At a Math Solutions retreat, Nicholas Branca posed this for us to investigate:
If only two numbers are marked on a number line, is it possible to figure out where all numbers are located?
Nicholas had us try for a few pairs of numbers, to be sure we understood the problem. For example, we tried a few using zero for on of the two points—0 and 1 (easy, peasy), then 0 and 3 (OK to eyeball), then 0 and 11 (we reached for rulers). Then we tried a few where neither of the pair of numbers marked was zero—1 and 2 (again, easy peasy), 3 and 5 (also OK to eyeball), 8 and 20 (we used rulers again).
We investigated in small groups, drawing lines, marking a pair of points, assigning them numbers, and seeing if we could figure out where all of the other numbers would go. We discussed different strategies.
I can’t remember exactly what I did then, so I played around with the problem a bit this morning. Then I explored various pairs of numbers:
- consecutive numbers (e.g., 6 and 7, or 10 and 11)
- one number a multiple of the other (e.g., 5 and 20, or 5 and 50)
- numbers with a common factor (e.g., 8 and 12, or 9 and 12)
- relatively prime numbers (e.g., 3 and 14, or 2 and 7)
I began to think about numbers other than whole numbers. And then negative numbers.
I stopped to write this blog post.
Musing As I Wrote
It’s always tempting to procrastinate when writing. It’s especially easy at the computer. As I’ve confessed recently, I’m new to blogging and tweeting and have been doing some catching up this summer, which is an easy way to procrastinate and still feel productive. I recently read Dan Meyer’s blog post, The Three Acts of a Mathematical Story. (OK, OK, I realize that he posted this on May 11, 2011, but I’ve already confessed to being a newbie to blogs and tweets.) Although I began as a secondary math teacher, I’ve been absorbed with elementary school math for most of my career and I’m interested in ways to apply Dan’s thinking about secondary math teaching to the elementary level.
I actually learned about Dan’s thoughts about three-act math from its application in a Kindergarten lesson that Joe Schwartz posted on his blog two months ago, Nobody Puts Kindergarten in the Corner! Joe got the lesson idea from Shark Bait, a post on Graham Fletcher’s blog.
I was especially curious about Joe’s post because Kindergarten teaching isn’t my thing. Ann Carlyle, who recently wrote a book with Brenda Mercado, Teaching Preschool and Kindergarten Math, once described Kindergarten to me as “every day is a birthday party.” That scared me off. But Ann and Brenda’s book (yes, books are still an important part of my professional learning) and wonderful accompanying videos made me think about reconsidering my attitude toward working with children this young.
Back to three acts and my morning thinking about number lines. Maybe Nicholas’s problem can be thought of as a three-act math investigation.
A Number Line Exploration in Three Acts
From Dan: “Introduce the central conflict of your story/task clearly, visually, viscerally, using as few words as possible.”
From me: Draw on the board a horizontal line with two marks. Explain: This is the start of a number line. So far I’ve made marks for two whole numbers. Now I’m trying to figure out how to make marks for all other whole numbers. Is this possible to do, no matter what two numbers I use to label these marks?
Do a few examples, as Nicholas did? Or not? (I think I would.)
Then explain: Try with other pairs of numbers. Find numbers for which it’s not possible to create the entire number line, or make a convincing argument for why it’s possible to do so with any two whole numbers.
From Dan: “The protagonist/student overcomes obstacles, looks for resources, and develops new tools.”
From me: Students work in pairs or small groups exploring pairs of numbers of their choosing.
From Dan: “Resolve the conflict and set up a sequel/extension.”
From me: Discuss as a class. If students have found pairs of numbers for which they think it’s impossible to create a number line, maybe give time over a day or two for others to investigate. If students have arguments for why it’s possible for them to create a number line from any two points, have them write their arguments on posters for the class to analyze.