KS3 · Maths

Place value of whole numbers

Three digit cards: 7, 1 and 2. Lay them in a row and you make a number, but the same three digits can make very different sizes. Why?

Try it first

Same three cards, very different numbers

Jacob has three digit cards: 7, 1 and 2. He lays all three in a row to make a number. Here are four numbers he could make. Before you check, slide each tag to where you think that number sits on the line.

Reading a big integer, one move at a time

Problem

Here is 6 180 254. What is the 8 worth, and how can we write the whole integer as a sum of place values?

Zeros

Which zeros matter?

Four learners are arguing about zeros in integers.

Which of these is closest to what you think right now?
How sure are you?

Rearranging digits

How many integers can three cards make?

You must use all three cards each time. Pick a first digit, then a second, then a third, and see where each path ends. Before you start: how many different integers do you think there will be?

First digit (hundreds) → Second digit (tens) → Third digit (ones)

3 × 2 × 1 = 6 different integers, and the tree ends 6 times.

On Digit cards 1, 2 and 3. 3 branches to choose from.

Three choices for the first digit, then two for the second, then one for the third. Walking the tree like this means no integer gets missed.

Watch out: Cards with a repeat behave differently. With 4, 2 and 2 the two 2s can swap places and the integer does not change, so the six paths make only three different integers: 422, 242 and 224.

Digit cards with a condition

Where does Mia's answer go wrong?

Mia uses the digit cards 9, 8, 7, 6 and 0, once each. She wants the smallest five-digit even integer. Read her working and tap the line where it first goes wrong.

Mia's answer — which line goes wrong?

Saying the same integer in different ways

Which integer does each name mean?

Every name here points to one integer. Pick a name, then place it under the integer it means. Watch the last digit: it sits in the column the unit names.

Still to sort

1500 (0)

One thousand, five hundred.

5000 (0)

Five thousand.

25 000 (0)

Twenty-five thousand.

Where the line is: Look at 2500 tens: the last digit of 2500 sits in the tens column, which gives 25 000.

250 000 (0)

Two hundred and fifty thousand.

Where the line is: Look at 25 ten thousands: the last digit of 25 sits in the ten thousands column, which gives 250 000. It is not the same as 2500 tens.

1 000 000 (0)

One million.

Not an integer (0)

A list of digits, not an amount.

9 of 9 still to sort.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

  • A digit is one of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. An integer is any positive or negative whole number, or zero.
  • A digit's worth depends on its place: 123 is 1 hundred, 2 tens and 3 ones, but 321 is 3 hundreds, 2 tens and 1 one.
  • Ask 'how much is it worth?', not 'which column is it in?': the 8 in 6 180 254 is worth 80 000.
  • A zero inside or at the end of an integer holds a column and must be written. A zero at the front changes nothing.
  • Any integer is the sum of its place values, such as 526 = 500 + 20 + 6, and this works for integers of any size.

The big picture

A digit's value depends on its place, so the same digits can make integers of very different sizes. This lesson shows how to read and write integers of any size, how zeros hold places, how to break an integer into place values, and how to build integers from digit cards.

Key points

1A digit's value is its worth in the place it sits: the same digits in different places make different integers.
2Columns go in groups of three (ones, thousands, millions, billions); with five or more digits we leave a gap between the groups.
3A zero inside or at the end of an integer is a placeholder; a leading zero is not needed. Partition by value (107 = 100 + 7), not by digit.
4Largest integer: biggest digit in the highest place. Smallest with a set number of digits: zero cannot lead. Conditions such as odd or even decide the ones digit first.
5To read 'so many tens, thousands...', put the last digit of the number in the named column: 25 ten thousands is 250 000, not 25 000.

Worked example

Problem

Write 23 283 as a sum of place values. The digit 3 appears twice. What is each 3 worth?

⚠ Watch out

Treating a zero as 'worth nothing' and dropping it, so 145 307 becomes 14 537, or splitting 107 into its digits 1, 0 and 7 (which add to 8) instead of by value, 100 + 7.

🧠

Memory hook

A zero is a seat-holder, not a nobody. Take it away and everyone shuffles along into the wrong seat. (Like all analogies, it stops working at the front: nobody sits to the left of a leading zero, so there is nobody to shuffle.)

✓

Check yourself

From memory: write 3207 as a sum of place values, then say why dropping its zero gives a very different integer. (Answer: 3000 + 200 + 7; without the zero it is 327.)

Flashcards

(15)
What is a digit? What is an integer?
A digit is one of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. An integer is any positive or negative whole number, or zero, such as −2, 0 and 153.
123 and 321 use the same digits. Why are they different integers?
Each digit is worth what its place says. 123 is 1 hundred, 2 tens and 3 ones. 321 is 3 hundreds, 2 tens and 1 one.
How do you find the value of a digit?
Use a place value chart and ask 'how much is it worth?', not just 'which column?'. In 6 180 254 the 8 is in the ten thousands column, so it is worth 80 000. In 285 the 8 is worth 80.
What does a placeholder zero do? Give an example.
It holds an empty column so the other digits keep their value. 145 307 needs its zero in the tens column: without it you would have 14 537.
Does a zero at the front of an integer change its value?
No. Leading zeros are not written, so 057 is the two-digit integer 57. A zero inside or at the end does change the value: 2240 and 2204 are not 224.
Partition 107 using place value. What goes wrong if you just split it into 1, 0 and 7?
107 is 100 and 7 (the zero tens is left out). Splitting into digits only gives 1, 0 and 7, which add to 8, not 107.
Write 5 083 050 as multiples of its place values.
5 × 1 000 000 + 8 × 10 000 + 3 × 1000 + 5 × 10. The zero digits are left out of the sum.
How are place value columns grouped? How do we write big integers?
In threes: ones, tens, hundreds; then thousands, ten thousands, hundred thousands; then millions, and so on. With five or more digits we leave a gap between the groups (21 000). A four-digit integer such as 4000 has no gap.
How do you make the largest integer from digit cards?
Put the biggest digit in the highest place, the second biggest in the next place, and so on. From 9, 8, 7, 6 and 0 the largest five-digit integer is 98 760.
From the digits 7, 3, 0 and 8, what is the smallest four-digit integer, and why?
3078. Zero cannot go in the highest place, because the integer would then have fewer digits. So the smallest digit that can lead is 3, then 0, 7, 8.
Largest two-digit odd integer from the cards 9, 4 and 2?
49. The last digit must be odd, so the 9 has to go in the ones place and the 4 goes in the tens place. (The largest two-digit integer from those cards is 94, but it is even.)
How many different integers can 1, 2, 3 make? What about 4, 2, 2?
1, 2, 3 make six: 123, 132, 213, 231, 312, 321. 4, 2, 2 make only three: 422, 242, 224, because swapping the two 2s changes nothing.
Say 1 000 000 in three ways.
1 million, 1000 thousands, or 10 hundred thousands.
Write 40 thousands and 25 ten thousands as integers.
40 thousands puts the 4 in the ten thousands column: 40 000. 25 ten thousands is 250 000, which is not the same as 2500 tens (25 000): the digits are not in the same places.
What do 5k, 10 grand and 'a quarter of a million pounds' mean? Is a phone number an integer?
5k is 5000. 10 grand is £10 000. A quarter of a million pounds is £250 000. A mobile phone number is a list of digits, not an integer.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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