Students will learn how to find the volume of cylinders, spheres, and cones. They will solve word problems and mathematical ones. Khan Academy covers this topic completely. Next, give your students the following two challenges:

The setup to the problem is here. Call the width \(w\) and the height \(h.\) Ask students to express \(w \mathop{?} h\) as a formula. After a bit of play, students should see the dots exist on two grids, one shifted. The more numerous set of dots is \(wh,\) while the less numerous is \((w - 1)(h - 1).\) Thus, the answer is

$$w \mathop{?} h = wh + (w - 1)(h - 1)$$Then use the setup found here, calling depth \(d.\) Ask for the formula \(w \mathop{@} h \mathop{@} d.\) The answer here is similar, it's

$$w \mathop{@} h \mathop{@} d \ = whd + (w - 1)(h - 1)(d - 1)$$Finally, give your students this Pythagorean theorem puzzle.

Conclude by leading this investigation:

Treefolk Tribes (symmetry, sorting, sequence)

by MathPickle