016i - Multiplying and Dividing Fractions and Decimals

Lesson 16 — Multiplying and Dividing Fractions and Decimals

Strand: Number | Descriptor: AC9M8N04 | Duration: 45 minutes

Learning Intentions

• To multiply and divide fractions, including with cancellation, efficiently and accurately.
• To multiply and divide decimals by reasoning about place value, and to relate dividing by a decimal to dividing by a whole number.

Success Criteria

I can:

1. Multiply fractions by multiplying numerators and denominators, simplifying by cancelling common factors first.
2. Divide fractions using the “keep, change, flip” (reciprocal) method, and explain why it works.
3. Multiply decimals by counting decimal places, and estimate first to check the answer’s size.
4. Divide a decimal by a decimal by converting the divisor to a whole number.

Warmup

(6 minutes — estimate first, mini whiteboards)

Without calculating exactly, estimate whether each answer is bigger or smaller than , and roughly how big:

Discussion: Question 3 is a common trap — many students expect multiplying to always make numbers bigger. Ask: “Why might multiplying two numbers less than 1 give an answer smaller than either of them?” Hold the discussion open; Activity 1 will formalise it.

Activities

Activity 1 — Explicit Instruction: Multiplying and Dividing Fractions (10 min)

I do / We do / You do.

I do: Multiply straight across, then simplify — or cancel common factors first for efficiency.

Faster, with cancellation before multiplying:

I do (division): Explain “keep, change, flip”: keep the first fraction, change ÷ to ×, flip (reciprocate) the second fraction.

Why it works: Dividing by asks “how many halves fit into ?” — multiplying by the reciprocal answers exactly that, since dividing by a number and multiplying by its reciprocal are always equivalent operations.

We do: Evaluate together: , , .

You do: Evaluate, simplifying fully: , , .

Activity 2 — Guided Practice: Multiplying and Dividing Decimals (10 min)

I do: Multiply decimals by ignoring the decimal point, multiplying as whole numbers, then placing the point using the total number of decimal places in the question.

This resolves the warmup: multiplying two numbers less than 1 gives a smaller result, because you are finding a fraction of a fraction.

I do (division): To divide by a decimal, multiply both numbers by the same power of 10 so the divisor becomes a whole number — the value of the quotient is unchanged.

We do: Estimate first, then calculate exactly: , , , .

You do: Estimate first, then calculate: , , , .

Activity 3 — Applied Task: Connecting Fractions and Decimals (13 min)

Pairs. Every answer must include an estimate before the exact calculation.

Problem 1. A recipe uses L of stock in total. How much stock is needed for of a batch that would normally serve people?

Problem 2. A roll of fabric is m long. It is cut into pieces m long. How many complete pieces can be cut, and how much fabric is left over?

Problem 3. A tank holds of its capacity in water, which is L. A tap drains water at L per second. How long, in minutes, will it take to drain the tank completely?

Socratic scaffolding for Problem 3:

Prompt	Purpose
Understand: what is being asked?	The total time to drain the full tank, not just the water currently in it — but we are only told the amount for of capacity.
Devise a plan: what do you need first?	The full capacity of the tank.
How can you find the full capacity from L?	Divide: gives the whole (capacity for “1”).
Carry out that step	L.
Now devise the next step	Time to drain L at L/s: divide .
Carry it out	seconds.
Convert to minutes and look back	minutes. Does this seem reasonable for draining a L tank at half a litre per second? Yes — roughly 6–7 minutes is sensible.

Answers: 1 — Estimate: about L; exact: L. 2 — Estimate: about 14 pieces; exact: complete pieces, no leftover. 3 — Full capacity L; draining time s minutes.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Evaluate , simplifying fully.
2. Evaluate .
3. Evaluate .
4. Evaluate .

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

Common Misconceptions

Misconception	How to pre-empt it
”Multiplying always makes numbers bigger.”	Use the warmup and Activity 2’s place-value explanation: multiplying by a number less than 1 finds a fraction of the original.
Flipping the first fraction instead of the second when dividing.	Chant “keep, change, flip” in order every time; explicitly mark which fraction is being reciprocated.
Multiplying decimal digits correctly but misplacing the decimal point.	Always estimate first (as in Activity 2/3), and count total decimal places across both numbers, not just one.
When dividing decimals, moving the decimal point in only one of the two numbers.	Model explicitly that both dividend and divisor must be multiplied by the same power of 10.
Cancelling numerators with numerators (or denominators with denominators) instead of across the fractions.	Show cancellation must occur between a numerator and a denominator, never numerator-numerator.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Evaluate .

Answer

This telescopes: every numerator cancels with the next denominator, leaving .

E2 (Kangaroo style). A number divided by gives the same result as the number multiplied by which whole number?

Answer

Dividing by is the same as multiplying by its reciprocal, .

E3 (Challenge). Find a decimal such that .

Answer

(A nice extension link to irrational numbers from Lessons 6–9.)

E4 (Investigation). Two fractions less than 1 are multiplied together. Can the result ever be larger than both original fractions? Investigate and explain.

Answer

No. Multiplying two proper fractions (each less than 1) always gives a result smaller than either — you are taking “a fraction of a fraction”. Testing confirms: , which is smaller than both and .

Homework

1. Evaluate, simplifying fully: (a) (b) (c) .
2. Evaluate: (a) (b) (c) (d) .
3. Estimate, then calculate: .
4. A container holds L of paint, enough to cover per L. What total area can be painted with the full container? Show your working.
5. Reasoning. Explain why dividing by has the same effect as multiplying by , using place value.
6. Challenge. A number is multiplied by and the result is then divided by . If the final answer is , find the original number.

Answers: Q1 — (a) (b) (c) . Q2 — (a) (b) (c) (d) . Q3 — . Q4 — lots of L, each covering : . Q5 — dividing by asks “how many tenths fit into the number”, which shifts every digit one place value up, identical to multiplying by 10. Q6 — working backwards: , then . The original number is .