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Problem Solving

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

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Problem Solving

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

  2. Problem Solving • Top down These strategies apply not only to mathematical problems but also problems in everyday life! e.g.??? 2. Bottom up Strategies and Presentation Techniques 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.

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

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

  5. 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 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. A combination of “top down” and “bottom up” strategies

  6. 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

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

  8. 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 downstrategy. Otherwise, rephrase it until it is equivalent to a simpler one you can readily prove.

  9. 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 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 Analogy: cross the river conclusion

  10. 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 is easier to understand (hence easier to • recall for future use) if we identify the crux moves • andhow they are proved. • Remember, frequent practice is essential. We can reproduce these • solutions in future only when the solution becomes familiar to us.

  11. To be a good problem solver • know the RIGHT moves • efficiency depends on your exposure and experiences accumulated from frequent practice • Acquire • Retain • Transfer • know the WRONGmoves • 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!

  12. 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 – A combination Back

  13. 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. 熟能生巧 • Transfer what you learnt to solve new problems. • Efficient recollection depend on how well you understand and organisethem. Look forkey stepsandpatterns. Analyze good solutions until they become familiar and naturalas if they were your own ideas! 得心應手

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

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