Tag Archives: cross multiplication

Setting norms

So here are the things that are important to me in my classroom and on my blog (in no particular order):

  1. Seeking to understand,
  2. Basing our arguments and claims on evidence,
  3. Asking honest questions,
  4. Looking for meaning, and
  5. Using only names and labels that make useful distinctions.

Despite my intentions, I do not honor each of these on a 24/7 basis.

example 1

See, for example, my claim that “Everyone knows that there are 2 Pop Tarts in a pack.gasstationwithoutpumps called me on it. I had no evidence for that claim (principle 2 above), and making that claim seemed to suggest that I wasn’t seeking to understand the perspectives of readers (principle 1). I took my well-deserved lumps.

example 2

I recently found myself having to backpedal in College Algebra when a student was explaining her thinking. I basically cut her off and finished what I thought was her thought. It was not. I was not really seeking to understand her in that interaction, nor was I asking her honest questions. I realized these things, apologized and told her that I was going to stop guessing what she was thinking and listen instead.

Why this matters

We teachers tend not to think about how much power we really have to set the tone in our classrooms. I was reminded of this when a College Algebra student recently asked plaintively, “But what does it mean?” And when one student asked another in another course, “Wait…did you get your question answered?”

And, by contrast, when two students in the next class (not mine) were watching me erase the previous class’s whiteboard, on which was written “1/16+1/16+1/32=5/32”. The first student couldn’t fathom how the right-hand side of the equation resulted from the left. The second student explained, and told the first, “You have to kind of dumb yourself down to see it.”

Those first two examples are the result of my hard work setting the tone in my classroom. The last examples are reminders that this tone doesn’t just happen on its own.

And I am sure the same is true of our virtual forums as well.

Getting down to business

I knew I’d get a few readers riled up about cross-multiplying this weekend.

If an algorithm can be political, this is the one.

Cross-multiplication is to algorithms what Herman Cain, Michael Moore and Ron Paul would be to politics if they were all one political entity. Its alleged misdeeds are numerous. Its defenders are passionate. It never, ever retires and goes away.

gasstation wrote to defend the algorithm:

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

I wanted to encourage principle 2, Basing our arguments and claims on evidence, and did so with a light-hearted jab.

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.

Michael Paul Goldenburg provided an example:

Say we’ve got 3/x = 1/6.

Now we have something to work with. But I’m not buying that this is the sort of thing gasstation meant.

It can’t be, can it? There was nothing about the algorithm I proposed that stipulates we need to compare denominators. Nope, my algorithm is couched in making comparisons among numbers with meaning. That proportion above? It represents two triangles, one with a short side of length 3 units and an unknown side length for the long side; the other with short side of length 1 unit and long side of length 6 units.

Scale factor from small triangle to large is 3.

If what we’re building is an algorithm based on meaning (principle 4 above), this hasn’t convinced me to change my blueprint. I need better evidence. Someone’s got something chambered, I’m sure. Let’s see it.

postscript

In the spirit of principle 5 above, I hereby and respectfully express my request to refrain from the following practices:

  1. Taking or implying sides in the Math Wars of yore,
  2. Lumping together groups of teachers in order to make claims about origins of problems in the teaching and learning of mathematics, and
  3. Using the term constant of proportionality as though it were something different from slope.

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.