Not even kidding; I just needed to figure this out.
So, it takes a while, but I like to cook my own beans. To make my life easier (and to get rid of even more of that pesky free time), I pack 1 cups of cooked beans in freezer bags, along with their juice, because most recipes specify some number of the size of can that holds that amount of beans. But I’m making something that calls for 1 pound of dried cannellini beans, to be cooked separately and then used, and I just cooked and packed a pile of these last week. So, dear Number Rockets (if I had any at the moment), how would you solve this problem? I have some dried cannellini around my kitchen. I have cookbooks and basic kitchen tools. I want to use the beans I’ve already cooked and frozen. I need the amount of cooked beans that 1 lb. of dried beans would produce. What should I do? What are the steps? Do I know everything I’ll need to know? How does the knowledge I have shape the steps I can take? Will I ever eat this tasty soup or will I grab some Brtit-Indi Takeaway and just forget about the math. (Hint: as delicious as Brit-Indi Takeaway can be, Number Rockets never just forget about the math!)
“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 . 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 .” 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) . Is the GCD greater than 1? Then it can be simplified. Let’s see how and why that works.
2) . Find the GCD and use it.
3) • . Associative Property of Multiplication.
4) 1 • . Identity Property of Multiplication.
So, = . 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 + . 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 .
1) = 1 • . Multiplication by 1 changes nothing (Identity Property).
2) • is still 1 • . We’re good.
3) So, = = . Now we can add + .
What about writing as a percentage?
1) Is the GCD greater than 1? No? Then is in simplest form.
2) What do we need? One hundred as a denominator would be nice.
3) What’s the clever one? , or (really clever, this one).
4) Now 1 • = • = = .
5) So, = 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.
“But I thought a minus and a minus makes a plus. We learned that last year.”
One problem with rules is that they depend on the user’s memory. One little jurisdictional detail forgotten or misremebered and the whole shebang can go sour. Students may have memorized rules for sign management when it comes to doing arithmetic with signed numbers and variables, but chances are pretty good that time away from the game, like, say, Summer, has eroded the fidelity of those memories. By “and,” do you mean addition, in which case (–)+(–)=(–), or by “and,” do you mean multiplication, in which case (–)•(–)=(+)? Can we offer students an understanding here that is more than a rerun of those earlier rules of engagement?
Maybe. There are several ways to illustrate the outcomes of the addition of signed objects, and multiplying differently signed objects is just a few steps away. In these cases, intuition, to the extent that my number rockets will have math-flavored intuition, is fairly reliable. The 18 ft. Great White in the pool, however, is the multiplication of two negatives. Why oh why is negative 2 times negative 3 a positive 6? Who was on watch when they rolled that one out? Middle school is not the time to break out number theory proofs, so what kind of model can we ask students to handle that will feed their intuition and not overwhelm their budding abilities for formal, abstract thinking?
Here’s one idea towards helping to correct this signage misconception. It’s a work in progress, but it basically breaks down like this. Let’s say we’re playing games in an unusual casino, Casino Zero. Some of this is imaginary, of course, but some can certainly be acted out in small groups in class.
- You get your playing chips from the cashier. Everybody starts out with zero credits in the form of, say 20 positive credits and 20 negative credits. Let’s say all the chips have a value of either +5 or –5, so you’d have this set of 8 chips:(+ + + +,– – – –), four +5 credit chips and four –5 credit chips. And you certainly understand that the net value in your hands is zero credits. If you were to cash in these 8 chips, the cashier would give you nothing in return.
- Say you win your bet of 10 credits. You have two ways to act that will register this change in value. You can GRAB two +5 chips from the pot, or you can TOSS two –5 chips from your hand. In either case, if you go straight to the cashier and cash in your chips, you have clearly increased your wealth by 10 credits, which the cashier will happily pay you. Students see this because they are comfortable with the addition of differently signed numbers and because they are holding the chips in their hands.
- Of course, that’s one action that would register that change in value. Another action would be to GRAB one +5 chip and TOSS one –5 chip. Same outcome. Same change in wealth. Same huge haul at the cashier’s window. And students are becoming fluent with these kinds of equivalences.
- We need to be explicit about these actions. Every action has a multiplier, which is a positive number if you GRAB and a negative number if you TOSS. (I think that’s reasonably intuitive.) And you can GRAB and TOSS both kinds of chips, + chips and –chips. For example, you lose 15 credits: TOSS three +5 chips, written as –3•5. Or, you could GRAB three –5 chips, written as 3•–5. In both cases, your wealth has decreased by 15 credits and you can see that in your hands. You know exactly what will happen at the cashier’s window.
- Let’s say a player wins 25 credits but there are no +chips to GRAB. The player is clever and so she knows that there is another action besides GRABBING from the pot; she can TOSS from her hand. She TOSSES five –5 chips from her hand, getting rid of (– – – – –). She writes this as –5•–5. When she looks in her hands, she sees that her wealth has increased by 25 credits. She skips happily towards the cashier’s cage, secure in her understanding that a negative times a negative is a positive. And so rich, to boot.