To ask you how to explain the WHY of this maths question?

[CaptainSeven]

Ok

I hope someone can answer this. 0.08 x 0.08 = 0.0064

WHY is there an "extra zero" in the answer? I can’t find a way of explaining it to DD in a way that makes sense. As in why doesn’t 0.08 x 0.08 = 0.064??? I know it doesn’t BUT WHY?

It’s to do with the fact that the numbers are to the right of a decimal point. I know this. I was told that and accepted it but DD needs to understand why for it to make sense and I can’t find a way of explaining that makes sense to her.

Drat. It's messed up the asterisks that I've just typed, showing only a random two out of the whole sequence of 1, 2, 3, 4. Something odd going on with the formatting. I thought I'd done something wrong originally but obviously not.

0.08 x 0.08 is 8% of 8%. You could try demonstrating it using a cake example. 8% of a cake is small. 8% of that 8% is even smaller.

"why doesn’t 0.08 x 0.08 = 0.064??? I know it doesn’t BUT WHY?"

This is a good place to get her to ask what feels right. It's an important thing to do.

0.08 is 8%.
0.064 is 6.4%, in other words it's 80% of the original number. Now you know you only want 8% of it. So 0.064 should feel wrong. So you must want 0.0064.

Has she done powers yet? You could also explain it via adding the exponent in powers.

I used to just think about it in terms of the number of zeros, so 10 (one zero) 10 (one zero) gives two zeros (100). 100 x 100 gives four zeros (10,000). Works the same with 0.08 x 0.08 - there are two zeros after the decimal point in total, so the answer is 0.0064 (two zeros).

0.064 is smaller than 0.08 so this answer (given by more than one person) is clearly not right!

It is right to say that multiplying by 0.08 is the same as finding 8% of a number, so as a sense check we should be getting an answer which is much smaller than 0.08. And 0.064 is not much smaller than 0.08. It is more than three quarters of the size.

When you multiply by 0.08 you are kind of doing two things. You are multiplying by 8 and then dividing by 100, because 0.08 = 8/100. When you multiply 0.08 by 0.08 you are multiplying 8 by 8 and then dividing by 100 twice.

8 times 8 is 64
Divide by 100 once is 0.64
Divide by 100 again is 0.0064

Another example. 0.3 times 0.002
0.3 is 3 divided by 10
0.002 is 2 divided by 1000 (count how many time you move the decimal point to get from 2 to 0.002)
So to get 0.3 times 0.002 we have:
3 times 2 is 6
Divide by 10 is 0.6
Divide by 1000 is 0.0006 (move decimal point 3 times)

Multiplying two numbers where one of those numbers is less than one, give an answer which is less than one (and smaller than either of the original two numbers).
E.g.
0.5 x 0.5 = 0.25.
0.25 x 0.25 = 0.0625, etc.

Some basic maths principles:

0.08 means eight one hundredths of something.

'x' can mean 'of' as well as 'times' (2x3 is the same as 2 lots of 3).

So 0.08 x 0.08 reads as eight one hundredths of eight one hundredths.

Imagine a large square divided into 100 smaller ones (a 10x10 grid). The first 0.08 is therefore represented by 8 of those smaller squares. Now we move to eight one hundredths of those 8 small squares. Divide each small square into hundredths again. Mark out 8 in each small square. Count them up. There will be 64 tiny squares marked. You've divided each small square from the very first larger one into hundredths again. So each small square is one hundredths of the original. When you divide a small square into hundredths, each tiny square is one ten thousandths of the first large one. You have marked out 64 of them, sixty four ten thousandths. In decimals that is 0.0064.

When you have a small amount of a small amount, the answer is an even smaller amount, not a larger one as you might expect from multiplication. To make this simpler, do the squares thing with 0.5x0.5. Halve the original square (the first 0.5). Then halve it again. What you get is0.25, a quarter. A half of a half is a quarter, the answer is smaller than either of the two numbers (0.5x0.5) in the original question.

Multiplying any number by less than 1 makes the answer smaller.

8 x 8 = 64.0
8 x 0.8 = 6.4
0.8 x 0.8 = 0.64
and so on.

But that doesn't help explain why the answer isn't 0.064!!!!

"X divided by Y" is synonymous with "how many Ys in X"

Picture a group of people and some pizzas. If you have ten pizzas but they're tiny so everyone needs two each then you have 10/2 = 5 portions.
If they're medium sized and a portion of pizza is a half, then you have 10/0.5 = 20 portions. If you decide that a portion of pizza is a quarter then you have 10/0.25= 40 portions.

Because 8/100ths of anything is less than ten percent.

0.08 is 8/100. Or, times 8 divided by 100.

So 0.08 * 0.08 is 8/100 times 8/100 = (8 * 8) / (100 * 100) = 64 / 10000 = 0.0064
Or put differently, it's 8 times 8 divided by 100 divided by 100.

If you just wanted the calculation rule, you take 8*8=64, then count how many places you had to shift the . to get 8 and 8 (you had to shift it by 2 places to get the first 8 from 0.08, then another 2 places to get the other 8 from 0.08). Then you shift the . back by a total of 2+2=4 places, giving 0.0064

Yes. I know. I don't need any help understanding it. I'm just getting frustrated by people who think it is sufficient to point out that multiplying by 0.08 will give a result that is less than the original number. The OP wanted to know why the answer isn't 0.064 which IS less than 0.08. So it is not an adequate answer. Saying 8/100 is less than 10% is much better.