If I ask you to count, starting at 103, for five counts, you do so by keeping track of two quantities. You keep track of them in your head, by speaking out loud and listening to what you just said, or you may seek help in using fingers or other implements. One way or another, two quantities need to be kept track of. It isn’t enough to be able to count, in this case you also need to know where to start and when to stop.
Keeping track of multiple quantities in relationship to each other, that’s a key capacity to develop and expand. One place where it shows up in middle school math is with percentages and proportions. In this post, we’ll look at a simple but interesting scenario.
I often start this with a seventh grader, by presenting the following sketch on a blank piece of lined paper:
I say: “Imagine we have 48 dollars, and we’re calling that 100 percent. What other percentages can you figure out so we can fill out the sheet?” Note that though I only mention 100%, I’ve already filled out 0%, and I have put the 0 on top and the 100 at the bottom of the sheet, and separated and labeled the columns. Invariably, the student will pick 50%. If I ask what 50% of $48 is, the student will say half of $48: $24. I ask where that should go on the sheet, and the student will point to the middle of the sheet. I then write 50% | $24 there. If I ask them what other percentages they can figure out, I get an almost universal response of 25%. They then say that’s half of $24, $12, and we write 25% | $24 on the sheet. Typically, students offer 75% next, though a small number will mention 12.5%. How much is 75% of $48? Here I get divergent answers. Some students find the amount in the middle of $24 and $48. Some students say it is three times what they got for 25%. And some students say you need to add what they got for 25% to what they got for 50%.
Next, most students will try 12.5%. Very few will offer, on their own, 10% or 1%. At some point, I’ll ask: “What about 10%?”
Some kids will say, “Oh yes,” and arrive at 4.8 quickly. Others have a harder time with this. It’s interesting to see what they take as their starting point. Some will try to get to 10% from 12.5%, reasoning, correctly, that it must be less but close by. If I ask them to point to 10%, they will all locate it appropriately on the scale from 0 to 100. Sometimes, that is sufficient for the student to come up with $4.80, apparently “translating” the 10 on the left scale to 4.80 on the right scale. Some students will take 50% as their starting point, dividing it by 5. As you might expect, many students will find 10% from 100%, by dividing $48 by 10. Some students find 10% difficult, but then suddenly remember how to get 1%, and say it is 48 cents. If students say this, I add it to the display and ask again about 10%. One way or another, I make sure the student arrives at 10% | $4.80.
At this point, I ask what other entries they can figure out. Usually, they then add 20% and 5% – each is found from 10% by doubling or halving. When I ask for 30%, most will reason that since they already have 20% and 10%, they can add the corresponding amounts. Few will triple what they got for 10%. When I ask for 90%, almost nobody will work upwards by 10% from 50% – they seem to see that this takes too long, and almost every student will subtract the amount for 10% from 100%.
It may surprise you, as it surprised me when I started to do this with seventh graders, that so little use is made of multiplication. They could have gotten all these numbers through multiplication by 48 cents. None of them do.
At the same time, these students seem to have a very firm grasp on a rather sophisticated idea about scaling:
that if you add two numbers on the left, you can add the numbers on the right, and get a new pair of numbers that fits. Though not every seventh grader is able to articulate this well, I have yet to encounter one who didn’t use this idea – including students who otherwise seemed to reason at a fourth-grade level.
Here’s a relationship between six numbers that students seem to be able to hold on to, and use! Moreover, I very much doubt that this was a relationship they were taught once and happened to remember. They sure didn’t remember the more recently-taught material about multiplying by the 1% number. Instead, these kids seemed to work out the additive relationship on their own, and without any fanfare. Me, I think this is pretty darn cool.