Goals
This builds on the study of Euler's characteristics for 2-D and 3-D shapes
See how similar concepts can be applied to seeming very different objects
What To Do
Ensure that you complete the Trees lesson before this lesson to explain what trees are
It is not necessary to use manipulates for this lesson, in fact pen and paper are probably best
Euler's Characteristic is the value of (number of vertices) - (number of edges) + (number of faces). For a tree:
the vertices are the ends of the sticks
the edges are the sticks themselves
the faces are the spaces inside the grid - they must be fully enclosed to count as face. (Note that the outside space surrounding the graph is usually counted as a face but that can be ignored for simplicity)
Ask the student to draw 3-4 dots and join them up using straight lines. The only rules is that the straight lines can not cross each other. (When you cross lines you are really just adding another vertex. If you count this then there is no problem)
Count the vertices (dots), edges and faces.
Repeat the exercise for other graphs
Calculate the value of Euler's characteristic for each shape. Is it always the same
Try for larger number of dots. Does this change the Euler's characteristic?
Additional Work
Try playing the Brussels Sprouts game. Can you work out a strategy to win?
Background Information