Problem Solving How to start? from which we can logically deduce a useful result. Most mathematical problems begin with a set of given conditions, What to prove? How to go on?

Problem Solving Strategies and Presentation Techniques 1.Top down 2.Bottom up 3. A combination ot the above These strategies apply not only to mathematical problems but also problems in everyday life! e.g.??? 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 Step 1 Restate/rephrase the problem to a simpler one (bottom up) Step 2 The simpler problem can be proved more readily (top down) given conditions conclusion Equivalent but a simpler problem Example

Presentation Techniques - Top down Strategy EgShow that n(n+1)(n+2) is divisible by 6 for any natural number n. given conditions conclusion Back SolGiven 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

EgShow that Presentation Techniques – A combination for any positive number x. Sol To show Restate/rewrite the problem to an equivalent but simpler one by (*), we have proved that Since given conditions conclusion Equivalent but a simpler problem Example

Problem Solving Ex2For any natural number n, show that 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. Ex1Show that

Once the key obstacles are overcome, the rest of the solution can be completed easily. given conditions conclusion Equivalent to statement 1 Equivalent to statement 2 crux move In this case, the crux move is “given statement 1, prove statement 2” easy crux move crux move Problem solving usually involves some crux moves Analogy: cross the river

The solution is so long! I can never reproduce it in future! I would never have been able to come up with that “trick solution”! 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. crux move crux move 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 We are all allowed to make mistakes, but right the wrongs and never make the same mistake again! Develop good common sense! so as to AVOID common mistakes!!! know the WRONG moves Acquire Retain Transfer

Presentation techniques – A combination EgShow that given conditions conclusion Equivalent but a simpler problem Now (9!) 10 =(9!) 9 (9!) 1 =(9!) 9 1x2…x9 and (10!) 9 =(9!x10) 9 =(9!) 9 x10 9 =(9!) 9 10x10…x10 Sol (*) Hence (9!) 10 <(10!) 9 By (*), Restate/rewrite the result to an equivalent but simpler one that can be readily proved Back

The ART of learning Acquire new skills and knowledge from class work, books and exercises. Retain them through frequent practice and regular revision. Experiences in problem solving and better understanding are most essential. 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 !

M&A over time Acquire … exposure Merge Consolidate what you just learnt … practice Associate the new with the old … patterns, similarities Eureka! A good mix … becoming familiar and natural! Expand your “COMMON SENSE”! You will gradually go faster, further, higher!!!