Grant Wiggins on budding middle school mathematicians

Grant Wiggins has posted a dialog between himself and some middle school mathematicians. Here is an excerpt from one of the letters they sent to him.

With that problem conquered, we moved on to the three-rock episode. Drew didn’t like our chances here. With all his experience in adjusting the four rocks to the perfect weights, using just three didn’t look good. We then remembered an earlier part of your email when you commented that future texts should leave out material to make problems more interesting. Were you doing that to us here, we wondered? Probably so. Therefore, we assumed that we had poetic license to create a little backstory for Farmer John.

So Farmer John has his rocks returned from Farmer Joe and is, at first, heartbroken to see that his forty-pound rock has become a one-pound rock, a three-pound rock, and a thirty-six pound rock. The original rock was used to measure the perfect amount of hay, and can still do that as a trio… and now the rocks are now a bit more portable, for those days that are hard on the back. So, things are looking up.

Farmer John also realizes that he now has the capability to measure other weights of hay. Using both sides of the balance, he can accurately measure hay in the amounts compiled by Kelsey, Aidan, Kirby, Jon, and Kyle and shown on the next page…

The letter is an excellent example of students thinking mathematically, as they ponder some of the various ways they can adjust the problem given to make it more interesting. It seems clear from this exchange that a pro-tip when teaching mathematics is to let students modify the problems to explore other possible interpretations.

Colouring in maps

The purpose of this investigation is to explore what the minimum number of colours is needed to colour in a map so that no two adjacent countries share the same colour. In this case, adjacent countries share a border of more than a point.

Students can start off with a standard map, like a Map of Africa, and begin by experimenting to see what colours work on this map. They could then move to creating their own non-standard maps, and seeing if the minimum still holds. They can also try and investigate what special maps (and their properties) lead to maps that can be shaded with 1, 2, or 3 colours.

The answer to this question is well known, but you should lead students to try and answer a less well known question, why does it work?

Aside: The easiest way I’ve found to actually have students work on this is to download a blank copy of the map of Africa (with the country borders included) and let them use MS Paint (or any other software that works) and let them shade in countries with the program.

Open-ended investigation: Exploring factorials

In this investigation, students can explore factorials (like 5! = 5x4x3x2x1) using a program like Yacas to calculate the larger values when their calculators run out of steam. Students can come up with some of their own questions (Why does 6000! have that many zeroes?) or teachers can offer suggestions for problems (look at 1/0! + 1/1! + 1/2! + 1/3! + … to see if you spot a pattern).