# Problem Solving

Document Sample

```					Problem Solving

School Support Services
University of Waikato
Objectives du Jour

   To develop the language of PS
   To know what math thinking is
   To develop PS skills
   To explore a range of problems
   To source resources
   To be able to apply PS to learning
Acknowledgements

The content of this course is based heavily
on the detailed work in “Lighting
Mathematical Fires” 1999 by Prof Derek
Holton (Otago) and Charles Lovitt (CDU
Canberra).

The path forming work in problem solving of
George Polya (1945, Princeton).
Best Laid Plans…
9.0    Gidday Mate
9.2    The language of PS.
9.5    Math Thinking
10.0   Problem solving
10.5   A few problems?
12.5   Brain food…
13.0   More problems!
14.0   Oodles of Resources
14:7   Learning opportunities
15.0   The bubble bursts
A pleasant walk, a pleasant talk?

When you leave today:-
with a better understanding of PS
knowing where to get resources
knowing how to facilitate the learning
being a little better at PS
Is that OK?
Let’s GO!

The following discussion and
problems are suitable for all
observant people
but in particular those
with perseverance,
curiosity and wonderment.
The Language

Problem Solving is …
Being able to solve a well-defined
problem for which a method of
solution is not immediately obvious,
and telling someone.

What do these words mean?
Problem, solution, well-defined, telling.
The Language

Problem solving techniques include …
Experimenting, diagrams, recording, trials,
guessing and checking, hunting for
counter examples, listing possibilities,
finding patterns, working backwards,
trying smaller cases, having an “ah-ha!”.

What do all these words mean?
More Language
Investigations and projects
compared to
problem solving

Investigations are bigger more complex
problems!

Projects are like a literary search and
summarize what is known.
And more and more and more…
Proof
The concept of proof is specific to
mathematics and must be experienced.

Eg Demo Angle Sum triangle = 180
Eg Demo Sum N = n(n+1)/2
Eg There is an infinite set of prime numbers
and more and more and more …
Telling
The key competencies of
communication (and thinking) are
the reasons why we learn
mathematics.

When we solve a problem, tell
someone! Write, speak, to one, to
many, draw, publish, display!
Str-r-r-e-e-e-t-t-c-c-c-chhhhh.
Brain Gym Time

L-R wakeup call

Breathe

T5
Thinking Maths
What do you know about how
someone learns to think
mathematically?

- Revise research underpinning
numeracy. AC-EA-AA-AM-AP-U
- U is the understanding and
extension to the general case with
explanation
Thinking Maths
Examples
- Fish…general term
- Factor Down
- Pentominoe discovery
- Chords of Circle
- Black/white problem
- Green/red hat problem
- King Arthur problem.
The Artificial School

   Problem solving in the classroom is
artificial because the solution is already
known.
The Oncer
   A problem can only ever
be solved once. If the
technique is used on a
similar problem it is then a
technique.

   Mind you, every person
can experience this path.
What did Polya say?

The Four Step Approach
1. Understand the problem
2. Choose a strategy
3. Apply and solve
4. Reflect

Discuss each step.
Not all problems fit this framework!
Why is problem solving good?

 Life skill preparation
 Flexibility of mind
 Joy of discovery
 Develops creative thought
 Confidence and self esteem
 Develops cooperative skills
 Develops communication skills
Is there a problem with PS?

   Teacher discomfort
   Curriculum constraints
   Preparation time
   Assessment difficulties
   Resources
   Range of abiilities
   Time involved in the process
Just Do iT!

 Problem solving, investigations and
project work with appropriate reporting
an excellent approach to developing
thinking and communication skills.
 Other key competencies of socialising,
etc are also being developed.
 Not only for students!
noitcefeR/Refection

How are we going?
What have we noticed?
Where shall we go?
Do we need to change?
….

Shall we try a few problems?
Problem #1• The Farmyard

There are some pigs and chickens in
the farmyard. A worm counts there
are 15 animals and 48 legs. How
many pigs are there?

You Decide

Do you want to select a problem and
investigate it, with report back?

Or continue with the looking at each
problem all together, with discussion
and reporting?
Problem #2• Farmer Brown

Play power point.

Problem #3• Peter and Veronica

Peter is 40 and is eight times older
than his daughter Veronica. How old
will they both be when Peter is twice
as old as Veronica?

Problem #4• 457457

Think of a three digit number and write
it twice making a six digit number.
Now divide it by 7, the answer by 11
and the answer by 13. What do you
notice? Why does this happen?

Problem #5• Pentominoes

How many ways can you put 5
squares together, side to side?

How do you know you have all of
them?

Problem #6• Robin’s problem

The numbers A and B each have three
digits. Robin was asked to calculate AxB.
Instead he put A to the left of B to form a
six digit number D. His answer D was
three times the correct answer AxB. What
were the original number A and B?

Problem #7• Making 50c

How many ways can you have a total of 50c
in coins in your pocket? Guess first!

Problem #8• Making 1999

The sum of a collection of whole numbers is
1999. What are these numbers if their
product is as large as possible?

Problem #9• Postage Stamps

The local Post Office has run out of all
stamps except 3c and 5 cents stamps.
What amounts can be made up using just
these stamps?

Problem #10 • Powers of 2 and 3
Using the powers of 2 we can make up all other
Eg 13 = 1 + 4 + 8

Using the powers of 3 we need to use addition and
subtraction.
Eg 16 = 27 - 9 -3 +1

Why does this work?
Problem #11• Tennis anyone?

Six people turned up for tennis. How
many different singles games are
possible?

What if it was a knockout competition?
Problem #12• Squares!

In the left “right-isoceleles” triangle the
square has an area of 441cm2. What
is the area of the square in the same
triangle on the right?
Problem #13• Diagonal
Y has coordinates (0,1) and X (1,0)
How long is diagonal PQ?
Y

P

X
Q
Problem #14• Dominoes on the
Chessboard
If the two opposite corners are removed can the
board be covered with dominoes?

Explain
your
Problem #15• Reversing numbers
4297 and 7924
Subtract the smaller from the larger.

Why is the answer a multiple of 9?

Did it matter how we
rearranged the numbers?
Does it matter how many
digits there are?
Problem #16• Further?

Is the circular arc from A to B longer,
the same or shorter than the two
arcs from A to C then C to B?

A     B          C
Problem #17• Nim
Cross out 1, 2 or 3. The player to take the
last one loses. Take turns to start.

|||||||||||||      (13 matches)

Win three in a row to become
a NIM-MASTER.
Play Super-NIM.
Problem #18• Even consecutives

Any two consecutives numbers n and
n+1 is the sum of the first n even
numbers.
Eg 4 x 5 = 2 + 4 + 6 + 8

Why?
Problem #19• Odd Squares!

The nth square number is the sum of
the first n odd numbers.
Eg 4 x 4= 1 + 3 + 5 + 7

Show this is true.
Problem #20• Consecutive

I notice 9 = 2+3+4 and also 4+5. Are
there any numbers that can not be
expressed as the sum of a two or
more consecutive whole numbers?

Problem #21• Tyresome
The front tyre on my bike lasts 60,000km.
The rear lasts 40,000km. If I swap tyres
around before they are worn how far can
I get out of a set?

How many swaps do I need? When do I
swap?

Problem #22• Paint a Problem.
I mix 1L of yellow and 1L of red to make the
colour Raro. I also mix 1L of yellow and
2L of red to make the colour Tango.

I want the colour Mandarin which is exactly
halfway between these colours. How do I
mix it using 2L tins of Raro and Tango?

Problem #23• Write a problem
Create, adapt, reword, re-invent, design, discover,
encounter and new problem.

Here is mine.
In the land of truth tellers and liars I encounter a
truthteller and a liar at the fork in the road to their
villages. I ask one question to one of them and
walk confidently to the village of the truthtellers.
What might be the question I asked?

Practical Resources
   Lighting Mathematical Fires
   Cut the Knot
   NLVM
   NZAMT
   AMC

(Google these to find the source)

   How to Solve It, G Polya, 1945
   Australian Maths Problem Book
   Ahha and Gotcha, Martin Gardiner
   References in Lighting Math Fires
   NZAMT http:www.nzmaths.co.nz
   RIME
   http:www.mathsotago.ac.nz
   SNP Resources and www.nzmaths.co.nz
Facilitating Learning in PS
• everyday is a good time.
• every student is a good expectation
• a place in your classroom
• allowing students to explain problems
• is there another way?
• if you can not model it, don’t teach it!
• ask big questions…solving climate change
Ok Go Home…Bubbles Burst!
   Collect a source and develop a series of problems
that range across strategy and strand. Encourage
creative thought.
   Ask “Is there another way?”
   Solve problems in more than one way.
   Get students to make up problems for others.
   Reward creative thought
and originality.
Le Fini

 Thank you and
 Send cool resources to
jimhogan@clear.net.nz
 Visit my website at