Download Modeling in Mathematics

Modeling Mathematical Ideas
••• Using Materials •••
Jim Hogan • University of Waikato, NZ
Brian Tweed • Massey University, NZ
MAV
DEC 2008
Aim of this session
To make you a better modeler of
mathematical concepts
You will know you are a better modeler
when you use models to explain ideas
From good models will come deeper
and other understandings.
Famous Last Words!
If you can’t model it, then don’t
teach it!
-
J Hogan, 2004 at a WMA PD session.
I have since discovered that being able to represent the concepts
of
mathematics in different forms or representations is a very good
thing to be able do.
Reference to Paul Cobb and his work. A model is a representation.
A Model is…
• any representation
– A picture
– A physical model
– A sketch
– An equation
– A graph
– A mime
In the beginning there was one.
Make a model of one.
Make a model of one and a half.
Make a model of counting 1, 2, 3, 4,…
Make a model 1 + 2 + 3 + 4
= T4
It’s About Even
Make a model of an even number.
It’s About Even
Make a model of an even number.
Does your model express the
essence of being “even”.
Is any model better than others?
Why is it better?
What is the essence of being “even”?
Assessment
A is a straightforward solution, (Understands
concept, Thinking KC).
M is clearly expressing the ideas (Symbols/Text
KC)
E is other solutions, new solutions, generalizing
the solution. Creative. Thinking KC.
Assess your model
• Why does your model illustrate an even
number?
• Express that clearly
• What is the general form of an even number?
It’s All Odd
Make a model of an odd number.
Your model must contain the
essence of being odd.
Can you make more than one
model?
What is the general form of an
odd number?
Properties of Odd and Even
• Explore the addition properties for odd and
even numbers.
• Explore the multiplicative properties.
• Explore the power!
• Can you explain all these ideas?
The Operations
Model the operation of addition.
3+4=
AME?
Using counters show me what each of the basic operations
mean:
Addition
Multiplication
Subtraction
Division
The Operations
An operation is dynamic, an action.
3+4=7
The Operations
Model the operation of subtraction.
7-3=
3+?=7
The Operations
Model the operation of multiplication.
3 x 4 = 12
AME?
The Operations
Model the operation of division.
12 / 4 = 3
12 / 3 = 4
If I make up groups of 4 how many groups do I get?
If I make 4 groups how many in each group?
An Odd Connection
I notice 1 + 3 + 5 + 7 = 16 = 4 x 4
Can you model that?
AME?
An Odd Connection
I notice 1 + 3 + 5 + 7 = 16 = 4 x 4
AME?
An Triangular Connection
I notice T3 + T4 = 16 = 4 x 4
The model is T4
An Triangular Connection
I notice T3 + T4 = 16 = 4 x 4
AME?
Odd -> Square
Triangular -> Square
what is Odd -> Triangular connection
Everything in mathematics is connected. Where is the connection?
Odd -> Square
Triangular -> Square
what is Odd -> Triangular connection
Odd -> Even so what is even to Triangular?
Triangular again
• What does T4 + T4 look like?
What does T3 + T3 look like?
What does T2 + T2 look like?
AME?
How many blocks in Tn ?
V.I.P. formula!
• T3+ T3 make a 3 x (3+1) rectangle
• Tn+ Tn make a n x (n+1) rectangle
• Cut in half! Model this!
• # blocks = n(n+1)/2
Hence!
• This adds up n whole numbers.
• # blocks = n(n+1)/2
• Test …sum 1 to 100
Extension
• Add the first 10 even numbers
• Add the first 10 multiples of 3; 5; 7; n
• Add the first 10 odd numbers… AME?
One Problem
(A number) x2 + 3 gives me {5, 7, 9, 11, …}
(A number + 3) x 2 gives me {8, 10, 12, 14, …}
Why is the second set 3 more than the first?
Can you model this?
AME!!!
Two Problem
Take any three digits, eg 1, 2, 3.
Make up all the 2 digit, eg 12, 13, 23, 21, 32, 31
Notice 12+13+23+21+32+31 = 132
Notice (1 + 2 + 3 )x22 = 132.
Why is it 22 x the sum of the digits is this sum?
Three problem
1+2=3
4+5+6=7+8
9 + 10 + 11 + 12 = 13 + 14 + 15
16 + 17 + …
= 21 + 22 + … and so on.
Model the second or third lines
and see why.
Four Problem
What does 12 x 12 tell you about 13 x 13?
Simplify the problem and explore
Can you generalize from (3+1)2
What happens in three dimensions?
Five Problem
I notice 8 x 10 = 92 - 1
and 19 x 21 = 202 - 1
Can I model why this is so?
Is the product of two consecutive odd or even
numbers always one more than a square?
Six problem
What are the two square numbers
that have a difference of 9?
Is there an odd number that is not the difference
of two squares?
How do you know?
Seven problem
Make a model of (3 + 1)3
Using the blocks can you see all the parts of the
expansion of (x+1)3 = x3 + 3x2 +3x + 1
What would (x+n)3 look like?
Eight Problem
The formula for adding the first n whole
numbers has an interesting symmetry.
n ( n + 1) /2 = n/2 x (n + 1) = n x (n + 1) /2
One of the numbers n or n+1 must be even. I choose the even
one to halve first and the total is the product.
Also (n + 1) /2 is the average of n numbers. The total of course
being the product! This is another understanding.
Powerful Problems
Make a model of 2 = 21
Make a model of 2 x 2 = 22
Make a model of 2 x 2 x 2 = 23
What is 20 ?
Three Odd?
Multiples of three are also the sum of three
consecutive numbers.
Eg 27 which is triple 9 is also 8 + 9 + 10.
Make a model and show why.
Can you extend this model to show more?
Do other numbers have this property?
Powers of 2
These numbers are a very curious group and
have many special properties. They appear
like magic in many problems. Know them!
Can 64 be written as the sum of three
consecutive numbers?
Which increases faster 2n or n2? Can you
illustrate your answer?
That’s it Folks!
Sensible models can be made of all problems.
Good models show the “essence” clearly.
There is often more than one model.
There are many types of models.
Good models lead to understanding.
This and other resources on
• http:schools.reap.org.nz/advisor
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Jim Hogan, Sec Math Advisor
[email protected]
Brian Tweed, Kaitakawaenga ki nga kura tuarua
[email protected]
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