Transcript A power point file on problem solving strategies and

•Acquire
•Retain
The ART of learning
•Transfer
•Acquire new skills and knowledge from class work,
books and exercises
熟
能
生
巧
集
思
廣
益
•Retain them through frequent practice and regular revision.
Experiences in problem solving and better understanding are
most essential. Bit by bit, you will gain momentum and further
develop and fine tune problem solving skills on your own.
•Transfer your experiences based on past problem solving
and apply your skills.Efficient recollection of what you
have learnt depend on how well you understand and
organise them. Look for patterns .Convince yourself until
solutions to many problems are so understandable and
natural as if they are of your own!
得
心
應
手
Problem Solving
Most mathematical problems begin with
a set of given conditions,
from which we can logically deduce
a useful result.
Problem Solving
Strategies and Presentation Techniques
by focusing on the common properties among various methods,
patterns emerge and you can classify them into categories
1.
Top down
2.
Bottom up
3.
Mid way
These skills solve not only
mathematical problems but also
problems in everyday life!
It is rare that there is only one solution
to a problem. A problem usually has many
solutions and can be solved by a combination
of different strategies.
The above names are not official and 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;
better work from the conclusion if we don’t know
how to begin* from the given conditions
given conditions
conclusion
*no clue at all /too many ways to begin
Example
3. Mid way strategy
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
Step 2
The simpler problem can
be proved more readily
(top down)
Equivalent but a simpler problem
conclusion
Example
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 – Bottom up Strategy
1
Show that x  x  2
Eg
Sol
To show
 To show
 To show
 To show
for any positive number x.
1
x 2
x
Working backwards from the result
1
x 2  0
x
x 2  1  2x
 0
given conditions
x
( x  1) 2
(*)
 0
conclusion
x
( x  1) 2
Since it is obvious that
 0 when x is positive ,
x
by (*), hence x  1  2
x
Back
Note:Proof by contradiction also starts from the conclusion
Presentation techniques - Mid way strategy
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
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
the result until it is equivalent to a simpler result you can readily prove.
I would never have been
able to come up with
that “trick solution”!
The solution is so
long! I can never
reproduce it in future!
Solution to a problem can be made much easier and more
understandable (hence easier to recall for future use) if we realise
the crux moves. Remember these crux moves and how they are
proved. The rest of the solution will be easy as one step follows
another naturally.
Remember, additional practice is essential. We can reproduce these
solutions in the future only when the solution becomes familiar to us.
crux
move
crux
move
Problem solving usually involves some crux moves
crux
move
given conditions
easy
crux
move
Equivalent to statement 1
crux
move
Equivalent to statement 2
easy
conclusion
In this case, the crux moves is to
prove statement 2, given statement 1.
Once the key obstacles are overcome, the
rest of the solution can be completed easily.
Problem solving
For complicated problems, sometimes it is useful to
1(a) start with simple cases (drawing can be useful)
(b) organise data (tabulation is helpful)
(c) find a pattern and guess intelligently a general formula
(d) prove the general formula
Exercise(optional)
In a room with 10 people, everyone shakes hands with everybody
Else exactly once. How many handshakes are there?
2
look at the problem from a different point of view and
replace the problem with a simpler equivalent problem
end