In the movie Good Will Hunting the main character, Will, uses math trees to solve a problem on a hallway chalkboard at MIT.
It is doubtful that math trees had anything to do with a complex problem that took several professors two years to solve—the structure just provided a nice visual for the audience. In fact, math trees are fairly simple and easy to grasp. They are also an excellent conceptual device that offers fun math games for kids of all ages.
Math trees are a part of mathematics called graph theory. That’s not the kind of graph you used to plot the slope of a line back in the day. It’s a type of graph that is used in computer science and other areas of discrete mathematics where only specific quantities are allowed.1
To Get Started You Need:
- Q-tips (several dozen)
- Pennies or colored cotton balls (several dozen)
- Pencil and paper
A math tree consist of lines and end-points (edges and vertices). The rules are simple. I’ll summarize them here:
(These rules are for an irreducible tree–the kind you saw on the chalkboard in Good Will Hunting.)
1. Every line must have an endpoint on both ends.
2. Only one line can connect two endpoints.
3. Lines cannot intersect.
4. An endpoint can have one, or three or more lines coming from it, but it cannot have two. (This is the rule that makes the tree irreducible.)
(Use a penny or cotton ball for the endpoint, and a Q-tip for the line…)
1. Build a tree together. Begin by placing a single endpoint on your table-top. Now take turns doing the following: add one line and one endpoint. Watch the tree grow. Be sure to pay attention to the rules, especially rule #4. When you’re finished, examine the tree. Verbally identify, then correct any mistakes.
2. Completely remove all the lines from the tree you made in game one. Look at the array of endpoints that are left. Now add the lines back to the tree. When you’re finished, examine the tree. Identify and correct any mistakes.
got the hang
a graph out loud
without the use
of visual tools.”
3. See which player is first to determine how many combinations are possible when there are exactly five endpoints. (This is very easy.)
4. See which player is first to determine how many combinations are possible when there are exactly six endpoints.
5. See which player is first to illustrate three different graphs when the number of endpoints is exactly eight.
6. See which player is first to illustrate five different graphs when the number of endpoints is exactly ten (if necessary, use pencil and paper instead of Q-tips and pennies).
7. Using pencil and paper and a timer, in two minutes see which player can graph the most combinations when the number of endpoints is exactly twelve.
8. Use pencil and paper, see which player is first to graph the following: A robin lays four eggs. A year later, one of the four (now an adult) lays four eggs. A year later, two of those four each lays three eggs. Related: how many eggs were laid in total?
Math trees offer a useful device to chart combinations and relationships between items. They also provide an easy resource to play math games with kids. The games are flexible enough to invite participation on many levels. Once you’ve got the hang of things, try explaining a graph out loud without the use of visual tools. For instance, the answers to games three and four above–even game eight–could be explained verbally next time you’re in the family car on your way to the grocery store. And really, what could be better than pondering irreducible math trees while circling for a parking spot?
1Our natural (counting) numbers are an example of discrete mathematics—when kids play hide and go seek, one person counts ‘1,2,3,4,5,6,7,8,9,10, ready or not here I come.’ That’s a case where only discrete quantities are allowed. If the person counted using real numbers, where fractions, square roots and repeating decimals are allowed, it might take a very long time to start a game.