

Prompt	Purpose
Understand: what does “of what remains” mean?	Each fraction acts on the current remainder, not the original bar.
Ana takes . What remains?	.
Ben takes of that. How much is that?	of the original.
What remains now?	.
Cara takes of .	… check: .
What is left?	.
Check: do the four pieces total ?	✓
Looking back	Multiplication chains the stages; addition checks the whole. Both operations in one problem.

Answers: 1. Ana , Ben , Cara ; 2. ; 3. They sum to , and each successive share shrinks.

Checks for Understanding

(5 minutes — exit ticket)

1. Calculate: (a) (b) (c) of .
2. Calculate: (a) (b) (c) .
3. Calculate .
4. Reasoning. Explain why , using the “how many fit?” meaning.
5. Reasoning. True or false: multiplying always makes a number bigger. Give a counterexample.

Answers: 1. (a) (b) (c) ; 2. (a) (b) (c) ; 3. ; 4. It asks how many halves fit into — each whole contains two halves, so ; 5. False — , smaller than . Multiplying by a fraction less than shrinks.

Common Misconceptions

Misconception	How to pre-empt it
Needing a common denominator to multiply.	Common denominators are for adding. Multiplication works directly — show why via the area model.
Flipping the first fraction when dividing.	Only the divisor is flipped. Anchor: “dividing by = multiplying by ”, where the flip is visibly on the divisor.
Believing multiplication always enlarges and division always shrinks.	Confront both directly with and .
Multiplying mixed numbers part by part: .	Convert to improper fractions first, always. Test the wrong method against the area model.
Cancelling across an addition or within one fraction’s top and bottom incorrectly.	Cancelling pairs a numerator with a denominator, and only when multiplying.
Forgetting to simplify or convert improper answers.	Simplified mixed numbers as the final form, every time.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). What is ?

Answer

Everything telescopes — each numerator cancels the previous denominator:

E2 (AMC Junior style). A ribbon is m long. How many pieces of length m can be cut from it?

Answer

E3 (Challenge). Find the missing fraction: .

Answer

E4 (Challenge). A tank is full. Water is drawn off until it is full. The amount removed is L. What is the tank’s capacity?

Answer

E5 (Challenge). Which is larger: of , or of ? Explain what this shows.

Answer

Both equal . Multiplication is commutative — “a fraction of a fraction” can be taken in either order. Many students expect a difference; the area model shows the same rectangle either way.

Homework

1. Calculate, cancelling first where possible: (a) (b) (c) (d) .
2. Find: (a) of (b) of (c) of .
3. Calculate: (a) (b) (c) (d) .
4. Calculate: (a) (b) (c) (d) .
5. A recipe needs cup of sugar per batch. How much sugar for batches?
6. How many L bottles can be filled from L of juice?
7. A gardener waters of the garden in the morning and of the remainder in the afternoon. What fraction of the garden is still unwatered?
8. Reasoning. Explain why dividing by is the same as multiplying by .
9. Challenge. Find the value of without multiplying everything out.

Answers: Q1 — (a) (b) (c) (d) . Q2 — (a) (b) (c) . Q3 — (a) (b) (c) (d) . Q4 — (a) (b) (c) (d) . Q5 — cups. Q6 — bottles. Q7 — after morning remains; afternoon waters ; unwatered . Q8 — ten tenths fit in each whole, so the count of tenths is ten times the number of wholes. Q9 — telescoping: .