Regular and Irregular Polygons
Warm-up
Hold up a square. Is this a regular polygon? (Yes: 4 equal sides, 4 equal angles.) Hold up a rectangle. Is this regular? (No: angles are equal but sides are not all equal.) Hold up a rhombus. Regular? (No: sides equal but angles not all equal.) The distinction requires checking BOTH conditions.
Explore
Polygon classification challenge: 12 polygon cards including regular and irregular examples of each named polygon. Students sort into: regular (both conditions) / irregular (one or both conditions fail). For each, record: number of sides, are sides all equal? are angles all equal? Therefore: regular or irregular?
Formalize
Yup'ik border pattern analysis: show a traditional Yup'ik border design. Identify: what polygons are used? Are they regular or irregular? What is the repeating core? How does understanding polygon attributes help recreate the pattern? Students attempt to replicate a simplified version.
Regular and Irregular Polygons
Polygon properties table: for each named polygon (triangle through octagon), record: number of sides, sum of interior angles (not calculated, just observed), example of regular version. Notice patterns: as the number of sides increases, the regular version looks more like a circle.
Practice
Students sort 10 polygon cards, build regular versions of triangles, quadrilaterals, and hexagons on geoboards, and replicate one Yup'ik border pattern segment. Exit ticket: name 2 irregular quadrilaterals and explain why each is irregular.
Exit ticket
Students sort 10 polygon cards, build regular versions of triangles, quadrilaterals, and hexagons on geoboards, and replicate one Yup'ik border pattern segment. Exit ticket: name 2 irregular quadrilaterals and explain why each is irregular.
Step 1: Define REGULAR with both conditions loudly: all sides equal AND all angles equal. Two tests, both mandatory.
Step 2: Interrogate a square: sides — ruler says all 4 equal ✓. Angles — corner-of-paper test says all right angles ✓. Regular.
Step 3: Interrogate the trap shape — the rhombus (all 4 sides equal, angles leaning): sides ✓, angles ✗ (two sharp, two wide — the paper corner rocks). NOT regular. One condition isn't enough.
Step 4: Interrogate the other trap — the rectangle: angles ✓ all right, sides ✗ (long ≠ short). NOT regular either. The two traps fail OPPOSITE tests — pin them side by side on the anchor chart as the "one-condition impostors."
Step 5: Hunt the room: window (rectangle — impostor), stop-sign picture (regular octagon ✓), floor tile, clock face. Every claim needs both tests cited.
The habit: "regular" is a verdict AFTER two measurements, never a vibe from looking.
Step 1: Build the naming ladder with examples drawn UGLY on purpose (skinny, tilted, lopsided): 3 sides triangle, 4 quadrilateral, 5 pentagon, 6 hexagon, 8 octagon.
Step 2: Classify a lineup: a long thin 3-sided sliver (triangle — no matter how un-triangle-ish it feels); a wild 4-sided kite shape (quadrilateral); a 5-sided house outline (pentagon — even though only the regular pentagon "looks like" the name).
Step 3: The count-carefully case: a 6-pointed star outline. Trace and count ACTUAL sides: 12 (each point contributes two). Not a hexagon — a dodecagon. Counting beats assuming.
Step 4: Sort the same lineup a second way: convex vs. having a "dent" (concave) — the star has dents; the sliver doesn't. Shapes can be classified along independent axes: side-count AND dentedness AND regularity.
Exit: draw (a) an irregular pentagon, (b) a hexagon with a dent, (c) a quadrilateral with exactly one pair of equal sides. Drawing to specification is classification in reverse — and much harder.
The claim, heard in every Grade 4 room. Time to retire it.
Step 1: Hand the student a cardboard square sitting flat. Verdict: "square." Now rotate it 45° while they watch continuously. Ask at what exact moment it STOPPED being a square. No cut, no stretch, no measurement changed — the properties travelled with it.
Step 2: Re-run the two regularity tests on the tilted shape: sides all equal ✓, angles all right ✓ (the paper-corner test doesn't care about tilt). It passes everything a square must pass. It IS a square.
Step 3: Where "diamond" comes from: posters that always draw squares base-down teach the POSE. "Diamond" is a pose-name, not a shape-name — mathematics doesn't use it.
Step 4: Vaccinate with variety: for a week, draw every shape in random orientations. Triangles on their points, rectangles leaning, pentagons upside down.
The principle, posted: properties define shapes; position is just where they happen to be standing.