QMST VI: Cross-multiplying

A dear friend asks (slightly edited),

Do you think there is any way to do cross multiplying with meaning?  I remember discussing this with you last year and I know you will probably say “no”.  But if we want to have students solve problems using any methods available to them, why not make this a method (not THE method but A method) available to them (discussing common denominators)? Honestly, if you were going to solve 12/123 = x/768,392.614, would you “scale up” or cross multiply?  I’m not trying to sound critical – I’m just about to teach this stuff and just trying to pick your brain.

That “with meaning” part is key for me. Consider the example presented here: 12/123=x/768,392.614. First let’s consider why we’re solving this proportion. I’m struggling to come up with a good one due to the great difference in magnitudes across these two fractions. So let’s just say they’re similar triangles. The numerators are the short sides of these triangles in centimeters; the denominators are the long sides (also in centimeters).

Cross-multiplying gives the equation 9,220,711.368=123x. What is the meaning of 9,220,711.368 in terms of our triangles? Well, it’s the product of the short side of the small triangle and the long side of the large triangle. I suppose we could think of it as the area of a rectangle with these two side lengths. But why should that area be the same as the area of the rectangle formed by the short side of the large triangle and the long side of the small triangle? And is this a new theorem?

The yellow triangles are similar, so the blue rectangle has the same area as the red one.

In any case, contrast that with the meaning involved in how I would really solve this proportion (and yes, I would really do it this way). 768,392.614÷123=6247.094… Now 12•6247.094=74,965.13

Let’s not argue about significant digits or rounding; those are tangential issues that could be raised about the originally proposed proportion. We can hash those out on someone else’s blog.

No, I want to talk about the meaning of that 6247.094. It’s the scale factor. It tells me how many times bigger the large triangle is than the small one.

If you end up with a student who can talk about the meaning of that 9,220,711.368, then by all means have her cross multiply. But in my classroom, I’m gonna insist on meaning all the way through. We don’t have to think about meaning at all times, but we have to be able to think about it.


11 responses to “QMST VI: Cross-multiplying

  1. I think that the cross-mutiplying approach becomes more useful when there are variables in the denominators. People have a lot easier time manipulating linear equations and polynomials than rational functions.

    • I think that the cross-mutiplying approach becomes more useful when there are variables in the denominators

      I’m gonna go ahead and forward this to the OMT Dep’t of Specificity. They’ll work it over to decrease the vagueness of the claim. Maybe even give us an example to chew on.

  2. Nice post as usual, Chris, but I think it’s possible, too, to cut to the chase here with numerically simpler examples. And since I don’t want to see the OMT DoS get overworked, I’ll provide an example that I hope will satisfy you and your friend regarding understanding.

    Say we’ve got 3/x = 1/6. Now, someone with reasonable number sense might look at this and argue, correctly, that since 3 is 3 times 1, then x should be 3 times 6 and hence must be 18. This is, of course, correct. But since most students either have never developed much number sense to begin with, or had it suppressed through school math (a claim I’m hardly alone in making: see Constance Kamii’s work, for starters, though there are many sources for the argument that school mathematics destroys students’ natural proclivity to think mathematically, use mental arithmetic, etc.), this reasonable conclusion isn’t going to occur to them. Instead, they will mechanically go through the approach they’ve been taught in school. For most, if they recall it, that will be to “cross-muliply.” So they will write, “3 * 6 = 1 * x” and then conclude, correctly, that the answer is “x = 18.”

    The question teachers need to ask (unless – mirabile dictu! – one or more students whose curiosity hasn’t been utterly crushed by school mathematics asks it), is, “Why does that work?”

    The reason it isn’t obvious to students why cross-multiplying works (aside from the fact that it’s a definite no-no when multiplying fractions, a fact that may well underlie some of their confusion and insecurity about what to do when), is that like many algorithms, things are compressed and hidden in the process to make it faster and easier. In the example, when we ‘cross-multiply,’ it seems like we’re multiplying the right side by x and the left side by 6. But in fact, we’re multiplying both sides by x and both sides by 6. If we showed that, we’d see 6x*(3/x) = 6x*(1/6) which becomes (18x)/x = (6x)/6. With the necessary assumption that x wasn’t zero, this simplifies to 18 = x, the same result we got with less writing the first time.

    On my view, it’s the compression that makes it seem magical/mysterious. And that’s part of why “long multiplication” and particularly “long division” are so magical/mysterious for many students (and in the latter case in particular, very frustrating for some). We gain speed and brevity at the cost of clarity, and loss of clarity leads to mystification, unfortunately.

    What bothers me no end is the resistance on the part of some “Math Warriors” and many elementary school teachers (though for different motives, I suspect) to unpacking these algorithms in the process of teaching them. There is also a great deal of resistance to the idea of allowing students to develop their own methods before exposing them to whatever “standard” algorithm is the current gospel. Putting those two things together guarantees confusion for most kids. And of course, the cycle is continued in the next generation, because those resistant elementary school teachers never really understood WHY the algorithms work when they learned them (if they really learned them). That’s why they are opposed to unpacking: I’ve had many pre-service and in-service teachers tell me, “Oh, my students would only be confused by that,” where “that” means an alternate approach, a “deep” unpacking of the standard algorithm, a model that isn’t strictly a mechanical set of steps, all of which are familiar and none of which stray from pure manipulation of numerals and other symbols. (Which means, in part, that your nice triangle/rectangle model above is not going to encounter wide-scale acceptance among either fearful elementary math teachers/education students, or the hard-line Math Warrior types.

    As to why the latter group eschews such explanations, there are probably several reasons, but the argument I find most specious and annoying is that to teach these “simple” algorithms in ways intended to unpack them is to act as if some students can’t learn mathematics without “dumbing it down.” I’ve never found any variation on that argument even vaguely convincing. It’s usually couched in the new right-wing rhetoric that turns progressives into racists and sexists (better known as psychic jiu-jitsu). Turns out that those of us who think it’s worth taking a less mechanical approach to teaching elementary arithmetic to students and would-be teachers are actually biased against minorities and women: we are infantilizing them by not just doing good old symbolic manipulation. Or to put it slightly differently, “If this was good enough for those (mostly) white males like me who grew up to become professional mathematicians without any of your silly “crutches,” then it’s good enough for everyone else, and anyone who says differently must be a racist and/or sexist.”

    Right. And I’m a Chinese jet pilot.

  3. Great timing, Christopher. On Friday, I was looking for strategies for teaching proportional reasoning to struggling students. The last thing I read before I left the office from Marian Small’s Big Ideas from Dr. Small, Grades 9-12. She writes,

    “When students cross multiply, they are often following a rote procedure that skips important steps. Although this method does work, it should be discouraged, especially for struggling students, until they have a better conceptual understanding about what solving proportions is all about, as it does not promote the same level of number sense that other methods promote.”

    The other methods that she goes on to discuss are scale factors and ratio/rate tables. Your thoughts on these tables?

    There is no mention of cross multiplying in our curriculum. Also, in the approved textbooks, the numbers are chosen so that students with just a little bit of number sense can see the scale factor. Still, some teachers continue to have students cross multiply to solve something like x/4 = 15/20. The justification is that it is easier when the numbers don’t work out nice. But it’s not about nice. The scale factor method is as meaningful if the scale factor is 6247.094 as it is if the scale factor is 5.

  4. Sorry not to have been more specific. I believe that Michael Paul Goldenberg explained the point well, though I would have left out the irrelevant political references. Cross multiplying is a shortcut that is important to learn to be able to work quickly and efficiently with rational functions, but it is just a shortcut for the more fundamental operation of multiplying both sides of the equation by the same number.

    I’m not precisely sure what the “scale factor method” is when given an arbitrary equation of rational expressions, but the concept of multiplying both sides of an equation by the same thing is certainly the fundamental concept that needs to be taught, with various shortcuts available as consequences.

  5. I have said for a long time that I would like to excise the phrase “cross-multiply” from the vocabulary of math. My primary beef with it is that so many students find the motion of it hypnotically appealing, and try to apply it in instances where it doesn’t apply. (I taught developmental math to college students for five years, so I had a lot of mathematically weak students with strong misconceptions that had many years to get heavily ingrained.)

    And then there are the students who at least only use it for solving proportions, but insist that it’s always the best/fastest way. (Which in their mind, is usually the same thing.) For those, I always used to like to use (2x+1)/(x^2+3x+2) = (x+6)/(x^2+3x+2). I’ve even been known to hold races or stingy-pencil-lead contests for this this problem to show exactly how much slower cross-multiplication can be.

    I disagree that cross-multiplication is “important to learn to be able to work quickly and efficiently with rational functions.” It’s frankly not that short, and students who have it as an ingrained habit tend not to look for nicer ways to solve a particular problem. Also, even for the problems where there’s not an especially nicer way of solving a problem, cross-multiplication isn’t really faster than multiplying both sides by a common multiple of the denominators. In my opinion, if you’re going to use a shortcut that detaches your work from meaningful operations, then at least it ought to be significantly faster.

    I’m a little bit surprised that cross-multiplication seems to be the only algebra-based way of solving proportions that is commonly considered by middle school teachers (basing that on this discussion, which means it’s likely to be way false!). However, at the middle school level especially, I can totally get behind emphasizing methods that build number sense.

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  7. I think of cross multiplying as the shortcut for “get common denominators, then ignore ’em (because they’re the same).” Unlike gasstationwithoutpumps, I am less fond of cross multiplying when variables are involved because variables in the denominator can cause problems. E.g. a problem like 1/(x-3) = 2x/(2x-6). Cross multiplying leads to a quadratic with roots x = 1 or 3 (and then you’d better remember to plug those in and check ’em). Getting a common denominator leads to 2/(2x-6) = 2x/(2x-6) which sensibly leads to x = 1. But unlike Christopher, I do think cross multiplying has meaning. It’s all about making fractions easier to compare by getting common denominators, which is a really big idea in fractions. Does the common denominator itself have meaning? Not necessarily. But it’s still a big, important, sensible idea.

  8. Max:

    But unlike Christopher, I do think cross multiplying has meaning. It’s all about making fractions easier to compare by getting common denominators, which is a really big idea in fractions. Does the common denominator itself have meaning? Not necessarily. But it’s still a big, important, sensible idea.

    I can dig it, and I agree.

    But that’s not the message students receive about cross-multiplication in American classrooms. Indeed, I have pressed middle school teachers for years in workshops to justify this algorithm, and a substantial portion cannot. This troubles me.

    Indeed, I have in front of me Holt Algebra, copyright 1986 (renewed 1992). On p. 14 we have “The Proportion Property”:

    A proportion is an equation of the form a/b=c/d. Both 2/3=10/15 and 2*15=3*10 are true equations. This suggests a property of proportions that can be used to solve an equation such as (2x+8)/(3x-2)=5/4. Proportion Property If a/b=c/d then a*d=b*c.

    This is typical American textbook stuff. I think it’s going to take more than a paragraph’s worth of talk about invisible common denominators to shake teachers used to this stuff out of their torpor.

    • I am in full support of a ban on cross-multiplying (as an algorithm to teach). In my secret wishes my students would invent cross multiplying because I would insist they get good and fast at both scaling and getting common denominators to solve proportions, and some kid (call her Josephine Genius) would say, “couldn’t we just multiply each numerator by the other one’s denominator and not bother to write the common denominators?” and we could name it “Josephine Genius’s shortcut” and talk about when it is useful.

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