A complete lesson on the theme of volumes of spheres, best suited for more able students. Given that this can be a very dull, restricted topic if pupils just calculate volumes of spheres, hemispheres, etc given the volume formula, the focus is more on deriving a formula. I would teach this after pupils have met all other volume rules (cuboids, cylinders, cones, pyramids) - the derivations and activities require a knowledge of these other rules.
A question to get pupils thinking about the different volume rules they’ve already met (cube, pyramid, cylinder, cone).
Starting from an image of a square within a circle within a square, pupils are prompted to come up with an inequality for the area of the circle (ie use the inner square as a lower bound and outer square as an upper bound). This ‘leads’ to an estimate of pi=3, but the real purpose is to prepare pupils to do a similar process in 3D, to come with an estimate for the volume of a sphere…
Starting from an image of a cone and a sphere within a cylinder (see cover image), pupils are prompted to come up with an inequality for the volume of the circle (using the cone as a lower bound and cylinder as an upper bound). This ‘leads’ to a conjecture for the volume rule of a sphere!
Some simple examples using the rule. At this point, you could supplement with extra ‘basic’ questions if necessary.
Some questions on the theme of the solar system, looking at volumes of planets and reverse problems (finding radius or diameter given the volume). This also involves standard form as the volumes involved are huge, and could be followed up with some questions about scale and volume factor.
I’ve also thrown in a formal proof for the volume rule, that could be looked at with very able students.
A link to a short video showing a completely different (and fairly accessible) proof, that could be recreated using an orange, knife, and some messy cutting…
Please review if you buy as any feedback is appreciated!