AC9M4N05: Multiply and divide by multiples and powers of 10

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.

power of 10

10, 100, 1000, … formed by multiplying by 10 repeatedly

place-value shift

digits changing position when scaling by powers of 10

factor

a number multiplied by another number

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

1. 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.
2. Factor the multiple: Solve 18 × 20, 35 × 40 and 12 × 300. Name the unit first: 36 tens = 360; 140 tens = 1400; 36 hundreds = 3600.
3. Inverse-chain check: Explain why 28 × 50 = 1400 and 1400 ÷ 50 = 28. Use the same tens unit to show 140 tens ÷ 5 tens = 28.
4. 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.

Important questions and answers

1. Calculate 47 × 100. 4700: 4 tens become 4 thousands and 7 ones become 7 hundreds.
2. Calculate 3600 ÷ 100. 36: 3600 contains 36 hundreds. Check 36 × 100 = 3600.
3. Solve 24 × 30 using factors. 24 × 3 × 10 = 72 × 10 = 720.
4. Solve 4800 ÷ 60. Rename both quantities in tens: 480 ÷ 6 = 80.
5. 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.

Teach with the Topic Guide and Classroom View, then use Homework, Practice and Test.

Assessment-style questions and review hints

1. Calculate 29 × 200 and explain what the 2 represents. Review hint: use hundreds as the unit. Answer: 58 hundreds = 5800.
2. 5400 counters are packed in bags of 30. How many bags? Review hint: rename both numbers in tens. Answer: 540 ÷ 3 = 180 bags.
3. 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.

• “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.

Curriculum equivalents for Solve problems involving multiplying or dividing natural numbers by multiples...

Mapped skill: 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

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.

• QCAA: Australian Curriculum v9 Mathematics resources
• Victorian Curriculum F–10
• NSW Curriculum

• AC9M4N01
• AC9M4N06