2. The ART of learning Acquire Retain Transfer 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. 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 !

3. Problem Solving Most mathematical problems begin with a set of given conditions, from which we can logically deduce a useful result.

4. 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.

5. 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

6. 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

7. 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 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

8. 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

9. EgShow that given conditions conclusion Back Presentation Techniques – Bottom up Strategy Note:Proof by contradiction also starts from the conclusion for any positive number x. Sol To show (*) Working backwards from the result by (*), hence

10. Presentation techniques - Mid way strategy 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

11. 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 the result until it is equivalent to a simpler result you can readily prove. Ex1Show that

12. 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 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

13. 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 moves is to prove statement 2, given statement 1. easy crux move crux move Problem solving usually involves some crux moves

14. end 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? 2look at the problem from a different point of view and replace the problem with a simpler equivalent problem