EDMATH-111D presentation for problem solving
- 2. Work alone. Change the original problem below in different ways such that the word problem will have varying levels of difficulty. Andres received P67 from his Auntie Myrla and P28 from his Auntie Rhoda. How much money did he receive?
- 3. Stoyanova identified three categories of problem posing experiences that can increase students' awareness of different situations to generate and solve mathematical problems. FREE SITUATIONS. These refer to situations where students pose problems without any restriction. Some examples of the free problem posing situations are the tasks where students are encouraged to write problems for friends to solve or write problems for mathematical Olympiads. SEMI-STRUCTURED SITUATIONS. These refer to situations where students are asked to write problems, which are similar to given problems or to write problems based on specific pictures and diagrams. STRUCTURED SITUATIONS. These refer to situations where students pose problems by reformulating already solved problems or by varying the conditions or questions of given problems.
- 4. Silver, on the other hand, classified problem posing according to where problem posing takes place: PRE-SOLUTION. This is a situation prior to problem solving when problems are being generated from particular presented stimulus such as a story, a picture, a diagram, a representation, etc. WITHIN-SOLUTION. This is a situation during problem solving when students intentionally change the goals and conditions of problems. POST-SOLUTION. This is a situation after solving a problem when experiences from the problem solving context are applied to new situations.
- 5. The hardest thing about solving word problems is translating English phrases or sentences to mathematical symbols. Usually, once you transformed it into math equation, you are fine. The actual math involved is often fairly simple. But figuring out the actual equation is sometimes difficult. Word problem may come in a simple or complex form. This difficulty could be changed depending on the different levels or abilities of the learners.
- 6. The following are some methods by which a variety of word problems can be constructed at different levels for different abilities of students. Example: Vicky had P437. She spent P176 on a diskette. How much money does she have left? Way of increasing difficulty of word problems Sample problem Increase size of number Vicky had P9500.00 She spent P6799.00 on a tv set. How much money does she have left? Use decimal and fraction numbers Vicky had P437.60. She spent P176.85 on a diskette. How much money does she have left? Change context of problem to unfamiliar environments Vicky was engaged in stock exchange. At SMC, she had a profit of P437.00 while at Benpress, she lost P176.00. What was her total profit?
- 7. Way of increasing difficulty of word problems Sample problem Change language and meaning The VHS tape costs P437.00 This was P176.00 more than the diskette. How much did the diskette cost? Change conditions Vicky had P437.00. She bought a diskette. She had P176.00 left. How much did the diskette cost? Give numbers in different order to how they are implemented. Vicky spent P176.00 on a diskette. Before the purchase, she had P437.00. How much money does she have now? Increase the number of steps Vicky had P437.00. She spent P176.00 on a diskette and P27.00 on a ribbon. How much money does she have left? Involve more than one operation Vicky had P437.00. She bought 6 pieces of ribbons each costing P27.00. How much money does she have left? Give too much information Vicky had P437.00. The diskette was P176.00 and the cassette tape was P144.00. She bought the
- 8. Way of increasing difficulty of word problems Sample problem Do not give enough information Vicky had P437.00. She bought a diskette. How much money does she have left? Change some combination of the above Vicky was engaged in stock market. She lost P17.00 on each of 5 Benpress shares. She lost P86.00 on an SMC share. She made a profit on a Shell's share. Overall, she made a profit of P151.00 on the 7 shares. What was the profit she made on the Shell's share? Absurd problem Vicky had P437.00. The laser diskette costs P678.00. How much money does she have left?
- 9. In order to correctly solve a problem, it is important to follow a series of steps. This is referred to as a problem- solving cycle. While this cycle is portrayed sequentially, people rarely follow a rigid series of steps to find a solution. People often skip steps or even go back through steps multiple times until the desired solution is reached. STERNBERG identified the following steps in the problem solving cycle:
- 10. PROBLEM IDENTIFICATION. Assessing whether there is a problem that needs solving. Identifying the problem is not always as simple as it sounds. In some cases, people might mistakenly identify the wrong source of a problem, which will make attempts to solve it inefficient or even useless. 1 DEFINING OF PROBLEM. It is important to fully define the details of the problem and represent the problem well enough to understand how to solve it. 2
- 11. CONSTRUCTING A STRATEGY FOR PROBLEM SOLVING. Develop a strategy to solve the problem. The approach used will vary depending upon the situation and the unique preferences of the individual. Planning a strategy for solving a problem involves: 3 ANALYSIS - breaking down the whole of a complex problem into manageable elements SYNTHESIS - putting together various elements to arrange them into something useful DIVERGENT THINKING - trying to generate a diverse assortment of possible alternative solutions to a problem CONVERGENT THINKING - narrowing down the multiple possibilities to converge on a single, best answer
- 12. ORGANIZING INFORMATION ABOUT THE PROBLEM. Before coming up with a solution, organize first the available information. What do we know about the problem? What do we not know? The more information available, the better in coming up with an accurate solution. 4 ALLOCATION OF RESOURCES. It is important to decide which parts of the problem require the greatest allocation of resources. 5 MONITORING PROGRESS. Checking problems towards problem solution should be considered. Effective problem-solvers tend to monitor their progress as they work towards a solution. If they are not making good progress toward reaching their goal, they will reevaluate their approach or look for new strategies. 6 EVALUATING THE RESULTS. After a solution has been reached, it is important to evaluate the results to determine if it is the best possible solution to the problem. This evaluation might be immediate, such as checking the results of a math problem to ensure the answer is correct. 7
- 13. As part of his work on problem solving, GEORGE POLYA developed a four-step problem solving process. Step 1. UNDERSTAND THE PROBLEM Find, specify and clearly define the unknowns, data and conditions. Find out if it is possible to satisfy the condition Is the condition sufficient to determine the unknown? Or is it sufficient? Or redundant? Separate the various parts of the condition. Can you write them down?
- 14. As part of his work on problem solving, GEORGE POLYA developed a four-step problem solving process. Step 2. DEVISE A PLAN Find the connection between the data and the unknown. Decide if you have seen it before in slightly different form. Or have you seen the same problem in different situations/conditions? Do you know a related problem? Do you know a theorem that could be useful? Try to use the information, solution ideas, results and methods that were used on the related/similar problem you found in the previous steps. Try to restate the problem if that doesn't work. Could you restate it still differently? Go back to definitions.
- 15. As part of his work on problem solving, GEORGE POLYA developed a four-step problem solving process. Step 3. CARRY OUT THE PLAN Did you use all the data? Did you use the whole condition? Have you taken account all essential notions involved in the problem? Step 4. LOOK BACK Check each step. Can you see clearly that the step is correct? Can you prove that it is correct? Examine the solution obtained. Does it answer the problem?
- 16. The four steps outlined above can be summarized into four verbs: SE E PLA N DO CHECK 1 2 3 4 Example: Christine purchased P1,500 books using her P30/h earnings as an encoder. While she was saving, her uncle gave her P450. How many hours did Christine work to earn the total?
- 17. SOLUTION
- 19. Work alone. Answer the problem below using the four steps of George Polya. Carmela opened her piggy bank and she found she had P135. If she had only 10 centavo, 25 centavo, P1, P5 and P10, how many coins of each kind did she have?
- 20. Guess and check gives students opportunities to engage in some trial-and-error approaches to problem-solving. This is not a singular approach to problem-solving but rather an attempt to gather some preliminary data. This is also called guesstimate, trial-error or grope-and- hope.
- 21. Guess and check gives students opportunities to engage in some trial-and-error approaches to problem-solving. This is not a singular approach to problem-solving but rather an attempt to gather some preliminary data. This is also called guesstimate, trial-error or grope-and- hope.
- 22. Guess and check may be appropriate when: There are limited numbers of possible answers for testing. You want to gain a better understanding of the problem. You can systematically try possible answers. There is no other obvious strategy to try.
- 23. Carmela opened her piggy bank and she found she had P135. If she had only 10 centavo, 25 centavo, P1, P5 and P10, how many coins of each kind did she have? Let us solve the problem in the Lesson Starter using the 4-step Polya process. Understand We are not specifically given the number of coins for each denomination. As long as the sum is P135, we can have the option. Plan The strategy is to guess and check whether our guess is correct. Do We can assign our guess for the number of coins. 10 centavo - 50 coins 25 centavo - 40 coins P1-30 coins P 5 20 coins P 10 4 coins Multiplying each denomination with the number of coins and adding them, we found out we have a total of P185. We have missed the target of P135. So, we assign again guesses for the number of coins for each denomination until we arrive at the correct total.
- 24. STRATEGY: GUESS AND CHECK Carmela opened her piggy bank and she found she had P135. If she had only 10 centavo, 25 centavo, P1, P5 and P10, how many coins of each kind did she have? Let us solve the problem in the Lesson Starter using the 4-step Polya process. Check As long as the sum of the coins is 135.00 and all coins are represented, then the guess is considered as correct.
- 25. STRATEGY: GUESS AND CHECK The sum of the ages of a father and his son is 100. The father is 28 years older than the son. How old are they? Let us try another example. Understand The sum of the ages is 100. The father is 28 years older than the son. Plan The strategy is to guess and check if our guess is correct. Do We can assign our guess for the numbers Guess 1: Try 60 and 40 60-40-20 Since we want a difference of 28, the numbers should be further apart. Guess 2: Try 65 and 35. 65-3530 The difference is too big, so they should be a little closer. Guess 3: Try 64 and 36 64-3628 Therefore, the father is 64 years old and the son is 36 years old. Check 64 + 36 = 100 64 – 36 = 28
- 27. Work alone. Answer the problem below using the four steps of George Polya. The telephone are codes in certain country are three digit numbers. The first digit cannot be 0 or 1. The second digit can only be 0 or 1. The third digit is not 0. How many different area codes can start with digit 3?
- 28. Making a list is appropriate when: Listing the possible answers can help solve the problem. Listing the given information can help identify a pattern or similarities.
- 29. The telephone are codes in certain country are three digit numbers. The first digit cannot be 0 or 1. The second digit can only be 0 or 1. The third digit is not 0. How many different area codes can start with digit 3? Let us solve the problem in the Lesson Starter using the 4-step Polya process. Understand Given the conditions, the first digit can be 2, 3, 4, 5, 6, 7, 8 and 9; the second digit can only be 0 and 1, while the last digit is 1, 2, 3, 4, 5, 6, 7, 8, and 9. Plan The strategy is to list the possible three-digit area codes. Do Assuming that the first digit is 3, list all possible area codes with a second digit of 0. 301 302 303 304 305 306 307 308 309 List all possible codes with 1 as a second digit. 311 312 313 314 315 316 317 318 319 There are 18 possible codes that begin with the digit 3. Check Check if all answers satisfy the conditions in the problem.
- 30. STRATEGY: A student is taking a true-or-false test. In how many ways can the three questions be answered? Let us try another example. Understand Given the conditions, the three questions can be answered by True or False. Plan The strategy is to list the possible three-digit area codes. Do T-T-T T-F-T T-T-F T-F-F F-F-F F-F-T F-T-T F-T-F So, there are 8 ways that the three questions can be answered. Check Check if all answers satisfy the conditions in the problem.
- 32. Work alone. Answer the problem below using the four steps of George Polya. What is the next number in the list? 1, 2, 4, 8, 16, 32, ____.
- 33. A problem can be solved by looking into the relationship of the elements of a list and identifying the pattern. Looking for patterns is an important problem-solving strategy because many problems are similar and fall into predictable patterns. A pattern, by definition, is a regular, systematic repetition and may be numerical, visual, or behavioral.
- 34. Look for a pattern is appropriate when: A list of data is given. A sequence of numbers is involved. You are asked to make a prediction or generalization. Information can be expressed and viewed in an organized manner, such as a table.
- 35. What is the next number in the list? 1, 2, 4, 8, 16, 32, ____. Let us solve the problem in the Lesson Starter using the 4-step Polya process. Understand Given the conditions, the numbers are increasing. We are looking for a number which is greater than 32. Plan The strategy is to identify the pattern. Do We can look for an operation which relates each number with the next. The next number is the previous number multiplied by 2. 1 x 2 = 2 x2 = 4 x 28 x 2 = 16 x 2=32 So, 32 will be multiplied by 2, yielding 64. Another way of interpreting this is to convert the numbers into another expression. 1 = 20 2 = 21 4 = 22 8 = 23 16 = 24 32 = 25 Hence, the next number should be 26 or 64. Check Check if the answers satisfy the conditions in the problem.
- 36. Mae has written a number pattern that begins with 1, 3, 6, 10, 15, ..., if she continues this pattern, what are the next four numbers in her pattern? Let us have another example. Understand Given the conditions, the numbers are increasing. We are looking for a number which is greater than 15. Plan The strategy is to identify the pattern. Do Look at the numbers in the pattern. 3=1+2 (starting number is 1, add 2 to make 3) 6 = 3 + 3 (starting number is 3, add 3 to make 6) 10 = 6 + 4 (starting number is 6, add 4 to make 10) 15 = 10 + 5 (starting number is 10, add 5 to make 15) New numbers will be: 15+6=21 Check 21+7=28 28+8=36 36+9=45 Check Check if the answers satisfy the conditions in the problem.