This is my fist post on Mathematics, so I hope my question isn't too trivial - you won't need a degree to answer this one! - but it's messing with my head....
Allow me to explain.....
I'm helping my daughter prepare for her Eleven Plus exam in the UK and as a maths lover myself, I've been enjoying setting her written questions that require some thought in order to write them in an algebraic form and find the solution.
Years ago, I demonstrated negative numbers to her, using pocket money as an example. Every Friday she gets £5 and if she's been good and wants something before Friday, then I will advance her the £5, or however much.
She understood the concept of negative numbers in this way - if she was advanced £10, it would take her two pocket money payments to repay the amount she owed:
-£10 + £5 + £5 = 0. Simple hey!?
So, I've told her that if she does a chore around the house, such as washing the car, I will take an amount off her debt (£3), but if she's naughty, she will get 50 pence (-£0.50) added to the amount she owes me.
When I decided to be clever (clearly not) and set an example question about this, I got into a pickle.
She currently owes me £10 and from her previous lessons from me, knows that her pocket money could be said to stand at -£10.
When calculating how much pocket money she'd owe if she was naughty 5 times (a minus £2.50 charge!), she has baffled me with the following:
(-£10.00) + (-£2.50) = -£7.50
~ she has a charge of -£2.50 added (+) to her debt ~
She has deduced that because -£2.50 will be added to her pocket money debt, she will actually be better off = being naughty is profitable!
Conversely, washing the car will take away from her debt by £3.00. Therefore:
(-£10.00) - (+£3.00) = -£13.00
Therefore: Being helpful is not profitable!
~ I'm going to give her £3 (+) to take off her debt (-) ~
Doh!
Now, humour me please, as I know my error is extremely basic and as soon as someone answers I will face-palm, but regardless, I cannot find a way to verbally explain how taking amounts away and adding amounts to her pocket money debt, can literally be applied in this way; especially as she's now being pedantic!
$(-£10.00) + (-£2.50)$ is not equal to $-£7.50$ (it's equal to $-£12.50)$.
And although $$(-£10.00) - (+£3.00) = -£13.00$$ is technically correct, you shouldn't subtract the $£3$, but add it, giving a money balance of $(-£10.00) + (+£3.00) = -£7.00$, which is what you'd expect if she was being rewarded.
If it helps, draw the number line (for instance from $-15$ to $15$) and go right when you add and left when you subtract.