Is this the perimeter?
A categorical atom from mrbartonmaths.com
What this sequence teaches
What the word perimeter points at. A highlight is the perimeter when two things are true: it is only on the edge of the shape, and it goes all the way around. There are no lengths and no calculations here. This atom comes before any adding up.
The two errors it is built to prevent are the two most common: taking the space inside for the perimeter (mixing it up with area), and leaving part of the edge out.
How the highlight is drawn
The trace is the meaning of the word, so it is drawn slowly and always the same way. A ring shows where the trace starts. A dot leads the trace round the edge at one steady speed. When the dot arrives back in the ring, the ring pulses: it went all the way around. On the last example the dot never gets back, and the empty ring is the gap.
The shape always appears on its own first, so students see what the shape is before they judge the highlight. When the inside is shaded, the shape waits a second and a half before the shading arrives.
The teaching sequence
Example 1 — the inside is shaded (not the perimeter). The same colour as every later highlight, so “it’s highlighted” can’t decide anything. Only where the highlight is can.
Example 1 → Example 2. The shape stays. The verdict clears, the stage holds still for a second, the shading fades, and then the edge is traced.
Example 2 — the edge is traced (perimeter). A shape with four slanted sides, so the first perimeter students see is not a square.
Example 3 — an L-shape (perimeter). The trace goes in and out of the step. Every part of the edge counts, including the parts that face inwards.
Example 4 — a semicircle (perimeter). A curved edge and a straight edge, both traced. Perimeter is not only for shapes with straight sides.
Example 4 → Example 5. The shape stays. The verdict clears, the stage holds, and the whole highlight fades away. Then the edge is traced again from the same starting ring, over the curve exactly as before, but this time the dot stops at the end of the curve. The dot and the empty ring pulse together, at the two ends of the gap.
Example 5 — only the curve (not the perimeter). The mistake students most often make with a semicircle. This is the contrast to leave them with.
What this sequence doesn’t address
A highlight that goes all the way around and also somewhere else (a line across the inside, or a shortcut across an inward corner) is in the testing sequence. The rule decides both, through the word “only”.
Shapes drawn on squares, shapes with a line inside, and shapes with some lengths marked 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 the highlight is only on the edge of the shape, and goes all the way around, it is the perimeter.
- Example 1 — “This is not the perimeter. How do I know? Because the highlight isn’t on the edge — it’s the space inside.”
- Examples 2, 3 and 4 — “This is the perimeter. How do I know? Because the highlight is only on the edge, and it goes all the way around.”
- Example 5 — “This is not the perimeter. How do I know? Because the highlight is only on the edge — but it doesn’t go all the way around.”
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 the thing it names pulses once. On Example 1 that is the shaded inside; on Examples 2 to 4 the whole trace; on Example 5 a dashed line appears where the highlight is missing.
Vocabulary. The page says “edge” for the whole outline of the shape and avoids “sides”, because a curve is part of the perimeter too. For a circle, the perimeter has its own name, the circumference; introduce it when students are fluent.
Running the sequence
Wait for the trace to finish, then pause before revealing. The pause is the prediction moment.
Replay is available on every example. On Examples 2 and 5 it replays the change from the example before. On the others it replays the trace. Use it freely: “Watch the dot. Does it get back to the ring?”
About the testing sequence
Ten items in a new random order each time: five are the perimeter, 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.
Every item is traced in the same way as the teaching examples, because the trace is how this page says “perimeter”. Students can answer while the trace is still drawing.
What each item is diagnosing
The circle, the curved shape with no straight edges, the triangle, the arrowhead and the T-shape, each fully traced. The perimeter of an unfamiliar shape: all curve, a sharp corner pointing inwards, eight sides.
The circle with its inside shaded. Area taken for perimeter, on a shape with no sides to count.
The triangle with one side left out, and the quarter circle with its curve left out. Part of the edge missing. The quarter circle is the reverse of the last teaching example: there the straight edge was missing, here the curve is.
The rectangle with a line across it. The highlight goes all the way around, but it isn’t only on the edge.
The L-shape with a rectangle traced round it. The highlight goes all the way around something, but it leaves the edge of the shape where the shape steps in. Students who see an L-shape as “a rectangle with a bit missing” say yes.
Discussion prompts
- “Say the rule. Now use it on the rectangle with the line across it.”
- “On the L-shape: where does the highlight leave the edge?”
- “On the quarter circle: which part of the edge is missing?”
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 highlight is the perimeter.
Missing the shaded circle suggests area and perimeter are mixed up. Missing the triangle or the quarter circle suggests students aren’t checking the whole edge. Missing the rectangle with the line, or the L-shape, suggests “goes all the way around” is being used on its own, without “only on the edge”.
About the expansion sequence
Six items in three pairs, each 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 three pairs
In all six the rule still works. What is new is something on the diagram that the rule says nothing about, and that students will meet in perimeter questions.
A shape with a dashed line inside it. Traced round the outside only, it is the perimeter. With the dashed line highlighted too, it isn’t: that line is inside the shape, not on its edge.
A shape on squares. Traced round the edge, it is the perimeter; with the squares inside shaded, it isn’t. Squares invite counting, and counting squares is area.
A rectangle with two lengths marked. Traced all the way around, it is the perimeter. With only the two marked sides highlighted, it isn’t. This is the error of adding only the numbers you can see.
Discussion prompts
- “Two sides of the rectangle have no number. Are they part of the perimeter?”
- “Is the dashed line part of the edge of the shape? How do you know?”
Reading the summary
The same two channels as the testing sequence. A miss on one item of a pair and not the other shows which new feature unsettled the rule.