Blogpost #4: Downbeat Mathematics & Clever Ones

“It’s got to be on the one.” -James Brown

“Everything had to happen on the one. However your rhythm changes were made or how you played your patterns, you had to distinguish the one and that helped you stay focused on what you were really doing. You could do anything you want as long as you were on the one.” -John “Jabo” Starks

Let’s say we have the fraction \frac{12}{36}. I hear this sort of thing a lot: “When you simplify this fraction, twelve goes in here once and in here three times, so the answer is \frac{1}{3}.” True indeed, but I wonder what experiences these students have had along the way to that shortcut. Does it matter? If a student can explain, “See, you divide each number by the biggest one you can and then you get the answer,” isn’t that enough? I’m not so sure. (Maybe it depends on what we mean when we say “enough.”) Of course, I wasn’t there in the previous year or two, so I don’t know how they’ve come to be able to do what they can. I do, however, believe that math instruction should be both philosophically explicit and developmentally appropriate. I think students should first walk the long path and then find the shortcut.
I mean that it’s important to know that dividing a rational number’s numerator and denominator by their GCD is the way to express that number’s equivalent value in lowest terms; it’s also important to be able to use that language. Once you can comfortably explain all that, then, by all means, take the shortcut and talk about numbers that gazinta each other all you want. But first I think the instruction should be explicit and finely detailed.
1) \frac{12}{36}. Is the GCD greater than 1? Then it can be simplified. Let’s see how and why that works.
2) \frac{12x1}{12x3}. Find the GCD and use it.
3) \frac{12}{12}\frac{1}{3}. Associative Property of Multiplication.
4) 1 • \frac{1}{3}. Identity Property of Multiplication.
So, \frac{12}{36} = \frac{1}{3}. But we explicitly used 1 and some important concepts to get there.
Understanding the importance of 1 and using it comfortably can be  just as helpful when working with rational numbers in the other direction. Suppose we need \frac{2}{3} + \frac{7}{36}. Rather than leading with “Whatever we do to the bottom, we have to do to the top,” which is the shortcut, invoke the use of a clever one. A clever one is 1 in whatever form is most useful to us at the time. Because students understand GCD and are comfortable with the properties of 1, the clever one in this case is \frac{12}{12}.
1) \frac{2}{3} = 1 • \frac{2}{3}. Multiplication by 1 changes nothing (Identity Property).
2) \frac{12}{12}\frac{2}{3} is still 1 • \frac{2}{3}. We’re good.
3) So, \frac{2}{3}\frac{12x2}{12x3} = \frac{24}{36}. Now we can add \frac{24}{36} + \frac{7}{36}.
What about writing \frac{13}{16} as a percentage?
1) Is the GCD greater than 1? No? Then \frac{13}{16} is in simplest form.
2) What do we need? One hundred as a denominator would be nice.
3) What’s the clever one? \frac{6.25}{6.25}, or \frac{625/100}{625/100} (really clever, this one).
4) Now 1 • \frac{13}{16} = \frac{6.25}{6.25}\frac{13}{16} = \frac{6.25x13}{6.25x16} = \frac{81.25}{100}.
5) So, \frac{13}{16} = 81.25%.
I think there are ways to teach important aspects of number theory to students even though they are lurching through the developmental moors that lie between concrete and formal thinking. By exploring such pathways through explicity guided instruction, those students will be able to build significant understanding of what they are really doing as they breeze skippingly along those cheerfully named shortcuts, many of which they can he helped to discover for themselves.


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