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

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 Mate9.2 The language of PS.9.5 Math Thinking10.0 Problem solving10.5 A few problems?12.5 Brain food…13.0 More problems!14.0 Oodles of Resources14:7 Learning opportunities15.0 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…

ProofThe concept of proof is specific to

mathematics and must be experienced.

Eg Demo Angle Sum triangle = 180Eg Demo Sum N = n(n+1)/2Eg There is an infinite set of prime numbers

and more and more and more …

TellingThe 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 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!

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?

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 them?

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 subtraction.

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 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?

Your turn…

P

Q

Y

X

Problem #14• Dominoes on the Chessboard

If the two opposite corners are removed can the board be covered with dominoes?

Explain your answer.

Problem #15• Reversing numbers

4297 and 7924Subtract 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 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?

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 swap?

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 NZAMT 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 NZAMT http:www.nzmaths.co.nz RIME http:www.mathsotago.ac.nz SNP Resources and www.nzmaths.co.nz

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

jimhogan@clear.net.nz Visit my website at

http://schools.reap.org.nz/advisor