Symbols

My daughter freaked at the sample math symbols questions in her SAT practice book last week, so I spent a couple of hours this week working up a practice sheet -- somewhat difficult for me, as I'm not a math guy. But I had my wife check it over (she's the math major), and she pronounced it sound. On the chance it might be generically useful --

Plus signs, minus signs, all of that are all symbols.
A plus sign says to add. A minus sign says to subtract.

A train ticket costs $10. How much would three tickets cost?
(cost of one ticket) times (number of tickets wanted)
So the total cost would be $30, which can be factored into 10 and 3.

How much would tickets for some unknown number of people cost?
(cost of one ticket) times (number of tickets wanted)
Without knowing the number of people, you can’t figure out the total cost.
But just as the other one’s total cost was 10 times the known number of people, this one will be 10 times the unknown number of people. And, just as the other one could be factored into
10 and 3, this one can be factored into 10 and ?
The question mark is a symbol.

You can make your own symbols to mean whatever you want them to mean. A symbol can stand for a number, or for a math express (like plus or minus), or whatever you want it to stand for.

If you have to subtract a first number from a second number, and then add five to the result, you could say the symbol @@ means to do that.
So 10@@6 would mean ‘subtract 10 from 6, then add five to the result’
(10 from 6 = -4) (-4 + 5 = 1) so 10@@6 = 1
And 9@@17 would mean ‘subtract 9 from 17, then add five to the result’
(9 from 17 = 8) (8 + 5 = 13) so 9@@17 = 13

If you show what numbers to use with your symbol, it becomes a formula.
15@@6 is a formula.
How could you show what the formula is if you don’t know what numbers will be used?
You could say ? @@ ?
But if you wanted one question mark to stand for 10 and the other to stand for 6, how would you know which is which? If you do it backwards, you get the wrong answer!

If you mean that the first ? stands for 10, and the second ? stands for 6, then ?@@? would mean 10@@6, which means ‘subtract 10 from 6, then add 5’. That’s equal to plus one.

If you mean that the first ? stands for 6, and the second ? stands for 10, then ?@@? would mean 6@@10, which means ‘subtract 6 from 10, then add 5’. That’s equal to plus nine.

To make it obvious which one you mean, if you have more than one unknown number, you use more than one symbol. You can use any symbol you want to stand for any unknown number. You just can’t use the same symbol to mean more than one number at the same time.
For example: X@@Y is okay, but X@@X or Y@@Y isn’t.

The formula (X@@Z) stays the same – only the numbers that you plug into it, and the result you get out of it, change.
Your symbol can mean anything you want. It can mean ‘take the first number, multiply it by three, to get the first result, then take the second number, multiply it by two, to get the second result, then divide the first result by the second result, to get the third result’.