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Exploring Patterns in Square Coloring: A Systematic Investigation

This activity invites learners to explore a 3x3 square coloring challenge, focusing on the placement of colors (red) on the square's perimeter. Participants will analyze how many smaller squares within this grid have 0 to 4 red sides based on varying sizes of the main square (2x2, 4x4, etc.). The task encourages systematic investigation, pattern recognition, and generalization of results into larger grids (e.g., 20x20, 50x50). Students are prompted to document their findings in tables and discover overarching rules governing their observations.

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Exploring Patterns in Square Coloring: A Systematic Investigation

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  1. Finding position-to-term rules

  2. Finding position-to-term rules

  3. Finding position-to-term rules

  4. Finding position-to-term rules

  5. Finding position-to-term rules

  6. Finding position-to-term rules

  7. Finding position-to-term rules

  8. Finding position-to-term rules

  9. Find position-to-term rules for these sequences:

  10. Squares investigation

  11. Squares investigation Imagine a 3 x 3 square:

  12. Squares investigation Imagine a 3 x 3 square: Colour the outside red:

  13. Squares investigation Imagine a 3 x 3 square: Colour the outside red:

  14. Squares investigation How many of the small squares have • 0 redsides? • 1 red side? • 2 red sides? • 3 red sides? • 4 red sides?

  15. Squares investigation Now change the size of the starting square to • 2 x 2 ? • 4 x 4 ? • etc…

  16. Squares investigation How many of the small squares now have • 0 redsides? • 1 red side? • 2 red sides? • 3 red sides? • 4 red sides? INVESTIGATE!

  17. Generalising from Diagrams 8 by 8

  18. Generalising from Diagrams 20 by 20 50 by 50 2 red sides = ? 1 red side = ? 0 red sides = ? 100 by 100 n by n

  19. Generalising from Diagram 20 by 30 2 red sides = ? 1 red side = ? 0 red sides = ? 50 by 40 90 by 50 a by b

  20. Squares investigation Put your results into a table! (If you need to) Work methodically! Spot patterns! Try to find some general rules!

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