I’m trying to help my 14 year old daughter understand simple formulas. I suck at this and seem to end up confusing her more, the more I try and explain it.
A simple example:
If you know that 12 inches = 1 foot, how many inches = 2 feet?
12 in = 1 ft
x in = 2 ft
How would you explain how to solve that?
Dimensional analysis.
2Ft X 12IN/1FT
The dimension Ft cancels in the numerator and denominator, leaving
2 X 12 IN/1 = 24IN
Yep. Write everything in ratios along with their units and explain that the units can be treated as numbers that are multiplied to the numbers they’re next to. If the units are treated as numbers, the same units on top and bottom can be canceled out leaving whatever unit they wish to find.
As long as she’s good at basic fractions and capable of canceling out common numerator and denominator factors, it shouldn’t be too difficult. Albeit, I had a bit of trouble with it at first until I realized how simple it was…
Good examples here: http://www.chem.tamu.edu/class/fyp/mathrev/mr-da.html
‘There are 12 inches in a foot. How many inches in two feet? You don’t know! So lets call the number of inches in two feet “x”. x-inches equals two feet. How many inches in a foot? That’s right! Twelve! And if you have two feet, then you have 12 and 12 inches, right? You can write that as x=12+12. So how much is 12 plus 12? Twenty-four! So our mystery number “x” is 12 + 12, or 24.’
Okay, that’s a bit simple for a 14-year-old. But I think it breaks it down well enough.
I teach this. Easiest way is with a proportion-once I show a student to properly set up a proportion, and solve for it (or create a formula using one), such a conversion is easy and virtually foolproof: no agonizing over “should I multiply or should I divide?” Proportions also have the advantage of being applicable to a wide variety of applications.
Make sure the same unit is on both numerators, and both denominators. In this example we’ll put feet on top:
Feet…Feet
------- = --------
Inches…Inches
Now putting the numbers in, with the base conversion numbers on the left, the numbers you are working with on the right:
1 Foot…2 Feet
----------- = ----------
12 Inches…X Inches
In a simple example like this it then becomes obvious that if you double the feet you also have to double the inches, hence the answer is 24 inches. In something more complicated you just cross multiply.
Yeah, the way John DiFool explained it is how I tried to go.
But I usually ended up more like **Johnny LA’**s explanation, which would get us to the answer for the problem at hand, but not in a way she could understand and apply to another similar problem (if it wasn’t as simple as my example)
Trying to find a nice clear explanation of what it means to ‘cross multiply’ is a challenge. Examples I found on the web always seem too complex. (For example BrandonR’s link mentions chemistry, which is OK, but just adds some confusion I don’t want at this point.)
I’m not worried about it being too simple for 14 year-old. Yes, she should know this by now. But that’s a rant for another day. Clearly she doesn’t really understand how to do it and I’d like to fix that.
How about this? Set it up like John DiFool says.
'You know that 12 inches equals a foot. So an inch is 1/12 of a foot. But instead of using a slanted line (/) use a straight line (__) like this:
1
______
12
'What you want to know is how many inches in two feet. On instead of a 1 on top of the line, use a 2; and instead of 12 inches, use x inches:
2
_____
x
'Now put them together:
1 2
______ = ______
12 x
‘I like to say “One is to 12 as 2 is to x.” Just remember that phrase and it will help you set it up. Now we need to cross-multiply. All that means is that you multiply the top of one side with the bottom of the other, and the bottom of the first side with the top of the other.’ [You may want to actually draw crossed arrows to illustrate.]
‘So now we have 1x = 24. 1x is the same as x, so x = 24. Twenty-four inches.’
As I said, if you set up the ratio and actually draw crossed arrows, it’s pretty clear what you’re doing.
If the thing on one side is doubled, then the one on the other side needs to be doubled as well.