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 = T 4

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 = AME?

Using counters show me what each of the basic operations mean: Addition Multiplication Subtraction Division

The Operations An operation is dynamic, an action = 7

The Operations Model the operation of subtraction =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 rr r 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 = 16 = 4 x 4 Can you model that? AME?

An Odd Connection I notice = 16 = 4 x 4 AME?

An Triangular Connection I notice T 3 + T 4 = 16 = 4 x 4 The model is T 4

An Triangular Connection I notice T 3 + T 4 = 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 T 4 + T 4 look like? What does T 3 + T 3 look like? What does T 2 + T 2 look like? AME? How many blocks in T n ?

V.I.P. formula! T 3 + T 3 make a 3 x (3+1) rectangle T n + T n 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 = 132 Notice ( )x22 = 132. Why is it 22 x the sum of the digits is this sum?

Three problem = = = …= … 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 = and 19 x 21 = 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 = x 3 + 3x 2 +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 = 2 1 Make a model of 2 x 2 = 2 2 Make a model of 2 x 2 x 2 = 2 3 What is 2 0 ?

Three Odd? Multiples of three are also the sum of three consecutive numbers. Eg 27 which is triple 9 is also 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 2 n or n 2 ? 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.

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