Views and Nets of 3D Objects
Warm-up
Hold up a mug: "Draw what the camera sees from directly above. Now from the front. Now the side." Three sketches, three different truths — circle-with-handle-dot, a squat rectangle with an ear, a rectangle with a full handle profile.
Grade 8's 3D unit is about representing objects: orthographic views (top/front/side), isometric drawings, and nets — the drafting languages that let a person in another city build your object exactly.
Explore
Cube-city studio: pairs build small structures from linking cubes (max 8), then draft the three orthographic views on grid paper. Partners swap VIEWS ONLY and rebuild each other's structures — discovering that some view-sets rebuild uniquely while others admit multiple buildings (a hidden cube behind a tower leaves no trace in any view!).
Then isometric bootcamp on triangle-dot paper: the same structures drawn with depth, edges at 30°. Close with net-matching: six nets, four solids, two impostor nets that fold into overlaps — matched by counting faces and mental folding first, scissors as the referee.
Formalize
Formalize the three representation systems and their contracts:
Each system trades something: orthographic views keep true measurements but need three pictures and can still be ambiguous; isometric shows shape at a glance but every angle lies. Engineers ship BOTH — plus conventions (dashed lines for hidden edges) that patch the ambiguity. A drawing is a contract, and conventions are its fine print.
Practice
Practice: draft three views of a given cube structure; rebuild from given views (one ambiguous set — find BOTH buildings); one isometric drawing; one net-to-solid matching set with face-count justifications.
Exit ticket: a structure's top view shows an L of 4 squares; front shows a 2-high column plus 1-high tail; side shows 2-high, 2-deep. Minimum cubes to build it? (Draw, then count: 5.)
Exit ticket
Practice: draft three views of a given cube structure; rebuild from given views (one ambiguous set — find BOTH buildings); one isometric drawing; one net-to-solid matching set with face-count justifications.
Exit ticket: a structure's top view shows an L of 4 squares; front shows a 2-high column plus 1-high tail; side shows 2-high, 2-deep. Minimum cubes to build it? (Draw, then count: 5.)
The object: a 3-step podium built from cubes — heights 1, 2, 3 in a row, each step one cube deep and wide.
Step 1: TOP view: looking straight down, every step shows its footprint: a 1×3 row of squares. (Heights are invisible from above.)
Step 2: FRONT view: the staircase profile — columns of 1, 2, 3 squares.
Step 3: SIDE view (from the tall end): a 3-high, 1-wide column… care: depth is 1, so a 1×3 column of squares.
Step 4: The information audit: could someone rebuild from these three? Yes — this object is view-unique. The top view carried the layout, the front carried the heights, the side confirmed depth. Three flat truths triangulating one solid: that's orthographic projection doing its job.
The views: top — a 2×2 square; front — a 2-wide, 2-high square; side — same.
Step 1: The obvious build: a full 2×2×2 block (8 cubes). Check all views ✓.
Step 2: The hunt: can cubes be REMOVED without changing any view? The interior-back-bottom cube… every cube in a 2×2×2 touches every view. Try removing one cube: its absence dents some view? Removing the back-top-left cube dents the top view? No — the cube UNDER it still fills that top cell; dents the front? The cube beside fills it; the side? Filled too. It can go! 7 cubes, same three views.
Step 3: Verify systematically: each view-cell needs only SOME cube along its sight-line — one missing interior corner leaves every line covered.
Step 4: The drafting moral: views are shadows, and shadows undercount. Real blueprints add section cuts and hidden-line conventions precisely because of this example. Minimum-cube and maximum-cube questions (7 and 8 here) turn the gap into a puzzle — and into respect for the fine print.
The candidates: (a) a 1×6 strip; (b) the cross; (c) a 2×3 block; (d) an S/zigzag of 6.
Step 1: Face-count: all have six squares — count alone filters nothing here.
Step 2: Mental-fold (a): wrap the strip — four squares ring the cube, the last two land ON already-covered faces: overlap + two bare faces ✗.
Step 3: (b) the cross: folds cleanly, arms become walls, top flap caps ✓. (d) the zigzag: staircase folding covers all six exactly ✓ (surprising — test it). (c) the block: folds into a 2×3 slab of doubled squares ✗.
Step 4: Extract the working heuristics: no row of more than 4; squares must be able to "escape" each other when folding (the block's tight packing is its doom). Of the 35 hexomino shapes, exactly 11 are cube nets — the class has verified 4 cases toward that census, and the heuristics now predict the rest faster than scissors.