Is this the perimeter?

What this sequence teaches

Two critical features define the perimeter of a shape:

• CF1: the highlight is on the boundary, not the interior.
• CF2: it covers the whole boundary — no gaps, no extras.

The sequence is five Examples (NPPPN). The opening N→P minimal-change attacks CF1 — the highlight moves from area to boundary on a single shape. The closing P→N minimal-change attacks CF2 — the boundary highlight loses two segments and becomes incomplete. The three middle Ps push the “any closed shape” message: a regular polygon, an irregular shape with a concave (reflex) corner, and a shape mixing straight and curved edges.

The trace animation — where the highlight draws itself from a starting point and runs around the shape — is the visual language for what “perimeter” means: peri (around) + meter (measure). The same trace appears in the testing sequence so the language carries through end-to-end.

Frame by frame

Example 1 (N) — square with the area shaded. A familiar shape so the student attends to the highlight rather than to identifying the shape. The interior is shaded in the same colour family as the perimeter highlight that’s coming next — deliberately. The choice isn’t between two highlighters; it’s between two places to highlight.

1 → 2 animation. The area shading fades out, then the trace begins from the top-left vertex and runs clockwise around the boundary.

Example 2 (P) — same square, perimeter highlighted. The minimal change establishes CF1 directly.

Example 3 (P) — irregular pentagon with a reflex angle. First push of the “any closed shape” message. The trace runs into and back out of the inward corner.

Example 4 (P) — a sector. Two straight radii and a curved arc, all part of the perimeter. Pushes “curves count too” while keeping straight edges in the picture.

4 → 5 animation. The two radii portions of the highlight fade out, leaving only the arc highlighted.

Example 5 (N) — the sector, arc only. The highlight fails to cover the whole boundary. CF2 attacked.

What’s not covered here

The sequence teaches recognition of perimeter, not calculation. Adding side lengths and applying formulas like 2πr belong in follow-on resources.

The “perimeter and extras” sophistication (full boundary highlighted plus a stray interior line) is attacked in the testing sequence rather than here. Example 5 attacks missing pieces; the extras case is the mirror probe, and the testing pool is the right place for it.

Sequence-design notes

This sequence is NPPPN rather than the default NPPPNN for a two-critical-feature atom. The two CFs differ in character — one is the foundational area-vs-boundary distinction, the other the procedural completeness check — and each benefits from being the focal point of a minimal-change transition. Splitting them across opening N→P and closing P→N gives each CF its own boundary moment.

The testing pool is animated rather than static, unlike earlier categorical-atom builds. The trace animation isn’t a transition embellishment here — it’s the visual language for the concept itself, and static test items would undercut the language taught in the teaching sequence.

Running the sequence

Click forward through Examples 1 to 5. Animations on the two minimal-change transitions are replayable. The verdict is hidden until reveal — ask students to commit to a verdict and a reason before clicking.

About the testing sequence

Ten items, presented in randomised order on each load. Five positives, five negatives. The pool varies shape kind (polygon, circle, sector, concave hexagon), number of sides, side-length pattern, and the kind of failure being probed.

Every item’s highlight animates in — the shape outline appears first, then the candidate highlight traces or fills. Static end-states can let some sophisticated negatives read like positives at a glance; the animation closes that gap.

What each item is diagnosing

Positives:

• Regular hexagon. Probes that the rule generalises to a polygon with more sides than the teaching examples.
• Scalene triangle. Three sides of different lengths, tilted — no symmetry to lean on.
• Circle. Pure curve, no straight segments. The perimeter is the circumference.
• Trapezium. Irregular convex quadrilateral, slightly tilted.
• L-shape. A concave hexagon with right angles only — a different reflex configuration from the teaching pentagon.

Negatives:

• Circle with the area shaded. The area-as-perimeter confusion under a curved shape — the crude form, applied somewhere new.
• Triangle with one side missing from the highlight. The crude form of an incomplete boundary — one whole side absent.
• Sector with only the two radii highlighted. Incomplete boundary where the missing piece is the curve. Mirror probe of Example 5 in the teaching sequence (which had the radii missing instead).
• L-shape with a rectangular outline highlighted around it. The sophisticated form of the incomplete-boundary failure — the path goes “all the way around” something, just not the actual shape. Matches the look of textbook compound-shape figures.
• Rectangle with the full perimeter highlighted plus an interior diagonal. The sophisticated form of the area-as-perimeter failure — the boundary IS there; the catch is that there’s extra. Probes whether students understand “perimeter” as boundary and only boundary.

Common confusions

A student who misses the circle item with the area shaded is using a “perimeter is the part that looks highlighted” rule without checking which part. Send them to CF1: is the highlight on the boundary or on the inside?

A student who misses the triangle with a side missing isn’t checking closure. Send them to CF2: does the path return to where it started?

A student who misses the L-shape shortcut item is flattening concave shapes to their convex outline. Ask: does the highlight follow every side of the shape, or does it cut across in places?

A student who misses the rectangle-plus-diagonal item is treating “perimeter is highlighted” as sufficient rather than “perimeter is highlighted and nothing else is.” Ask: is everything that’s highlighted part of the boundary?

Discussion prompts

1. Two of the items show a circle. In one, the inside is shaded; in the other, the boundary is highlighted. Why is one a perimeter and the other isn’t? Could you say the same thing about a square?
2. Two of the items show an L-shape. The highlights look superficially similar — both go around the outside. What’s the difference between them, and which is the “real” perimeter?
3. One item highlights the full boundary of a rectangle AND a diagonal across the inside. Why does the diagonal stop this from counting as the perimeter, when everything else is correct?
4. Of the five negatives, which felt closest to a real perimeter? Why?

Reading the summary

Each item is annotated with two channels of information: a green or red tint indicates whether the student answered correctly; a ✓ or ✗ in the corner indicates whether the highlight is in fact the perimeter. The two channels can disagree — that disagreement is where the diagnostic information lives.