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).

Design project: Make Your Own Attributes Matching Game!

This is another project idea from Malke Rosenfeld. The basic idea is this – students create their own versions of an attribution matching game. In these games the idea is that each card in the game is unique, but that it has traits (like colour, shape, type, etc…) in common with other cards, and the objective of the game is usually to try and create matches based on those attributes.

Read her blog post to get a more complete description of how this project works.

Design project: Create a multiplication tower

This idea comes from Malke Rosenfeld. There are other ways of arranging the tower, so while the diagram above should give you some ideas as to what a multiplication tower is, you and your students should adapt this project. Additionally, once it is built, you can use the tower to look for patterns in multiplication.

Year long project: Look for math

This project can be done in any class, but lends itself well to elementary and middle school classes best. The task is simple, everyone in the class (teacher included) looks for examples of math in their day to day lives and records those examples as they find them. This is not intended to teach students the mathematics, although the examples students find may work as a good hook for a lesson, but to teach them that math is part of their world.

Here’s a Flickr group you can use to have your class share their examples with a wider group.

Open ended problem: Scoring investigation

Using shoes for keeping track of score

The problem: Design a scoring system for a game that can be used with shoes. Note that this problem may actually be different depending on what game you are playing.

This problem has  a lot of possible solutions, and might make assessment challenging, but my recommendation is: play a game with each of the scoring systems so that students can see them in action, and then decide which one they think makes the most sense.

Open ended problem: Parking lot

I thought of this investigation as I was walking home and happened to notice an unusually shaped parking lot. It made me wonder if the way the parking spots were arranged within the parking lots was the most efficient way to pack the cars in. Obviously the picture above is a sample, and you are free to choose whatever parking lot you would like to share with your students.

Question: What is the most efficient way of creating parking spots within a parking lot so that as many cars as possible are able to park in the parking lot, and so that each of these cars can leave whenever they want?

Here is a Geogebra applet I designed to allow students to explore different arrangements themselves.

Investigating repeating decimals

A number of years ago, I had a 9th grade math class where I decided we would investigate when the decimal representation of a fraction repeats, and when it terminates. We also decided to investigate to see if there was a pattern in the number of repeating digits when the decimal representation of a fraction does repeat. It took us about two weeks to conclude that we weren’t going to find a pattern without a lot more work, and so we abandoned the investigation, but I still remember the process quite keenly.

If you decide to try this investigation with your own students, you may find this arbitrary precision division calculator (we quickly ran into the limits of our computer’s calculator when doing this investigation) useful: http://davidwees.com/divider/

Open ended problem: Tables for a party

This problem allows for multiple possible solutions, and will require students to think about some assumptions they will have to make to solve this problem. You can easily vary the difficulty of the problem by changing the number of people invited to the party.

How many tables and how many chairs?

Ellen is planning a party for her friends. She has invited 100 of them, but she doesn’t know exactly how many of her friends will attend. She wants to put out tables for her friends, and she wants to put enough chairs at each table so that none of her friends has to sit alone. Assume that her friends will fill up each table as they arrive. How many tables should she put out, with how many chairs at each table?