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?
The following discussion and
problems are suitable for all
but in particular those
curiosity and wonderment.
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.
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?
Investigations and projects
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
What do you know about how
someone learns to think
- Revise research underpinning
- U is the understanding and
extension to the general case with
- 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
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
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?
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!
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?
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
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
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
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?
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?
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.
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
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?
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
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?
Lighting Mathematical Fires
Cut the Knot
(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
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
• 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
Ask “Is there another way?”
Solve problems in more than one way.
Get students to make up problems for others.
Reward creative thought
Thank you and
Send cool resources to
Visit my website at