B. Number · Operations: Rational Numbers, Integers, Fractions and Proportional ReasoningMultiplying and Dividing by Powers of Ten in Your Head
B2.3 A1About 40 minutes
Learning goals- I can multiply and divide whole numbers and decimals (to thousandths) by 10, 100, 1 000 and 10 000 in my head.
- I can explain my strategy using place value: digits move left or right, the decimal point stays put.
- I can multiply and divide by 0.1, 0.01 and 0.001 by thinking of the matching power of ten.
- I can use these strategies for metric conversions.
Words to know
power of ten
A number like 10, 100, 1 000 or 10 000 that can be written as 10 with an exponent: 10¹, 10², 10³, 10⁴.
place value
The value of a digit depends on its place. In 4.256 the 5 is worth 5 hundredths.
placeholder zero
A zero written to hold a place so the other digits stay in the right places, like the zeros in 0.058.
Remember from earlier grades
In earlier grades you multiplied and divided by 10, 100 and 1 000, and in Grade 7 you wrote numbers in expanded form with powers of ten, like 10³ = 1 000. In Grade 8 you will do this in your head with decimals to thousandths, and explain why your strategy works.
Let's start on a ski trail
A cross-country ski trail is 2.375 km long. How many metres is that?
There are 1 000 m in 1 km, so multiply by 1 000: 2.375 × 1 000 = 2 375 m.
You did not need to work it out digit by digit. Multiplying by 1 000 makes every digit 1 000 times as big, so every digit moves 3 places to the left.
Trail lengths are often given in kilometres. To change 2.375 km to metres, multiply by 1 000 and every digit moves 3 places to the left: 2 375 m. AI-generated imageMultiplying by 1 000 makes each digit 1 000 times as big, so each digit moves 3 places left. The decimal point stays where it is. | Thousands | Hundreds | Tens | Ones | . | Tenths | Hundredths | Thousandths |
|---|
| 4.256 | | | | 4 | . | 2 | 5 | 6 |
| × 10 = 42.56 | | | 4 | 2 | . | 5 | 6 | |
| × 100 = 425.6 | | 4 | 2 | 5 | . | 6 | | |
| × 1 000 = 4 256 | 4 | 2 | 5 | 6 | . | | | |
The strategy
- Multiply by 10ⁿ: each digit moves n places left (the number gets bigger). 10 → 1 place, 100 → 2 places, 1 000 → 3 places, 10 000 → 4 places. The number of zeros in the power of ten tells you the number of places.
- Divide by 10ⁿ: each digit moves n places right (the number gets smaller).
- Add placeholder zeros when you run out of digits: 0.045 × 1 000 = 45, and 3.2 ÷ 1 000 = 0.0032.
Multiplying by 0.1, 0.01 or 0.001 is the same as dividing by 10, 100 or 1 000, because 0.1 = 1/10. Dividing by 0.1, 0.01 or 0.001 is the same as multiplying by 10, 100 or 1 000 (that is the reciprocal relationship from the first lesson).
In this grade's Number Sense unit you will use the same idea for scientific notation.
Watch out!
"Just add a zero" only works for whole numbers. 2.5 × 10 is 25, not 2.50 (2.50 is the same as 2.5). Think about the digits moving instead.
Worked exampleExample 1: Multiply a decimal by 1 000
- × 1 000 moves each digit 3 places left.
- 0.072 → 0.72 → 7.2 → 72.
Worked exampleExample 2: Divide by 1 000
- ÷ 1 000 moves each digit 3 places right.
- 58 → 5.8 → 0.58 → 0.058. A placeholder zero holds the tenths place.
Worked exampleExample 3: Divide by 0.01
- Dividing by 0.01 is the same as multiplying by 100 (how many hundredths are in 3.6?).
- 3.6 × 100 = 360.
Worked exampleExample 4: Metric conversions
Change 1 250 g to kilograms and 0.75 L to millilitres.
- 1 000 g = 1 kg, so divide by 1 000: 1 250 ÷ 1 000 = 1.25 kg.
- 1 L = 1 000 mL, so multiply by 1 000: 0.75 × 1 000 = 750 mL.
Each step to a smaller metric unit multiplies by a power of ten; each step back divides by the same power. For example, 3.5 m × 100 = 350 cm and 350 cm × 10 = 3 500 mm.Explain your strategy
A good explanation names the place value change: "Multiplying by 100 makes each digit 100 times as big, so each digit moves two places to the left. The 3 in the ones place moves to the hundreds place." Also do a size check: multiplying by 1 000 should make a number much bigger.
Hands-on activityMetric label hunt
You need: food packages from the cupboard, a measuring tape, paper, a pencil.
- Find 5 packages with metric labels (g, kg, mL or L). For example, a box labelled 525 g or a jug labelled 1.89 L.
- Convert each label in your head: grams to kilograms (÷ 1 000), litres to millilitres (× 1 000).
- Measure something with the tape in centimetres. Convert it to metres (÷ 100) and to millimetres (× 10).
- Write the power of ten you used each time, with an exponent (for example 10³).
Model answers: 525 g = 0.525 kg; 1.89 L = 1 890 mL; a table 152.5 cm long = 1.525 m = 1 525 mm.
Math talk: "When you change to a bigger unit, does the number get bigger or smaller? Why?"
Just for funWhy did the decimal point join a dance class? It already knew the moves: three steps left for ×1000, three steps right for ÷1000. (Okay, technically the digits do the moving. Don't tell the decimal point.)
Your turn
347. Digits move 2 places left.
6.52. Digits move 2 places right.
0.0071. 7.1 → 0.71 → 0.071 → 0.0071.
350. 10⁴ = 10 000, so digits move 4 places left: 0.35, 3.5, 35, 350.
0.48. × 0.01 is the same as ÷ 100.
26. ÷ 0.1 is the same as × 10.
More practice: on your own
20.8. Each digit moves 2 places left: the 2 tenths become 2 tens.
0.153. Each digit moves 2 places right.
3A roll of tape is 3.5 m long. How many millimetres is that?
3 500 mm. 1 m = 1 000 mm, so 3.5 × 1 000 = 3 500.
4Fill in the blank: 6.4 × ___ = 6 400
1 000 (10³). The digits moved 3 places left: 6.4 → 64 → 640 → 6 400.
5Explain why 0.45 × 0.1 = 0.045.
Multiplying by 0.1 is the same as dividing by 10 (0.1 = 1/10). Dividing by 10 moves each digit one place right, so 0.45 becomes 0.045.
Challenge
1A box of 1 000 paper clips has a mass of 450 g. What is the mass of one paper clip? Of 100 paper clips?
0.45 g each; 45 g for 100. 450 ÷ 1 000 = 0.45. Then 0.45 × 100 = 45 (or 450 ÷ 10 = 45, since 100 is one tenth of 1 000).
2If 0.072 × 10ⁿ = 720, what is n?
n = 4. 0.072 → 0.72 (1 place) → 7.2 (2) → 72 (3) → 720 (4). The digits moved 4 places, so the power is 10⁴.
3Overnight, 0.025 m of snow fell. How many millimetres is that? How many centimetres?
25 mm and 2.5 cm. 0.025 × 1 000 = 25 mm. 0.025 × 100 = 2.5 cm. (Check: 25 mm ÷ 10 = 2.5 cm. ✓)
Quiz
3. Multiplying by 0.01 is the same as:
4. How many litres is 1 750 mL?
Big idea: Ontario's Grade 8 expectation B2.3 asks students to use mental math strategies to multiply and divide whole numbers and decimal numbers up to thousandths by powers of ten, and explain the strategies used. The explanation matters: the place-value idea (digits move, the point stays) is better than a rule about "moving the decimal point", because it explains why the strategy works.
Common difficulties:
- Moving the wrong direction. Ask: "Should the answer be bigger or smaller?"
- Forgetting placeholder zeros (58 ÷ 1 000 written as 0.58).
- "Adding a zero" to decimals (2.5 × 10 = 2.50).
- × 0.1 versus ÷ 0.1. × 0.1 makes a number 10 times smaller; ÷ 0.1 makes it 10 times bigger.
How to help: Do the metric label hunt in the kitchen or at the store. Ask your child to say each conversion out loud and explain which way the digits moved.