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Designing Rich Tasks to provoke Mathematical Thinking

Explore engaging math tasks that provoke critical thinking and problem-solving skills, designed by Charlie Gilderdale & Alison Kiddle from NRICH Mathematics Project.

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Designing Rich Tasks to provoke Mathematical Thinking

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  1. Designing Rich Tasks to provoke Mathematical Thinking Charlie Gilderdale & Alison Kiddle NRICH Mathematics Project

  2. Opposite Vertices

  3. Opposite Vertices Challenge If I give you a line, can you tell me straight away if that line could be: • the side of a square • the diagonal of a square If such squares CAN be drawn, can you find an efficient method for drawing them?

  4. What mathematical thinking did this task provoke for you? What aspects of the presentation of the task prompted this thinking?

  5. Different audiences – different presentations The problem as it appears on NRICH The Teachers’ Notes accompanying the problem A closely-related, more challenging alternative

  6. What’s Possible? • What do the cards have in common? • Do you notice anything interesting? • Could you create more cards for your set? • What questions might a mathematician ask themselves next?

  7. 3² − 2² = 5 7² − 6² = 13 10² − 9² = 19 Does it prompt any questions? What do you notice?

  8. 7² − 2² = 45 9² − 6² = 45 23² − 22² = 45 Does it prompt any questions? What do you notice?

  9. 4² − 2² = 12 8² − 6² = 28 11² − 9² = 40 Does it prompt any questions? What do you notice?

  10. 12² − 11² = 23 7² − 5² = 24 5² − 0² = 25 Does it prompt any questions? What do you notice?

  11. 7² − 2² = 45 9² − 6² = 45 23² − 22² = 45 12² − 11² = 23 7² − 5² = 24 5² − 0² = 25 3² − 2² = 5 7² − 6² = 13 10² − 9² = 19 4² − 2² = 12 8² − 6² = 28 11² − 9² = 40

  12. Can you find a way to write each number from 1 to 30 as the difference of two squares? Can you write any of them in more than one way?

  13. If time allowed… Choose one conjecture to work on, and be prepared to feed back on your findings

  14. Final Challenge If I give you a number, can you tell me all the possible ways to write it as the difference of two squares?

  15. What mathematical thinking did this task provoke for you? What aspects of the presentation of the task prompted this thinking?

  16. Different audiences – different presentations The problem as it appears on NRICH The Teachers’ Notes accompanying the problem A closely-related alternative task

  17. The two tasks… What was the same? What was different?

  18. Some final thoughts… NRICH views Mathematics as a creative discipline. In English lessons, students read novels, poems and plays but are also given opportunities to write for themselves. In Maths lessons, we expose students to interesting results and proofs, but we must also give them opportunities to create their own mathematical thinking, to work as mathematicians. NRICH tasks offer these opportunities.

  19. “Maths is not just a spectator sport. It’s about participation and collaboration” NRICH Teachers’ Notes suggest how students can work in small groups without the need for teacher ‘interference’. The teacher’s role is as an observer with big ears and a small mouth, who is happy to stand back and let mistakes be made…

  20. … but who draws the class’s attention to insights and thinking where appropriate and helps students to see how their discoveries fit into the bigger mathematical picture. This model of teaching offers the opportunity for lots of ‘lightbulb moments’ across the class rather than one lightbulb moment from a student that blinds everybody else!

  21. Other NRICH tasks See our Stage 3 and 4 Curriculum Mapping Document for other tasks whose design has been informed by the same thinking. See Tilted Squares for video footage of these ideas being put into practice.

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