Reasoning · Grade 2-1 Making Shapes

Problem

Overlap shapes and count figures

Several flat shapes are drawn on top of one another so their edges cross. Only the region two shapes share is coloured, and its name comes from counting its straight sides. Count how many coloured regions are triangles, quadrilaterals, pentagons and hexagons.
Your answer
How to solve
Strategy Draw a Diagram — The whole question is about a picture, so I trace each shape's outline carefully (Draw a Diagram) and shade in just the parts where two shapes sit on top of each other. To make sure I miss none, I go through the shapes pair by pair in a systematic list and ask 'do these two cross, and where?' for each pair. Each shaded overlap is then its own small subproblem: once it is colored, I only have to walk its boundary and count the straight sides to name it. I must NOT stop after finding four -- the shapes are densely stacked, so I keep checking until every crossing is colored. I also do not assume the four answer blanks mean exactly one region each; the blanks are just the four shape names that might appear, and any of them can hold zero, one, or several.
1STEP 1

Go through the pairs and find where they cross

Pair by pair the outlines cross in many places, fencing off several shared regions.

many crossings → several overlap regions (not 4)
2STEP 2

Most overlaps fenced off by two straight edges are triangles

A slanted edge clipping a corner leaves three sides — a triangle.

3 straight sides = triangle
3STEP 3

Overlaps fenced off by four edges are quadrilaterals

Where four sides survive the overlap is a quadrilateral, and that happens twice.

4 straight sides = quadrilateral
4STEP 4

The big right-hand overlap is a pentagon

Only the big right-hand overlap keeps five sides — a pentagon.

5 straight sides = pentagon
5STEP 5

Tally the colored overlaps by type

The tally is triangles 3, quadrilaterals 2, pentagons 1, hexagons 0.

triangles 3, quadrilaterals 2, pentagons 1, hexagons 0
Answer
triangles 3, quadrilaterals 2, pentagons 1, hexagons 0
When straight-edged shapes are stacked, most crossings cut off a corner or a strip, which fences off three- or four-sided pieces -- so triangles and quadrilaterals should dominate, exactly as found. A six-sided overlap would need two shapes to cross in six places, which is rare here, so finding zero hexagons is reasonable. The earlier idea of 'exactly one triangle, one quadrilateral, one pentagon, one hexagon' is too tidy: the picture has many more than four crossings, so it cannot be one of each.
Takeaway

Every place two shapes cross makes a new little shape -- and when straight-edged shapes pile up, you get LOTS of them, mostly triangles and four-sided pieces. Just color each overlap and count its sides to name it!

  • Go through the pairs and find where they cross
  • Most overlaps fenced off by two straight edges are triangles
  • Overlaps fenced off by four edges are quadrilaterals
  • The big right-hand overlap is a pentagon
  • Tally the colored overlaps by type