The ART of learning

1. Acquire • Retain • Transfer The ART of learning 集思廣益 • Acquirenew 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. 熟能生巧 • 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 naturalas if they are of your own! 得心應手

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

3. Problem Solving • Top down These skills solve not only mathematical problems but also problems in everyday life! Strategies and Presentation Techniques by focusing on the common properties among various methods, patterns emerge and you can classify them into categories 2. Bottom up 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. 3. Mid way The above names are not official and many other strategies are not mentioned here.

4. Top down Strategyworking forward from the given conditions ;a natural way to solve straightforward problems given conditions Strategies and Presentation Techniques conclusion Example

5. 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 Strategies and Presentation Techniques conclusion *no clue at all /too many ways to begin Example

6. Step 2 The simpler problem can be proved more readily (top down) given conditions Equivalent but a simpler problem Step 1 Restate/rephrase the problem to a simpler one (bottom up) conclusion Example 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 3. Mid way strategy

7. given conditions conclusion Eg Show that n(n+1)(n+2) is divisible by 6for 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 Presentation Techniques - Top down Strategy Back

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

9. Sol (*) given conditions Now (9!)10=(9!)9(9!)1 =(9!)91x2…x9 and (10!)9=(9!x10)9=(9!)9x109 =(9!)910x10…x10 Equivalent but a simpler problem Hence (9!)10<(10!)9 By (*), conclusion Eg Show that Restate/rewrite the result to an equivalent but simpler one that can be readily proved Presentation techniques - Mid way strategy

10. Ex1 Show that Problem Solving Ex2 For 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.

11. crux move crux move 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.Rememberthese crux moves andhow 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.

12. crux move crux move Problem solving usually involves some crux moves given conditions easy Equivalent to statement 1 crux move 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. Equivalent to statement 2 easy conclusion

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