Students split polygons into triangles from one corner, find that a shape with n sides makes n − 2 triangles, and use (n − 2) × 180° for the angle sum (AC9M7M05, node 2).
Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.
Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min
Do the Run board and the Make task yourself first. Students need the last lesson: a triangle's angles add to 180°.
The method is to draw lines from one corner only (the board's Split into triangles). Students who split from a point in the middle get n triangles and must take off the full turn of 360° round that point: that is Ben's mistake, planted after the method.
The board only splits convex shapes (every corner points out). A concave shape (a corner that points in, with a reflex angle) is the edge case in Modify: the angle sum is still (n − 2) × 180°.
| What you will see | A question to ask | The fix |
|---|---|---|
| A student counts n triangles (Ben's split from the middle). | Which angles in your triangles are not corners of the polygon? | Point at the angles round the middle point: a full turn, 360°. Take it off, or split from a corner instead. |
| A student draws lines from A to the corners next to it. | Do those lines make a new triangle? | Those lines are sides. Join A only to the corners that are not next to it. |
| A student thinks each angle of any pentagon is 108°. | Is the pentagon on the board regular? What happens to one angle when you drag a corner? | 108° is each angle of a regular pentagon only: 540° shared equally. The total is 540° for every pentagon. |
One possible solution for each task. Other correct programs are fine.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Open task: any program that does what the step asks.
Stretch: Why can no polygon have angles adding to 1000°? Each interior angle of a regular polygon with 20 sides? (162°.)
Support: Use the choose board (More sides) to count triangles for each shape. A printed table with the first two columns filled helps students see the pattern.
For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.
You are a patient maths coach for a Year 7 student in Australia. Help me understand ONE idea only, from the lesson "Angles in a polygon". The idea: Interior angle sum of polygons by decomposition: I can Students use decomposition of polygons into triangles to determine the interior angle sum formula for n-sided polygons. What understanding looks like: - Correctly decomposes a polygon into triangles and calculates the interior angle sum. - States the general formula (n−2) × 180° and applies it correctly. - Can verify their result by measuring. Mistakes students often make here: - A student counts n triangles (Ben's split from the middle). - A student draws lines from A to the corners next to it. - A student thinks each angle of any pentagon is 108°. How to coach me (rules from the Waypoints Kit coach): - Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question. - Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit). - If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go. - If I seem to be missing the idea that comes before this one (Demonstrating the triangle angle sum), check it quickly, then come back. - Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back. - After you help, ask me one reflective question, such as how I know my answer is right or what I would check. - Praise my effort and method, not my ability. - You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so. - Use simple words. - Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Angles in a polygon".
Find the angle sum of a heptagon (7 sides), and explain your method.
900°. From one corner it splits into 7 − 2 = 5 triangles, and 5 × 180° = 900°.
Next in class: finding unknown angles in triangles and polygons, giving the angle rule as the reason (node 3).
Where it leads: exterior angles (they add to 360° for any convex polygon), which regular polygons tessellate, and the turtle: a closed shape turns 360° in total, which the coding pathway uses for its polygons.