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Problem Solving


									Problem Solving

 Jim Hogan, Sec Maths Advisor
 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

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

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

Projects are like a literary search and
  summarize what is known.
And more and more and more…
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 …
The key competencies of
  communication (and thinking) are
  the reasons why we learn

When we solve a problem, tell
 someone! Write, speak, to one, to
 many, draw, publish, display!
Brain Gym Time

L-R wakeup call


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

- Revise research underpinning
  numeracy. AC-EA-AA-AM-AP-U
- U is the understanding and
  extension to the general case with
Thinking Maths
 - 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
The Oncer
   A problem can only ever
    be solved once. If the
    technique is used on a
    similar problem it is then a

   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
   Student/teacher readiness
   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!

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?

                         Your turn…
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.

                    Your turn…
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?

                          Your turn…
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?

                           Your turn…
Problem #5• Pentominoes

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

How do you know you have all of

                          Your turn…
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?

                                 Your turn…
Problem #7• Making 50c

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

                              Your turn…
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?

                               Your turn…
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?

                             Your turn…
Problem #10 • Powers of 2 and 3
Using the powers of 2 we can make up all other
  numbers by only adding.
Eg 13 = 1 + 4 + 8

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

Why does this work?
                                      Your turn…
Problem #11• Tennis anyone?

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

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?

Your turn…


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

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?

Explain your answer.

Your turn…
                    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
Play Super-NIM.
Problem #18• Even consecutives

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

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?

Your turn…
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

Your turn…
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?

Your turn…
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?

Your turn…
Practical Resources
   Lighting Mathematical Fires
   Cut the Knot
   NLVM
   AMC

    (Google these to find the source)
Reading Resources

   How to Solve It, G Polya, 1945
   Australian Maths Problem Book
   Ahha and Gotcha, Martin Gardiner
   References in Lighting Math Fires
   RIME
   SNP Resources and
Facilitating Learning in PS
I advise
• everyday is a good time.
• every student is a good expectation
• a place in your classroom
• allowing students to explain problems
• ask the answer
• 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
 Visit my website at

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