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PROBLEM SOLVING
Objectives At the end of the course, you must have:
- Solved any type and level of problems correctly;
- Employed different strategies in problem solving;
- Enhanced your problem solving skills;
- Performed problem solving with willingness and perseverance;
- Improved your self-concept with respect to ability to solve problems; and
- Been aware that many problems can be solved in more than one way. Problem Solving "Problem Solving means engaging in task for which the solution method is not known in advance. In order to find a solution, students must draw on their knowledge and through this process; they will often develop new mathematical understandings. Solving Problem is not only a goal of knowing mathematics, but also a major means of doing so."
- According to the National Council of Teachers in Mathematics (NCTM, 2000, p.52) Problem solving is a process that comprises of three components, which are: The Goal, states what is to be accomplished. Obstacle presents a real problem that hinders one from easily arriving at the solution. And lastly, Solution, presents the achievement of the goal after the strategy has been employed. It’s the time to reflect to find out if the problem solving process has been successful. Routine and Non-Routine Problems Problem solving may be categorized into routine and non-routine. It is good to identify these two basic types of problem solving as purposes and strategies used for solving these problems are different. Routine Problems Routine problem solving involves using at least one of the four arithmetic operations to solve problems that are practical in nature. It stresses the use of algorithms to solve the problem. Routine problems are typically done even by young children. Adults also often do routine problems in their daily normal activities. For students to be successful in solving routine problems, they first need to understand the meaning of an arithmetic operation. As to which operation to use to solve the problem is the skill that has to be learned next. Armed with these two skills and a good grasp of the problem situation makes solving routine problems much easier. Using steps/phases in problem solving will be of great help, too. One strength of this type of problems is that it can be assessed easily with paper-and-pencil tests typically focusing on the algorithms being used. However, by its nature, routine problems are not that good in fostering 1
Goal Obstacle Solution
higher order thinking skills and creativity since they can simply be solved with the use of algorithms. Further, routine problems do not promote divergent thinking as each problem usually leads to one correct solution. Polya’s Problem-Solving Strategy One of the foremost recent mathematicians to make a study of problem solving was George Polya (1887–1985). He was born in Hungary and moved to the United States in 1940. The basic problem-solving strategy that Polya advocated consisted of the following four steps. Polya’s Four-Step Problem-Solving Strategy
- Understand the problem.
- Devise a plan.
- Carry out the plan.
- Review the solution. EXAMPLE : Apply Polya’s Strategy A baseball team won two out of their last four games. In how many different orders could they have two wins and two losses in four games? Solution
- Understand the Problem There are many different orders. The team may have won two straight games and lost the last two (WWLL). Or maybe they lost the first two games and won the last two (LLWW). Of course there are other possibilities, such as WLWL
- Devise a Plan -make an organized list of all the possible orders.
- Carry Out the Plan 1. WWLL (Start with two wins) 2. WLWL (Start with one win) 3. WLLW 4. LWWL (Start with one loss) 5. LWLW 6. LLWW (Start with two losses)
- Review the Solution The list has no duplicates and the list considers all possibilities, so we are confident that there are six different orders in which a baseball team can win exactly two out of four games. Examples: Number- Relation Problem
- The sum of three consecutive numbers is 42. What are the numbers? Solution: Let x = the first consecutive number x + 1 = the second consecutive number 2
- Two train start from the same station and run in opposite directions. One runs at an average rate of 40 miles per hour, while the other at 65 miles per hour. In how many hours will they be 315 miles apart? Solution: A table showing the relationship of the information and condition in the problem is as follows: Formula: Distance = Rate × Time Train Time Rate/Hour Distance Covered First X 40 40x Second X 65 65x The equation is: 40x + 65x = 315 105x = 315 x = 3 Thus, the two trains will be 315 miles after 3 hours. Check: First train travelled 40x = 40(3) = 120 miles; second train, 65x = 65(3) = 195 miles. Hence, 120 miles + 195 miles = 315 miles Work Related Problems
- Arthur can weed a garden in 3 hours while his sister Jane can do the same in 5 hours. How long will it take them to finish the work if they work together? Solution: For purposes of analyzing the problem, prepare a table showing the relationships of the information and conditions cited. Information Arthur Jane No. Of hours required to finish the job alone
Part of job done in one hour 1/3 1/ No. of hours worked x x Fractional part of job x/3 x/ Hence, the equation is:
x
x
= 1 ¿] (15)
Clearing the equation of fractions, we shall have: 5x + 3x = 15 8x = 15 4
x = 1
hours Hence, the number of hours required for Arthur and Jane to do the work if they work together id 1
hours. Check: The fractional part of job done by: Arthur is 1
÷ 3 =
, while for Jane is 1
÷ 6 =
. Hence,
= 1 or
of 1
hours = 1
hours added to
of 1
hours equals 1
hours Investment Problem
- Mr.Rivera invests a part of ₱ 85,000 in a mortgage which pays 6% interest; and the rest in bonds paying 5% interest. The annual income from the bonds is twice the income from the mortgage. How much is invested in each of the 2 kinds of investment? Solution: Let x = amount invested in mortgage 85,000 - x = amount invested in bonds .06x = income from the mortgage .05(85,000 - x) = income from the bonds Hence, the equation shall be: .05(85,000 - x) = 2(0.06x) 4,250 - 0.05x = 0.12x -0.05x - 0.12x = -4,250. -.0.17x = -4,250. x = ₱ 25,000. Exercises: Solve the routine problems.
- The sum of two numbers is 25. One of the numbers exceeds the other by 9. Find the numbers.
- The sum of three consecutive multiples of 8 is 888. Find the multiples.
- The length of a rectangle is twice its width. If the perimeter is 72 meter, find the length and width of the rectangle.
- Aaron is 5 years younger than Ron. Four years later, Ron will be twice as old as Aaron. Find their present ages.
- A number is divided into two parts, such that one part is 10 more than the other. If the two parts are in the ratio 5 : 3, find the number and the two parts. 5
Therefore, since there are 6 zeroes in 1 000 000, then its factors which do not contain any zeroes are or 26
x 56 which are 64 and 15 628.
Problem # 2 Becky bought five pencils and pens at a total cost of P29. A pencil costs P4 while a pen costs P7. How many pens did Becky Buy? For guess and check to be more effective, one has to use his number sense or logical reasoning to come up with a guess that is, if not correct, would at least yield an answer which is close to the correct one In this problem, we can start by considering the minimum and maximum values. A pencil costs P4. If all 5 items were all pencils, then the total amount would be P4 x 5, which is P20. If they were all pens, the total price would be P7 x 5, that is P35. We can say, then, that P29 is closer to P35 than to P20, so there may be more pens than pencils. Let’s try 3 pens and 2 pencils. 3 pens at P7 each is P 2 pencils at P4 each is P8. The total of P21 and P8 is P29, which is the correct answer. It means Becky bought 3 pens Problem # 3 During their class game, teacher Angie had her six students stand in a circle from 1 and continue it as necessary. So the first student says "1" the second says "2" the third says "3" and so on. A student drops out if his/her number is a multiple of 4. In what number will the last student drop out? Assume the following students Stand in a circle in the following order, and counting in clockwise direction: Collin, Irene, Lorjie, Machel, Pen, and Shine Collin says, "1", Irene says, "2", Lorjie says, "3", Machel says, "4". Since 4 is a multiple of 4, then she drops out of the game. The counting continues. Pen says "5" and Shine says, "6". The counting goes back to Collin. He says, "7", Irene says "8". Again, 8 is a multiple of 4 so Irene drops out. This counting continues and those counting 12, 16, 20, and so on drop out. If we continue, it's Pen who will drop out last and that it the count of 24. Using logical reasoning and finding patterns, this answer can be expected without actually finishing the process of counting. Problem # 4 A coach for table tennis has the following players: April, May, June, Julio, and Agusto. How many different two-player doubles team can be formed from the players of the team? What do we know? 7
-There are five players. -Teams of two players each will be formed. What do we want to know? -The number of two-player doubles team that can be formed. If being solved in class, the best strategy to use is acting out as it will show the students the possible pairing of the five players. Using drawing or diagram are appropriate to use, too. To act out, 5 students may be asked to stand in front of the class to represent the five players. Then show that Player 1 may be paired with all other players, that is April & May, April & June, April & Julio, April & Agusto. Then pair May with all other players except April since their pair was already counted. So May will have May & June, May & Julio, May & Agusto. To continue we have, June & Julio, June & Agusto. And finally, Julio & Agusto. To sum up, there are 10 possible two-player doubles team. Problem # 5 Find the difference when the sum of the first 100 positive odd integers is subtracted from the sum of the first 100 positive even integers. When written as an expression, the problem would look like this (2 + 4 + 6 + ... + 198 + 200) - (1 + 3 + 5 + ... + 197 + 199). We can spend the whole day solving this problem using long method. We can, however, solve this fast by starting with a simpler version, find the pattern that emerges, develop a rule, then use the rule to solve the original problem. Starting with the difference between the first 2 positive odd and even integers, we have 8
78 Since 78 is the middle number, the median is 78. 82 93 Strategy 2: Draw a Diagram, Picture, or Model Drawing a diagram, picture or model is one of the most helpful strategies for understanding a problem and of obtaining ideas for a solution. Pupils represent problem situations with drawings to help them see the relationships between the components in a problem. The use of drawing provides a method for organizing information that could lead to the selection of another problem solving strategy. Students are encouraged to draw only the essential details that would make the problem clearer and easier to understand. Problem #
- Jeena bought 8 marbles from Roden. Roden told Jeena, "these eight marbles look alike but one is slightly heavier than the others. I will give you these three marbles for free if by using a balance scale, you can determine the heavier one in exactly 3 weighing." Jeena was so happy to take home her free marbles. How did she do it? Start by presenting the 8 marbles with small circles numbered 1 to 8. For the first weighing, group the marbles into two groups by placing marbles 1-4 at the left of the balance scale and marbles 5-8 at the right. If the left side of the scale goes down then the heavier marble is among marbles 1-4. If the right side goes down, then the heavier one is among marbles 5-8. Assuming that the left side of the balance scale goes down, then we separate marbles 1 and 2 from 3 and 4 and place them on the opposite ends of the balance scale Let’s assume, this time, that the right side of the balance scale goes down. Then the heavier marble is between 3 and 4. Finally, we place 3 and 4 on opposite sides of the balance scale. 10
a
If the left side goes down, then marble 3 is the heavier marble. On the other hand, if the right side goes down, then its marble 4 which is the heavier one. So in three weighing, we were able to identify the heavier marble among eight marbles. Example: Fortune Problem: a man died and left the following instructions for his fortune, half to his wife; 1/7 of what was left went to his son; 2/3 of what was left went to his butler; the man’s pet pig got the remaining $2000. How much money did the man leave behind altogether? Strategy 3: Guess and Check Also known as Guess, Check, Revise and Trial and Error, this strategy is a primitive way to solve problems. Although unconsciously at times, this strategy had been used for years. Children, and even adults, have a natural affinity for this strategy and should be encouraged to use it when appropriate. It is not only useful across age groups, using the strategy also helps improve one’s number sense as the process of repeated guessing and checking makes one aware of the correctness or incorrectness of his/her answers or whether the answer is close to or still far from the correct one. Moreover, good exposure to this strategy enhances one's skill in estimation Although using this strategy does not always yield a correct solution immediately but it provides information that can be used to better understand the problem. To use the guess-check strategy, one follows these steps: A. Making a logical guess at the answer. In the process of guessing, we learns more about the problem. B. Checking the guess. Does it satisfy the problem? C. Using the information obtained in checking to make another guess if necessary. The student is left to make his guess skip around so he can bracket the right answer. D. Continuing the procedure until the right answer is obtained. Problem # Mang Tomas owns goats and ducks. Counting heads there are 39. Counting legs there are 110. How many goats and how many ducks has Mang Tomas? What do we know? 11
Logically, we first think of the minimum and the maximum possible scores any competitor may garner. Minimum: if all five darts would land on 2-point area the score would be 5 x 2 = 10 Obviously, the score of 6 is not possible Maximum: if all 5 darts land in, the 10-point area the score would be 5 x 10 = 50. This time, a score of 57 is not possible. Since all possible points a dart may score are even numbers then a total score which is odd is not possible. It means that 17 is not a possible score. That leaves 14, 38, and 42 as possible scores and those can be checked by looking for at least a combination of points that will give these total. Let's take 14 first. 14 = four 2 points + one 6 points = 8+ 38 = three 10 points + one 2 points + one 6 points =30 + 2 + 6 42 = four 10 points + one 2 points = 40 + 2 We may use an organized list to determine all the combinations of points that total 14, 38, and 42. Example: At the Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog. Of these people, 12 swim and play tennis, 19 play tennis and jog, and 13 jog and swim. Nine people do all three activities. How many members are there altogether? Solution: Hint: Draw a Venn Diagram with 3 intersecting circles. Strategy 5: Act Out Some word problems are best solve when students act out the situation. Although it may be time- consuming and may need more materials, yet doing it can bring many benefits to the whole class and the individual pupils. For one, it may enliven class discussion and eliminate boredom as excitement may build up as students get to interact as they act out or dramatize some problem situations. Acting out can have the same effect as drawing a picture. What's more, acting out the problem might disclose incorrect assumptions being made. Using this strategy, pupils visualize and stimulate the actions described in the problem. Acting out makes the relationship between variables and clearer as they experience how the 13
problem goes. Pupils' creativity in presenting the series of actions as they act out may be helpful in visualizing the problem at hand. Problem # 1 There are 39 children in a classroom. The teacher assigned each one number from 1 to 39. She then told them to stand. Next she said: All those whose number is odd sit down. The children followed. Then she again assigned those who are standing the numbers 1, 2, 3, and so on up to the last child. She said; Those whose number is odd sit down. At this point, how many children remain standing? What do we want to know? -The number of children who remain standing after a series of instructions from the teacher. If there is enough students and if there is sufficient time and space, this problem can best be executed through acting out To do that, we assign 39 students a number each, from 1 to 39. Have them stand, then those whose number is odd sit. By this time, it will be obvious that 20 will sit and 19 will remain standing. Then, the remaining 19 will be assigned to numbers 1 to 19. If those numbers are odd will sit then the only ones standing will be 9 children as 10 will sit. There will be nine children who remain standing. Strategy 6: Find a Pattern A pattern is a regular, systematic repetition, and which often occurs in problems where there is a progression of data. Patterns may be numerical, verbal, spatial/visual, patterns in time or patterns in sound. When pupils use this problem solving strategy, they are encouraged to analyze patterns in data by decoding rules that create the pattern and make predictions and generalizations based on their analysis. The rules or formulas developed or discovered point to the solution. Pupils then must check the generalization against the information in the problem and possibly make a prediction from, or extension of the given information. By identifying and pattern, one can predict what will come next and will happen again and again in the same way. Looking for patterns is a very important strategy in problem solving, and is used to solve many different kinds of problems. Sometimes one can solve a problem just by recognizing a pattern, but often he or she will have to extend a pattern to find a solution. Making a table often reveals patterns, and for this reason it is frequently used in conjunction with this strategy. Problem # 1 What is the 12th term in the sequence: 0, 1, 1, 2, 3, 5, ...,? This number series is called Fibonacci sequence. The Fibonacci sequence is a series of numbers, defined by the fact that each term is the sum of previous two terms. The sequence is named after Leonardo of Pisa, also known as Fibonacci (c. 1170-1240), an Italian mathematician who was a major figure in spreading Arabic numerals to the rest of the world in his book Liber Abaci. The 7th^ to 12th^ terms are 14
Strategy 8: Simplify the problem This strategy is often used along with other problem-solving strategies. When a problem is too complex to solve in one step, it often helps to divide it into simpler problems and solve each one separately. Also, when solving difficult problems with large numbers and complicated patterns that require a series of actions, a simpler but similar problem may be worked on first, then the pattern or the solution is applied to solve the original problem. Creating a simpler problem from a more complex one may involve rewording the problem; using smaller, simpler numbers; or using a more familiar scenario to understand the problem and find the solution. Some helpful tips on using this strategy are: a. helping pupils think about the information given and what they need to find b. clarifying new or unfamiliar terms and confusing conditions or requirements; c. letting them break the problem into manageable parts and encourage them to answer one question first before they proceed to the next; and d. restating the problem in their own words for greater understanding. Problem # 1 Find the difference when the sum of the first 100 positive odd integers is subtracted from the sum of the first 100 positive even integers. When written as an expression, the problem would look like this (2 + 4 + 6 + ... + 198 + 200) - (1 + 3 + 5 + ... + 197 + 199). We can spend the whole day solving this problem using long method. We can, however, solve this fast by starting with a simpler version, find the pattern that emerges, develop a rule, then use the rule to solve the original problem. Starting with the difference between the first 2 positive odd and even integers, we have Even: 2 + 4 -Odd: 1 + 3 1 + 1 Let's jump to the first five positive odd and even integers. Even: 2 + 4 + 6 + 8 + 10 -Odd: 1 + 3 + 5 + 7 + 9 1 + 1 + 1 + 1 + 1 Do you now see the pattern? Yes, if each positive even integer is subtracted by its corresponding odd integer, the difference would always be 1. Since there are 100 pairs of odd and even integers, the difference would be 100 ones or 100. The answer, then, is 100 16
Exercises: Solve the following non-routine problem. Use any strategy which you think is most appropriate.
- Sixteen toothpicks are arranged as shown. Remove four toothpicks so that only four congruent triangles remain.
- Suppose that you buy a rare stamp for $15, sell it for $20, buy it back for $25, and finally sell it for $30. How much money did you make or lose in buying and selling this stamp?
- The figure below shows twelve toothpicks arranged to form three squares. How can you form five squares by moving only three toothpicks?
- Lori Ann Marie wants to know where you will be when you take: 3 /5 of a CHICK, 2/ 3 of a CAT, and 1 /2 of a GOAT?
- Three apples and two pears cost 78 cents. But two apples and three pears cost 82 cents. What is the total cost of one apple and one pear?
- How many posts does it take to support a straight fence 200 yards long if a post is placed every 20 yards?
- Raymond got an 85, 88, and 93 on his first three tests. What must he get on his fourth test so that his average on the four tests is 90 (the lowest ‘A’)?
- A bag of marbles can be divided in equal shares among 2, 3, 4, 5, or 6 friends. What is the least number of marbles that the bag could contain?
- Emily spent two thirds of her money. Then she lost two thirds of the money that was left. Four dollars remained. How much money did Emily have to begin with?
- Sarah went to a store, spent half of her money, and then spent $10 more. She went to a second store, spent half of her remaining money, and then spent $10 more. Then she had no money left. How much money did she have in the beginning when she went to the first store? 17
