Problem
Different ways to finish a net
Part of a triangular prism's net is drawn, with a right-triangle base. One face is missing. Nets that match after a flip count as one. Find every way to draw the missing face in.
Reasoning · Grade 6-1 Using Nets
Your answer
Watch out A matching length looks enough, but a spot that overlaps when folded fails.
How to solve
Strategy Make a Systematic List — Drawing completions until no new one comes to mind is how spots get missed, so instead I settle exactly which face is missing, then list the faces it has to touch on the folded prism, and turn each of those into one place on the boundary of the drawing. That gives a list that is complete by construction. Only then do I bring in the flip rule, comparing the finished drawings in pairs — and I check which drawings the flip leaves unchanged rather than assuming the flip halves the count.
1STEP 1
Work out which face is missing
Five faces are needed, so one rectangle is missing.
2 + 3 = 5 faces, 5 - 4 = 1 missing
Matching each rectangle already drawn to the side of the triangle it sits over leaves exactly one side of the triangle uncovered, and that names the missing face and both of its measurements.
6.G.A.4Identify Subproblems2STEP 2
List the faces the missing rectangle touches
List the faces it must touch.
A net keeps some of the solid's edges joined and cuts the rest, so the only places a missing face can be attached are the edges it really has along that face on the solid.
6.G.A.4Identify Subproblems3STEP 3
Find those four places on the drawing
The net offers four places.
4 places
Walking the boundary in order and checking each free edge against the two measurements of the missing face is a list you can see is complete, unlike thinking of completions one at a time.
6.G.A.4Draw A Diagram4STEP 4
Rule out the edges that only look possible
One of them overlaps when folded.
An edge can only be a fold if the two faces meeting there are the same length along it, so comparing lengths throws out the wrong spots before any drawing is done.
6.G.A.4Make A Systematic List5STEP 5
Use the flip rule on the four drawings
Flip-matching drawings count as one.
Tracing a drawing and turning the tracing paper over is exactly what the flip rule means, so you can see which two pictures land on each other and which land on themselves.
4.G.A.3Create A Physical RepresentationTwo drawings that are mirror images of each other are the same net, so one of each pair is struck out.
Why?
Flipping a net keeps every face and every fold, so the flipped drawing folds into exactly the same solid the same way.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
Grouping the drawings into mirror pairs puts each in exactly one group, so counting the groups counts the genuinely different nets.
🧱Disjoint cases addSplit the possibilities so no case overlaps another and none is left out, and the case counts simply add.
6STEP 6
Count the different nets
That leaves 3 different nets.
1 + 1 + 1 = 3
Counting the pictures that stay put separately from the pictures that swap is the safe way to use a 'flips count as one' rule; dividing by 2 quietly assumes nothing stays put.
6.G.A.4Make A Systematic List7STEP 7
Draw and fold the three
Folding all three shows each closes up.
Folding is the honest test that a picture really is a net, and all three fold into the same prism, just cut open along different edges.
6.G.A.4Create A Physical RepresentationAnswer
3 ways
5 − 4 = 1
Each completed picture has 5 faces — 2 congruent right triangles and 3 rectangles of the same width, one for each side of the triangle — which is exactly the face list of a triangular prism, and every added edge matches its partner in length (about 6 units against the height edges, about 3 units against the short legs). Each of the 4 rectangles is drawn on empty paper, so nothing overlaps, and each folds up to close the prism. The count itself is sensible: 4 places, of which one mirror pair collapses, must give 3, and 3 is neither more than 4 nor less than the 2 completions that are their own mirror images.
Takeaway
List every spot first, then see which pictures a flip swaps and which it leaves alone — never just cut the count in half.
- Work out which face is missing
- List the faces the missing rectangle touches
- Find those four places on the drawing
- Rule out the edges that only look possible
- Use the flip rule on the four drawings
- Count the different nets
- Draw and fold the three