Maximum number of regions created by straight line and planar cuts
The figure below uses five lines to make five triangular regions. Is there an arrangement of seven lines that makes eleven triangular regions?
I would start this lesson by emulating Pólya's instruction, as seen here. That is, discuss the maximum number of regions you can create in the plane using just straight line cuts. Then do the same for planar cuts in 3D.
Next, ask your students this question: How many regions can you divide a cross shape into using just two straight cuts? The answer: 6 regions.
After that, here's another closely related problem to give your students: How many regions can you divide a crescent shape into using just two straight cuts? What about three straight cuts? Answer: With two straight cuts, a crescent can be cut into 6 regions, with three cuts, 10 regions.
Can we generalize our strategy for any shape? Sure. Notice that two lines cut the plane into 4 regions. Thus, whether our shape be a crescent or otherwise, making cuts that intersect somewhere inside the shape will produce at least 4 regions. For convex shapes, you'll get exactly 4 regions with 2 cuts, and that's the best you can do. That's because it's impossible to draw a line in a convex shape that contacts more than two of its boundary points. With concave shapes, such as the crescent, our goal must be to contact as many boundary points as possible. For polygons, that means looking for reflex interior angles, and lines that can cut across their sides. Making this even more general, that is, to allow for curved shapes too, we can start by creating a convex hull. Basically, the convex hull is a rubber band that has been stretched around the shape. Wherever the rubber band leaves the shape, the shape must be curving inward, and so that alerts us to a candidate position for making a cut.