Classification of Prisms and Pyramids
Warm-up
Mystery shape: I have 5 faces. Two of my faces are triangles. Three are rectangles. What am I? (Triangular prism.) New: I have 5 faces. One is a square. Four are triangles. What am I? (Square pyramid.) The number and type of faces determine the shape.
Explore
Prism net construction: students construct nets for rectangular and triangular prisms, cut and fold to verify. For each: identify the base, count faces/edges/vertices, write the name. Then: find 3 prisms in the classroom or school environment.
Formalize
Quadrilateral hierarchy diagram: build a sorting tree. Start: quadrilateral. Branch: parallelogram (2 pairs parallel sides) vs. trapezoid (1 pair). Parallelogram branches: rectangle (4 right angles) and rhombus (4 equal sides). Rectangle and rhombus overlap: square (both conditions). Students place shape names and examples on the tree.
Classification of Prisms and Pyramids
Honeycomb hexagonal prism: why do bees build hexagonal prisms? A honeycomb cell has a hexagonal cross-section, which is the polygon that tiles a flat surface with no wasted space AND has the greatest area for a given perimeter (among polygons that tile). This is geometry serving biology.
Practice
Students build nets for 2 prisms, complete the quadrilateral hierarchy chart, and identify 5 prisms/pyramids in the environment with justification. Exit ticket: name a 3D shape with 6 rectangular faces.
Exit ticket
Students build nets for 2 prisms, complete the quadrilateral hierarchy chart, and identify 5 prisms/pyramids in the environment with justification. Exit ticket: name a 3D shape with 6 rectangular faces.
Step 1: State the sorting questions: (1) Does it have TWO identical parallel faces (a matched top-and-bottom pair)? → prism family. (2) Does it have ONE base and a point (apex) where all other faces meet? → pyramid family.
Step 2: Run the machine on a crate of solids: cereal box — two identical rectangles top and bottom → rectangular prism. Toblerone box — two identical TRIANGLES at the ends → triangular prism (lying on its side fools people; the twin faces are the ends, not top/bottom). Egyptian souvenir — square base, four triangles to a point → square pyramid. Dice/cube — two identical squares (six ways to pick them!) → a prism so symmetric every face can be the base.
Step 3: Name by the base: [base-shape] + [family]: pentagonal prism, triangular pyramid, hexagonal prism.
Step 4: The imposters: cylinder (two parallel circles but no flat side rectangles — an honorary prism with curved sides; not a polyhedron) and cone (pyramid's curved cousin). Sort them into a "curved relatives" annex.
Exit: "a solid has 2 hexagonal faces and 6 rectangles — name it and count its edges." (Hexagonal prism: 6+6 around the hexagons + 6 verticals = 18.)
Step 1: Gather data from known prisms into a table. Triangular prism: F 5, E 9, V 6. Rectangular: F 6, E 12, V 8. Pentagonal: F 7, E 15, V 10.
Step 2: Hunt the patterns against the base's side-count n: faces = n + 2 (the n side-rectangles plus 2 bases). Vertices = 2n (each base corner, twice). Edges = 3n (n around each base = 2n, plus n vertical posts).
Step 3: Predict for an octagonal prism SIGHT UNSEEN: F = 10, E = 24, V = 16. Build or sketch to verify ✓.
Step 4: Run the same census on pyramids: base n-gon → F = n + 1, V = n + 1, E = 2n. Predict for a hexagonal pyramid: 7, 7, 12 ✓.
Step 5: The magic check that always works: F + V − E = 2 for every one of these (5+6−9 = 2 ✓, 10+16−24 = 2 ✓ …). Name-drop Euler; promise them this innocent-looking fact is deep enough that mathematicians still write books about it. Counting became algebra; algebra found a law.
Step 1: Present a net: one square in the middle, four identical triangles attached to its four sides, points outward.
Step 2: Interrogate before folding — the analysis IS the skill: count faces: 1 square + 4 triangles = 5 → the solid has 5 faces. A square base with triangles rising… square pyramid (predicted).
Step 3: Mental-fold: triangles hinge UP; their apexes must MEET at one point above the square's centre. For that to work the triangles must be identical and tall enough — degenerate flat triangles would collide into the base.
Step 4: Fold the physical copy: square pyramid ✓.
Step 5: Reverse detective work: sketch a net for a triangular prism WITHOUT a template. Requirements harvested from the naming work: 2 triangles + 3 rectangles, hinged so the triangles cap the rectangle-tube. Common failure: attaching both triangles to the SAME rectangle edge-side (they'd collide). Peer-test each net by mental fold before scissors touch paper.
The exit standard: given any net of ≤8 faces, name the solid BEFORE folding, with the face-census as evidence.