Transcript An introduction to problem solving

Problem Solving
Most mathematical problems begin with
a set of given conditions,
from which we can logically deduce
a useful result.
How to start?
How to go on?
What to prove?
Problem Solving
Strategies and Presentation Techniques
1.
Top down
These strategies apply not only to
mathematical problems but also
problems in everyday life! e.g.???
2.
Bottom up
3.
A combination ot the above
A problem usually has many solutions and can be solved by different
strategies. Many other strategies are not mentioned here.
Strategies and Presentation Techniques
1.
Top down Strategy
working forward from the given conditions ;
a natural way to solve straightforward problems
given conditions
conclusion
Example
Strategies and Presentation Techniques
2.
Bottom up strategy
working backwards from what we need to prove;
works especially well if we don’t know how to
begin* from the given conditions
given conditions
conclusion
*no
clue at all /too many ways to begin
3. A combination of “top down” and “bottom up” strategies
Restate/Rephrase the problem until it is replaced by an
equivalent but a simpler one that can be readily proved
from the given conditions
given conditions
Equivalent but a simpler problem
Example
conclusion
Step 2
The simpler problem can
be proved more readily
(top down)
Step 1
Restate/rephrase the
problem to a simpler one
(bottom up)
Presentation Techniques - Top down Strategy
Eg Show that n(n+1)(n+2) is divisible by 6
for any natural number n.
Sol
Given that n is a natural number,
consider the 3 consecutive numbers n, n+1, n+2.
At least one of the 3 numbers is even. (Why?)
At least one of the 3 numbers is divisible by 3. (Why?)
It follows that the product n(n+1)(n+2)
is divisible by 2x3, i.e.6
given conditions
conclusion
Back
Presentation Techniques – A combination
1
Show that x  x  2
Eg
Sol
for any positive number x.
1
To show x   2
x
1
 To show x   2  0
x
Restate/rewrite the problem
to an equivalent but simpler one
Since
1
x 2  2 x  1 ( x  1) 2
x 2 

0
x
x
x
by (*), we have proved that x  1  2
x
Example
given conditions
Equivalent but a simpler problem
conclusion
Problem Solving
Ex1 Show that
1
sin x
 tan x 
, where 0  x  .
sin x
1  cos x
Ex2 For any natural number n, show that
1
2n  1
1

.
3n  1 2n  2
3n  4
Hint: Choose one strategy/a combination of strategies when solving a
problem. If it is straightforward, try top down strategy. Otherwise,
rephrase it until it is equivalent to a simpler one you can readily prove.
Problem solving usually involves some crux moves
crux
move
given conditions
easy
crux
move
Equivalent to statement 1
crux
move
In this case, the crux move is
“given statement 1, prove statement 2”
Once the key obstacles are overcome, the
rest of the solution can be completed easily.
Equivalent to statement 2
easy
conclusion
Analogy: cross the river
I would never have been
able to come up with
that “trick solution”!
crux
move
The solution is so
long! I can never
reproduce it in future!
crux
move
•Solution to a problem is easier to understand (hence easier to
recall for future use) if we identify the crux moves
and how they are proved.
•Remember, frequent practice is essential. We can reproduce these
solutions in future only when the solution becomes familiar to us.
To be a good problem solver
• know the RIGHT moves
efficiency depends on your exposure and experiences
accumulated from frequent practice
•Acquire
•Retain
• know the WRONG moves
•Transfer
so as to AVOID common mistakes!!!
We are all allowed to make mistakes, but right the wrongs and
never make the same mistake again!
Develop good common sense!
Presentation techniques – A combination
Eg
Show that 9
Sol
To show
9
9!  10 10!
9!  10 10!
 To show (9 9!)90  (10 10!)90
 To show (9!)10  (10!)9
Restate/rewrite the result
(*) to an equivalent but simpler
one that can be readily proved
Now (9!)10=(9!)9(9!)1 =(9!)91x2…x9
and (10!)9=(9!x10)9=(9!)9x109 =(9!)910x10…x10
Hence (9!)10<(10!)9
By (*),
9
9!  10 10!
given conditions
Equivalent but a simpler problem
Back
conclusion
The ART of learning
•Acquire new skills and knowledge from class work,
books and exercises.
集
思
廣
益
•Retain them through frequent practice and regular
revision. Experiences in problem solving and better
understanding are most essential.
熟
能
生
巧
•Transfer what you learnt to solve new problems.
Efficient recollection depend on how well you understand
and organise them. Look for key steps and patterns.
Analyze good solutions until they become familiar and
natural as if they were your own ideas!
得
心
應
手
M&A over time
搜 Acquire … exposure
集
整 Merge
合 Consolidate what you just learnt … practice
Associate the new with the old … patterns, similarities
融
會 Eureka!
貫 A good mix … becoming familiar and natural!
通
Expand your “COMMON SENSE”!
You will gradually go faster, further, higher!!!