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.

0.08 = 8/100
8/100 x 8/100 = (8x8)/(100x100) = 64
/10000 = 0.0064

If 0.08 x 0.08 = 0.064 then the answer (0.064) would be bigger than each of the things you are multiplying (0.08).

But when you multiply by 0.08 things get smaller. When you multiply X by 0.08 you get 8% of X.

Multiplying something by two makes it bigger.
Multiplying something by a half makes it smaller. “A half” is 0.5.
Multiplying something by anything less than one makes it smaller.
So if you multiply one thing by 0.5 it gets smaller. If you multiply one thing by 0.08 it gets a lot smaller.
The extra zero shows that it’s smaller.

There are better explanations but this should show you why the “extra zero”’ is there - it’s showing it’s smaller.

Think of it as 8% of 8%. 8% of a number is smaller than the number... so 8% of 8% will be smaller than 8%.

It’s because decimals are fractions, just written in a different way.

0.08 is 8/100.

8/100 x 8/100 is 64/10000 or 0.0064

Can you see that 0.08 and 0.064 are very close in size to each other?if you multiplied each by 1000 they would be 80 and 64. You are trying to do 0.08 of something so that’s got to be less than a tenth of the thing. So it cannot be so close in size, but must be a tenth of that hence the extra zero.

the simple answer is there are two zeros after the decimal point in your inputs to the sun so there have to be two zeros in the answer, but I hope the above helps you feel the why.

Multiplying by 0.08 is the same as finding 8% of it. So the answer has to be a lot smaller than your original number. 0.064 is pretty close to 0.08, so that can’t be right.

I hope one of the ways of explaining works for her!

If it were 0.1 x 0.1 then it would be a tenth of a tenth, which is a hundredth or 0.01.

🫠

Multiply the number as you would any number.

add up the number of digits after the decimal points in the question.

Simplest way to remember it is to combine how many numbers after the decimal point to give your new numbers needed after the decimal point. You had 0.08 x 0.08, so combined you had 4 numbers after the decimal point. Try it out with a few scenarios, it works.

Do they understand percents?

0.08 = 8%

So 8% of 0.08 has to be a smaller number.

This is the one I'd go for.

Do they understand the additional 0s in 80x80 = 6,400

if yes use their understanding of this to help

There's lots of great answers here. Thanks. I'll test them on DD.

I'm quite impressed she wants to understand why. When I was learning I just accepted it!

This.....brilliant 😊

I still don't know why a - x - is a plus tbh, don't think anyone ever explained.

Despite being fairly numerate, I usually fall back on a visual for this, and look at an area, so

A 1m by 1m Square is 1m^2
I then picture smaller numbers in the square.
A 0.5m by 0.5m Square is 0.25m^2

It's not quite a good "why" but the visual helped the idea be less abstract.

You’re basically taking a tiny fraction of a tiny fraction, which leaves you with an even tinier fraction. So the answer has to be smaller than the original numbers. Like half of a half has got to be smaller than a half, IYSWIM.

On the flip side - Why does dividing by numbers less that one result in a number bigger than 1?

Because, for example, dividing is a way of asking how many groups of x can I get out of this number. So if you ask how many halves can I get out of a whole thing the answer would be 2 ie double the original number

I find this concept easier to grasp with fractions. Use a number smaller number to do the illustration to start. Show the fraction multiplication and then its companion decimal multiplication. So 1/2x1/2. 1/10x1/10 and work your way up.

There isn't an 'extra' 0. It's the original number. .08 x .08 has two. The answer has two. .0064. Why?
If you had only one 0 to the right of the decimal point that would be multiplying 0.8 x 0.08 (= 0.064). which would obviously be wrong because .8 is ten times more than .08. The 0 in .08 tells you that it is ten times smaller than .8 .

8.0 x 8.0 = 64.00
8.0 x 0.8* = 6.40
0.8 x 0.8 ** = 0.64
0.8 x 0.08 = 0.064
0.08 x 0.08* = 0.0064

*[0.8 is 1/10 of 8 = 8 /10 of 1 (less than 1, so the answer is less than 8 x1)]:
8 x 8/10 = 64/10 = 6 4/10 = 6.40 in decimal form.

** 1/10 of 8 x 1/10 of 8 = 8/10 x 8/10 = 64/100 = 0.64

* 1/10 of 8 x 1/100 of 8 = 8/10 x 8/100 = 64 /1000 = 0.064

1/100 of 8 x 1/100 of 8 = 8/100 x 8/100 = 64 / 10000 = 0.0064

Note 4 digits after the decimal point in .08 x .08 and 4 digits after the decimal point in your answer .0064.

And 2 0s after the decimal point in .08 x .08 and 2 0s after the decimal point in your answer .0064. You must keep your 0s to keep you right!

@Abitofalark I realise it doesn't have an "extra zero" I was repeating DD's way of explaining the area of her concern.

I tried drawing pictures and number lines and a few other explanations but none of them worked. We think in different ways.

Her way of doing maths is more like yours so I think your step by step might help. In any event I'm going to print all of these out and see which resonates with her.

Thanks all you've given me some great material. I really appreciate it.

Thanks CaptainS. Just noticed some asterisks were lost along the way, which can only be confusing. They're meant to read in sequence * * * *. There definitely shouldn't be a single asterisk at the last one. At least the vital 0s are present and correct!

Meant to say Well done to your daughter, but forgot. She's thinking and engaging with it, which is great. Sign of an enquiring mind.