Is this a triangle?
A categorical atom from mrbartonmaths.com
What this sequence teaches
A shape is a triangle when three things are true: it has three sides, the sides are straight, and they are all joined up. Each of the three non-examples breaks exactly one of these.
It also works against the picture many young children hold: a triangle points up, sits on a flat base and has equal sides. None of the three triangles shown here looks like that.
The teaching sequence
Example 1 — three corners, gently curved sides (not a triangle). It has three sides and they are joined up. Only the straightness is missing. The curve is slight on purpose, so the shape looks like a triangle at a glance.
Example 1 → Example 2. The verdict clears, the stage holds still for a second, then the three sides straighten. The corners do not move.
Example 2 — a triangle. Tilted, no flat base, no equal sides, so the first triangle students see is not the textbook one.
Example 3 — a long, thin triangle with three different sides. A new example, chosen to look odd.
Example 4 — a triangle with one very wide angle. A new example.
Example 4 → Example 5. The verdict clears, the stage holds, then one corner is cut back and a short side appears across it.
Example 5 — a corner snipped off (not a triangle). It still looks like a triangle at a glance, but it has four sides.
Example 5 → Example 6. The verdict clears and the stage holds still for a second. Then the short fourth side fades away and leaves a gap. Nothing else moves.
Example 6 — a gap (not a triangle). Three straight sides, and still not a triangle. This is the contrast to leave them with.
What this sequence doesn’t address
Shapes with four or more sides that look nothing like a triangle, a shape with only one curved side, and a gap in the middle of a side are in the testing sequence. The rule decides all of them.
Shaded and coloured-in triangles, rounded corners, a zigzag side and dots on the corners are in the expansion sequence.
The rule and the script
The rule, exactly as the Show rule button displays it: For the following examples: if a shape has three straight sides, all joined up, it is a triangle.
- Example 1 — “This is not a triangle. How do I know? Because it has three sides, all joined up — but the sides are not straight.”
- Examples 2, 3 and 4 — “This is a triangle. How do I know? Because it has three straight sides, all joined up.”
- Example 5 — “This is not a triangle. How do I know? Because the sides are straight and all joined up — but there are four sides, not three.”
- Example 6 — “This is not a triangle. How do I know? Because it has three straight sides — but they are not all joined up.”
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 points at what it names. On the triangles the three sides light up one at a time, so the class can count them. On Example 1 the curved sides light up; on Example 5 the short fourth side; on Example 6 the two loose ends either side of the gap.
Vocabulary. The page says “joined up” where a textbook would say “closed”. Introduce “closed shape” when students are fluent.
Running the sequence
Pause before each reveal. The pause is the prediction moment.
Replay appears on Examples 2, 5 and 6 and replays the change from the example before, with the same pauses.
About the testing sequence
Ten items in a new random order each time: five are triangles, five are not. The rule decides every one. There is no Show rule button here. Why? works as in the teaching sequence, after each answer.
What each item is diagnosing
The triangle with three equal sides on a flat base, the one pointing down, the very thin one, the one with a square corner and the very small one. Whether students accept triangles that look different from each other. The flat-based one is here on purpose: the familiar triangle is a triangle too.
The four-sided shape, and the triangle with a corner snipped off. Counting the sides. The first is easy to reject; the second looks like a triangle until the sides are counted.
The shape with three curved sides bending inwards, and the shape with one curved side. Straight sides. In the second, two sides are straight, so every side has to be checked.
The shape with a gap in the middle of a side. All joined up. In the teaching sequence the gap was at a corner; here it is somewhere else.
Discussion prompts
- “Say the rule. Now use it on the shape with one curved side.”
- “On the shape with the snipped corner: count the sides with me.”
- “On the shape with the gap: which part of the rule does it break?”
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 the shape is a triangle.
Saying no to the thin, small or upside-down triangles suggests students are matching to a picture, not using the rule. Saying yes to the snipped corner suggests they are not counting. Saying yes to the gap or the single curved side suggests they are not checking every side.
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.
The two kinds of item
Something new on the shape that the rule says nothing about. A shaded triangle, a triangle coloured in with no outline, and a triangle with dots on its corners are all triangles. A shaded shape with curved sides is not: shading doesn’t turn it into one.
The rule needs careful use. A triangle with rounded corners has sides that are straight most of the way, but they curve where they meet, so it is not a triangle. A shape with a zigzag side has only straight lines, all joined up, but six of them.
Discussion prompts
- “On the rounded one: run your finger along a side. Is it straight all the way?”
- “On the zigzag one: count the sides with me.”
Reading the summary
The same two channels as the testing sequence. Saying no to the shaded, coloured-in or dotted triangles suggests the new feature unsettled the rule. Saying yes to the rounded corners or the zigzag suggests students are judging by the overall look.