Introducing place value

Place value is a concept that we take for granted as adults. It’s obvious to us that the digit 3 means something entirely different in the numbers 23 and 32, and different again in 320 or 32000. But it’s a surprisingly tricky concept for many children to grasp, although others pick it up intuitively. If your child is one who struggles with this concept, then it’s vital to help him or her understand how place value works. If not, then any kind of arithmetic is going to be extremely difficult.

The concept of place value

One way to introduce the idea of place value is to have a large number of objects – bricks, cars, or anything else of interest to the child, which he can count reasonably easily. You need at least fifty or sixty items, preferably more. If your child cannot yet count as far as twenty, then he’s not ready to understand the idea of place value. It’s also best if your child can recognise the individual digits at least as far as ten or twelve – perhaps as a result of learning to read clocks, or just general interest in numbers. If this hasn’t yet nappened, then don’t worry about place value yet. He may still grasp it without any help.

So, assuming your child can count at least to twenty, and knows what some numbers look like, you could start by telling him that there are only ten digits – digits being the individual numbers 0 up to 9. We have ten fingers, and they’re also sometimes known as digits, and it’s probably because of our ten fingers that we use a ‘decimal’ system of counting – ie we have exactly 10 different digits, and have to use the place value system to show if a number is more than 9.

(If you have older children, they may like to know about other counting systems used by computers such as the binary system that uses only 1s and 0s, or the hexadecimal system which has 16 digits, using the letters A-F as well as the ten regular digits.)

Ten digits from 0-9

So, we can represent only ten different quantities with our ten individual digits, including zero to represent nothing at all. Perhaps you and your child could write the individual digits on post-it notes or similar, and place them next to groups of objects. If you can’t find fifty items to use, you could draw objects instead. Show that 4 represents four objects, that 7 represents seven objects, and so on. Right up to 9.

Then make a group of ten objects, count them with your child, and ask him how on earth you would show ten in numbers, when you’ve used up all the digits. He should be able to tell you that we use 10, but may not have thought about the reason why. Perhaps you could brainstorm together as to what it means to use a 1 followed by a 0, discussing other two-digit numbers he recognises, such as 11 and 12 (and others, if he knows them).

Place value in the decimal system

Don’t push for a response – your child may realise by himself that we use the 1 to represent ten items, and the other number to represent how many more than ten we have. If he doesn’t, you could suggest this to him and see how he reacts. If he doesn’t get it, leave the topic for now and come back to it another time. A lot of learning goes on after a ‘lesson’ and you may find that a few days later he understands this first significant part of the place value system for himself. Or he may get it straight away, and want to know more.

When that happens, you can ask him if he can guess how to write thirteen, fourteen and so on (if he doesn’t already know), and show how logically that works. Then ask how he would write twenty, since you’ve run out of digits again by that stage.

He may spot instantly that it would be a 2 followed by a zero, or he might suggest a 1 followed by another 1 and then a zero – which is a perfectly good suggestion, and shows that his mind is working along the right lines. But you could point out that if we used a system like that, then it would take ages to write out a very long number so we use a 2 followed by another digit to represent twenty (or two sets of ten). If he understands this, then he’ll quickly see that three lots of ten will be represented by 30, and so on.

If your child is still interested and enthusiastic, you could write out more numbers (it would take too long to count objects at this stage) showing them up to 99…. then ask what comes next. It’s quite a leap to see that after 99 we have to have 100, ie a 1 representing ten lots of ten, then TWO zeroes. But it makes sense and sooner or later your child will see it.

This is such an important concept that it’s worth talking about it casually every so often, to ensure it’s clear in your child’s mind. It’s entirely possible to do basic arithmetic and other maths without a good understanding of place value, but when the numbers get higher and the calculations more complex (and even more when you introduce the decimal point) a child who hasn’t got the place value idea firmly in his mind will probably get very confused and make a lot of mistakes.

Place value games

There are various games involving place value online which can help a child who understands the idea of numbers up to at least the thousands, but is still struggling with the idea of place value. Dino place value is a useful game showing a pile of rocks with numbers on, and a dinosaur waiting to eat them. A two-digit number is given and the child has to click on the rocks representing what it means – so for instance if 19 is given, the child should click on 10 and 9. Shark place value is a bit more complicated; on the boat at the left of the screen are shown some blocks – some in units of ten and some as single blocks. The child has to click the bubble that represents the number of blocks in all, although there does not seem to be a time limit.

More articles about introducing children to maths informally:

Number bonds – beginning addition
Algebra for six-year-olds
Prime numbers and factors
Basic division

Mathematics for toddlers

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