Is this an exterior angle?
A categorical atom from mrbartonmaths.com
What this sequence teaches
An exterior angle is between one side of a shape and the next side carried straight on. That is one condition, and the two non-examples break it in the two ways students most often do: by shading the angle inside the corner, and by shading the whole outside of the corner.
There are no letters, numbers or measurements. The decision is made from position alone. “Exterior angles add up to 360°” and “interior plus exterior is 180°” are facts that follow this decision, and they are deliberately not mentioned here.
The teaching sequence
Example 1 — the angle inside the corner (not an exterior angle). One side is already carried on with a dashed line, so the only thing wrong is which angle is shaded.
Example 1 → Example 2. The verdict clears, the stage holds still for a second, then the shading closes to nothing against the side the two angles share and opens on the other side of it. No line moves.
Example 2 — the exterior angle at the same corner. It is small on purpose, so students do not fix on exterior angles being large.
Example 3 — a triangle, a large exterior angle. A new example. A different shape, a different corner, and the other one of the two sides is carried on.
Example 4 — a six-sided shape with both sides carried on. A new example. There are two dashed lines at the corner and one exterior angle is shaded. Students may ask about the other one: it is an exterior angle too.
Example 4 → Example 5. The verdict clears, the stage holds, then the shading fans open, past both dashed lines, the long way round the outside of the corner. No line moves.
Example 5 — the whole outside of the corner (not an exterior angle). It is outside the shape and it is still a no, because it is between two sides. This is the contrast to leave them with.
What this sequence doesn’t address
The angle between two carried-on lines is in the testing sequence. The rule decides it, and Example 4 has already shown both dashed lines at one corner.
Carried-on sides drawn solid, every side carried on, and a dashed line that is not a side carried straight on are in the expansion sequence.
Every shape here has all its corners pointing outwards.
The rule and the script
The rule, exactly as the Show rule button displays it: For the following examples: if the angle is between one side and the next side carried straight on, it is an exterior angle.
- Example 1 — “This is not an exterior angle. How do I know? Because it is at a corner of the shape — but it is between two sides, not one side and the next side carried straight on.”
- Examples 2, 3 and 4 — “This is an exterior angle. How do I know? Because it is between one side and the next side carried straight on.”
- Example 5 — “This is not an exterior angle. How do I know? Because it is at a corner of the shape — but it is between two sides, the long way round. It isn’t between one side and the next side carried straight on.”
Whether students see the words is your call. Nothing appears unless you click. Show rule puts the rule above the stage and it stays until you turn it off. Why? appears after each reveal: it shows the justification and plays the check again.
What Reveal shows. Before the tick or cross appears, the check plays. Each arm of the shaded angle lights up in turn, then an orange line runs round the angle. A side lights up from the corner outwards. A carried-on side lights up from its far end, along the side, through the corner and straight on along the dashed line. On a yes students see one of each. On Examples 1 and 5 they see two sides and nothing carried on.
“Carried straight on” is used in place of “extended” so the rule needs no new vocabulary. Once students are secure, tell them the word: “You know how I kept saying ‘carried straight on’? The word for that is extended.”
The dashed lines mark a side carried on. The expansion sequence draws them solid, as exam diagrams usually do.
Running the sequence
Pause before each reveal. The pause is the prediction moment.
Replay appears on Examples 2 and 5 and replays the change from the example before, with the same pauses. After Reveal it plays the check again as well.
About the testing sequence
Ten items in a new random order each time: five are exterior angles and five are not. The rule decides every one. There is no Show rule button here. After each answer the same check plays as in the teaching sequence: each arm of the angle, then the angle. Why? shows the justification and plays it again.
What each item is diagnosing
The two items with the angle inside the corner. A dashed line at the corner does not make the shaded angle exterior. One has one dashed line; the other has two.
The two items with the shading all the way round the outside (one on a triangle). “Exterior” does not mean the whole outside of the corner.
The angle between two dashed lines. It is outside the shape and next to carried-on lines, but neither arm is a side.
The very thin exterior angle and the exterior angle that is a square corner. Size does not matter. On the square corner the angle inside and the angle outside are the same size, so only position separates them.
The four-sided shape with the other side carried on, and the seven-sided shape with both sides carried on. Either side can be carried on, and a second dashed line does not change the answer.
The plain exterior angle on the regular five-sided shape. The baseline. If this one is wrong, re-teach.
Discussion prompts
- “Say the rule. Now use it on the item with the shading all the way round the outside.”
- “On the angle between the two dashed lines: which arm is a side?”
- “On the square corner: which side has been carried on?”
Reading the summary
Each cell has two channels: a green or red tint shows whether the student answered correctly; the ✓ or ✗ in the corner shows whether it is an exterior angle.
Saying yes to the shading all the way round suggests students think exterior means outside. Saying yes to the angle inside the corner suggests they are answering from the dashed line. Saying no to the thin angle or the square corner suggests they are judging by size.
About the expansion sequence
Six items a little beyond what the teaching and testing sequences showed, to see how far students’ understanding transfers. It works exactly like the testing sequence. Teaching and testing always drew a carried-on side dashed; here that help is taken away or turned against them.
The two kinds of item
The rule still works, on something unfamiliar. A carried-on side drawn solid, as in most exam diagrams. A triangle sitting on a line, with one side carried on both ways. A shape with every side carried on. All three are exterior angles. Two more are not: the shading all the way round the outside with a solid line, and the angle between two carried-on lines on a triangle with every side carried on both ways.
The dashed line stops helping. In one item a dashed line leaves the corner but bends away, so it is not the side carried straight on. A student who has learned “one solid arm and one dashed arm” says yes. After the answer, a dotted orange line shows where the side would go if it were carried straight on.
Discussion prompts
- “Say the rule. Does it say anything about dashed lines?”
- “On the dashed line that bends away: how can you check whether a side has been carried straight on?”
Reading the summary
The same two channels as the testing sequence. Saying no to the solid-line items suggests students were relying on the dashes. Saying yes to the dashed line that bends away suggests the same thing from the other direction.