Place value to one million
Reading, writing, comparing and rounding whole numbers to 1,000,000
About four lessons of 45 to 60 minutes
Some numbers are far too big to count one at a time
How many people live in your nearest city? How many views does a popular video have? How far is it to the Moon in kilometres? These answers run into the hundreds of thousands and beyond, and nobody counts them one by one. We read them, compare them and round them using place value, the same idea that told you a 3 could be worth 3, or 30, or 300.
Today the places keep going: past the hundreds into thousands, ten thousands, hundred thousands, all the way to one million. By the end you will read a six-digit number aloud without stumbling, say exactly what each digit is worth, decide which of two big numbers is greater, and round a giant number to a friendly one for a quick estimate.
- A city of 452,367 peoplefour hundred fifty-two thousand, three hundred sixty-seven, read in periods of three digits
- A video with 380,000 viewsthe 3 is worth 300,000, ten times more than the place to its right
- About 384,000 km to the Moonrounded from 384,400 to the nearest thousand for a quick sense of the distance
- A prize of $250,000a quarter of a million dollars, easy to compare with $205,000 once you read the places
What students will be able to do
Students will read and write whole numbers to one million in numerals, words and expanded form, state the value of any digit using place value, compare two multi-digit numbers with the symbols >, = and <, and round a multi-digit number to any place using a number line and the digit rule.
- I can read and write a number up to 1,000,000 in numerals, in words, and in expanded form.
- I can say the value of any digit, and explain that each place is ten times the place to its right.
- I can compare two large numbers and record it with >, = or <.
- I can round a large number to the nearest ten, hundred, thousand, ten thousand or hundred thousand.
- I can use rounding to make a quick estimate and check whether an answer is reasonable.
Standards this unit teaches
- 4.NBT.A.1Common Core (US)Place value of multi-digit numbers
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.
- 4.NBT.A.2Common Core (US)Read, write and compare large numbers
Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
- 4.NBT.A.3Common Core (US)Round multi-digit numbers
Use place value understanding to round multi-digit whole numbers to any place.
- AC9M3N01Australian Curriculum v9 (ACARA)Order numbers beyond 10,000 (Year 3 anchor)
Recognise, represent and order natural numbers using naming and writing conventions for numerals beyond 10 000. Australia introduces reading and ordering large whole numbers a year earlier than the US, so this unit maps to Australian Year 3 for its reading and comparing work.
- AC9M4N06Australian Curriculum v9 (ACARA)Estimate to check answers (Year 4)
Estimate quantities and use estimation when solving problems to decide whether calculations are reasonable. ACARA does not name rounding to a chosen place as a separate Year 4 descriptor, it frames it as estimation, so the rounding work here supports this Year 4 code.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Place value of tens, hundreds and thousandsthe starting point this unit extends leftward
- Expanded formwriting a number as the sum of the value of each digit
- Comparing and ordering numberscompare place by place from the largest place
- Roundingthe nearest friendly number, from a number line
- The number lineequal spacing, used here to round large numbers
Words to teach and display
- Digit
- one of the symbols 0 to 9 that a number is written with
- Place value
- the value a digit has because of its position in the number
- Period
- a group of three digits, ones, thousands and millions, set off by commas
- Standard form
- the normal way of writing a number, such as 452,367
- Expanded form
- a number written as the sum of the value of each digit
- Round
- replace a number with a nearby friendly number ending in zeros
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. The places keep going: thousands to millions
ConcreteDraw a place-value chart and read it from right to left: ones, tens, hundreds, then thousands, ten thousands, hundred thousands, and finally millions. Each new place to the left is worth ten times the place before it. Ten ones make a ten, ten tens make a hundred, ten hundreds make a thousand, and the pattern never stops. The single most important rule of our number system is this: a digit in one place is worth ten times what the same digit would be one place to its right.
So the digit 4 is worth 4 in the ones place, 40 in the tens place, 400 in the hundreds place, 4,000 in the thousands, 40,000 in the ten thousands, and 400,000 in the hundred thousands. The digit did not change, its place did, and the place decides its value.
Group the six places into two periods of three: the thousands period and the ones period. The comma between them is not decoration, it marks the end of the thousands. Reading the chart in periods is what makes a huge number sayable.
- In 452,367, what is the digit 5 worth?
- How many times bigger is the thousands place than the hundreds place?
- Which two periods make up a six-digit number, and what separates them?
2. Reading and writing big numbers
PictorialTurn the chart into a reading routine. Split the number into its periods at the comma, read the thousands period, say the word thousand at the comma, then read the ones period. 452,367 becomes four hundred fifty-two, thousand, three hundred sixty-seven. Writing works the same way in reverse, and expanded form makes every digit's value show.
Watch the zeros: they hold a place open so the other digits stay put. In 208,540 the zero in the ten-thousands place and the zero in the ones place are doing real work, and you do not say them, you say two hundred eight thousand, five hundred forty.
Read 452,367, then write it in expanded form.
- Split at the comma into periods: 452 and 367.
- Read the thousands period, then say thousand: four hundred fifty-two thousand.
- Read the ones period: three hundred sixty-seven.
- Expanded form is the value of each digit added: 400,000 + 50,000 + 2,000 + 300 + 60 + 7.
Answer: Four hundred fifty-two thousand, three hundred sixty-seven. Expanded: 400,000 + 50,000 + 2,000 + 300 + 60 + 7.
- Read 380,000 aloud. Which places hold a zero?
- Write four hundred sixty thousand, five hundred nine in standard form.
3. Comparing large numbers
AbstractTo decide which of two numbers is greater, compare them place by place starting from the largest place, not from the right. Line the numbers up so their places match, then move left to right until you find the first place where the digits differ. The bigger digit there wins, and you record the result with > (greater than), < (less than) or = (equal).
This is why lining up matters: 452,367 and 452,637 share the same hundred thousands, ten thousands and thousands. The first place that differs is the hundreds, where 3 is less than 6, so 452,367 is the smaller number. Everything after that first difference is irrelevant to the comparison.
Compare 452,367 and 452,637 with >, = or <.
- Line up the places. Hundred thousands: 4 and 4, equal. Ten thousands: 5 and 5, equal. Thousands: 2 and 2, equal.
- Move to the hundreds, the first place that differs: 3 and 6.
- 3 is less than 6, so 452,367 is less than 452,637. The later digits do not change this.
Answer: 452,367 < 452,637.
- Compare 318,905 and 318,950. Which place decides it?
- Why is it wrong to compare two whole numbers starting from the ones digit?
4. Rounding big numbers
AbstractRounding swaps a number for a nearby friendly one that ends in zeros, which makes it quick to say and easy to estimate with. The number line makes it honest: mark the two friendly numbers your target sits between, mark the halfway point, and see which side of halfway your number falls on. Then the digit rule gives the same answer fast: look at the digit just to the right of the place you are rounding to, 5 or more rounds up, 4 or less rounds down.
To round 452,367 to the nearest ten thousand, the two candidates are 450,000 and 460,000, and halfway is 455,000. Since 452,367 is left of 455,000, it rounds down to 450,000. The digit rule agrees: the digit to the right of the ten-thousands place is the thousands digit, 2, and 2 is less than 5, so round down.
The place to the right of your rounding place decides everything, and every digit after it becomes zero. Rounding never changes the digits to the left of the rounding place unless a round-up carries.
Round 452,367 to the nearest ten thousand.
- The ten-thousands place holds 5, so the candidates are 450,000 and 460,000.
- Look at the digit to its right, the thousands digit: 2.
- 2 is less than 5, so round down. All places after the ten thousands become zero.
Answer: 452,367 rounds to 450,000.
- Round 452,367 to the nearest hundred thousand. Which digit do you check?
- Round 6,482 to the nearest thousand. What happens to the hundreds, tens and ones?
Common misconceptions and how to address them
MisconceptionThe commas in a big number are just decoration, or they mark a decimal point.
Why it happens: Students meet commas as pauses in writing and do not yet see them as place-value markers.
How to address it: Name the comma as the end of the thousands period. Every comma, reading right to left, closes a group of three: ones, thousands, millions. Read 452,367 as four hundred fifty-two thousand, then the last three.
MisconceptionRead the digits one at a time, so 452,367 is four, five, two, three, six, seven.
Why it happens: It works for a phone number, and nobody has shown that a number is read in periods of three.
How to address it: Cover all but the thousands period and read it, say thousand at the comma, then uncover and read the ones period. Practise with a place-value chart until the periods are automatic.
MisconceptionThe 5 in 452,367 is just five.
Why it happens: The digit looks the same wherever it sits, so its place is easy to ignore.
How to address it: Point to the place-value chart: the 5 is in the ten-thousands place, so it is worth 50,000. Say the value in full, five ten-thousands, not five.
MisconceptionTo compare two numbers, compare the ones digits, or the number with the bigger last digit wins.
Why it happens: Students carry over habits from small numbers and forget that the largest place decides size.
How to address it: Compare place by place from the left. Stop at the first place where the digits differ, and the bigger digit there wins. In 452,367 versus 452,637 the ones digit 7 is bigger, but the hundreds place already decided it the other way.
MisconceptionWhen rounding, change the digits to the left as well, so 452,367 to the nearest ten thousand is 500,000.
Why it happens: Round-up gets over-applied, or students round several places at once.
How to address it: Rounding only zeros the places to the right of the rounding place. Check the single digit just to the right: here it is the thousands digit 2, so round down and the answer is 450,000. Digits to the left stay put unless a carry forces them up.
MisconceptionAfter rounding, keep the leftover digits, so 452,367 to the nearest thousand is 452,367 with the last part unchanged.
Why it happens: Students round the target place but forget that the smaller places must become zero.
How to address it: Rounding produces a friendly number ending in zeros. To the nearest thousand, 452,367 becomes 452,000, and every place below thousands is zero.
Guided practice (with answers)
1. What is the digit 4 worth in 452,367?
Answer: 400,000, four hundred thousand. It sits in the hundred-thousands place.
2. Write 600,000 + 70,000 + 3,000 + 200 + 50 + 1 in standard form.
Answer: 673,251. Each part goes in its place.
3. Read 208,540 in words.
Answer: Two hundred eight thousand, five hundred forty. The zeros are not spoken but hold their places.
4. Compare 318,905 and 318,950 with > or <.
Answer: 318,905 < 318,950. They agree until the tens place, where 0 is less than 5.
5. Round 6,482 to the nearest thousand.
Answer: 6,000. The hundreds digit is 4, which is less than 5, so round down and zero the lower places.
6. Round 452,367 to the nearest hundred thousand.
Answer: 500,000. The digit to the right of the hundred-thousands place is 5, so round up.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Begin with place value and reading, then move to comparing and rounding once reading is secure.
Differentiation
- Keep a printed place-value chart on every desk and have students write each digit into its column before reading.
- Start with five-digit numbers (ten thousands) before moving to six digits.
- For comparing, have students underline the first place that differs before choosing the symbol.
- For rounding, keep the number line in front of them so it is a distance question, not a rule to recall.
- Round the same number to several different places and discuss why the answers differ.
- Order a set of five six-digit numbers from least to greatest and justify the order.
- Extend past a million: read and write seven-digit numbers, adding the millions period.
- Pose an estimation problem, such as roughly how many seats in a stadium, and defend a rounded answer.
Assessment: exit ticket
A three-question exit ticket in the last five minutes. It samples value of a digit, comparing, and rounding, the three pillars of the unit.
1. What is the value of the digit 3 in 439,120?
Answer: 30,000, three ten-thousands.
2. Fill in > or <: 74,905 __ 74,950.
Answer: 74,905 < 74,950. The tens place decides it.
3. Round 68,214 to the nearest ten thousand.
Answer: 70,000. The thousands digit is 8, which is 5 or more, so round up.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 the places and value of a digit (section 1), Lesson 2 reading, writing and expanded form (section 2), Lesson 3 comparing (section 3), Lesson 4 rounding plus the exit ticket (section 4 and assessment).
- Language to keep saying: what is it worth, ten times the place to its right, read it in periods, compare from the largest place. These phrases pre-empt most of the misconceptions.
- The part-whole bars are drawn to scale, so the hundred-thousands part dwarfs the thousands part. That is honest and reinforces where most of a number's size lives.
- The rounding number line uses big numbers on the ticks. Only the two ends are labelled to keep it readable, and the marks show the target and the halfway point. Teach it as a distance question first, then introduce the digit rule as the fast version of the same decision.
- Curriculum note and a US and AU divergence: US Grade 4 (4.NBT.A) reads, compares and rounds numbers to one million. ACARA orders large whole numbers a year earlier, in Year 3 (AC9M3N01), and treats rounding as part of estimation (AC9M4N06 in Year 4) rather than as a separate skill. So this unit maps to Australian Year 3 for reading and comparing, and Year 4 for the estimation strand.
- Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.