This unit starts inside solids. A cone or a pyramid hides a right-angled triangle, and finding it gives the slant height that the surface needs. The same unit then gives the volume of pyramids, cones and spheres, the surface of spheres and joined solids, and the way area and volume change when a solid is enlarged.
The second half moves shapes on a grid. Students describe translations, rotations, reflections and enlargements in full, then see that two reflections make a translation or a rotation and that the order of transformations can change the result.
A graphic display calculator (GDC) is optional all the way through. Lessons 1 to 4 have a closed “With a GDC” section to open if you have one; lessons 5 and 6 are done by hand.
The lessons, in order
- Lesson 1 · Slant height and surface area The slant height from Pythagoras, the curved and total surface area of a cone (πr² + πrs), and the surface area of a square-based pyramid. Move the radius and height of a cone, then the base and height of a pyramid, and watch the slant length and the surface area change. GDC optional
- Lesson 2 · Volume of pyramids, cones and spheres Volume of pyramids and cones (a third of base area times vertical height), the volume of a sphere and a hemisphere, and composite solids. Compare a cone with the cylinder that has the same base and height, then a sphere with the tube that just holds it. GDC optional
- Lesson 3 · Spheres and composite solids The surface area of a sphere (4πr²) and a hemisphere (3πr² with its flat face), and the surface area and volume of solids made from a cylinder with a hemisphere or a cone. Set a great circle beside the sphere it belongs to, then build a cylinder with a dome or a spike on top and total its faces. GDC optional
- Lesson 4 · Similar solids and scale factors Similar solids with linear scale factor k: areas scale by k² and volumes by k³; finding k from an area or volume ratio; scale models. Enlarge a brick by a scale factor and compare its length, area and volume, then turn two lengths into three ratios. GDC optional
- Lesson 5 · Describing transformations fully Translation by a column vector, rotation about a centre, reflection in a mirror line (x = p, y = q, y = x) and enlargement from a centre, with images found from coordinates. Move a triangle by a vector or turn it about a centre, then reflect it in a mirror line you choose. No calculator
- Lesson 6 · Combining transformations Two reflections in parallel lines (a translation of twice the gap) and in intersecting lines (a rotation through twice the angle), combined transformations, and the order of transformations. Reflect a triangle in two mirrors and name the single move that does the same, then apply a turn and a slide in both orders. No calculator
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.