065i - Multiplying and Dividing Decimals

Lesson 65 — Operations with Decimals: Multiplication and Division

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

Learning Intentions

• To multiply and divide decimals, including by powers of ten.
• To place the decimal point using estimation rather than digit-counting alone.

Success Criteria

I can:

1. Multiply and divide decimals by , and .
2. Multiply two decimals, placing the decimal point correctly.
3. Divide a decimal by a whole number.
4. Divide by a decimal by rewriting as an equivalent whole-number division.

Warmup

(6 minutes — powers of ten, mini whiteboards)

(Answers: ; ; ; ; ; .)

The rule, stated precisely: multiplying by moves every digit one place left (the number grows); dividing moves digits right. Say “the digits move”, not “the decimal point moves” — the point marks a fixed boundary between wholes and parts.

Activities

Activity 1 — Multiplying Decimals (14 min)

The method: compute with whole numbers, then place the point by estimation.

I do.

Step 1 — estimate: roughly , so the answer is near .

Step 2 — whole-number product: .

Step 3 — place the point using the estimate: the answer must be near , so it is (not or ).

The digit-count check (secondary, not primary). has one decimal place, has one; the product has . This agrees: ✓. Teach it as a cross-check — estimation should lead, because digit-counting is applied blindly and breaks down with trailing zeros (e.g. ).

I do — small numbers.

Estimate: both are small, so the answer is very small. Whole numbers: . Digit count: places, giving .

This is where “multiplication makes bigger” dies completely: the product is smaller than both factors.

We do: ; ; ; .

(Answers: ; ; ; .)

You do:

(Answers: ; ; ; ; ; .)

Activity 2 — Dividing Decimals (14 min)

Case 1 — dividing a decimal by a whole number. Divide as normal; the decimal point in the answer sits directly above the point in the dividend.

(Short division: r ; r ; .)

Case 2 — dividing by a decimal: rewrite first. The key identity:

Multiplying both numbers by the same power of ten leaves the quotient unchanged.

Why it works: the fraction is scaled by . Same idea as equivalent fractions (Lesson 55).

Sense-check: “how many s fit into ?” — lots, since is small. is plausible ✓

I do.

(Both scaled by to clear the two decimal places of the divisor.)

I do — mixed case.

(Scale both by — enough to make the divisor whole. The dividend may stay decimal.)

You do:

(Answers: ; ; ; ; ; .)

Discussion of Q5: dividing by a small decimal gives a large answer — from . Ask why. (Very many s fit into .) This mirrors Lesson 63’s fraction division.

Activity 3 — Inquiry: the Supermarket Unit Price (7 min)

Pairs.

Which is better value?

• Brand A: kg of muesli for 6.30$
• Brand B: kg of muesli for 9.48$

Socratic scaffolding:

Prompt	Purpose
Understand: what does “better value” mean?	Lower cost for the same amount — compare unit prices.
What unit should you price?	Dollars per kilogram.
Set up Brand A.	.
That is division by a decimal. What first?	Scale both by : .
Brand B.	.
Compare.	A costs 8.40$7.90$/kg. B is better value.
Looking back — estimate check	A: for three-quarters of a kg is like per kg ✓. Reasonable.

Answers: A 8.40= $7.90$/kg; Brand B.

Extension: supermarkets print unit prices by law. Why per g rather than per kg for small items? (Smaller, more comparable numbers for light products.)

Checks for Understanding

(5 minutes — exit ticket)

1. Calculate: (a) (b) .
2. Estimate, then calculate: .
3. Calculate: (a) (b) (c) .
4. Reasoning. A student computes . Use estimation to show this is wrong, and give the correct answer.
5. Which is better value: kg for 4.200.8$6.40$?

Answers: 1. (a) (b) ; 2. Estimate ; exact ; 3. (a) (b) (c) ; 4. Half of must be less than — the answer is ; 5. 8.40$8.00$/kg — the second.

Common Misconceptions

Misconception	How to pre-empt it
Placing the decimal point by digit-count alone, without estimating.	Estimation leads; digit-count is the cross-check. Both should agree.
”Multiplying makes bigger, dividing makes smaller.”	and kill both halves of this.
Moving only one number’s digits when dividing by a decimal.	Both numbers are scaled — it is an equivalent fraction. Show the fraction form.
Saying “the decimal point moves”.	The digits move through the fixed place value columns. Phrasing matters for later work.
Treating and as different answers.	Trailing zeros after the point do not change the value — but keep them for money.
Forgetting that shifts all digits, mangling e.g. into .	Use a place value chart for any number containing an internal zero.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Given , write down the values of and .

Answer

(estimate: ). (both are about a quarter and a half — the product is about a tenth).

E2 (AMC Junior style). Petrol costs 1.8532.4$ L cost, to the nearest cent?

Answer

59.942 \times 32 = 64$, so plausible.)*

E3 (Challenge). A stack of sheets of paper is cm tall. How thick is one sheet, in millimetres?

Answer

E4 (Challenge). , where both digits are the same. Find the digit.

Answer

, so the digit is .

E5 (Challenge). A runner’s stride is m. How many complete strides to cover km?

Answer

So strides are needed to cover the distance ( complete strides fall just short).

Homework

1. Calculate: (a) (b) (c) (d) (e) .
2. Estimate, then calculate: (a) (b) (c) .
3. Calculate: (a) (b) (c) (d) .
4. Calculate: (a) (b) (c) (d) (e) .
5. Apples cost 3.802.4$ kg.
6. A L bottle is shared equally among glasses. How much in each, in litres and in millilitres?
7. Which is better value: kg of cheese for 5.200.7$8.75$?
8. A wall is m long. How many m tiles fit along it exactly?
9. Reasoning. Explain why has the same answer as .
10. Challenge. Given , write down: (a) (b) (c) .

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) ; (b) ; (c) ; . Q3 — (a) (b) (c) (d) . Q4 — (a) (b) (c) (d) (e) . Q5 — 9.120.25= 250$13$12.504.32 \div 0.24 = 1810\tfrac{10}{10} = 119.0419.04340$.