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

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

but may be

Copymasters of dot papers

Copymaster of the problem (Māori)

Copymaster of problem (English)

4x4 geoboards or 4x4 dot paper

### Problem

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

**Teaching sequence**

- Introduce the problem as a "treasure hunt". How many squares can be made on this board (or grid)? Let the search begin!
- 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.
- Share the squares found by the class. Notice whether the squares displayed are different by size, location or both.
- Pose the problem: How many squares do you think that you can find?
- 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? - 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.

#### Extension

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

Other contexts for the problem

#### Solution to extension

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, 2^{2}, 3^{2})