Expanded notation with powers of 10: a week of ready-to-teach maths
Five lessons for Year 7 Number. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus a place value mat, digit cards, power of ten match cards, a mini-check and every answer with its working.
Start here: five minutes to Monday
- Skim the week at a glance on the next page.
- Print the five days. Each day is two A4 sheets: a plan and a worksheet.
- Print the place value mat and the card sheet, one copy per pair, and cut out the cards before Day 1. Both come back all week.
- Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.
- Teach straight from the plan. Timings, talk prompts, misconceptions and worked answers are all on the one page.
The idea in plain words
Every place in a whole number is worth ten of the place to its right, so the places are the powers of ten: ones 10⁰ = 1, tens 10¹ = 10, hundreds 10² = 100, thousands 10³ = 1000, and so on. A digit is worth the digit times its place. Expanded notation writes those worths as a sum: 3705 = 3 × 10³ + 7 × 10² + 5 × 10⁰. A zero adds nothing, so its term drops out of the sum, but in the number it still holds the other digits in their places. Multiplying by ten moves every digit one place to the left.
One day, one lesson
The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.
How we write big numbers and powers
Numbers with five or more digits are grouped in threes with a small space: 47 000, 2 604 315. Four-digit numbers are written without a space: 3705. Read 10⁴ as “ten to the power of four”; the exponent counts the zeros in 10 000. 10⁰ = 1 is the ones place, and the term for the ones is often written as just the digit: 3705 = 3 × 10³ + 7 × 10² + 5. No calculators this week.
The week at a glance
One lesson a day for a week. Each day stands on the day before, so run them in order.
| Day | Lesson | Students learn and do | On screen |
|---|---|---|---|
| 1 | Every place is a power of ten | Name the places to the millions as powers of ten, say what each digit is worth, and see why the exponent counts the zeros | Each place is a power of ten; The exponent counts the zeros |
| 2 | Expanding a number | Write numbers in expanded notation with place values and with powers of ten, and explain what the zeros do | Expanding a number |
| 3 | Times ten shifts the digits | Multiply and divide by powers of ten as a shift of the digits, and find a missing power | Times ten shifts the digits |
| 4 | Assembling from the parts | Rebuild a number from its parts in any order, with zeros holding the empty places, and regroup parts bigger than 9 | Assembling from the parts |
| 5 | Big numbers at work | Read, write, compare and order large numbers from real life in expanded notation | Self-check quiz as a class game |
How the week builds
Day 1 names every place as a power of ten; Day 2 takes numbers apart into those places; Day 3 sees multiplying by ten as a shift of place; Day 4 puts numbers back together from their parts; and Day 5 uses all of it on large numbers from real life. It builds on the place value of primary school and on the exponents of the last two units, and it prepares the exponent laws of Year 8 and the very large and very small numbers of later years.
Materials for the week (one trip)
- From the classroom: pencils, scissors, small envelopes or paper clips for the cards, this pack printed.
- Cut-out sheet 1, one per pair: the place value mat, ones to millions (all week).
- Cut-out sheet 2, one per pair: 20 digit cards for the mat (Days 2 to 4) and 14 power of ten match cards (Day 1).
- No calculators and no other equipment to buy.
Words for the word wall
place, place value, digit, ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, power of ten, exponent, index, expanded notation, placeholder, regroup
Dear families
This week in maths, Year 7 is writing numbers in expanded notation with powers of ten. Each place in a number is a power of ten (hundreds = 10² = 100, thousands = 10³ = 1000), so 3705 = 3 × 10³ + 7 × 10² + 5. We also see why multiplying by ten moves every digit one place to the left.
Try this at home
- Find a big number in the news or on a sign (a crowd, a price, a distance). What is each digit worth?
- Ask: how many zeros are in 10⁶, and what is it called? (Six; a million.)
- Ask: what is 35 × 1000? What happens to the digits?
- Ask: why is the 0 in 3705 needed, if it adds nothing?
Big numbers this week
Fill one row a day with a large number you found and its expanded form.
| Day | Where I found it | The number | In powers of ten |
|---|---|---|---|
| Monday | |||
| Tuesday | |||
| Wednesday | |||
| Thursday | |||
| Friday |
Printed from the free seegongsik Expanded notation with powers of 10 teaching pack · seegongsik.com/au/y7/number/AC9M7N03/pack
Every place is a power of ten
The places of a whole number are the powers of ten. Students label the places to the millions, say what a digit is worth, and see that the exponent counts the zeros.
We are learning to
- name each place from the ones to the millions as a power of ten,
- say what a digit is worth from its place,
- explain why each place is ten times the place to its right.
Success criteria
- I can say that the 6 in 2 604 315 is worth 6 × 10⁵ = 600 000.
- I can write 10⁴ as a number and say why it has four zeros.
You need
The power of ten match cards from cut-out sheet 2 and the place value mat (one each per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Columns of ten Show the board (see the box). Write 10³ = 10 × 10 × 10 = 1000 on the mat. Ask: “The thousands place is 10³. Why is the next place to the left 10⁴?” |
| 15 min | Power match Pairs lay the 14 match cards face down and take turns to turn two, keeping a power and its value as a pair. Ask: “How did you know 10⁶ goes with the card that has six zeros?” |
| 35 min | Places and worth Students complete Parts A to C; early finishers do Part D. Share Part B, naming each place as a power. |
Two shorter sessions instead? Split after the match game; start the second with Part A.
Watch for these ideas
- Naming a place by counting from the left. Count from the ones: the 5 in 35 120 is in the thousands, 10³.
- Thinking 10⁰ is 0. It is the ones place, worth 1.
- Reading 10⁴ as 10 × 4 = 40. It is 10 × 10 × 10 × 10.
Answers
- Part A: ones 10⁰ = 1, tens 10¹ = 10, hundreds 10² = 100, thousands 10³ = 1000, ten thousands 10⁴ = 10 000, hundred thousands 10⁵ = 100 000, millions 10⁶ = 1 000 000.
- Part B: 7 × 10³ = 7000 (thousands); 5 × 10⁵ = 500 000 (hundred thousands); 8 × 10⁴ = 80 000 (ten thousands); 9 × 10³ = 9000 (thousands); 4 × 10⁵ = 400 000 (hundred thousands); 6 × 10⁴ = 60 000 (ten thousands).
- Part C: 10 000; 1 000 000; 1; 10³; 10⁵; 7.
- Part D: the first 7 is worth 7000 and the last is worth 7, so 1000 times, or 10³; 10 × 10 × 10 × 10 × 10 = 10⁵ = 100 000.
Every place is a power of ten
A. Label the places
Fill in each power of ten and its value. Two are done for you.
| millions | hundred thousands | ten thousands | thousands | hundreds | tens | ones |
|---|---|---|---|---|---|---|
| 10³ | 10⁰ | |||||
| 1000 | 1 |
B. What is the underlined digit worth?
| Number | Place | Digit × power of ten | Worth |
|---|---|---|---|
| 47 382 | |||
| 506 913 | |||
| 2 080 640 | |||
| 9135 | |||
| 3 400 027 | |||
| 61 205 |
C. Powers of ten both ways
D. Stretch
In 7707, how many times as much is the first 7 worth as the last 7? Write your answer as a power of ten. ____________
Write 10 × 10 × 10 × 10 × 10 as a power of ten and as a number. ____________________
Expanding a number
Expanded notation writes a number as the sum of what its digits are worth. Students expand numbers with place values and with powers of ten, see the zero terms drop out, and fix expansions that have gone wrong.
We are learning to
- write a number in expanded notation with place values and with powers of ten,
- explain why a zero term drops out of the sum but the zero stays in the number,
- check an expansion by adding it back.
Success criteria
- I can write 50 408 as 5 × 10⁴ + 4 × 10² + 8 × 10⁰.
- I can say why 50 408 needs its zeros.
You need
The place value mat and digit cards (one set per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Take it apart Show the board (see the box). Then expand 50 408 both ways together. Ask: “The zeros in 50 408 add nothing. What would happen to the number if we left them out?” |
| 15 min | Build and expand One partner builds a number on the mat with digit cards; the other writes it in expanded notation with powers of ten. Swap. Ask: “Which cards did you need for the empty places?” |
| 35 min | Expand and check Students complete Parts A to C; early finishers do Part D. Share Part C: how did adding back catch each mistake? |
Two shorter sessions instead? Split after the mat; start the second with Part A.
Watch for these ideas
- Matching a digit to the wrong power after a zero, as in 7 × 10³ + 2 × 10² for 7020. Count places from the ones.
- Leaving the zeros out of the number as well: 50 408 is not 548.
- Thinking every place needs a term. 0 × 10² may be written or left out.
Answers
- Part A: 6482 = 6 × 10³ + 4 × 10² + 8 × 10¹ + 2 × 10⁰; 30 507 = 3 × 10⁴ + 5 × 10² + 7 × 10⁰; 704 060 = 7 × 10⁵ + 4 × 10³ + 6 × 10¹; 1 250 009 = 1 × 10⁶ + 2 × 10⁵ + 5 × 10⁴ + 9 × 10⁰.
- Part B: 30 507 drops 0 × 10³ and 0 × 10¹; 704 060 drops 0 × 10⁴, 0 × 10² and 0 × 10⁰. The zeros hold the other digits in their places; without them 30 507 would read as 357.
- Part C: 4060 wrong, as it adds to 4600: 4 × 10³ + 6 × 10¹; 52 300 right; 9005 right; 18 040 wrong, as it adds to 18 400: 1 × 10⁴ + 8 × 10³ + 4 × 10¹; 700 200 wrong, as it adds to 702 000: 7 × 10⁵ + 2 × 10².
- Part D: 10 101 = 1 × 10⁴ + 1 × 10² + 1 × 10⁰; 1 010 101 = 1 × 10⁶ + 1 × 10⁴ + 1 × 10² + 1 × 10⁰. Each 1 sits two places from the next, so the powers go up in twos.
Expanding a number
A. Expand it two ways
| Number | With place values | With powers of ten |
|---|---|---|
| 6482 | ||
| 30 507 | ||
| 704 060 | ||
| 1 250 009 |
B. The zeros
Which terms drop out of 30 507? ________________________________
Which terms drop out of 704 060? ________________________________
Why must the zeros still be written in the number? ________________________________________
C. Right or wrong?
Add each expansion back. Tick the right ones; write the correct expansion for the others.
| 4060 = 4 × 10³ + 6 × 10² | |
| 52 300 = 5 × 10⁴ + 2 × 10³ + 3 × 10² | |
| 9005 = 9 × 10³ + 5 × 10⁰ | |
| 18 040 = 1 × 10⁴ + 8 × 10³ + 4 × 10² | |
| 700 200 = 7 × 10⁵ + 2 × 10³ |
D. Stretch
Expand 10 101 and 1 010 101 with powers of ten. What pattern do you see in the powers, and why? ________________________
Times ten shifts the digits
Each place is ten times the place to its right, so multiplying by ten moves every digit one place to the left, and dividing by ten moves it one place to the right. Students multiply and divide by powers of ten as shifts, and find a missing power.
We are learning to
- multiply a whole number by a power of ten by moving its digits to the left,
- divide by a power of ten by moving the digits to the right,
- find the missing power of ten in a multiplication or a division.
Success criteria
- I can work out 36 × 10⁴ = 360 000 and say how far the digits moved.
- I can find the power in 51 × 10? = 51 000.
You need
The place value mat and digit cards (one set per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Slide left Show the board (see the box). Ask: “47 became 4700. Did we add two zeros, or did the 4 and the 7 move? Which is the better way to say it?” |
| 15 min | Slide on the mat Pairs put 36 on the mat with digit cards and slide the cards one place left for each × 10, filling the empty places with zero cards; then slide right to divide. Ask: “For 36 × 10⁴, how many places do the cards move? Which cards fill in behind?” |
| 35 min | Shift both ways Students complete Parts A to C; early finishers do Part D. Share Part C: how did you count the places? |
Two shorter sessions instead? Split after the mat; start the second with Part A.
Watch for these ideas
- Learning “add a zero” as the rule. It breaks with decimals (4.7 × 10 is not 4.70); the digits move one place left.
- Moving the digits the wrong way when dividing. Dividing by 10³ moves them three places right.
- Losing a zero inside the number: 407 × 10² is 40 700, not 4700.
Answers
- Part A: 63 × 10¹ = 630; 63 × 10² = 6300; 63 × 10³ = 63 000; 305 × 10² = 30 500; 6 × 10⁵ = 600 000; 82 × 10⁴ = 820 000.
- Part B: 5800 ÷ 10¹ = 580; 5800 ÷ 10² = 58; 72 000 ÷ 10³ = 72; 900 000 ÷ 10⁴ = 90; 1 300 000 ÷ 10⁵ = 13.
- Part C: 26 × 10³ = 26 000; 4 × 10⁵ = 400 000; 870 000 ÷ 10⁴ = 87; 3 × 10¹ = 30.
- Part D: 4700 is 470 tens, 47 hundreds, or 4 thousands and 700 more; 2 500 000 ÷ 10³ = 2500.
Times ten shifts the digits
A. Multiply
B. Divide
C. The missing power
Write the exponent in the box.
How did you work out the missing powers? __________________________________________
D. Stretch
How many tens are in 4700? ______ How many hundreds? ______ How many thousands, and how much is left? ________________
A number times 10³ is 2 500 000. What was the number? ______________
Assembling from the parts
Reading expanded notation backwards rebuilds the number. Students assemble numbers from parts given in any order, let zeros hold the empty places, and regroup a part with more than 9 in one place.
We are learning to
- write the number for an expansion given in any order,
- use zeros to hold the empty places,
- regroup a part with more than 9 in one place.
Success criteria
- I can turn 6 × 10⁴ + 8 × 10¹ into 60 080.
- I can work out 13 × 10² + 6 × 10⁰ = 1306.
You need
The place value mat and digit cards (one set per pair), the worksheet. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Back together Show the board (see the box). Then build 6 × 10⁴ + 8 × 10¹ on the mat: 60 080. Ask: “Which places did the zeros fill, and what would the number be without them?” |
| 15 min | Mat race One partner reads out three or four terms in a jumbled order; the other builds the number on the mat and reads it aloud. Swap. Ask: “The terms came out of order. How did you know where each card goes?” |
| 35 min | Assemble and regroup Students complete Parts A to C; early finishers do Part D. Share Part C on the mat: where does the extra ten go? |
Two shorter sessions instead? Split after the mat race; start the second with Part A.
Watch for these ideas
- Writing the digits side by side without the zeros: 6 × 10⁴ + 8 × 10¹ written as 68.
- Placing the terms in the order they are listed. Start from the biggest power, whatever the order.
- Writing 13 in one place. 13 × 10² is 1300, so 1 goes into the thousands.
Answers
- Part A: 4029, 70 300, 506 001, 8 040 020, 9200, 60 103.
- Part B: 70 300 is seventy thousand, three hundred; 506 001 is five hundred and six thousand and one; 8 040 020 is eight million, forty thousand and twenty.
- Part C: 1200 + 50 = 1250, or 1 × 10³ + 2 × 10² + 5 × 10¹; 3000 + 1500 = 4500, or 4 × 10³ + 5 × 10²; 250 + 7 = 257, or 2 × 10² + 5 × 10¹ + 7 × 10⁰; 60 000 + 11 000 + 4 = 71 004, or 7 × 10⁴ + 1 × 10³ + 4 × 10⁰.
- Part D: biggest 9 × 10⁴ + 6 × 10³ + 2 × 10² = 96 200; smallest 9 × 10⁰ + 6 × 10¹ + 2 × 10² = 269. The biggest digit takes the biggest power for the biggest number, and the smallest power for the smallest.
Assembling from the parts
A. Assemble the number
Some come out of order. Start with the biggest power.
4 × 10³ + 2 × 10¹ + 9 × 10⁰ = ________________
7 × 10⁴ + 3 × 10² = ________________
5 × 10⁵ + 6 × 10³ + 1 × 10⁰ = ________________
8 × 10⁶ + 4 × 10⁴ + 2 × 10¹ = ________________
2 × 10² + 9 × 10³ = ________________
3 × 10⁰ + 6 × 10⁴ + 1 × 10² = ________________
B. In words
70 300: ______________________________________________
506 001: ______________________________________________
8 040 020: ______________________________________________
C. Regroup
A part may have more than 9 in one place. Work out what it is worth, then add.
12 × 10² + 5 × 10¹ = ________________
3 × 10³ + 15 × 10² = ________________
25 × 10¹ + 7 × 10⁰ = ________________
6 × 10⁴ + 11 × 10³ + 4 × 10⁰ = ________________
D. Stretch
Use the digits 6, 2, 9, each times a different power of ten from 10⁰ to 10⁴. What are the biggest and the smallest numbers you can make? Write each in expanded notation and as a number. ____________________________
Big numbers at work
Large numbers from real life put the week together: reading them, writing them in expanded notation, comparing them by their highest place, and multiplying and dividing by powers of ten.
We are learning to
- read and write large numbers in figures, in words and in expanded notation,
- compare and order large numbers by their highest place,
- use powers of ten to solve problems.
Success criteria
- I can write two million and thirty thousand as 2 030 000 = 2 × 10⁶ + 3 × 10⁴.
- I can put 5 × 10⁴, 48 × 10³ and 50 001 in order, smallest first.
You need
The worksheet and the unit’s self-check quiz on the board. No calculators.
Lesson flow (about 60 minutes)
| 10 min | Numbers in the news Write a large city of about 5 million people as 5 000 000 = 5 × 10⁶, and a town of 500 000 as 5 × 10⁵. Ask: “Which is quicker to compare at a glance: 5 000 000 and 500 000, or 5 × 10⁶ and 5 × 10⁵?” |
| 30 min | Problems Students work through problems 1 to 6, writing each step; 7 and 8 are the stretch. |
| 20 min | Class quiz game Run the self-check quiz as a class game (see the box), then wrap up. Ask: “What does the exponent tell you about a number before you even read its digits?” |
Two shorter sessions instead? Split after problem 3; start the second with problem 4.
Watch for these ideas
- Comparing by the first digit. A term such as 39 × 10⁴ has to be worked out before it can be compared.
- Miscounting the zeros in a million: 1 000 000 has six, so it is 10⁶.
- Writing 27 million as a single power, such as 10²⁷. It is 27 × 10⁶.
Answers with working
- 27 million = 27 000 000 = 2 × 10⁷ + 7 × 10⁶.
- 384 000 km = 3 × 10⁵ + 8 × 10⁴ + 4 × 10³.
- 9 × 10⁴ = 90 000 and 1 × 10⁵ = 100 000; 1 × 10⁵ is bigger by 10 000.
- 27 000 000 ÷ 100 000 = 270 stadiums.
- 39 × 10⁴ = 390 000, 4 × 10⁵ = 400 000, 400 001, 9 × 10⁵ + 4 × 10⁴ = 940 000.
- $1 000 000 ÷ $100 = 10 000 = 10⁴ notes.
- 1 000 000 × 1000 = 1 000 000 000, 9 zeros, 10⁹ (a billion).
- biggest 5 × 10⁶ + 5 × 10⁵ = 5 500 000; smallest 5 × 10⁶ + 5 × 10⁰ = 5 000 005.
Big numbers at work
Show each step. Real-world figures are rounded.
Problems (7 and 8 are the stretch)
- Australia had about 27 million people in 2024. Write 27 million as a number and in expanded notation with powers of ten.
- The Moon is on average about 384 000 km from Earth. Write this distance in expanded notation with powers of ten.
- Which is bigger, 9 × 10⁴ or 1 × 10⁵? By how much?
- A stadium holds about 10⁵ people. How many stadiums that size would hold 27 million people?
- Put in order, smallest first: 4 × 10⁵; 39 × 10⁴; 400 001; 9 × 10⁵ + 4 × 10⁴.
- How many $100 notes make $1 000 000? Write your answer as a number and as a power of ten.
- Stretch. Write 10⁶ × 10³ out in full. How many zeros does it have? Write it as a single power of ten.
- Stretch. A 7-digit number has just two terms in its expanded notation, and both digits are 5. What are the biggest and the smallest such numbers?
Place value mat
One mat per pair, kept all week. On Day 1, write the power of ten under each place name. From Day 2, lay digit cards in the big boxes to build numbers; slide them left or right to multiply or divide by ten.
| millions | hundred thousands | ten thousands | thousands | hundreds | tens | ones |
|---|---|---|---|---|---|---|
Row 2: the power of ten. Row 3: a digit card. Every empty place between digits needs a zero card.
Digit cards and power of ten match cards
Digit cards for the mat (Days 2 to 4)
Power of ten match (Day 1)
Lay all 14 cards face down. Take turns to turn two; keep them if a power matches its value.
Teacher note: the pairs are 10⁰ with 1, 10¹ with 10, 10² with 100, 10³ with 1000, 10⁴ with 10 000, 10⁵ with 100 000, 10⁶ with 1 000 000.
What we know: Expanded notation with powers of 10
Work on your own, without a calculator. Show your working.
- In 3 862 041, what is the 6 worth? Write it as a digit times a power of ten. ______________________
- Write 10⁵ as a number. ______________________
- Write 40 716 in expanded notation with powers of ten. ______________________________________
- What number is 9 × 10⁵ + 3 × 10³ + 8 × 10⁰? ______________________
- 58 × 10³ = ______________
- 6 200 000 ÷ 10⁴ = ______________
- The zero in 40 716 drops out of its expanded notation. Why must it still be written in the number? ______________________
- Which is bigger, 7 × 10⁴ or 1 × 10⁵? By how much? ______________________
How did this week feel? Circle one: I can teach it · I am getting there · I need more help
Answers and marking guide
Answers with working
- The 6 is in the ten thousands: 6 × 10⁴ = 60 000.
- 100 000, a 1 and 5 zeros.
- 40 716 = 4 × 10⁴ + 7 × 10² + 1 × 10¹ + 6 × 10⁰.
- 903 008.
- 58 000: the digits move 3 places left.
- 620: the digits move 4 places right.
- It holds the thousands place, so the 4 stays in the ten thousands. Without it the number would read as 4716.
- 1 × 10⁵ = 100 000 and 7 × 10⁴ = 70 000, so 1 × 10⁵ is bigger by 30 000.
A quick three-level guide
| Idea | Working towards | At standard | Beyond |
|---|---|---|---|
| Places as powers (Q1, Q2) | names a place by counting from the left, or reads 10⁴ as 40 | names each place to the millions as a power of ten and says what a digit is worth | explains why each place is ten times the one to its right |
| Expanding and assembling (Q3, Q4, Q7) | drops the zeros from the number as well as from the sum | expands with powers of ten and assembles a number, zeros holding empty places | regroups a part with more than 9 in one place |
| Shifting by powers of ten (Q5, Q6) | adds zeros at the end when dividing, or moves the wrong way | multiplies and divides by a power of ten as a shift of the digits | finds a missing power from the two numbers |
| Comparing big numbers (Q8) | compares by the first digit only | compares two numbers written with powers of ten by working out both | compares at a glance from the highest place |
Eight questions, four ideas. A student at standard answers most questions with working and checks an expansion by adding it back.
Weekly class record
Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.
| Name | Places as powers | Expanding | Shifting by ten | Assembling | Big numbers |
|---|---|---|---|---|---|
The five columns are the five days: places as powers, expanding, shifting by ten, assembling, and big numbers.