Look at this price tag: a bag of apples is €0.25 on offer. We say "twenty-five cent", but a quarter of a euro is also written as 0.25. So which fraction is hiding inside 0.25? Have a think before anyone answers.

0.25

Here is our first decimal, built on the place-value mat. Read it with me: two tenths and five hundredths. The last digit sits in the hundredths column, so the bottom of our fraction is one hundred. All the decimal digits become the top: twenty-five. That gives us 25/100. Now we simplify, and I want you to watch how I find the number to divide by. I ask: what number goes into both 25 and 100? I test five first — five goes into 25 (five times) and into 100 (twenty times), so five works. But can I do better? Twenty-five also goes into both: 25 ÷ 25 = 1 and 100 ÷ 25 = 4. Twenty-five is the biggest that fits, so I divide top and bottom by 25 in one step and 25/100 becomes 1/4. A quarter of a euro, exactly like the price tag said.

0.6

Our second decimal is shorter. The last digit is in the tenths column, so the bottom is only ten. Six tenths is 6/10. Now I hunt for the biggest number that goes into both 6 and 10. Five goes into 10 but not into 6, so five is out. Two goes into both: 6 ÷ 2 = 3 and 10 ÷ 2 = 5. Nothing bigger fits, so I divide both by 2. Predict with your neighbours: before I show you, what do you think the answer will be?

0.125

Now a longer one. The last digit reaches all the way to the thousandths column, so the bottom must be one thousand. One hundred and twenty-five over a thousand — 125/1000. This is the tricky one, so we hunt for the highest common factor slowly. Two does not go into 125 (it is odd), so I try five: five goes into both, giving 25/200. I keep going — five again gives 5/40, and five again gives 1/8. So dividing by five three times is the same as dividing by 125 in one go, because 5 × 5 × 5 = 125. Either way, 125/1000 simplifies right down to 1/8. That is the trick: if you cannot spot the biggest factor at once, keep dividing by a factor you can see until nothing else fits.

Now we work through these three decimals together on the mat: 0.4, then 0.35, then 0.06. For each one, before anyone builds it, everyone predicts two things: how many decimal places it has (so what the bottom of the fraction will be) and what number we should divide by to simplify. Then a pupil builds it on the board and we check our predictions together. Watch the columns carefully on 0.06 — that zero in the tenths column catches people out.