Modeling Mathematical Ideas••• Using Materials •••

Jim Hogan • University of Waikato, NZ Brian Tweed • Massey University, NZ

MAVDEC 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.

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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 = 312 / 3 = 4

rr rrr 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 ConnectionI 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 4AME?

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 -> SquareTriangular -> Square

what is Odd -> Triangular connection

Everything in mathematics is connected. Where is the connection?

Odd -> SquareTriangular -> 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 = 132Notice (1 + 2 + 3 )x22 = 132.

Why is it 22 x the sum of the digits is this sum?

Three problem

1 + 2 = 34 + 5 + 6 = 7 + 89 + 10 + 11 + 12 = 13 + 14 + 1516 + 17 + … = 21 + 22 + … and so on.

Model the second or third lines and see why.

Four ProblemWhat 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 - 1and 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 ProblemThe 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

• Jim Hogan, Sec Math Advisor• jimhogan@clear.net.nz• Brian Tweed, Kaitakawaenga ki nga kura tuarua• b.tweed@massey.ac.nz

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