Problem Solving (Lecture)
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The document discusses problem solving and provides examples of different types of problems. It outlines the 5 steps to problem solving as: 1) defining the problem, 2) making a model, 3) analyzing the model, 4) finding a solution, and 5) checking the solution. Examples are provided to demonstrate applying these steps to solve word and logic problems.
Problem Solving (Lecture)
- 2. Problem Solving … Problems have always existed We don’t generally need a computer to solve them But, the computer often makes problem solving easier and faster This is because the computer is a useful tool for solving problems
- 3. Problem Solving (cont’d) There are many ways to classify problems For example: Easy vs. Difficult Mathematical vs. non- mathematical Another way to classify problems is by their answers or solutions…
- 5. 1. Problems with well- defined solutions The problem – “How do you bake a cake?” The solution (of course) – “Follow a recipe”
- 6. 2. Problems with many solutions The problem – “What should I do tonight?” Some possible solutions – Do my homework Go to the library Go fishing Watch TV Visit Mr. Wachs at home?”
- 7. 3. Problems with an optimal (or best) solution The problem – “How much tin should be used for a can to hold apple juice?” There are many solutions – The best solution – “Use a can that uses the smallest amount of tin while holding the most juice
- 8. 4. Problems with no solution The problem – “How many ancestors do you have?” The solution – ?????? A “No” solution will mean a solution that today we cannot verify as true or false
- 9. What happens when we solve problems? 1. We learn new ideas about what the problem asks 2. We practice things we know 3. We transfer ideas from one area to another 4. Our curiosity is stimulated
- 10. Watch out for Problems When following an outline, watch out for things like Einstrillings (a mind-set or mind-fix) Read this.. PARIS IN THE THE SPRING
- 11. Say out loud, the COLOR of each word Yellow Red Green Black Black Green Red Yellow Red Yellow Black Green Green Black Yellow Red Yellow Red Green Black Another example…
- 12. Can you read this? Your mind might be a pitfall in solving a problem Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae. The rset can be a total mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe....... amzanig huh?
- 13. The 5 Steps to Problem Solving
- 14. The 5 Steps to Problem Solving There are 5 steps you can use to solve any problem 1.What is the problem? 2.Make a model of the problem 3.Analyse the model 4.Find the solution 5.Check the solution
- 15. Let’s Try it… (1.) The problem: A salmon swims 3 km upstream and the current brings her back 2 km each day How long does it take her to swim 100 km?
- 16. Solution… 1. What is the problem? Does the salmon swim the 3 km during the day and then at night drift back 2km when resting? Or, does she swim continuously all day, slowed down by the current, so that she swims only 1 km during the day? Does “day” mean daylight or 24 hours?
- 17. We’ll define the problem as … How many 24 h. days does it take for a salmon to swim 100 km if in the 24 h. period she swims upstream and the current brings her back 2 km. while she is resting? What is the problem (cont’d)
- 18. 2. Make a model of the problem For this problem, the model will be a “picture” of what happens Start 3 km Upstream 24 h. 2 km back Etc. This gap is here so that we can easily see her path. It is not counted
- 19. 3. Analyse the model Make a table to keep track of her path: Day Number traveled forward Maximum number reached (end of day) Number pushed back Total distance travelled by next day 1 2 3 • • • 97 98 3 km 3 km 3 km • • • 3 km 3 km 3 km 4 km 5 km • • • 99 km 100 km 2 km 2 km 2 km • • • 2 km 2 km 1 km 2 km 3 km • • • 97 km 98 km
- 20. 4. Find the solution While you are making up the table, look at the model In this case the “picture” to check the numbers When the table headings are written down, fill in the table to get the solution We can see by the table that by the end of day 98 she will reach the 100 km point The salmon will take 98 days to swim 100 km
- 21. 5. Check the solution Check the table for errors Sometimes, ask others to check your solutions Even if errors are not found, however, this does not prove the solution is correct!
- 22. Let’s Try another one … (2.) The problem: One day, a cabdriver picked up three young couples and took them to a dance One girl was dressed in red, one in green, and one in blue The boys all wore the same three different colours (red, green, blue) When the three couples were dancing, the boy in red danced with his female partner over to the girl in green and said…
- 23. The problem (cont’d) “Isn’t it weird, Mary? Not one of us is dancing with a partner dressed in the same colour” Given this information, can you deduce the colour of the partner the girl in red is wearing?
- 24. Solution… (5 steps) 1. What is the problem? To use the information given to find out what colour the partner of the girl in red is wearing
- 25. 2. Make a model of the problem For this problem, the model will incorporate another type of model called a logic chart Drawn as follows… Red Girl Green Girl Blue Girl Red Boy Green Boy Blue Boy
- 26. 3. Analyse the model Make sure you have all the boys and all the girls? And all the colours? Red Girl Green Girl Blue Girl Red Boy Green Boy Blue Boy
- 27. 4. Find the solution We know the Red boy is not dancing with the Green girl (because he talked to her) Red Girl Green Girl Blue Girl Red Boy Green Boy Blue Boy X The Red boy says “Not one of us is dancing with a partner of the same colour” (this applies to all colours) X X X
- 28. Find the solution (cont’d) The Red boy must be dancing with the Blue girl Red Girl Green Girl Blue Girl Red Boy Green Boy Blue Boy X X X X Yes
- 29. Find the solution (cont’d) Because the Blue girl is already dancing with someone, she cannot be dancing with the Green boy Red Girl Green Girl Blue Girl Red Boy Green Boy Blue Boy X X X X Yes X The Green girl must be dancing with the Blue boy Yes
- 30. Find the solution (cont’d) Because the Blue boy is already dancing with someone, he cannot be dancing with the Red girl Red Girl Green Girl Blue Girl Red Boy Green Boy Blue Boy X X X X Yes X The Green boy must be dancing with the Red girl Yes X Yes The Solution
- 31. 5. Check the solution Check for errors Ask others to check your solutions Even if errors are not found, however, this does not prove the solution is correct! For Example: We are assuming that the boys are always dancing with the girls!
- 32. Let’s Try one more … (3.) The problem: How many different games are possible in a best of three volleyball playoff between the “Huskies” and the “Bunnies”
- 33. Solution… (5 steps) 1. What is the problem? How many different games are possible if a team must win two games in a best-of-three playoff This means that a team must win two games to win the playoffs
- 34. 2. Make a model of the problem For this problem, the model will incorporate another type of model called a tree diagram Drawn as follows…
- 35. Make a model of the problem (cont’d) H B H B H B B H H H B B H B Huskies win Bunnies win Game 3 Game 2 Game 1
- 36. 3. Analyse the model Does the model give a clear picture of what happens as each game is played?
- 37. 4. Find the solution Use the tree diagram to count up the number of different possibilities The answer is 6
- 38. 5. Check the solution Check over the solution for errors Write down the possibilities… 1. Huskies win, Huskies win 2. Huskies win, Bunnies win, Huskies win 3. Huskies win, Bunnies win, Bunnies win 4. Bunnies win, Huskies win, Huskies win 5. Bunnies win, Huskies win, Bunnies win 6. Bunnies win, Bunnies win
- 39. Using Analogy as Your Model Sue is a good basketball player Sue is 6 ‘ 2 ‘’ tall and slim Will Sue be a good gymnast? Ingrid is 6 ‘ 2 ‘’ tall, slim, a good basketball player, and a good gymnast Using analogy we observe similarities and conclude that Sue will be a good gymnast However, analogy does not guarantee our conclusions are correct
- 40. Using Analogy as Your Model (cont’d) For instance… What if I told you that even though Sue is a good basketball player She is a wheelchair athlete So does this change her ability in gymnastics? You decide if your analogy still stands… You must make a judgment on the validity of your analogy
- 41. Let’s Recap the 5 steps to problem solving
- 42. 1. What Is the Problem What am I trying to find? What do I know? What information is given? State in your own words What information do I need? Take out vague words Interpret the wording Restate the problem
- 43. 2. Make a Model of the Problem Get what you need (e.g. Paper) Draw (e.g. logic chart, picture, tree diagram) Build it Is there a pattern? Use analogy (where two things are alike) Look for something familiar Use top-down design (next time)
- 44. 3. Analyze The Model Look at the model to see if it does what you want Could try trial and error If it doesn’t work … make it again For number problems, use a chart or table If you have time, “sleep on it” If it seems to difficult to solve as is, change it to a slightly different problem; now solve the new problem Then see if the new solution can help solve the original problem
- 45. 4. Find The Solution Fill out the chart Do what the problem states Use the model
- 46. 5. Check The Solution Is it reasonable? Do the problem again Use a team approach
- 47. Conclusion: There is definitely an advantage to using a structured approach to problem solving This applies to the various types of problems encountered You can always incorporate the use of the 5 steps to problem solving