B. Number · Operations: Divisibility, Decimals, Fractions, Ratios and Percents

Multiplying Whole Numbers by Decimal Tenths

B2.7 B2.1About 45 minutes

Learning goals

I can explain why multiplying by a decimal less than 1 gives a smaller answer.
I can multiply a three-digit whole number by a decimal in tenths using an algorithm.
I can place the decimal point correctly and check with an estimate.

Words to know

decimal tenths
A decimal with one digit after the decimal point, like 0.7 or 3.4.

product
The answer when you multiply.

estimate
A close answer found quickly, used to check if an exact answer is reasonable.

area model
A rectangle split into parts to show how a multiplication is broken into smaller products.

Remember from Grade 5

In Grade 5 you multiplied by 0.1 in your head (74 × 0.1 = 7.4) and multiplied 2-digit numbers with an area model and an algorithm. Now you'll multiply three-digit whole numbers by decimal tenths, like 324 × 0.7 or 215 × 3.4.

Let's start with a walk

Each of Amir's steps is about 0.7 m long. He takes 324 steps to walk to the rink. About how far is that?

0.7 is 7 tenths. So 324 × 0.7 means 324 groups of 7 tenths.

324 × 7 = 2 268, so that's 2 268 tenths. And 2 268 tenths = 226.8.

Amir walks about 226.8 m. That makes sense: 0.7 is a bit less than 1, so the answer is a bit less than 324.

Each tenth of 324 is 32.4. Seven of them make 226.8, a bit less than 324, because 0.7 is less than 1.

The algorithm

Estimate first.
Multiply as if there were no decimal point. (For 324 × 0.7, find 324 × 7.)
You multiplied by tenths, so the product has one decimal place. Put the decimal point so there is one digit after it.
Check with your estimate.

Bigger or smaller?

Multiply by a number less than 1 (like 0.7): the product is smaller than the whole number.
Multiply by a number greater than 1 (like 3.4): the product is bigger.

Worked example

Example 1: Multiply by a number less than 1

324 × 0.7 = ?

Estimate: 0.7 is close to 3/4. A bit less than 3/4 of 324 (which is 243). So a bit less than 243.
Multiply without the decimal: 324 × 7 = 2 100 + 140 + 28 = 2 268.
One decimal place: 226.8.
226.8 is a bit less than 243. ✓

324 × 0.7 = 226.8

An area model for 215 × 3.4. The widths are drawn to scale. 215 × 3 = 645 and 215 × 0.4 = 86, so 215 × 3.4 = 645 + 86 = 731.

Worked example

Example 2: Multiply by a number greater than 1

215 × 3.4 = ?

Estimate: 200 × 3 = 600, and the real answer is more. About 700.
Area model: split 3.4 into 3 + 0.4. 215 × 3 = 645. 215 × 0.4 = 215 × 4 tenths = 860 tenths = 86. 645 + 86 = 731.
Algorithm: 215 × 34 = 215 × 30 + 215 × 4 = 6 450 + 860 = 7 310. One decimal place: 731.0 = 731.
731 is close to 700. ✓

215 × 3.4 = 731

Worked example

Example 3: A garden problem

A community garden is a rectangle 148 m long and 2.5 m wide. What is its area?

Area = length × width = 148 × 2.5.
Estimate: 150 × 2.5 = 375 (two and a half groups of 150: 150 + 150 + 75).
148 × 25 = 3 700. One decimal place: 370.0 = 370.
370 is close to 375. ✓

The area is 370 m².

Estimate to place the point

Always ask: should the answer be bigger or smaller than the whole number? If you multiply 512 by 0.9 and get 4 608, the estimate tells you something is wrong: 512 × 0.9 must be a bit less than 512. The right answer is 460.8.

Just for fun

What if Amir took tiny mouse steps of just 0.1 m? His 324 steps would get him only 32.4 m. He'd need 2268 steps to reach the rink!

Hands-on activity

How far is it in steps?

You need: a measuring tape or metre stick, paper and a pencil. Ask a grown-up to help you pick a safe route, like from your door to the end of the driveway or along a hallway.

Take 10 normal steps. Measure how far you went, in metres. Divide by 10 to find one step, and round to the nearest tenth of a metre (for example, 0.6 m).
Walk your route and count your steps. If you have more than 100 steps, great!
Multiply: number of steps × step length. Estimate first.
If you can, measure the route with the tape to check.

Math talk: "Why is the answer smaller than the number of steps?"

Your turn

400 × 0.3 = ?

123 × 0.2 = ?

250 × 0.6 = ?

318 × 0.4 = ?

142 × 1.5 = ?

205 × 2.3 = ?

Is 512 × 0.9 = 4 608 reasonable? If not, fix it.

More practice: warm-up

350 × 0.1 = ?

120 × 0.5 = ?

3 × 0.4 = ?

100 × 0.7 = ?

20 × 0.6 = ?

More practice: on your own

436 × 0.8 = ?

275 × 1.2 = ?

509 × 3.6 = ?

A store has 125 bags of apples. Each bag has a mass of 2.4 kg. What is the total mass?

186 × 0.5 = ?

Challenge

Mr. Leblanc drives 4.8 km to work and back every day. How far does he drive in 365 days?

Without calculating, put these in order from least to greatest: 250 × 1.1, 250 × 0.5, 250 × 0.9. Then check.

Which is greater: 432 × 0.6 or 216 × 1.2? Explain without calculating first.

Check your understanding

Quiz

What is 300 × 0.4?

What is 214 × 0.3?

What is 125 × 1.6?

Which product is less than 480?

What is the best estimate for 297 × 4.1?

What is 150 × 2.4?

For parents and teachers

Big idea: B2.7 asks Grade 6 students to represent and solve problems involving the multiplication of three-digit whole numbers by decimal tenths, using algorithms. The core idea is that 0.7 is 7 tenths, so 324 × 0.7 = 2 268 tenths = 226.8. Estimation tells you where the decimal point goes.

Common difficulties:

Thinking multiplication always makes a number bigger. Multiplying by 0.7 makes it smaller.
Forgetting the decimal point, or putting it in the wrong place. Always estimate.
Dropping a trailing zero too early: 215 × 34 = 7 310, so 215 × 3.4 = 731.0 = 731 (not 73.1).

How to help: Use real measurements: step lengths, the length of a hockey rink, fuel use in litres. Ask "Should the answer be more or less than the whole number?" every time.