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Michael Drake

MMP Day 2008. Teaching algebra for meaning. Michael Drake. Algebra. What is algebra?. Think…. Discuss in pairs…. If you ask parents (or Y11 students), what would they say?. How do we use letters in mathematics?. Have a chat about it…. Tell me what you have come up with….

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Michael Drake

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  1. MMP Day 2008 Teaching algebra for meaning Michael Drake

  2. Algebra • What is algebra? Think… Discuss in pairs… • If you ask parents (or Y11 students), what would they say?

  3. How do we use letters in mathematics? • Have a chat about it… • Tell me what you have come up with…

  4. How do we use letters in mathematics? 1) A letter can be used to name something • In the formula for the area of a rectangle, the base of the rectangle is often named b

  5. The development of formulae Area = base × height A = b × h A = bh 3 cm A = 5 × 3 A = 15cm2 5 cm

  6. How do we use letters in mathematics? 2) A letter can be used as a variable • This happens in patterning

  7. Developing patterning • Write the next three terms of the sequence 3, 6, 9, 12, … Write a sentence explaining how to get from one term to the next. Start with “To get the new term …” •  3 •  6 •  9 •  12  Down rule  Across rule 

  8. Number of triangles Number of matches •  3 •  6 •  9 •  12 • Write a sentence for the across rule. Start it with “the number of matches is found by taking the number of triangles and…”

  9. How many terms? • What is the next term of this sequence? 1, 2, _ • What is the next term of this sequence? 1, 2, 4, _

  10. Differentiating… • Write the next four terms of the sequence 10 507, 10 627, 10 747, 10 867, • Write a sentence that explains how to get from one term in the sequence to the next • Can we write this more briefly, and still know what we mean?

  11. How do we use letters in mathematics? 3) A letter can be used to stand for a specific unknown number that needs to be found In a triangle, x is often used for the angle students need to find • This happens when we solve equations

  12. When do we need to use letters to solve problems? • Jemima has some sweets. She eats 5 and has 1 left. How many did she start out with? • Rewi has 47 super 14 rugby cards. 19 of them are repeats that he can swap. How many different cards does he have?

  13. In numeracy schools, student may have seen an algebraic form of recording problems since Year 1… 6 +  = 10 …and may have even learned to record problems in this format

  14. How do we use letters in mathematics? 4) A letter can be used as a variable that can take a variety of possible values (This also happens with equations) + = 10

  15. How do we use letters in mathematics? 5) A letter is really a number c = speed of light e π Or if you are a student, in substitution a = 1…

  16. How do we use letters in mathematics? 6) In the equation y = 2x + 1, x can be any number, a whole number, a fraction, integer or decimal 7) In y = mx +c, m and c act as parameters

  17. Lessons • Letters mean different things in different situations • Algebra is not something that should be taught in a single block… • Algebra often makes sense when it arises from a context…

  18. Number property generalisation Patterns and relationships Equations Formula development Introduction to the conventions of algebra The algebra domains

  19. Time to do some mathematics…

  20. Starters • Think of a number • Add 3 to it • Double it • Subtract two • Halve it • Subtract the number you started with • Add 6 • Multiply the number by 3

  21. Think of a number… • How do we write down that we are thinking of a number – without giving away the number we are thinking of? • What would one more than the number look like? • What would 3 less than the number look like? • What would double the number look like?

  22. Starters What is? 2  4 3  3 52 6  4 6  6 5  7 7  7 8  6 102 9  11

  23. Generally generalising • 6 + 6 + 6 + 6 + 6 + 6 + 6 = 7  6 • Explain why the sentence is true • Give me another example Thinking time: discuss in pairs • How can we show that this always works – no matter the number?

  24. Did we always have to use 7 of them? • What other generalisations could we have made? • How can we show this always works – regardless of the number of things we are multiplying?

  25. First order abstraction Second order abstraction Generalising number properties Generalising algebra properties Using and exploring number properties The development of algebraic thinking

  26. What is algebra? • What are your answers now? • Generalised number A way of describing trends and patterns we find in sets of similar problems • The structure of the number system A way of looking behind (underneath) a set of number problems to see what is really going on

  27. Write me some… • Write me a story problem which has the answer x = 4. Try using a context that tells us something about your culture or your family Sharing time…

  28. So how does this algebra work? • Why does it help us to work out what is going on?

  29. Teaching volume • How do we each teach volume? • How might these resources be useful? • How can we use learning about solids as a context for algebra development? • What prior knowledge is needed for this approach to be useful?

  30. What about 16  16? What is 4  4? • 250  250? • Does this always work? • Explain why this works using a drawing or some equipment from the box • Can you write this as a rule in words? • Write a sentence with symbols to show your rule

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