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

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Teresa Maguire, Alex Neill February 2006Background ARB resources Why algebraic thinking?

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Teresa Maguire, Alex Neill February 2006 Historical development of algebra Ordinary language - rhetorical Unknowns – finding value for a letter or letters Givens – relationships, functions, generalised number, variables, parameters 56 - = 29 = 4 +

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Teresa Maguire, Alex Neill February 2006Pre-algebra How early can you start teaching algebra? Pre-algebraic thinking

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Teresa Maguire, Alex Neill February 2006 Research Question: What do students need to know and need to be able to do in order to think algebraically?

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Teresa Maguire, Alex Neill February 2006 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) ]

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Teresa Maguire, Alex Neill February 2006 Pre-algebraic concepts Symbolic relationships If x = 5 then 3x = 15 not 3x = 35

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Teresa Maguire, Alex Neill February 2006 Pre-algebraic concepts Relationships/relational thinking 14 + x = 25 + 17 Solving equations + = 6

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Teresa Maguire, Alex Neill February 2006 Our research A local Wellington school – decile 9, roll of 200 Who? Year 4s What? 6 lessons over 3 weeks Where?

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Teresa Maguire, Alex Neill February 2006Methodology 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

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Teresa Maguire, Alex Neill February 2006 Additional information Interviews videotaped Lessons Teacher – observer Books

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Teresa Maguire, Alex Neill February 2006Equality = What does the equals sign mean to you?

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Teresa Maguire, Alex Neill February 2006 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

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Teresa Maguire, Alex Neill February 2006 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)

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Teresa Maguire, Alex Neill February 2006 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

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Teresa Maguire, Alex Neill February 2006 Student number sentences 50 = 25 + 258 + 8 = 15 + 1 19 = 9 + 1 + 5 + 1 + 1 + 1 + 18 = 1 + 2 + 5 3 + 8 = 3 + 8162 = 162 70 + 100 = 160 + 10 6 – 3 + 2 – 1 = 4 + 1 – 1 10 = 4 + 4 + 25 x 5 = 5 x 5 50 = 10 + 10 + 10 + 10 + 102 x 6 = 3 x 4 168 + 26 = 163 + 3118 + 3 = 14 + 7

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Teresa Maguire, Alex Neill February 2006Equality Cuisenaire Rods (8 = …) Journal entry: I think the equals sign means…

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Teresa Maguire, Alex Neill February 2006 Journal Entries – Equality (n =27) Response Number The same as14 The answer/the total/the equation ends17 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. its helpful) 2

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Teresa Maguire, Alex Neill February 2006 Equality – post-test ideas (n = 28) Response Number The same as12 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

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Teresa Maguire, Alex Neill February 2006 Equality – post-interview (n =6) 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. The same as or here is the answer.

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Teresa Maguire, Alex Neill February 2006 True/False Number sentences Pre-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)

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Teresa Maguire, Alex Neill February 2006 True/False Number Sentences Pre-interviews (n = 6) 5 + 2 = 4 + 34 correct (T) – 2 incorrect Its backwards. Its the same one except its 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 its got a, it equals 3, it should be 7. 2 + 4 = 6 all 6 correct (T) 8 = 3 + 5 5 correct (T) – 1 incorrect 2 + 4 = 6 + 34 correct (F) – 2 incorrect

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Teresa Maguire, Alex Neill February 2006 Pre-interviews continued 5 + 3 = 8 5 + 3 = 8 but + 2, it should equal 10. 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

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Teresa Maguire, Alex Neill February 2006 Open Number Sentences Pre-test (n = 28) Problem % Correct 2 + = 8 96 5 = 2 + 79 = 1 + 7 64 6 + 2 = + 5 61 8 + 1 = 3 + 61 + 4 = 2 + 5 61

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Teresa Maguire, Alex Neill February 2006 7 = 4 + 5 + = 9 All correct Open Number Sentences Pre-interview (n = 6) 11. Because 7 + 4 = 11. 2. Because 2 = 2. 4. 4 + 2 = 6. Also it can be the other way around – 6 + 2 = something. Equals 8. = 2 + 6

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Teresa Maguire, Alex Neill February 2006 Open Number Sentences Pre-interview (n=6) 9. Because 4 + 5 = 9. 2. I added on to 4 to get to 6. 4 + 5 = + 3 7 + 2 = 4 + All correct + 4 = 6 + 1

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Teresa Maguire, Alex Neill February 2006 Balancing Equations Concrete to abstract Balance scales and multiblocks 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. ===

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Teresa Maguire, Alex Neill February 2006 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

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Teresa Maguire, Alex Neill February 2006 Interesting response 5 + 2 + 3 + 4 = 14 4 + 4 + 5 + 3 = 16 5 + 4 + 6 + 3 = 18 7 + 3 + 9 + 1 = 20

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Teresa Maguire, Alex Neill February 2006 Open Number Sentences Worksheet C e) 8 + 3 = 6 + a) 6 + 2 = + 5 b) 7 + = 10 + 2 c) + 1 = 3 + 4 d) 5 + 7 = + 9

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Teresa Maguire, Alex Neill February 2006 Did it make a difference? (n = 28) 96100 61 93 61 96 6189 64 93 79 100 Pre-test % correct Post-test % correct Problem 3 + = 9 + 4 = 2 + 6 9 + 1 = 3 + 4 + 3 = + 5 = 4 + 6 6 = 2 +

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Teresa Maguire, Alex Neill February 2006 Additive Identity 14 + = 14 18 – 18 = 42 + 67 – 67 – 23 + 23 = 17 + 48 – 48 = 29 + 38 – 29 = 21 + 14 – 14 = 35 + 12 – 12 + 23 – 23 =

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Teresa Maguire, Alex Neill February 2006 What the students said: Student 1 48 + 48 + 17. 16 + 7 = … 17 + 48 – 48 = I see youre doing some calculating there. When you look at that number sentence can you think of an easier way to do this that doesnt 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 theres a takeaway in there. So thats 17 plus 48 basically. So if you put another 48 on the end, youre just taking it away. Well, no, thats not going to work.

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Teresa Maguire, Alex Neill February 2006 What the students said: Student 2 21 + 14 – 14 = 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.

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Teresa Maguire, Alex Neill February 2006 What the students said: Student 2 35 + 12 – 12 + 23 – 23 = What did you do that time? 35 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 theres not much point in adding. Right, because what happens if you add something then take it away? Um, sometimes it gets confusing.

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Teresa Maguire, Alex Neill February 2006 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.

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Teresa Maguire, Alex Neill February 2006 What the students said: Student 3 21 because if you take away 14 from 14 its a zero then theres a 21 at the beginning. 21 + 14 – 14 =

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Teresa Maguire, Alex Neill February 2006 Additive Identity True/False number sentences 49 + 0 = 49 64 + 23 = 64 123,456 + 0 = 123,456 Rules/conjectures about zero

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Teresa Maguire, Alex Neill February 2006 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. Its a trick/Its a confusion. Youre still using the same number. Pretend it isnt there.

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Teresa Maguire, Alex Neill February 2006 More ideas about zero I think of it as nothing, but if its at the end of a number you have to take notice of it. When you add zero to a number it doesnt really change anything. But 05 would be the same as 5. If its before a number you are just filling in the gaps. Multiplicative identity.

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Teresa Maguire, Alex Neill February 2006 Conjecture 1 about zero When you add zero with another number it doesnt change the number you started with. a + 0 = a

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Teresa Maguire, Alex Neill February 2006 Conjecture 2 about zero When you take away zero from a number it doesnt change the number you started with. a – 0 = a

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Teresa Maguire, Alex Neill February 2006 True/False number sentences about zero Worksheet D Look at each number sentence below. Circle if it is True or False. i)8 + 0 = 8True or False ii)11 - 11 = 11True or False iii)0 + 95 = 0True or False iv)53 - 0 = 53True or False v)50 + 0 = 500True or False

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Teresa Maguire, Alex Neill February 2006 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.

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Teresa Maguire, Alex Neill February 2006 Students true number sentences 1 + 50 000 + 10 = 11 + 40 000 + 10 000 - 0 1 000 + 0 = 999 + 1 11 – 0 – 11 = 0 33 + 8 – 0 = 33 + 8 + 0 150 + 50 = 200 + 0 20 + 0 + 5 = 25 2 + 8 = 10 + 0 11 = 11 - 0 500 + 0 = 500 + 100 - 100

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Teresa Maguire, Alex Neill February 2006 Finding the zero 6 + 5 – 5 = 12 + 7 – 7 = 6 + 5 = 11, minus 5: take 5 back again. Just like adding zero. Add 7 then take away 7, its the same. Same as what? 12 + 0 = 12.

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Teresa Maguire, Alex Neill February 2006 Finding the zero 38 + 27 – 27 = I started with 38 + 27 then I said Oh, no, then I noticed youd added 27 then taken away 27. 38 + 0 = 38 You could swap the plus and the minus around to make 38 – 27 + 27 =

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Teresa Maguire, Alex Neill February 2006 Finding the zero 85 + 44 – 85 = 79 + 23 – 79 = 44. Because 85 + 44 – 85 doesnt make a difference because its got 85 – 85. I took out the 23 and got 79 – 79 then I put the 23 back in.

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Teresa Maguire, Alex Neill February 2006 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 =

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Teresa Maguire, Alex Neill February 2006 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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Teresa Maguire, Alex Neill February 2006 Did it make a difference? 8 + = 8 48 + 79 – 79 – 35 + 35 = 85 + 95 – 28 – 85 + 28 = 6 + 0 + 8 = 27 + 64 – 27 = 79 + 107 – 68 – 79 + 68 =

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Teresa Maguire, Alex Neill February 2006 Post-interview conversation How about this one? [79 + 107 – 68 – 79 + 68] 107. Because 79 – 79 equals zero and 68 take, oh, wait, no, it doesnt. That [points to 79 – 79] equals zero and um….[pauses] Youre thinking about those 68s are you? What are you thinking about them? That the 68 was there [points to –68], but then… The 68 got tooken away but then it came back here [points to +68] so I need to add 107 and 68. 175. So if you take that 68 off and then you add it back on again, you have to add that 68 onto the 107? Yeah.

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Teresa Maguire, Alex Neill February 2006Commutativity And now over to Alex… The Commuter Tiz Property a + b = b + a

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Teresa Maguire, Alex Neill February 2006 Commutativity – Prior Ideas (n=6) Recognition (Type 1) What sign should you put in the box ( )? 3 + 8 8 + 3100% 9 + 12 21 + 9 67% 35 + 27 27 + 35100% 83 + 47 74 + 38100% 254 + 326 236 + 542 83%

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Teresa Maguire, Alex Neill February 2006 Commutativity – Prior Ideas (n=28) Variable – as yet unknown (Type 2) Fill in each box to make the equation true. Correct Wrong + 24 = 24 + 1936% (10) 0 42 + 31 = 31 + 46% (13)73, 104 53 + = 74 + 5336% (10)21 12 + 69 = + 1236% (10)81 Reason for errors: = means and the answer is

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Teresa Maguire, Alex Neill February 2006 Intervention (Type 1) True or False? Discuss whether these sentences are true of false. 3 + 8 = 8 + 3 10 + 12 = 12 + 10 31 + 42 = 42 + 13 25 + 46 = 46 + 25

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Teresa Maguire, Alex Neill February 2006 Commutativity – Class (n=28) Its actually true because the numbers are just swapped around. The same numbers are on each side. Its like a reflection of each other (uses hands to show this). 3 + 8 = 11 and 8 + 3 = 11 It needs to be 3 + 8 = 3 + 8

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Teresa Maguire, Alex Neill February 2006 Commutativity – Class (n=28) Write your own True or False number sentences. 72 + 83 = 83 + 72T 44 + 61 = 16 + 44F 35 + 82 + 34 = 82 + 34 + 35T 231 + 123 = 321 + 132F 905 + 509 = 509 + 905T 36 + 80 – 80 + 0 = 0 + 80 – 80 + 36T 70046 + 70064 = 70064 + 70046T

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Teresa Maguire, Alex Neill February 2006 Commutativity – Class (n = 28) Make a general rule It doesnt matter if the numbers are swapped around on each side of a number sentence, its still the same. It doesnt matter if the numbers are swapped around on each side of the number sentence. If the numbers are the same, the number sentence will still balance If the numbers on each side of a number sentence are the same, the number sentence will balance.

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Teresa Maguire, Alex Neill February 2006 Number line Introduce the number line as a model: Write the number sentence that this number line shows. Draw the number line to show 12 + 6 = 6 + 12

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Teresa Maguire, Alex Neill February 2006 Commutativity - Class (n=28) Introduce a variable as yet unknown (Type 2) 28 + 15 = 15 + Did anyone start calculating? Did you need to? 67 + = 58 + 67 372 + 183 = + 372 All students did a worksheet based on this, and everyone could get it, even those without the required computational skills.

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Teresa Maguire, Alex Neill February 2006 Did it make a difference? (n=28) 36100 46 100 36 96 36 96 Pre-test % correct Post-test % correct Problem 75 + = 89 + 75 61 + 48 = 48 + + 35 = 35 + 27 15 + 58 = + 15

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Teresa Maguire, Alex Neill February 2006 Students own representations Show why 6 + 8 = 8 + 6. Swapping – commutativity (7) The numbers have just been swapped around. Because what ever is on this side has to be on that side. 6 + 8 = 8 + 6 because it is Jorh sort arend.

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Teresa Maguire, Alex Neill February 2006 Students own representations Equal – computational (4) Number line (1) 6 + 8 = 14 and 8 + 6 = 14

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Teresa Maguire, Alex Neill February 2006 Students own representations Balance models (5) Ordered (3) Unordered (2)

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Teresa Maguire, Alex Neill February 2006 Students own representations Others It is the same!

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Teresa Maguire, Alex Neill February 2006 Associativity – Prior Ideas (n=28) Variable – as yet unknown (Type 2) Fill in each box to make the equation true. Correct (2 + 7) + = 2 + (7 + 5)36% (10) 18 + (13 + 15) = (18 + 13) + 46% (13) (285 + ) + 176 = 285 + (88 + 176)36% (10)

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Teresa Maguire, Alex Neill February 2006 Who can think associatively? Answers Freq 5, 15, 88 7All but 1 understood = 5, 15, - 3Mainly understood = 2, 15, 88 3Mainly understood = Wrong or missing 15 6 mainly understood = 9 did not understand =

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Teresa Maguire, Alex Neill February 2006 = is necessary To think associatively, students need to have a good idea of what = means. (6 of the 7 who got all the pre-test questions on associativity correct had a correct or largely correct view about what = means) To think commutatively, students need to have a good idea of what = means. (all 10 who got the pre-test questions on commutativity correct had a correct or largely correct view about what = means)

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Teresa Maguire, Alex Neill February 2006 = is not sufficient Several students who seemed to understand what = means could not correctly answer the questions on commutativity or associativity. = is necessary, but not sufficient. Teach meaning of = first, and commutativity and associativity fall into place easily.

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Teresa Maguire, Alex Neill February 2006 What are the culprits? 3 + 4 = 28 - 5 = 26 + 5 – 7 = 4 5 = 18 6 = x + 3 = 7 Black boxes

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Teresa Maguire, Alex Neill February 2006 What are some solutions? 3 + 4 = 5 + 2 7 = 6 + 1 7 = ? + 1 What does = mean? Does Three plus four equals make sense?

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Teresa Maguire, Alex Neill February 2006 Calculators can help! Scaffolds learning by removing lower level skills to help discover higher level ones. 486 + 368 = 368 + 486 T/F Allows exploration, discovery and understanding to develop. 81 = 9, 9 9 = 81 90 9.467, 9.467 9.487 90 Works for any number! White Box

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Teresa Maguire, Alex Neill February 2006 Calculators can help! Models the correct use of = 3 + 4 = 3+4 7 Balance model 3+4=5+2 7=7 3+4=7+2No solution

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Teresa Maguire, Alex Neill February 2006Identity 7 + 0 = 0 No solution 7 + 0 = 77=7 7 – 7 = 00=0 x + 0 = 0 No solution x + 0 = 1x=1 x + 0 = xx=x 314 * 1 = 1 No solution 314 * 1 = 314 314=314 x * 1 = xx=x

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Teresa Maguire, Alex Neill February 2006 Commutativity / Associativity 4 + 8 = 8 + 4 12=12 479 + 368 = 368 + 479 847=847 479 + 368 = 378 + 469 847=847 x+ y = y + x x+y=x+y (47+86)+26= 47+(86+26) 159=159 (x+y)+z=x+(y+z) x+y+z=x+y+z 7+34+48-34 = 48-7 41=41

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Teresa Maguire, Alex Neill February 2006 Solving equations Provides scaffolding on applying identity properties x+3 = 7 (+,-,, or by 3 or 7) 3x=7 (+,-,, or by 3 or 7) (7x+6) 3 = 5x (+,-, or by 3, 5, 6 or 7)

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