Lots of squares
Purpose
This problem solving activity has a geometry focus.
Achievement Objectives
GM2-7: Predict and communicate the results of translations, reflections, and rotations on plane shapes.
Student Activity
How many squares can you find that have dots from this array as their corners?
Specific Learning Outcomes
- Describe the properties of a square.
- Work systematically when problem solving.
- Devise and use problem solving strategies (act it out, use equipment, draw a picture).
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 
but may be 
.
Required Resource Materials
- Copymasters of dot papers
- Copymaster of the problem (Māori)
- Copymaster of problem (English)
- 4x4 Geoboards and rubber bands or 4x4 dot paper and pencils (a digital tool could also be used - search for digital geoboard)
Activity
The 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? Let the search begin!
- Ask the students, in pairs, to make a square on the Geoboard with a rubber band (or draw one on the dot paper). It can be any size and in any location. Ensure students understand that a square is a flat shape with 4 sides of equal length, and 4 interior angles of equal size. If necessary, differentiate between what a square is and isn't (by drawing or with geoboards).
- Share the squares found by the class. Notice whether the squares displayed are different (e.g. by size, location, or amount of rotation).
- 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
Solution
20 squares
| Size of square (in squares) | 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)
Solution to the extension
Solution to the Extension
50 squares could be made using a 5x5 Geoboard or copymaster 5 x 5 dot paper. See if you can find all of these using the system shown above.
Attachments
LotsSquares.pdf80.82 KB
HeTapawhaRite.pdf124.4 KB
LotsOSquaresDotPaper5x5.pdf30.9 KB
LotsOSquaresDotPaper4x4.pdf21.11 KB