VA ClassesWhat Are Decimals?
The decimal point is not punctuation. It is the border between whole things and pieces of one, and almost every mistake in this topic comes from a child who has never been told that.
What a decimal is
A decimal is a way of writing an amount that is not a whole number.
Money already works exactly this way, which is worth leaning on, because children believe money. A price of 3.45 dollars means 3 whole dollars and 45 cents, and nobody has ever confused those cents for extra dollars, because the point sits between them keeping the two apart.
Every decimal behaves the same way. Left of the point, whole things. Right of it, pieces of a single one.
The columns
Every column to the right of the point is worth a tenth of the one before it, which is the same pattern the whole number columns follow, just running in the other direction.
So 3.45 is 3 wholes, plus 410, plus 5100. Children who can say that sentence about any decimal you show them will sail through the next three topics, and children who cannot will keep tripping over the same thing in different disguises for years.
The rule
The first column after the point is tenths. The second is hundredths. The third is thousandths.
A digit's position decides what it is worth. The digit itself does not.
On a number line
Between 0 and 1 sit ten equal steps, and each step is one tenth.
Hand your child the line and ask them to point at 0.7, then at 0.35. If they put 0.35 further along, stop and go back to the columns, because everything on the next few pages is built on top of that idea and none of it will hold without it.
Saying them aloud
How a decimal gets said out loud matters more than it looks, because the wrong reading plants the wrong idea.
- 3.45 is three point four five. Not "three point forty five".
- 0.7 is zero point seven, and it is fine to call it seven tenths.
- 0.06 is zero point zero six. The zero is doing real work, so say it.
Saying "forty five" invites the brain to treat the digits after the point as an ordinary number, which is exactly the mistake that follows.
Decimals are fractions in disguise
Every decimal can be written as a fraction, and the column name tells you the denominator.
- 0.7 is 710, because 7 sits in the tenths column.
- 0.45 is 45100, because the last digit sits in the hundredths column.
- 0.125 is 1251000, and that simplifies to 18.
Common mistakes
1. Reading the part after the point as a whole number
0.35 gets chosen over 0.7 because 35 is a bigger number than 7. Those digits are not 35 and 7 though, they are 35 hundredths and 7 tenths, and once you write seven tenths as 70 hundredths the comparison is no longer close at all. This is the single most common error in the whole topic and it comes straight from reading the tail of a decimal as an ordinary number.
2. Thinking more digits means a bigger number
For whole numbers this is perfectly sound, since 1000 does beat 99 every time. After a decimal point it stops being true, and 0.5 comfortably beats 0.4999 despite looking like the shorter, simpler number.
3. Dropping a zero that is holding a place
Three hundredths gets written as 0.3 instead of 0.03, because the zero looks like it is doing nothing. It is doing the only job that matters: holding the tenths column open so that the 3 lands in hundredths. Take it out and the number becomes ten times bigger.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
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For parents
Use money, because your child already believes money in a way they do not yet believe decimals. Ask what 0.7 of a dollar comes to, then 0.35 of a dollar. Seventy cents against thirty five cents ends the argument in about a second, and the reasoning transfers straight back to the numbers on the page without anyone having to make a speech about place value.
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