Year 4 Mathematics · AC9M4N05

Multiply and divide by multiples and powers of 10

Solve problems involving multiplying or dividing natural numbers by multiples and powers of 10 using place-value relationships

Ready to project and teach

Learning goalsSay it simply

Multiplying by 10, 100 or 1000 makes each digit’s value 10, 100 or 1000 times as large. Dividing reverses this relationship when the quotient is a natural number. A multiple such as 30 means 3 tens; 300 means 3 hundreds. Use these units to multiply and divide without a calculator.

Learning intention: I can use place value to solve multiplication and exact division problems involving multiples and powers of 10, and explain why my method works.

Before starting: Know the ones, tens, hundreds and thousands places; use multiplication facts to 10 × 10 and their division facts. These activities use natural-number inputs and whole-number answers.

Learning routine: Identify the total or group size → Choose a place-value unit → Calculate → Interpret the answer → Check with the inverse operation.

Success looks like

Multiply and divide by 10, 100 and 1000.
Use factors such as 30 = 3 × 10 and 400 = 4 × 100.
Explain how a digit’s value changes, including when a number contains zeros.
Solve grouping and sharing problems, then check the answer in context.

Clean visual examplesOne-page board

1. Multiply by powers of 10

Place-value chart: the digits keep their order; their values change.
Number	Ten-thousands	Thousands	Hundreds	Tens	Ones
406	—	—	4	0	6
406 × 10 = 4060	—	4	0	6	0
406 × 100 = 40600	4	0	6	0	0

In 406, the 4 is worth 400 and the 6 is worth 6. Multiplying by 10 changes these values to 4000 and 60, giving 4060. Multiplying the original 406 by 100 changes them to 40000 and 600, giving 40600. The zero between the 4 and 6 must still mark the empty place.

Multiplying by 100 has the same effect as multiplying by 10 twice, since 10 × 10 = 100. Multiplying by 1000 has the effect of three factors of 10: for example, 8 × 1000 = 8000.

2. Divide by powers of 10

3600 ÷ 100 = 36. Rename 3600 as 36 hundreds. There are 36 groups of 100. The 3 changes from a value of 3000 to 30, and the 6 from 600 to 6. Each digit’s value becomes one hundredth as large.

Check: 36 × 100 = 3600. Similarly, 3600 ÷ 10 = 360 and 36000 ÷ 1000 = 36. Explain the place-value relationship before using a written shortcut.

3. Multiply by a multiple of 10

24 × 30 = 720. Rewrite 30 as 3 tens. First calculate 24 × 3 = 72. This means 72 tens, so 72 × 10 = 720.

For 12 × 300, use 12 × 3 = 36, so 12 × 300 = 36 hundreds = 3600. When both factors contain tens, account for both: 40 × 60 = 4 × 6 × 10 × 10 = 2400. Four tens multiplied by six tens make 24 hundreds.

The same reasoning works in thousands: 5 × 6000 = 30 thousands = 30000, because each group contains 6 thousands.

4. Divide by a multiple using the same unit

4800 ÷ 60 = 80. The total is 480 tens and each group has 6 tens. There are 480 ÷ 6 = 80 groups. This is why 4800 ÷ 60 and 480 ÷ 6 have the same answer.

Another valid chain is 4800 ÷ 10 ÷ 6 = 480 ÷ 6 = 80, because 60 = 10 × 6. Check with 80 × 60 = 4800. For hundreds, 8400 ÷ 400 = 84 hundreds ÷ 4 hundreds = 21. Dividing both numbers by the same power of 10 preserves the number of groups; changing only one of them does not.

For a larger group size, 42000 ÷ 6000 = 42 thousands ÷ 6 thousands = 7. Check: 7 × 6000 = 42000. Use thousands as the common unit, just as you used tens or hundreds.

5. Choose the operations in a practical problem

A club has 14 cartons containing 50 pencils each. It puts the pencils into packs of 20. How many full packs can it make?

Find the total: 14 × 50 = 14 × 5 tens = 70 tens = 700 pencils.
Find the new number of groups: 700 ÷ 20 = 70 tens ÷ 2 tens = 35 packs.
Check in context: 35 packs × 20 pencils = 700 pencils; every pencil is packed.

For a fixed total, making each pack 10 times as large gives one tenth as many packs. For example, 600 items fill 30 packs of 20 or 3 packs of 200.

Curriculum examplesCopied content

Content description: solve problems involving multiplying or dividing natural numbers by multiples and powers of 10 without a calculator, using the multiplicative relationship between the place value of digits.

E1: Materials: Build 3 ones, then make ten copies. Regroup the 30 ones as 3 tens. Repeat with 3 tens to make 3 hundreds. Explain why each place is worth 10 times the place to its right.
E2: Charts and sliders: Use the chart above to represent 406, 4060 and 40600. Read the value of each nonzero digit, then reverse the process with division. Include the ten-thousands column when multiplying 406 by 100.
E3: Patterns to mental calculation: Record 27 × 10, 27 × 100 and 27 × 1000 as 270, 2700 and 27000. A teacher may use a calculator or spreadsheet to explore and check the pattern initially; students then explain it and solve new multiplication and division problems without a calculator.

These teaching activities address the descriptor and its materials, chart and pattern elaborations. Decimal multiplication and division, remainder problems and formal index notation are not required here.

Build, explain and generalise

Digit-value tracker: Draw a place-value chart with columns from ten-thousands to ones. Place 406, then show 406 × 10 and 406 × 100. Say what each nonzero digit is worth.
Factor the multiple: Solve 18 × 20, 35 × 40 and 12 × 300. Name the unit first: 36 tens = 360; 140 tens = 1400; 36 hundreds = 3600.
Inverse-chain check: Explain why 28 × 50 = 1400 and 1400 ÷ 50 = 28. Use the same tens unit to show 140 tens ÷ 5 tens = 28.
Predict before checking: Record 4200 ÷ 10 = 420, 4200 ÷ 100 = 42 and 42000 ÷ 1000 = 42. Create a new exact division pattern and justify it without a calculator.

Support: Use bundled counters or labelled place-value cards and facts such as 3 × 4 before working in tens.

Core: Choose between multiplying and dividing by multiples of 10 and 100, including numbers with internal zeros.

Extend within this skill: Change a pack size, compare the numbers of packs and explain how keeping the total fixed affects the quotient.

Questions and answersWith answers

Important questions and answers

Calculate 47 × 100. 4700: 4 tens become 4 thousands and 7 ones become 7 hundreds.
Calculate 3600 ÷ 100. 36: 3600 contains 36 hundreds. Check 36 × 100 = 3600.
Solve 24 × 30 using factors. 24 × 3 × 10 = 72 × 10 = 720.
Solve 4800 ÷ 60. Rename both quantities in tens: 480 ÷ 6 = 80.
Explain why 406 × 10 is 4060. The 4 is now worth 4000 and the 6 is worth 60. The zero between them still marks an empty place.

Practice and reviewReady for practice

“Add zeros” without meaning: In 406 × 10, explain 4000 + 60. Keep the original placeholder zero as well as the new empty ones place.
30 treated as 3: 30 is 3 tens, so 24 × 30 gives 72 tens, not 72 ones.
Only one number scaled in division: For 4800 ÷ 60, convert both numbers to tens. 480 ÷ 6 gives the same number of groups; 4800 ÷ 6 does not.
Division moves digits the wrong way: Dividing by 10 makes each digit’s value one tenth as large, so the digits occupy places one column to the right.
Units ignored: A total number of items and a number of packs answer different questions. State which one your result describes.

Assessment-style questions and review hints

Calculate 29 × 200 and explain what the 2 represents. Review hint: use hundreds as the unit. Answer: 58 hundreds = 5800.
5400 counters are packed in bags of 30. How many bags? Review hint: rename both numbers in tens. Answer: 540 ÷ 3 = 180 bags.
A student writes 4200 ÷ 70 = 600. Correct the answer and check it. Review hint: multiply the proposed quotient by 70. Answer: 60, because 60 × 70 = 4200.

Exit ticket

Without a calculator, solve 308 × 100 and 7200 ÷ 90. Explain a digit-value change in the first answer and check the second with multiplication.

Expected evidence: 30800; the 8 changes from 8 ones to 8 hundreds. Then 7200 ÷ 90 = 720 tens ÷ 9 tens = 80, and 80 × 90 = 7200.

Review route: Revisit the place-value chart for a misplaced zero; revisit the tens/hundreds unit for a factor-of-10 error; revisit the practical example if the calculation does not answer the story.

Curriculum alignmentStart here

E1: Materials: Build 3 ones, then make ten copies. Regroup the 30 ones as 3 tens. Repeat with 3 tens to make 3 hundreds. Explain why each place is worth 10 times the place to its right.
E2: Charts and sliders: Use the chart above to represent 406, 4060 and 40600. Read the value of each nonzero digit, then reverse the process with division. Include the ten-thousands column when multiplying 406 by 100.
E3: Patterns to mental calculation: Record 27 × 10, 27 × 100 and 27 × 1000 as 270, 2700 and 27000. A teacher may use a calculator or spreadsheet to explore and check the pattern initially; students then explain it and solve new multiplication and division problems without a calculator.

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