Level 6: Solids and sizes

Measurement · level 6 of 8 · rough age 13-14 (a guide for parents; anyone can start anywhere)

In level 5 every solid had the same shape all the way through. In this level the solids come to a point or curve all over: you prove why a cone is exactly one third of its cylinder and why a ball is two thirds, unroll a cone into a slice of a circle, and measure the volume and surface of Mars. Then you search for the best shapes, the ones that use the least material, measure time by the turning of planets, and carry units through rate problems so that they check your answers for you.

Lesson	You will be able to
Volume of pyramids and cones	Explain with three pyramids in a cube why a pyramid or cone is one third of base × height, and use slant and vertical heights
Volume of a sphere	Prove \(V = \tfrac43 \pi r^3\) by slicing a hemisphere, a cone and a cylinder, and find the volume of Mars
Surface area of cones, pyramids and spheres	Unroll a cone into a sector to get \(\pi r l\), find pyramid surfaces, and explain why a sphere's surface is \(4\pi r^2\)
Best shapes	Show that the square is the best rectangle, find the best pen, box and can, and explain why real cans differ
Time across the planets	Use time zones, explain why the Earth spins once in 23 h 56 min, and convert between days, sols and Mars years
Rates of measurement	Solve filling, fuel and rainfall problems with units that cancel, and combine rates that work together or against each other

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