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.
Each lesson works two ways. Students can work through the online tutorial at their own pace (it checks their answers and saves their progress in their browser), or you can teach it to the whole class from the slides, then send students into the tutorial for the Modify and Make steps.
Every lesson guide has plain-language notes, the answers to every check, a solution for every task, and slides and a worksheet for Waypoints Prepare. These guide pages are not linked from student pages, but they are not secret either: keep printed Teacher editions for answers you do not want students to find.
Whole-course cheat sheet: Open to print (every command, grouped by unit: a handout for the final challenge)
The angle pairs a transversal makes with parallel lines, and how one angle gives all eight.
A transversal crossing two parallel lines makes eight angles, four at each crossing. Because the lines go the same way, the bottom crossing is a copy of the top one, so the angles come in pairs:
With vertically opposite angles and angles on a straight line (Year 6), one angle gives all eight. Each angle found comes with a reason.
The pairs are the first real geometric reasoning students do: a chain of facts, each justified. They also prove that a triangle's angles add to (draw a line through one corner, parallel to the opposite side: two alternate pairs), and they return in construction, surveying and anything built with parallel parts (scissor lifts, roof trusses, shelving).
parallel: going the same way, never meeting (marked >>). transversal: a line crossing two or more lines. corresponding, alternate, co-interior: the three pairs above. supplementary: adding to . vertically opposite: opposite each other at a crossing.
Cheat sheet: Open to print (A4, with board pictures) · Markdown
Why a triangle's angles always add to 180°, and what that gives for any polygon.
The three angles of any triangle add to 180°. Change the shape and one angle grows while the others shrink by the same amount; change the size and the angles do not change at all. Why: tear off the three corners and put their points together, and they fill a straight line, which is 180°.
Any polygon splits into triangles from one corner: join that corner to every corner not next to it. A shape with sides makes triangles, and their angles fill the polygon's corners exactly, so the angle sum is : 360° for a quadrilateral, 540° for a pentagon, 720° for a hexagon.
Angle sums give reasons in geometry: a missing angle can be worked out, not measured, and the answer is exact. They also explain which regular shapes tile a floor, and link to the turtle in the coding pathway, which turns 360° in total round any closed shape.
interior angle: an angle inside a shape at one of its corners. angle sum: the interior angles added together. polygon: a closed shape with straight sides. regular: all sides and all angles equal.
Cheat sheet: Open to print (A4, with board pictures) · Markdown
Triangles by sides and angles, the quadrilateral family, and polygons: regular or not, convex or concave.
Shapes are named by their properties, not by how they look.
A triangle gets two names: by its sides (scalene, isosceles, equilateral) and by its biggest angle (acute, right-angled, obtuse). Equal sides sit opposite equal angles, so an equilateral triangle has three 60° angles and can never be right-angled.
A quadrilateral's name lists properties: trapezium (at least one pair of parallel sides), parallelogram (two pairs), rhombus (four equal sides), rectangle (four right angles), square (both), kite (two pairs of equal neighbouring sides). A shape with all of a name's properties belongs to that family: a square is a rectangle and a rhombus.
A polygon is named by its number of sides, and is regular only when its sides and angles are all equal. It is concave when a corner points in.
Definitions by properties are where mathematical reasoning starts: "it is a rectangle, so its diagonals are equal" is an argument. Pythagoras (next in Measurement) needs right-angled triangles named reliably.
scalene, isosceles, equilateral, acute, right-angled, obtuse, trapezium, parallelogram, rhombus, kite, regular, convex, concave, and the polygon names pentagon to decagon.
Cheat sheet: Open to print (A4, with board pictures) · Markdown
Nets that fold into solids, the views of a pile of cubes, and (Year 8) points in 3D.
A solid can be drawn flat in three ways, and each shows something different.
A net unfolds the solid: every face is drawn at its true size, so a net gives the surface area. A cube's net is six squares that cover six different faces; six squares alone is not enough. A triangular prism's net is two triangles and three rectangles.
An isometric drawing shows what the solid looks like in 3D, but squashes the faces and hides what is behind.
Views (top, front and side) are what you would see looking from one side. A front or side view shows the tallest pile in each column, so cubes behind it are hidden. A top view with the height written on each pile shows every cube.
Engineers, builders and designers pass solids around as drawings. Choosing the drawing that shows what you need is a reasoning skill: a net for cutting and costing, views for building.
A third axis, z, points up from the x-y plane, so a point in space is (x, y, z): along x, along y, then up. The order is always x, y, z, and points on the floor have z = 0. A box from the origin to (x, y, z) has edges x, y and z long. Real systems do the same with other names: east, north and up for a drone, metres up in a building, latitude, longitude and altitude for a plane. The common slip: writing the height first.
net, prism, face, surface area, isometric drawing, view, top view (plan), front view, side view (elevation), z-axis, 3D coordinates, altitude.
Cheat sheet: Open to print (A4, with board pictures) · Markdown
Sliding, flipping and turning a shape on the coordinate plane.
A transformation moves a shape on the coordinate plane. The original is the object and the result is the image; A becomes A′. The size and shape do not change.
Moving shapes by coordinates is how computers draw games and animations, and it leads to symmetry, tessellations and (later) matrices.
transformation, object, image, translation, reflection, rotation, centre of rotation, clockwise, anticlockwise.
Cheat sheet: Open to print (A4, with board pictures) · Markdown
Year 8: the tests that make triangles congruent, similar shapes and scale factors, and proving and using properties of quadrilaterals.
Congruent shapes are the same shape and size: one fits exactly on the other. Triangles do not need all six parts checked. Each congruence test fixes the triangle: SSS (three sides), SAS (two sides and the angle between them), ASA or AAS (two angles and a matching side) and RHS (a right angle, the hypotenuse and another side). Two sides and an angle that is not between them (SSA) can make two different triangles, so it is not a test.
Similar shapes are the same shape, maybe a different size: every length is multiplied by the same scale factor k = copy ÷ original, and the angles stay the same. Triangles are similar by AA, by SSS (sides in one ratio) or by SAS (two sides in one ratio, the angle between them equal). For other shapes, check angles and side ratios: all squares are similar, but not all rectangles.
Congruent triangles prove properties of quadrilaterals. Split the shape with a diagonal, show two triangles are congruent, and matching parts are equal: a rectangle's diagonals are equal, a parallelogram's opposite angles are equal and its diagonals cut each other in half. Those properties then solve problems, with a reason for each step.
Builders and manufacturers check parts are identical with a few measurements; maps, models and shadows use similar shapes to measure what can't be reached. Proof by congruent triangles is the first real proof most students write.
congruent (≡), similar, scale factor, enlargement, SSS, SAS, ASA, AAS, RHS, included angle, hypotenuse, diagonal, bisect, proof.
Cheat sheet: Open to print (A4, with board pictures) · Markdown
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 precise set of steps and decisions that gives the right result every time. A classification algorithm sorts shapes by asking yes/no questions about their sides and angles.
Two things make it work. Order: ask the most specific question first, or an equilateral triangle is called isosceles and a square is called a parallelogram. Coverage: every yes and every no must lead on to another question or a name.
To check one, trace it on one shape of every class. A flowchart, a decision tree and numbered steps are three ways to show the same algorithm.
Every if and else in a program is a decision like these. Writing the steps clearly first is how programmers avoid bugs.
The same ideas sort pairs of triangles. Ask the congruence tests first (SSS, SAS, ASA or AAS, RHS), then the similarity conditions (AA, sides in one ratio), and end with "not enough to tell". Answer each question from what is known to be equal, not what could be worked out. Test it on one pair of every kind, including SSA, which must not be called congruent; when you fix a gap, test every pair again.
algorithm, attribute, decision, flowchart, decision tree, trace.
Cheat sheet: Open to print (A4, with board pictures) · Markdown