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

Teresa Maguire, Alex NeillFebruary 2006

Algebraic Algebraic Thinking through Thinking through

NumberNumberWorkshop presented at

National Numeracy Facilitators Conference

February 2006Teresa Maguire and Alex Neill


BackgroundBackground

ARB resources

Why algebraic thinking?


Historical development of Historical development of algebraalgebra

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 +


Pre-algebraPre-algebra

How early can you start teaching algebra?

Pre-algebraic thinking


Research Question:Research Question:

What do students need to know and need to be able to do in order to think algebraically?


Pre-algebraic conceptsPre-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) ]



Symbolic relationships

If x = 5

then 3x = 15

not 3x = 35



Relationships/relational thinking

14 + x = 25 + 17

Solving equations

+ = 6


Our researchOur research

A local Wellington school – decile 9, roll of 200

Who?

Year 4s

What?

6 lessons over 3 weeks

Where?


MethodologyMethodology

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


Additional informationAdditional information

Interviews videotaped

Lessons

Teacher – observer

Books


EqualityEquality

=What does the equals

sign mean to you?


Prior ideas Prior ideas (from pre-interviews (from pre-interviews and classroom discussion)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


Prior ideasPrior 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)


First lessonFirst 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


Student number sentencesStudent 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


EqualityEquality

Cuisenaire Rods (8 = …)

Journal entry:

“I think the equals sign means…”


Journal Entries – Equality Journal Entries – Equality (n =27)(n =27)

Response Number

The same as 14

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. it’s helpful) 2


Equality – post-test ideas Equality – post-test ideas (n = 28)(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


Equality – post-interview Equality – post-interview (n =6)(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.


True/False Number sentencesTrue/False Number sentencesPre-test Pre-test (n = 28)(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)


True/False Number SentencesTrue/False Number SentencesPre-interviews Pre-interviews (n = 6)(n = 6)

5 + 2 = 4 + 3 4 correct (T) – 2 incorrect

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)8 = 3 + 5 5 correct (T) – 1 incorrect

2 + 4 = 6 + 3 4 correct (F) – 2 incorrect


Pre-interviews continuedPre-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


Open Number SentencesOpen Number SentencesPre-test Pre-test (n = 28)(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


7 = 4 +

5 + = 9 All correct

Open Number SentencesOpen Number SentencesPre-interview Pre-interview (n = 6)(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


Open Number SentencesOpen Number SentencesPre-interview Pre-interview (n=6)(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


Balancing Equations 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.

= = =


More balancingMore 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


Interesting responseInteresting response

5 + 2 + 3 + 4 = 14

4 + 4 + 5 + 3 = 16

5 + 4 + 6 + 3 = 18

7 + 3 + 9 + 1 = 20


Open Number SentencesOpen Number Sentences

Worksheet C

e) 8 + 3 = 6 +

a) 6 + 2 = + 5

b) 7 + = 10 + 2

c) + 1 = 3 + 4

d) 5 + 7 = + 9


Did it make a difference? Did it make a difference? (n = 28)(n = 28)

96100

6193

6196

6189

6493

79100

Pre-test

% correct

Post-test

% correctProblem

3 + = 9

+ 4 = 2 + 6

9 + 1 = 3 +

4 + 3 = + 5

= 4 + 6

6 = 2 +


Additive IdentityAdditive Identity

14 + = 14 18 – 18 =

42 + 67 – 67 – 23 + 23 =

17 + 48 – 48 = 29 + 38 – 29 =

21 + 14 – 14 = 35 + 12 – 12 + 23 – 23 =


What the students said: Student 1What the students said: Student 1

48 + 48 + 17. 16 + 7 = …

17 + 48 – 48 =

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.



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.



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 there’s not much point in adding.

Right, because what happens if you add something then take it away?

Um, sometimes it gets confusing.


More from Student 2More from Student 2Right. 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.



21 because if you take away 14 from 14 it’s a zero then there’s a 21 at the beginning.

21 + 14 – 14 =


Additive IdentityAdditive Identity

True/False number sentences

49 + 0 = 49

64 + 23 = 64

123,456 + 0 = 123,456

Rules/conjectures about zero


Ideas about 0Ideas about 0

Basically 0, it is nothing (like in 8 + 0).

Maybe you can just leave it out when you have a plus zero.

It’s a trick/It’s a confusion.

You’re still using the same number.

Pretend it isn’t there.


More ideas about zeroMore ideas about zero

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.

Multiplicative identity.


Conjecture 1 about zeroConjecture 1 about zero

When you add zero with another number it doesn’t change the number you started with.

a + 0 = a


Conjecture 2 about zeroConjecture 2 about zero

When you take away zero from a number it doesn’t change the number you started with.

a – 0 = a


True/False number sentences about zeroTrue/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 Falseii) 11 - 11 = 11 True or Falseiii) 0 + 95 = 0 True or Falseiv) 53 - 0 = 53 True or Falsev) 50 + 0 = 500 True or False


Conjecture 3 about 0Conjecture 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.


Student’s true number sentencesStudent’s 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


Finding the zeroFinding 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, it’s the same.

Same as what?

12 + 0 = 12.



38 + 27 – 27 =

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 =



85 + 44 – 85 =

79 + 23 – 79 =

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.



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 =


““Tricky” number sentencesTricky” 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


Did it make a difference?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 =


Post-interview conversationPost-interview conversation

How about this one?[79 + 107 – 68 – 79 + 68]

107. Because 79 – 79 equals zero and 68 take, oh, wait, no, it doesn’t. That [points to 79 – 79] equals zero and

um….[pauses]

You’re 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

.


CommutativityCommutativity

And now over to Alex…

The Commuter Tiz Property

a + b = b + a


Commutativity – Prior IdeasCommutativity – Prior Ideas (n=6)(n=6)

Recognition (Type 1)

What sign should you put in the box (<,=,>)?

3 + 8 8 + 3 100%

9 + 12 21 + 9 67%

35 + 27 27 + 35 100%

83 + 47 74 + 38 100%

254 + 326 236 + 542 83%


Commutativity – Prior Ideas Commutativity – Prior Ideas (n=28)(n=28)

Variable – “as yet unknown” (Type 2)

Fill in each box to make the equation true.

Correct Wrong + 24 = 24 + 19 36% (10) 0

42 + 31 = 31 + 46% (13) 73, 104

53 + = 74 + 53 36% (10) 21

12 + 69 = + 12 36% (10) 81

Reason for errors: = means ‘and the answer is’


Intervention (Type 1)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


Commutativity – Class Commutativity – Class (n=28)(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


Commutativity – Class Commutativity – Class (n=28)(n=28)

Write your own True or False number sentences.

72 + 83 = 83 + 72 T

44 + 61 = 16 + 44 F

35 + 82 + 34 = 82 + 34 + 35 T

231 + 123 = 321 + 132 F

905 + 509 = 509 + 905 T

36 + 80 – 80 + 0 = 0 + 80 – 80 + 36 T

70046 + 70064 = 70064 + 70046 T


Commutativity – Class Commutativity – Class (n = 28)(n = 28)

Make a general rule

It doesn’t matter if the numbers are swapped around on each side of a number

sentence, its still the same.

It doesn’t 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.


Number lineNumber 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


Commutativity - Class Commutativity - Class (n=28)(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.


Did it make a difference? Did it make a difference? (n=28)(n=28)

36100

46100

3696

3696

Pre-test

% correct

Post-test

% correctProblem

75 + = 89 + 75

61 + 48 = 48 +

+ 35 = 35 + 27

15 + 58 = + 15


Students’ own representationsStudents’ 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.



Equal – computational (4)

Number line (1)

6 + 8 = 14 and 8 + 6 = 14



Balance models (5)

Ordered (3)

Unordered (2)



Others

It is the same!


Associativity – Prior Ideas Associativity – Prior Ideas (n=28)(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)


Who can think associatively?Who can think associatively?

Answers Freq

5, 15, 88 7 All but 1 understood “=“

5, 15, - 3 Mainly understood “=“

2, 15, 88 3 Mainly understood “=“

Wrong or missing 15 6 mainly understood “=“

9 did not understand “=“


= is necessary= 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)


= is not sufficient= 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.


What are the culprits?What are the culprits?

3 + 4 =

28 - 5 =

26 + 5 – 7 =

4 5 =

18 6 =

x + 3 = 7

Black boxes


What are some solutions?What are some solutions?

3 + 4 = 5 + 2

7 = 6 + 1

7 = ? + 1

“What does ‘=‘ mean?”

Does “Three plus four equals” make sense?


Calculators can help!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


Calculators can help!Calculators can help!

Models the correct use of =

3 + 4 =

3+4 <exe> 7

Balance model

3+4=5+2 7=7

3+4=7+2 No solution


IdentityIdentity

7 + 0 = 0 No solution

7 + 0 = 7 7=7

7 – 7 = 0 0=0

x + 0 = 0 No solution

x + 0 = 1 x=1

x + 0 = x x=x

314 * 1 = 1 No solution

314 * 1 = 314 314=314

x * 1 = x x=x


Commutativity / AssociativityCommutativity / 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


Solving equationsSolving 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)

Documents

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