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Lots of Squares

Achievement Objectives:

Achievement Objective: GM2-7: Predict and communicate the results of translations, reflections, and rotations on plane shapes.
AO elaboration and other teaching resources

Specific Learning Outcomes: 

Describe the properties of a square

Work systematically when problem solving

Devise and use problem solving strategies (act it out, draw a picture)

Use equipment appropriately

Description of mathematics: 

This problem explores the concept of "squareness". It reinforces the understanding that a square has 4 equal sides and 4 right angles. The students need to recognise that a square is not always  dotsSquare.
but may be dotsSquare2.


Required Resource Materials: 
4x4 geoboards or 4x4 dot paper
Copymaster of problem (English)
Copymaster of the problem (Māori)
Copymasters of dot papers


How many squares can you find that have dots from this array as their corners?


Teaching sequence

  1. Introduce the problem as a "treasure hunt". How many squares can be made on this board (or grid)? Let the search begin!
  2. Ask the students, in pairs, to make a square on the geoboard with a rubberband (or draw one on the dot paper). It can be any size and in any location.
  3. Share the squares found by the class. Notice whether the squares displayed are different by size, location or both.
  4. Pose the problem: How many squares do you think that you can find?
  5. As the students work ask:
    How do you know when something is a square?
    What did you find out during this activity?
    Are you organising your search for the squares? How?
  6. Share answers. If there are different answers look at why this is so. Get the students to share the ways that they organised their search.


Squares in a 5x5 geoboard or copymaster 5 x 5 dot paper 

Other contexts for the problem


18 squares


Size of square Number
1x1 9
2x2 4
3x3 1
tilted (one dot enclosed) 4
tilted (4 dots enclosed) 2


Some students may notice that there is a pattern in the number of the non-tilted squares (1, 22, 32)





Lots of Squares.pdf34.21 KB
Lots of Squares dot paper Maori.pdf47.49 KB
Lots of Squares dot paper.pdf21.11 KB
Lots of Squares dot paper 5 x5.pdf30.9 KB