Angles at parallel lines, then angle sums, then naming shapes. Corresponding and alternate angles are equal and co-interior angles add to 180°; a triangle's angles add to 180°, and any polygon's to 180° for each triangle it splits into; triangles, quadrilaterals and polygons are classified by their sides and angles.
The angle pairs a transversal makes with parallel lines, and how one angle gives all eight.
A line crossing two parallel lines makes eight angles. Pairs in the same place at each crossing are corresponding; between the lines on opposite sides, alternate; between the lines on the same side, co-interior.
When the lines are parallel, corresponding angles are equal, alternate angles are equal, and co-interior angles add to 180°. From one angle you can find all eight, giving a reason for each.
Find unknown angles in steps, giving a reason for each: what you know, the rule that links it to the next angle, and the reason in words. Parallel-line rules join straight-line and vertically opposite angles.
Why a triangle's angles always add to 180°, and what that gives for any polygon.
However you change a triangle, its three angles add to 180°. Tear off the corners and they make a straight line.
Split a polygon into triangles from one corner. A shape with n sides makes n − 2 triangles, so its angles add to (n − 2) × 180°.
Find unknown angles in triangles and polygons in steps, with a reason for each: the angle sum of the shape, angles on a straight line, equal angles in an isosceles triangle. An exterior angle of a triangle equals the sum of the two interior opposite angles.
Triangles by sides and angles, the quadrilateral family, and polygons: regular or not, convex or concave.
Name a triangle twice: by its sides (scalene, isosceles, equilateral) and by its biggest angle (acute, right, obtuse). Equal sides sit opposite equal angles, so some pairs are impossible: no right-angled equilateral triangle.
Quadrilaterals are named by their properties: parallel sides, equal sides and right angles. A shape that has all of another's properties belongs to it too: every square is a rectangle and a rhombus, and every rectangle is a parallelogram.
A polygon is named by its number of sides (pentagon 5, hexagon 6, octagon 8, decagon 10). It is regular when all its sides and all its angles are equal, and concave when a corner points in.
Nets that fold into solids, the views of a pile of cubes, and (Year 8) points in 3D.
A net is a flat pattern that folds into a solid. Six squares make a cube only if they land on six different faces; a triangular prism has two triangles and three rectangles.
A pile of cubes can be drawn in 3D or as top, front and side views. A front or side view shows only the tallest pile in each column; a top view with numbers shows every pile.
A third axis, z, gives a point in space three coordinates (x, y, z). Real systems do the same with other names: a drone, a building, a plane's latitude, longitude and altitude, and a 3D printer.
Sliding, flipping and turning a shape on the coordinate plane.
A translation slides every point of a shape the same distance in the same direction. Right adds to x, left takes away; up adds to y, down takes away. The shape does not change size or turn.
A reflection flips a shape over a mirror line. In the x-axis, (x, y) becomes (x, −y); in the y-axis, (x, y) becomes (−x, y). Each point lands as far on the other side as it started.
A rotation turns a shape about a centre by an angle in a direction. About the origin, a quarter turn anticlockwise changes (x, y) to (−y, x) and a half turn changes it to (−x, −y).
Year 8: the tests that make triangles congruent, similar shapes and scale factors, and proving and using properties of quadrilaterals.
Congruent triangles are the same shape and size. You don't need all six parts to check: some sets of three parts fix the triangle (the congruence tests: SSS, SAS, ASA or AAS, RHS), and one set of three can fit two different triangles.
Similar shapes are the same shape but can be different sizes, like a photo and its enlargement. Find the scale factor, use it to work out missing sides, and learn the conditions that make two triangles similar.
A diagonal cuts a quadrilateral into two triangles. When those triangles are congruent, their matching sides and angles are equal, and that proves a property for every shape of the kind: a rectangle's diagonals are equal, a parallelogram's cut each other in half, a kite's cross at 90°.
Use the properties of parallelograms, rhombuses, rectangles, kites and trapeziums to find unknown angles and lengths, with a reason for each step, and check with the angle sum of 360°.
Writing a clear set of steps to sort shapes into classes, and (Year 8) to decide whether two triangles are congruent, similar or neither.
An algorithm is a clear set of steps and decisions. A classification algorithm asks yes/no questions about a shape's sides and angles, in an order that gives each shape its own name, and you test it on one shape of every class.
An algorithm can decide whether two triangles are congruent, similar or neither by asking about their equal parts in a careful order: the congruence tests first, then the similarity conditions, and 'not enough to tell' when nothing fits. You test it on one pair of every kind, find what it misses, and fix it.
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For teachers: Teacher guide (notes, answers, solutions, slides and worksheets)