Reasoning · Grade 4-2 Counting Figures Formed by Joining Dots
Problem
Upright ones have sides 1 to 4: 4 kinds.
Starting from the smallest and growing by one unit at a time is the natural order, and the board's own edge tells you exactly where to stop.
4.G.A.2Make A Systematic ListUpside-down ones turn onto those, so they are not new.
Two triangles with the same three side lengths are the same shape and size, so which way the point faces cannot make them different.
4.G.A.2Visualize Spatial RelationshipsAn upside-down triangle is the same shape as an upright one turned over, so it is not a new kind to count.
Turning a triangle over keeps every side length and every angle, so the flipped copy is the same triangle in a new position.
Grouping the triangles into families that can be moved onto each other puts every triangle in exactly one family to be counted.
Sides taken at a slant also make equilateral triangles.
You do not have to measure the slanted sides: the third-of-a-turn spin sends each slanted side onto the next one, so all three must be the same length.
4.G.A.1Draw A DiagramThe tilted ones number 2.
Every tilted triangle is trapped inside a smallest upright triangle, so checking upright triangles of side 2, 3 and 4 one at a time really does cover all of them.
4.G.A.2Make A Systematic ListAltogether 4 + 2 = 6.
The six blank boards are the problem quietly telling you when your list is complete.
4.G.A.1Make A Systematic ListSort your search first (straight ones, then tilted ones) and go smallest to largest — that way you find all six and draw none of them twice.