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.