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Algebraic Thinking through Number

Algebraic Thinking through Number. Workshop presented at National Numeracy Facilitators Conference February 2006. Teresa Maguire and Alex Neill. Background. ARB resources Why algebraic thinking?. Historical development of algebra. 56 - = 29. = 4 + .

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Algebraic Thinking through Number

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  1. Algebraic Thinking through Number Workshop presented at National Numeracy Facilitators Conference February 2006 Teresa Maguire and Alex Neill

  2. Background • ARB resources • Why algebraic thinking?

  3. Historical development of algebra 56 - = 29 = 4 + • Historical development of algebra • Ordinary language - rhetorical • Unknowns – finding value for a letter or letters • Givens – relationships, functions, generalised number, variables, parameters

  4. Pre-algebra • How early can you start teaching algebra? • Pre-algebraic thinking

  5. Research Question: What do students need to know and need to be able to do in order to think algebraically?

  6. Pre-algebraic concepts • Equality • Number properties • Identity [ a + 0 = a, a – a = 0 ] • Commutative [a + b = b + a] • Associative [ a + (b + c) = (a + b) + c] • Distributive [ a  (b + c) = (a  b) + (a  c) ]

  7. Pre-algebraic concepts • Symbolic relationships If x = 5 then 3x = 15 not 3x = 35

  8. Pre-algebraic concepts + = 6 • Relationships/relational thinking 14 + x = 25 + 17 Solving equations

  9. Our research • A local Wellington school – decile 9, roll of 200 Who? • Year 4s What? • 6 lessons over 3 weeks Where?

  10. Methodology • 2 lower achievers, 2 average achievers, 2 higher achievers Pre- and post-tests: • 30 pupils, including some Year 5s Pre- and post-interviews: • Six students • 3 girls, 3 boys

  11. Additional information • Interviews videotaped • Lessons • Teacher – observer • Books

  12. Equality = What does the equals sign mean to you?

  13. Prior ideas (from pre-interviews and classroom discussion) You use it at the end of an equation to show what two numbers added together is. A bit of a pause then it gives the answer. Tells you the answer The total A pause before the answer

  14. Prior ideas You put it there between the answer so it separates it and shows that 1 + 1 = 2 not 1 + 1 2. It has two meanings – the same and what the two numbers equal together. To separate the equation from the answer When two numbers are the same (like 2 and 2 are the same or 1 + 1 is the same as 2)

  15. First lesson • Challenge beliefs • True/False number sentences • Different formats: 85 + 75 = 160 7 = 4 + 3 6 = 6 3 + 5 = 3 + 5 3 + 2 = 5 + 3 3 + 5 = 4 + 4 • Students write their own

  16. Student number sentences 50 = 25 + 25 8 + 8 = 15 + 1 19 = 9 + 1 + 5 + 1 + 1 + 1 + 1 8 = 1 + 2 + 5 3 + 8 = 3 + 8 162 = 162 70 + 100 = 160 + 10 6 – 3 + 2 – 1 = 4 + 1 – 1 10 = 4 + 4 + 2 5 x 5 = 5 x 5 50 = 10 + 10 + 10 + 10 + 10 2 x 6 = 3 x 4 168 + 26 = 163 + 31 18 + 3 = 14 + 7

  17. Equality • Cuisenaire Rods (8 = …) • Journal entry: “I think the equals sign means…”

  18. Journal Entries – Equality (n =27) ResponseNumber The same as 14 The answer/the total/the equation ends 17 The break/splits the eqn. from the answer 2 Can go any/everywhere in a maths eqn. 2 Many different things 3 Other (e.g. it’s helpful) 2

  19. Equality – post-test ideas (n = 28) Response Number The same as 12 The same as or the answer to the problem 9 The answer 4 A break before the total 1 A lot of things 1 No answer 1

  20. Equality – post-interview (n =6) The same as or here is the answer. The same as or it can separate the equation from the answer. (2 students) Two meanings. What two numbers equal together or the numbers are the same. In number sentences it can mean the answer. Like 2 + 3 is the same as 4 + 1. The answer or it has to be balanced. The numbers have to be the same or equal the same on both sides.

  21. True/False Number sentencesPre-test (n = 28) Problem % Correct 5 + 4 = 9 100 (T) 7 = 1 + 6 75 (T) 5 + 4 = 9 + 3 79 (F) 7 + 2 = 3 + 6 68 (T) 6 + 1 = 7 + 5 79 (F) (2 + 6) + 3 = 2 + 9 54 (T)

  22. True/False Number SentencesPre-interviews (n = 6) It’s backwards. It’s the same one except it’s got plus 3. Because 5 does not equal 4! False. No, true because 5 + 2 = 7 but that equals 4 so it must be false because it’s got a, it equals 3, it should be 7. 2 + 4 = 6 all 6 correct (T) 5 + 2 = 4 + 3 4 correct (T) – 2 incorrect 8 = 3 + 5 5 correct (T) – 1 incorrect 2 + 4 = 6 + 3 4 correct (F) – 2 incorrect

  23. Pre-interviews continued 5 + 3 = 8 but + 2, it should equal 10. 5 + 3 = 8 3 + 4 = 7 not 3. 5 + 3 = 8 + 2 5 correct (F) – 1 incorrect (3 + 4) + 5 = 3 + 9 4 correct (T) – 1 incorrect 1 unable to answer

  24. Open Number SentencesPre-test (n = 28) 2 + = 8 96 5 = 2 + 79 = 1 + 7 64 6 + 2 = + 5 61 8 + 1 = 3 + 61 + 4 = 2 + 5 61 Problem % Correct

  25. Open Number SentencesPre-interview (n = 6) 5 + = 9 All correct 7 = 4 + 11. Because 7 + 4 = 11. 2. Because 2 = 2. = 2 + 6 4. 4 + 2 = 6. Also it can be the other way around – 6 + 2 = something. Equals 8.

  26. Open Number SentencesPre-interview (n=6) 4 + 5 = + 3 9. Because 4 + 5 = 9. 7 + 2 = 4 + All correct 2. I added on to 4 to get to 6. + 4 = 6 + 1

  27. Balancing Equations Worksheet A – Balance Pans In each diagram of scales below, draw in the number of black blocks needed on the right-hand side to make the scales balance. = = = • Concrete to abstract • Balance scales and multiblocks

  28. More balancing For each balance pan diagram below, write an equation to show that the sides are the same as each other (equal). = = = = Worksheet B – Balance Pan Number Sentences

  29. Interesting response 5 + 2 + 3 + 4 = 14 4 + 4 + 5 + 3 = 16 5 + 4 + 6 + 3 = 18 7 + 3 + 9 + 1 = 20

  30. Open Number Sentences a) 6 + 2 = + 5 b) 7 + = 10 + 2 c) + 1 = 3 + 4 d) 5 + 7 = + 9   e) 8 + 3 = 6 + Worksheet C

  31. Did it make a difference? (n = 28) Post-test % correct Problem Pre-test % correct 3 + = 9 100 96 100 6 = 2 + 79 93 = 4 + 6 64 4 + 3 = + 5 89 61 9 + 1 = 3 + 61 96 + 4 = 2 + 6 61 93

  32. Additive Identity 14 + = 14 18 – 18 = 42 + 67 – 67 – 23 + 23 = 29 + 38 – 29 = 17 + 48 – 48 = 35 + 12 – 12 + 23 – 23 = 21 + 14 – 14 =

  33. What the students said: Student 1 48 + 48 + 17. 16 + 7 = … I see you’re doing some calculating there. When you look at that number sentence can you think of an easier way to do this that doesn’t involve any calculating? Can you see any relationship between the numbers that might make it easier to find the answer? That some of them are even. That there’s a takeaway in there. So that’s 17 plus 48 basically. So if you put another 48 on the end, you’re just taking it away. Well, no, that’s not going to work. 17 + 48 – 48 =

  34. What the students said: Student 2 21. 21 + 14 is 35, takeaway 14… is the number that you added to 14. Right. OK, so did you actually do some calculations there? Did you add up the numbers in your head or did you just see that there was some relationship? Yeah. I added up the numbers, then I took the 4 away, the 14 away. 21 + 14 – 14 =

  35. What the students said: Student 2 35 + 12 – 12 + 23 – 23 = 35 What did you do that time? I rushed through the other numbers to see if you could do anything. What could you do? Did you find you could do anything? No, because it goes plus 12, takeaway 12, plus 23, takeaway 23. So there’s not much point in adding. Right, because what happens if you add something then take it away? Um, sometimes it gets confusing.

  36. More from Student 2 Right. So in this one you knew that if you added 12 then took away 12 and added 23 and took away 23 you would get what? Um, 35. Which is what? The number… At the very start.

  37. What the students said: Student 3 21 because if you take away 14 from 14 it’s a zero then there’s a 21 at the beginning. 21 + 14 – 14 =

  38. Additive Identity • True/False number sentences 49 + 0 = 49 64 + 23 = 64 123,456 + 0 = 123,456 • Rules/conjectures about zero

  39. Ideas about 0 • Basically 0, it is nothing (like in 8 + 0). • Maybe you can just leave it out when you have a plus zero. • You’re still using the same number. • Pretend it isn’t there. • It’s a trick/It’s a confusion.

  40. More ideas about zero • Multiplicative identity. • I think of it as nothing, but if it’s at the end of a number you have to take notice of it. • When you add zero to a number it doesn’t really change anything. But 05 would be the same as 5. If it’s before a number you are just filling in the gaps.

  41. Conjecture 1 about zero When you add zero with another number it doesn’t change the number you started with. a + 0 = a

  42. Conjecture 2 about zero When you take away zero from a number it doesn’t change the number you started with. a – 0 = a

  43. True/False number sentences about zero • Worksheet D • Look at each number sentence below. Circle if it is True or False. • i) 8 + 0 = 8 True or False • ii) 11 - 11 = 11 True or False • iii) 0 + 95 = 0 True or False • iv) 53 - 0 = 53 True or False • v) 50 + 0 = 500 True or False

  44. Conjecture 3 about 0 11 – 11 = 0 or a – a = 0 If you take away the same number from the one you started with you get zero.

  45. Student’s true number sentences 1 + 50 000 + 10 = 11 + 40 000 + 10 000 - 0 11 – 0 – 11 = 0 1 000 + 0 = 999 + 1 2 + 8 = 10 + 0 33 + 8 – 0 = 33 + 8 + 0 20 + 0 + 5 = 25 11 = 11 - 0 500 + 0 = 500 + 100 - 100 150 + 50 = 200 + 0

  46. Finding the zero 6 + 5 = 11, minus 5: take 5 back again. Just like adding zero. Add 7 then take away 7, it’s the same. Same as what? 12 + 0 = 12. 6 + 5 – 5 = 12 + 7 – 7 =

  47. Finding the zero I started with 38 + 27 then I said “Oh, no”, then I noticed you’d added 27 then taken away 27. 38 + 0 = 38 You could swap the plus and the minus around to make 38 – 27 + 27 = 38 + 27 – 27 =

  48. Finding the zero 44. Because 85 + 44 – 85 doesn’t make a difference because it’s got 85 – 85. I took out the 23 and got 79 – 79 then I put the 23 back in. 85 + 44 – 85 = 79 + 23 – 79 =

  49. Finding the zero Worksheet E – Can you find the zero?? Find the answers to each of the following problems without doing any calculating. a) 25 + 16 – 16 =e) 28 + 36 – 36 + 52 – 52 = b) 33 + 41 – 41 = f) 28 – 28 + 95 + 15 – 15 = c) 50 + 37 – 50 =g) 78 – 44 + 44 = d) 62 + 74 – 62 =h) 67 – 67 + 55 – 23 + 23 =

  50. “Tricky” number sentences 5 + 500 000 – 500 000 = 225 – 25 + 200 + 25 – 200 = 225 80 + 60 + 70 + 266 – 60 + 20 – 70 – 266 = 7000 + 20 – 20 + 30 = 7030 100 000 000 + 450 – 100 000 000 = 86 + 72 – 6 – 80 = 72 263 433 222 611 – 0 + 1 – 0 = 433 222 611 + 263 000 000 000 +0 +0 +0 + 1

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