Place value is how much each digit in a number is worth and relates to its position, or place, in that number. For example, 627 comprises 600, 20, and 7, or 6 hundred, 2 tens, and 7 ones – rather than six, two, and seven. If you swap these digits around, their place value changes. For example, 762 is made of the same three numbers, but each has a different value now.Top of Form
As you may have noticed, places further to the left are worth more, getting smaller as you move to the right. It doesn’t only start at hundreds; you can keep going higher by adding more and more digits for thousands, tens of thousands, hundreds of thousands, and so on. It also doesn’t end at the ones. After a decimal point, you’ll keep going down in size with tenths, hundredths, and thousandths. Again, these can carry on infinitely.
How do we know that 600 is more significant than 60 or six? After all, zero means that there is nothing there, so surely they’re all the same. Of course, that’s not the case. Zero is a placeholder, telling us the place value of the other digits in a number. The same goes for 0.5 and 0.05 or 0.005. The zeros tell us that the numeral 5 is worth something different in each and that these numbers are getting smaller.
What is a place value holder?
You may hear this term mentioned when teaching place value to your children, so you might wonder, “What is a place value holder?”
A place value holder is a numeral – 0 – used to ensure all other digits are in the correct place. It has no value and helps us identify which number is in the ones, tens, or hundreds column.
When considering place value holders, it’s best to look at an example. So let’s look at the number 308. We can look at this as 3 × 100, 0 × 10, and 8 × 1.
But why is there a number in that column if there are no tens?
In this instance, the 0 in the tens column of 308 is a place value holder. We know there are no tens in this number, but we still need a 0 – a place value holder – in that column. Otherwise, we’d end up with 38, which is a different number.
Can you spot the place value holder in each of these numbers?
Why do we use place value in Maths?
We use place value to understand how to read numbers, recognizing strings of digits as the specific numbers they represent. But we don’t know we’re doing it until we step back and think about it. Further use of place value requires much more thought, which is partitioning. It is where we take a conscious look at place value and split numbers into their different units, just like we did above to show that 627 = 600 + 20 + 7.
To clarify this, we can use place value columns or grids like the one below, where the constituent digits are placed according to whether they’re ones, tens, hundreds, etc.
Using a grid allows children to understand the different values of digits when separated and acts as a great visual representation. Try giving children a number and ask them to split it into the correct columns. The more confident a child becomes, the bigger the number they can try. This skill is vital for much of the Maths work they’ll be doing in primary school.
A secure understanding of place value provides the essential number knowledge needed to complete calculations, including addition, subtraction, multiplication, and division. It also lets us work with decimals and understand how to round numbers.
Place value and the four operations
A sum like 842 + 531 would be tricky to do straight away; we certainly can’t start taking 531 leaps on a number line. That’s where partitioning and place value come in to make things easier. We can split complicated questions into more straightforward calculations by breaking up the numbers and taking each place value in turn.
For example, we can work out this addition problem by adding the ones, the tens, and the hundreds, then putting it all together. 2 + 1 = 3, 40 + 30 = 70, and 800 + 500 = 1300, making the total 1373. Place value makes multi-digit calculations easier with manageable chunks.
Subtraction works in a very similar way. For 842 – 531, we can take away the numbers by their place value. So, 2 – 1 = 1, 40 – 30 = 10 and 800 – 500 = 300. Then, we have our answer, 311.
Multiplication and division have a few more steps for numbers this large, as we need to times or divide the whole number by the entirety of the other number, not just the ones by the ones or the tens by the tens. But let’s look at an example with 3-digit and 1-digit numbers to see how the same idea applies.
We can break down 972 × 3 into 900 × 3 = 2700, 70 × 3 = 210 and 2 × 3 = 6. Adding these together gives us 2916.
972 ÷ 3 works the same way. 900 ÷ 3 = 300, and 70 ÷ 3 = 20, with ten left over to add to the final sum. 10 + 2 = 12, and 12 ÷ 3 = 4. When we add those up, we get 324.
Place value is vital for children to use the four operations with large numbers. It lets them follow more manageable steps to reach their answer. There are helpful ways to make this even more accessible, including the column method for addition, subtraction, and multiplication and the bus stop method for division. They all rely on looking at the different digits and understanding how their place value affects the overall result.
Even when a calculation has a number that seems easy to use, like ten, 100, or 1000 (the powers of ten), we have to ensure we’re always aware of place value. So, for example, we need to focus on the tens column when we add or take ten from a number. Then, we can use the zeros at the end of these numbers for multiplication and division to determine the answer.
10 has one zero, so to multiply by then, we move everything one space over to the left and put a zero in the one’s space. If we’re working with decimals, this space will already be filled by the digit in the tenth column, and we don’t need to worry about the final step. 100 has two zeros, so we move everything two spaces to the left and put a zero in any empty places. Getting the hang of it?
For division, we go the opposite way and move the number however many spaces to the right as there are zeros. Sometimes this will mean using decimal places we weren’t using before, so don’t forget to include the decimal point if you need to.
Remember, this only works when the first digit is 1. So if we wanted to multiply or divide by 20, for example, we couldn’t use this method, but the place value would still be significant.
Decimal place value names
So, now that we’ve covered the basics of place value and decimals, what are the decimal place value names? Initially, it can be a little confusing, and the names themselves are tricky to pronounce! If you’d like a summary to hand while you’re teaching this, then here’s a helpful table of decimal place value names for you:
|0.1||1 ⁄ 10||One tenth|
|0.01||1 ⁄ 100||One hundredth|
|0.001||1 ⁄ 1,000||One thousandth|
|0.0001||1 ⁄ 10,000||One ten-thousandth|
It’s unlikely that you’ll ever need to use a decimal smaller than one ten-thousandth. To give you an idea of how small one ten-thousandth is, a single human hair usually weighs about 0.0003-0.0006 grams (that’s three to six ten-thousandths). Many high-tech lab equipments can’t handle measurements as small as this!
Of course, this raises a tricky question: What’s its name if we’ve got a number like 1.032? Well, luckily, there’s a handy formula to help make this easier:
- Write the number before the decimal point, just as you usually would.
- Then, write the decimal point as “and”.
- After this, you’ll need to consider how many decimal places there are. Again, the example above tells us that we’re dealing in thousandths.
- Since there are 32 thousandths, we arrive at the following name:
One and thirty-two thousandths
It’s that simple!
Place value and rounding numbers
Many tests, including the NAPLAN Numeracy test, will ask for answers to be rounded. Whether we’re rounding decimals or whole numbers, we rely on place value once again. Children must understand which digit to look at when rounding to a certain number of decimal places, or the nearest ten, hundred, thousand, etc.
They’ll have to look at the appropriate column and the one to the right. If it’s lower than five, they round down, changing the remaining places to zero or removing them if the placeholder isn’t needed. If it’s five or higher, they round up, adding one to the place value column they’re rounding to and, again, dealing with any placeholders they may need.
Here are a couple of examples. To round 8437.52 to the nearest ten, you look at the tens column, which has the numeral 3, and the column to the right with the numeral 7. It is higher than five, so we add one to the tens column, making 8440. If we were rounding the same number to one decimal place, we’d look at the tenth column (5) and the one to its right (2). This time, the digit is minor than five, so we round down to 8437.5.