Problem
Cut and fold edges of a net
A closed prism cut along edges and opened flat gives a net. Every edge is either cut or left as a fold line. The net is drawn on the right. Find how many edges are cut and how many stay joined.
Reasoning · Grade 5-2 Rectangular Prisms
Your answer
Watch out The count looks net-dependent, but there are always five folds.
How to solve
Strategy Create a Physical Representation — The surest way to see this is to build it: tape a small box together, run a finger along all 12 edges to count them, then cut with scissors until the box flops open flat, and count how many cuts it took. Doing it once makes the two counts obvious and shows why they must add to 12. To be sure I have not miscounted, I also read the numbers straight off the picture — the dashed lines in the printed net are the uncut edges, so I count those — and then explain with a picture argument why 5 fold lines is the only number that could ever work for a 6-face solid.
1STEP 1
Count the edges of the prism
A prism has 12 edges.
4 + 4 + 4 = 12
Splitting the edges into top ring, bottom ring and uprights makes them easy to count without pointing at the same edge twice, which is exactly what the hidden dashed lines in the sketch are there to help with.
6.G.A.4Draw A Diagram2STEP 2
Read the uncut edges off the net
The net shows 5 fold lines.
3 + 1 + 1 = 5
The picture already separates the two kinds of edge for you: dashed means still joined, solid means cut open, so counting dashes is the whole job.
6.G.A.4Make A Systematic List3STEP 3
Subtract to get the cut edges
Subtracting, 7 edges are cut.
12 - 5 = 7
Every edge belongs to exactly one of the two groups, so once you know one group the other is just a subtraction — no need to count the cuts separately and risk a different mistake.
2.OA.A.1Make A Systematic List4STEP 4
See why 5 folds is the only possible number
Joining six faces always needs 5 folds.
6 faces → 6 - 1 = 5 folds, 12 - 5 = 7 cuts
Joining 6 things into one connected group always needs one link fewer than the number of things — the same reason 6 children need 5 handshakes to form a single chain — and a child can check it by cutting a real box open.
6.G.A.4Create A Physical RepresentationA net of a box always has exactly five fold lines, whatever shape the net takes.
Why?
The six faces must all hang together in one piece, and joining six pieces into one takes exactly five joins.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each fold line is one uncut edge joining exactly two faces, so folds and those joins match up one for one.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Answer
7, 5 edges
12 − 5 = 7
The two counts add back to 12, which is the number of edges a rectangular prism has, so nothing has been lost or double-counted. The answer also passes a common-sense test: most of the edges have to be cut, since a box opens out only when it is well and truly slit, but a few must stay joined or the net would fall into separate pieces instead of staying in one flat shape. Folding the printed net back up confirms it: 5 creases stand the 6 faces up into a closed box, and the 7 cut edges are exactly the seams that then meet edge to edge. Since the number 5 came from counting the dashes in the picture and also, independently, from the 6-faces-need-5-links argument, the answer is checked two different ways.
Takeaway
Every box has 12 edges, and opening one flat always means keeping 5 of them as folds and cutting the other 7!
- Count the edges of the prism
- Read the uncut edges off the net
- Subtract to get the cut edges
- See why 5 folds is the only possible number