Arithmetic with Polynomials and Rational Expressions exponents Rational Expressions Seeing Structure in Expressions


What’s the mistake? Diagnose the disease, and find the cure in the comments.

Thanks to Anna Blintsein for the submissions. Go follow her on twitter!

8 replies on “Cross-Multiplying”

The student makes ready analogies between new problems and ones that are visually most similar. Analogies are dangerous in math (as in, the integral of 1/cos(x) is ln(cos(x))). The student needs to get to a point where instead he recognizes major functional elements in notation and uses them to decide a course of action. Here, all the student sees are the two big fractions even though the proper course of action is determined by the little symbol between them (or if juxtaposition had been used instead, by the absence of said little symbol!)

I don’t have great ideas for fixing this. An obvious suggestion is to make sure the student realizes you only rarely introduce an equals sign except when you define a new symbol. I’d bet this student uses the words “expression” and “equation” interchangeably. One cannot think clearly without thinking in precise terminology, but it’s hard to get students to care about word choice. The way I do it is by requiring that the explain what they do in writing and give feedback whenever their writing gets imprecise.

I’ll admit, I didn’t even notice the dot between the two fractions myself. Granted the first thing I noticed were the big cross-multiplication “butterfly” wings which probably weren’t there initially. This seems like a classic case of a student *assuming* what to do versus doing what’s asked.

For some reason the set up of the problem reminded the student of all those cross-multiplying problems he/she likely solved at some point in the past, and that overrode what was actually written on the board. This is the type of mistake that can be helped by asking other students if they agree/disagree with this answer and to explain why. The lesson here is to pay attention to the actual problem in front of you, not the one you’re imagining in your mind. If absolutely no one notices, then I would give the correct answer and challenge the students to figure out why this one is incorrect.

I think that a big part of the lesson here is the strength of patterns in activating student knowledge. If those patterns are primarily visual, then what students will remember is the visuals. But if we vary the visuals while keeping the abstract structure the same then I think we’d see fewer mistakes like this one.

I’d say: don’t teach cross-multiplication. I’d rather see the students change both fractions to common denominators and then compare the numerators. (“If the denominators are the same, and the fractions are equal, then what does that tell you about the numerators?”)

But if we are going to teach cross-multiplication, then vary the visuals a bit and throw in some multiplication problems to the problem sets.

I’d add on to the “Don’t Teach Cross Multiplication” for this reason –

THE underlying principle of solving equations the way we do is that we are undoing operations one at a time. Remind students that division is the inverse of multiplication. Finding a common denominator and comparing numerators is well and good. I would say it is more powerful to simply multiply each side of the equation by this common denominator to simplify the equation we are dealing with. The underlying problem this child has is this – s/he has no idea of the difference in dealing with equations versus expressions.

I agree whole-heartedly with the not teaching cross-multiplication for other reason as well. A key principal in solving an equation is that we may add the same number to both sides or multiply both sides by the same non-zero number. Cross-multiplying makes it seem like its okay to multiply each side of the equation by different amounts. Of course, that’s not what is actually happening, but visually it looks that way. I’d rather focus either as Michael says on a common denominator, or else multiplying both sides by the same non-zero number. (When cross-multiplying we are really multiply both sides by the product of the denominators, but it’s even better if we use the least common denominator).

I second barryrsmith’s diagnosis. I’m a TA at a large university, and I get a lot of intro calculus students (and also students in deeper courses) who don’t seem to distinguish between expressions and equations. In other words, they don’t distiguish between noun phrases (“My favorite animal”) and sentences (“My favorite animal is my cousin’s dog.”).

If I had to suggest a way to fix this, I would say something like, “Make students understand that mathematical notation is a language, and then teach them how to speak it”… but that isn’t terribly practical or constructive.

I think cross multiplication is fine for elementary students who are adding basic fractions like 5/8 + 3/4. Once you start making it complex and doing algebra all that should be thrown out the window and never looked at again.

I would say, then, that it is not reasonable to even mention this technique. If it is so limited in its usefulness, why grant it the privilege of a name and some memory space. I’ll repeat what I said above and claim that the principle of undoing is what is important to remember in solving equations. Cluttering heads with specialized techniques that mask the important general principle at hand does the students no good, in fact may harm them. Remember the Hiipocratic oath – First, do no harm

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