This unit starts with the surfaces and volumes of cones, pyramids and spheres, and asks why the formulas are what they are: the curved surface of a cone opens into a sector, and a sphere fills two thirds of the cylinder around it. The unit then asks what happens to length, area and volume when a solid is enlarged, and how to use that in both directions.
The second half moves points and shapes on a grid. Students dilate from a centre with fractional and negative scale factors and find the centre from an image, rotate about the origin and about other points, find perpendicular lines, and see why two reflections make a slide or a turn and when the order of transformations matters.
A graphic display calculator (GDC) is optional all the way through. Lessons 1 to 3 have a closed “With a GDC” section to open if you have one; lessons 4 to 6 are done by hand.
The lessons, in order
- Lesson 1 · Cones and pyramids: slant height and surface area The slant height of a cone and of a pyramid face by Pythagoras, the curved surface as a sector, total surface area, the sector angle, and working back from an area to a height. Set a radius and a height and watch the cone and the sector it opens into change together. GDC optional
- Lesson 2 · Volume of pyramids, cones and spheres Volume of pyramids, cones and spheres, why a pointed solid is a third of its prism, the sphere as two thirds of its cylinder, and the volume of composite and recast solids. Set a radius and compare the volumes of a cone, a sphere and a cylinder that fit together. GDC optional
- Lesson 3 · Similar solids: lengths, areas and volumes Similar solids and the scale factor k, lengths by k, areas by k squared and volumes by k cubed, finding k from area and volume ratios, percentage change, and cutting a cone. Enlarge or reduce a cuboid and watch its surface area and volume change against the ghost of the original. GDC optional
- Lesson 4 · Dilations: centres and negative scale factors Dilation of a point from a centre by a positive, fractional or negative scale factor, finding the centre and the scale factor from a point and its image, and the area factor. Move the centre and the scale factor and watch the image and its rays. No calculator
- Lesson 5 · Rotating about the origin and perpendicular lines Rotation of a point about the origin by 90 degrees either way and 180 degrees, the gradient of a perpendicular line, equations in the form ax + by + c = 0, and rotation about another point. Set a point and a number of quarter turns and compare the gradients of the two radii. No calculator
- Lesson 6 · Two reflections and the order of transformations Two reflections in parallel mirrors (a translation of twice the gap) and in intersecting mirrors (a rotation of twice the angle), three reflections, a glide reflection, and the order of transformations. Reflect in two mirrors, parallel or meeting at a point, then swap the order and compare. 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.