Count the Shapes

Free puzzles to print

Each set: a Learn sheet with a worked example, 10 puzzles from easiest to hardest, and the answers.

• Level 1

  Ages 4–6 · Pre-K–K

  

  The easiest puzzles: a few separate shapes in a row, nothing touching. Count the triangles or the squares (up to 5) and check the right number card.

  The middle puzzles: a little picture of 3 to 6 squares and triangles that touch; each one is still counted on its own.

  The hardest puzzles: one shape cut by one line, so two small triangles make a big one, and the big one counts too.

  On the sheet: “How many? Big ones made of small ones count too.”

  Print Level 1 (Letter)Other paper size: A4

• Level 2

  Ages 6–8 · Grades 1–2

  

  The easiest puzzles: separate shapes, nothing crossing; find the triangles among other shapes.

  The middle puzzles: one triangle or rectangle cut by one or two lines, 3 to 8 triangles.

  The harder puzzles: shapes cut by two or three lines (up to 13 triangles), or a small grid of squares.

  The hardest puzzles: shapes cut by three or four lines (14 to 20 triangles). Or a bigger grid, such as 3 by 4, with 11 to 20 squares.

  On the sheet: “How many? Big ones made of small ones count too.”

  Print Level 2 (Letter)Other paper size: A4

The thinking it works on

• Seeing the whole and the parts at once

  Switches between looking at the small pieces and the whole outline.

  Mondloch and colleagues (2003) found that 6- and 10-year-olds see a whole shape before its parts even more strongly than adults do. This pull toward the whole settles to the adult level between ages 10 and 14. Source

• Systematic counting

  Counts small shapes first, then two-piece shapes, then bigger ones.

  English (1991) gave children aged 4½ to 9 tasks of finding every combination and saw six strategies, from random picking to a fixed, systematic order. Source

• Knowing what makes a triangle or a square

  Checks sides and corners, so a tilted triangle counts and a slanted box doesn't.

  Clements and colleagues (1999) interviewed 97 children aged 3 to 6. The children judged shapes mostly by overall look, so a tilted or stretched triangle was often not called a triangle. Source

• Taking shapes apart and putting them together

  Sees which small triangles join to make a bigger one, and which don't.

  Clements, Wilson and Sarama (2004) found that, for children aged 3 to 7, seeing several small shapes as one new, bigger shape comes late, after trial and error. Source

• Seeing rows and columns in a grid

  Finds 2-by-2 squares inside a bigger grid of squares.

  Many students do not yet see a grid of squares as rows and columns; Battista and colleagues (1998) described several levels in how children structure such grids. Source

• Keeping a running record

  Holds the count so far, or writes it down, while hunting for the next shape.

  Gathercole and colleagues (2004) found that the mental space for holding a count while searching grows roughly in a straight line from age 4 to 15. Source

Where is this thinking used?

• Floor plans and maps: One outline holds rooms inside rooms.
• Inventories: Checking a list without missing or double-counting anything.
• Diagrams: Spotting a hidden pattern inside a bigger picture, as in science or circuit drawings.