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What is Algebra? Pete Griffin S.W. Regional Coordinator

What is Algebra? Pete Griffin S.W. Regional Coordinator. Shady Circles. A natural sense or “power” Pre-symbolic Pattern and structure Using reasoning to predict “what if?”. Shade every 3 rd circle. Ian Stewart – “Nature’s Numbers”.

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What is Algebra? Pete Griffin S.W. Regional Coordinator

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  1. What is Algebra? Pete Griffin S.W. Regional Coordinator

  2. Shady Circles • A natural sense or “power” • Pre-symbolic • Pattern and structure • Using reasoning to predict “what if?” Shade every 3rd circle

  3. Ian Stewart – “Nature’s Numbers” • “Mathematicians are forced to resort to written symbols and pictures to describe their world – even to each other. • But the symbols are no more that world than musical notation is music”.

  4. What do the symbols symbolise? 3 feet = 1 yard 3 f = 1 y ?

  5. Fruit salad algebra Given the number of tables, how many chairs are there? 6 tables so that times 2 plus two chairs: 6t  2 = 12t + 2c Dave Hewitt, Birmingham University

  6. Do these in your head and writedown the answer • (7 × 6) + (6 × 3) • (4 × 9) + (9 × 6) • 82 + double 8 • 32 + (7 × 3) • Why are the answers all multiples of 10?

  7. Linking symbols to what they symbolise 5x + 5y = 5(x + y) • Pick an example of some written algebra and ask yourself what is being expressed. 5x + 6x = 11x (x + 1)2 = x2 + 2x + 1 x2 + xy = x + y x

  8. Matchsticks Dave Hewitt – MT 163 “Approaching Arithmetic Algebraically”: “The algebra is not the statement but it is the work you have to do in order to get yourself in a position where you could make a statement”. Draw the 17th one and attend to how you are drawing it

  9. Dave Hewitt – MT 163: “There’s a difference between counting and watching yourself counting”. “Algebra is concerned with awareness of awareness”. How many match sticks?

  10. Think of a number Double it and add 1 Add 1 and then double it What’s the difference? Working with the “grammar” of the symbols

  11. Working with the “grammar” of the symbols “To-ing and Fro-ing”, Don Steward - Median

  12. Working with the “grammar” of the symbols Multiple Expressions.

  13. Working with the “grammar” of the symbols Multiple Expressions. 1 + 3n

  14. Working with the “grammar” of the symbols Multiple Expressions. 3n + 1

  15. Working with the “grammar” of the symbols Multiple Expressions. 4 + 3(n – 1)

  16. Working with the “grammar” of the symbols Multiple Expressions. n + n + (n + 1)

  17. Working with the “grammar” of the symbols Multiple Expressions.

  18. Working with the “grammar” of the symbols Multiple Expressions.

  19. Working with the “grammar” of the symbols Multiple Expressions.

  20. Working with the “grammar” of the symbols Multiple Expressions.

  21. Working with the “grammar” of the symbols Multiple Expressions.

  22. Working with the “grammar” of the symbols Multiple Expressions.

  23. Working with the “grammar” of the symbols Multiple Expressions.

  24. Working with the “grammar” of the symbols Multiple Expressions. 4n - (n – 1)

  25. Looking at individual cases (particular examples) Seeing something the same in these individual cases (seeing the general in the particular). Expressing this sameness in words and/or pictures (generalising) Expressing using symbols (first your own and then using the conventions of others) Manipulating the symbols arising from comparing different ways of expressing (Using algebra to name the unknown and operate on it) Algebra – the full picture

  26. Spotting patterns versusspotting structure.

  27. Spotting patterns versusspotting structure.

  28. Spotting patterns versusspotting structure.

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