Reasoning · Grade 6-1 Prisms and Pyramids
Problem
A triangle's side can be one or two sticks.
Naming the attribute first — three equal sides, each a whole number of sticks — turns a vague 'make a triangle' into a short list of sizes you can actually try.
2.G.A.1Make A Systematic ListOne big triangle with two-stick sides shows 1.
Making a shape bigger instead of making more shapes is the move here: doubling the side length is what lets one triangle swallow all six sticks.
1.G.A.2Create A Physical RepresentationTwo separate small triangles show 2.
Six splits neatly into two threes, and three sticks is the smallest triangle there is, so this arrangement almost builds itself.
1.G.A.2Create A Physical RepresentationFour triangles need each stick in two triangles.
Counting the same thing two ways — once as triangle sides, once as matchsticks — is a standard way of pinning down what an arrangement must look like before you try to draw one.
4.OA.A.3Make A Systematic ListFlat, the outermost stick makes that impossible.
A flat figure always has an outside, and whatever is on the outside has empty space on one side of it. That single observation rules out every flat arrangement at once, so you can stop searching.
4.G.A.1Draw A DiagramFour triangles cannot be made flat on the table, because the sticks would have to serve more slots than there are.
Each stick can serve at most a limited number of triangle sides, so too few sticks cannot fill all the sides four triangles demand.
Counting the slots and finding the demand outruns the supply refutes every flat arrangement at once, without trying any.
Built upwards it becomes a triangular pyramid with 4.
The puzzle never said 'flat'. Once you allow the sticks to be edges of a solid rather than lines on paper, every stick automatically has a face on each side of it — exactly the condition the counting demanded.
6.G.A.4Visualize Spatial Relationships4 vertices, 4 faces and 6 edges check out.
Counting edge-slots face by face and then halving is the reliable way to count a solid's edges without missing the hidden ones at the back — and here it lands exactly on the six sticks you were handed.
6.G.A.4Make A Systematic ListFour triangles from six sticks needs every stick to have a triangle on both sides — and only a solid can do that, so build a pyramid instead of a picture.