Decimal numbers are just fractions where the denominator is a power of ten.
So we have tenths, hundredths, thousandths, ten thousandths, … and so on.
Each position is worth a certain fraction of a whole, just like each position to the
left of the decimal point is worth a multiple of a whole.
The nine in this number is worth nine times ten, or ninety.
In exactly the same way, the two is worth two divided by ten, or two tenths.
The four is worth four divided by a thousand, or four thousandths.
We tend to say decimal numbers by simply reading off the digits.
“Ninety-one point two five four eight.”
But if we do want to refer to a specific digit, we use its place value, which is just the
name of a fraction.
“Five hundredths,” “Round off to the nearest thousandth,” and so on.
Represent decimal fractions using anything you can easily divide into ten, a hundred,
a thousand.
Square grids work well.
And if you have physical objects that can be held and manipulated, all the better.
Do you have some square blocks that could be arranged into squares or built up into
solid blocks?
Here’s how to do it.
Zero point four.
(Incidentally, I encourage you to always write that leading zero, so that the decimal point
doesn’t get lost when you read it back later.)
Draw a fraction as usual: point four is just four tenths.
Then if I need hundredths, I can divide up my fraction into hundredths like this, and
I have forty hundredths: point four oh.
In fact, we can use this model to convert a decimal fraction into a common fraction.
For example, suppose I have zero point one six.
Well, it’s got two decimal digits, so I’m going to need hundredths: draw a square grid
with a hundred squares, ten by ten.
Shade sixteen squares: sixteen hundredths.
So write it as a common fraction: sixteen over a hundred.
Okay, now simplify the fraction if possible.
Do sixteen and a hundred have any common factors?
Yes: four.
Divide both top and bottom by four to get four twenty-fifths.
This is zero point one six as a common fraction in simplest form.
What about going the other way?
How do I convert a common fraction into a decimal number?
Just look for an equivalent fraction with a denominator that’s a power of ten.
Three fifths.
Well, five goes into ten twice, so use tenths.
Three times two is six.
Six tenths, which is zero point six.
Seven twentieths.
Well, I can’t use ten, because I’d have to have half of seven on the top, and two
doesn’t go evenly into seven.
So try a hundred instead.
Ah, twenty times five is a hundred.
Seven times five is thirty-five.
Thirty-five hundredths, which is zero point three five.
Three eighths.
Hmm, ten won’t work: eight doesn’t go into ten.
A hundred won’t work: eight doesn’t divide a hundred either.
I have to go all the way to a thousand.
You may want a calculator here: eight goes into a thousand one hundred and twenty-five
times, so I have three times a hundred and twenty-five, which is three hundred and seventy-five.
Zero point three seven five.
Okay, now a hard one.
Two elevenths.
Have a think about it: What can I use on the denominator?
Eleven won’t go into ten.
What about a hunded?
a thousand?
In fact, there’s no power of ten that I can put on the denominator of this fraction.
It’s simply impossible.
Does that mean I can’t write two elevenths as a decimal?
Well, kind of.
Let’s try working this out a different way.
Remember the fraction line, the vinculum, just means “divide”.
Do you know the written algorithm for dividing whole numbers without a calculator?
We’re going to extend that into decimals.
It works like this.
Two divided by eleven.
How many times does eleven go into two?
It doesn’t.
Zero, remainder two.
Hmm, well, we could write two as two point zero, putting a decimal point in our answer
as well, and keep going from there.
Carry the remainder two.
Eleven into twenty goes once, remainder nine.
Write another zero and keep going.
Eleven into ninety?
Eleven eights are eighty eight, leaving two left over.
Carry the remainder.
Eleven into twenty goes once, remainder nine.
Eleven into ninety goes eight times, remainder two.
Eleven into twenty goes…
Can you see the pattern developing here?
What’s happening?
We’re getting the same digits repeating again, and again, and again.
It’s never going to stop, is it?
We call this a “recurring decimal”.
The pattern repeats forever.
We write it like this, with a bar over the repeating pattern.
Be careful with these.
You cannot write this number accurately as a decimal without including the recursion
symbol, the bar (or sometimes just a dot, if it’s only a single digit that repeats).
You can round it off, but you must remember that that would then be only an approximation.
It won’t be exactly equal to the fraction.
As a teacher, there are some common fraction decimal equivalents that you should just know
by heart.
Draw yourself up a table like this, halves, thirds, quarters and so on, all the way to
tenths.
Now go back and write next to each one its decimal equivalent.
Then, learn the table.
No, I mean it.
You really should know these.
You will need them in the classroom, and you don’t want to be reaching for a calculator
every time.
Finally, calculating with decimals.
Just extend the written algorithms you already know for whole numbers, like I did earlier
with that decimal division.
Work out the meaning by remembering that they’re really just another way of writing fractions.
That’s all there is to it.
But I will warn you to be careful with multiplication.
The number of decimal places in your answer may not be what you expect.
Look carefully at your answer in the context of the original question to make sure that
your answer makes sense.