percents

Is over Of = % over 100

(alas, computers don’t do too well with ytping fractions)

I work on the concept for a while — equivalent fractions they probably don’t really grasp yet. There are lots of math sites that have good ideas for this. If they don’t know how to “cross multiply” this is just going to be a hard one to teach.

And I’d review often and well and then some more after that and then once a week after that the idea that you can’t add fractions of different denominators. Half these guys will graduate not remembering it. The ones that end up getting tutored by me will have to relearn it.

Percents are fractions with a denominator of 100.
Nothing more, nothing less.

Since nearly every student I meet is hopelessly lost with fractions, you have to start there and make sure that fractions are actually understood. This is time-consuming but the time is invested in something that will actually do some good; trying to do percents otherwise is trying to carry water in a sieve.
Fractions (and percents) need to be understood concretely, in terms of models and/or pictures; verbal rules of “just divide both by the same thing” are useful summaries, but they are not the meaning.

Many people, including a lot of misguided teachers and text writers, will tell you to just ignore fractions and use decimals; unfortunately, there is no meaningful way of explaining decimals without tenths, which leads you straight back to fractions.

I disagree with Sue about “cross-multiplication”; this is second only to “cancellation” as a meaningless term which I refuse to hear. It is a quick trick that works in one very very special situation. However, students try to generalize it and use it every time they see a fraction. Nine times out of ten this is inefficient, and half the time it is dead wrong and guaranteed to mess up.

Examples:

x/3 = 5/7
correct and easy method to solve: x is divided by 3, so “undo” by multiplying both sides by 3. One step, logical, simple.
The “cross” method requires three steps, in which you first multiply by seven and then divide it back out. Inefficient and highly error-prone.

5 + x/4 = 7
correct and easy method: “undo” the division of four by multiplying the entire equation by 4. Simple, logical, and builds on previous knowledge.
The “cross” method is dead wrong here and leads to a wrong answer every time. Sure, students are supposed to know not to use it — but *how* are they supposed to know that, when they haven’t been taught the logical underpinnings, but only the “cross” quick trick, in the first place?

(x^2 + 3x - 4)/(x^2 + 5x + 4) = 3x/(2x + 2)
correct method (not so easy, but easier than anything else): *first* simplify the fractions, then look at multiplying.
The “cross” method here is correct, but will give you a cubic problem which is far beyond most high schoolers (heck, most college students) to solve.

Gosh I am going to seem like a repeating whatever, but I really really really agree on learning things on a concrete level. Little tricks and sayings like the one for dividing fractions (invert the divisor and multiply) well they work but it would really really help to know WHY they work.

I am very fond of my new On Cloud Nine manual, think there are a lot of great ideas in there. For decimals actually. Also there are some nice concrete materials out there for teaching decimals/percentages.
I can’t recommend anything real specifically as I have not had occassion to teach that high up. This may change.

—des

Percentages are one of those thigns that I teach the concept for… but then also teach the mnemonics and strategies for doing the symbols easily for. I”ve found that even when a student can make complete sense of the concept when talking about it, there just isn’t enough time in the day to reconstruct the concept with every problem.
Now, I also find that an *awful* lot of college students are utterly and completely clueless about things like equivalent fractions. Miss Jones *longs* to grab a good visual thinker with design skills and a programming team and develop some vivid and vivacious instructional stuff…