Problem Solving

April 5, 2007 [MATH_PROBLEM SOLVING] Question 1

Problem Solving

Problem solving has long been recognized as one of the hallmarks of mathematics. The greatest goal of learning mathematics is to have people become good problem solvers. We do not mean doing exercises that are routine practice for skill building.

Definition of problem solving

What does problem solving mean? Problem solving is a process. It is the means by which an individual uses previously acquired knowledge, skills, and understanding to satisfy the demands of an unfamiliar situation. The student must synthesize what he or she has learned and apply it to the new situation.

Compare and contrast characteristics of routine and non-routine problems 1

April 5, 2007 [MATH_PROBLEM SOLVING]

There are two kinds of problems in our life. There are routine problem and nonroutine problem. There is comparing of routine and non-routine problems:

Routine problems Real life problem Involve one mathematical operation

Non-routine problems Complex problem Require more than one mathematical

Basic skills and sequence steps

operation Using critical and creative thinking

Need

understanding,

skills retrieve Need

understanding,

retrieve

information, choose the operation and information, choose the operation and algorithms algorithms Solving through story telling and relate Various strategies and methods to it to real situation Example problem:

solve it Example problem:

Each puppet act takes 15 minutes.

Adult : RM8.29 each

How long do 4 acts take?

Youth : RM5.49 each

The Vaughn family takes a train trip. They buy 3 adult tickets and 3 youth tickets. How much does the Vaughn family spend for tickets?

Solution: 2

April 5, 2007 [MATH_PROBLEM SOLVING] 4 acts take 1 hour Solution: Adult Ticket costs RM8 Youth Ticket costs RM5

Try the simpler problem: Adult Ticket : 3 x RM8 = RM24 Youth Ticket : 3 x RM5 = RM15 Total ticket costs:RM24+RM15 = RM39

Now, solve the problem the same way: 3 x RM8.29 = RM24.87 3 x RM5.49 = RM16.47 Total ticket cost: RM41.34

The Vaughn family spends RM41.34 on train tickets.

Strategies / heuristics for problem solving

There are many ways to solve word problems. Several word problems will be given and students can choose the method they feel most comfortable with. Some methods are:

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April 5, 2007 [MATH_PROBLEM SOLVING]

(i)

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Choose the operation

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Guess, check and improve

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Look for a pattern

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Draw a diagram

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Draw a table

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Logical reasoning

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Make an analogy

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Act a situation

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Working backwards

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Make a graph

Choose the operation

Sample problem: The second grades pick 32 pumpkins. The first graders pick 17 pumpkins. How many more pumpkins do the second graders pick than the first graders?

Solution:

The question asks how many more, so we can subtract. 32 – 17 = 15 The second graders pick 15 more pumpkins than the graders.

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April 5, 2007 [MATH_PROBLEM SOLVING] (ii)

Guess, check and improve

In the strategy guess and check, we first guess at a solution using as reasonable a guess as possible. Then we check to see whether the guess is correct. If not, the step is to learn as much as possible about the solution based of the guess before making the next guess. This strategy can be regarded as a form of trial and error, where the information about the error helps us choose what trial to make next.

Sample problem:

A worker at Putra Minigolf has a basket of 30 balls for a minigolf party. There are more them 5 people in the party. The golf balls are passed equally to people in the party. After the golf balls are passed around, 6 are left.

How many people are in the party? How many golf balls does each person get? Solution:

One way to solve the problem is to make a guess, check it, and revise you guess until you find the correct answer.

Guess: 6 people in the party Check: 30/6 = 5

There are no golf balls left. 5

April 5, 2007 [MATH_PROBLEM SOLVING] Try are different number.

Guess: 7 people in the party Check: 30/7 = 4 R2

There are 2 golf balls are left. Try a different number.

Guess: 8 people in the party Check: 30/8 = 3 R3 6 golf balls are left

One possible answer is that 8 people in the party. Each person gets 3 golf balls and 6 balls are left.

(iii)

Look for a pattern

When using the look for a pattern strategy, on usually lists several specific instances of a problem and then look to see if a pattern emerges that suggest a solution to entire problem.

Sample problem:

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April 5, 2007 [MATH_PROBLEM SOLVING] Sara arranged pine cones into different groups. She started with one pine cone in the first group. In the next group she put 2 pine cones. She put 4 pine cones in the third group, 8 pine cones in the fourth, and so on. If the number in each row continues to increase in the same way, how many pine cones does she put in the eighth group?

The number of pine cones doubles each time. x2 1

x2 2

x2 4

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Continue the pattern to find how many pine cones will be in the eight group.

x2 1

x2 2

x2 4

x2 8

x2 64

128

7

x2 16

x2 32

April 5, 2007 [MATH_PROBLEM SOLVING] Sara puts 128 pine cones in the eight group.

(iv)

Draw a diagram

In geometry, drawing a picture often provides the insight necessary to solve a problem. The following problem is a non-geometric problem that can be solved by drawing a diagram or could be done using a piece of paper.

Sample problem:

Ahmad and his father are talking a tour of his state. He lives in Taman Jaya. The list show the distances to Taman Aru, Taman Damak, and Uda City. He wants to travels to the cities in order from east to west. Going from east to west, put the cities in order. Taman Aru is 4.5 km east of Taman Jaya Taman Damak is 3.5 km east of Taman Aru Uda City is 3.1 km east of Taman Aru

We want to know the distance between the cities and the order of the cities.

Solution:

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We use the problem solving is to draw a diagram.

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Use the centimeter ruler to draw a diagram. 8

April 5, 2007 [MATH_PROBLEM SOLVING] •

Draw a diagram of the distance from Taman Jaya to Taman Aru using 1 cm for each km.

•

Then, draw the distances of Uda city and Taman Damak from Taman Aru. You must know that 3.5 km is equivalent to 3.5 cm.

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The cities in order from west to east Taman Jaya, Taman Aru, Uda City and Taman Damak.

•

The cities from east to west are Taman Damak, Uda City, Taman Aru, and Taman Jaya.

4.5 cm Taman Jaya

(v)

3.1 cm Taman Aru

3.5 cm Uda City Taman Damak

Draw a table

A table can be used to summarize data or to help us see a pattern. It also helps us to consider all possible cases in a given problem.

Sample problem:

Mr. Singh’s class voted on the type of fish they would to buy to go into the new aquarium. They will buy the fish chosen by the greatest number of student. Which type of fish is most popular?

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April 5, 2007 [MATH_PROBLEM SOLVING]

Solution:

Find which type of fish received the most votes. Organize the votes in a table.

Type of fish Angelfish Clownfish Puffer

Tally

Mr. Singh’s class will choose the clownfish for the aquarium.

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Number 8 11 5

April 5, 2007 [MATH_PROBLEM SOLVING] (vi)

Simplify the problem

Sample problem:

Fatimah is making a deli platter for a party. She wants to have the same amount of meat and cheese on the platter. Look at her list. Does she have the same amount of meat and cheese?

Solution:

Add to find how much meat. 1 + 2 + 1 +1 = 5 pounds

Add to find how much cheese. 11

April 5, 2007 [MATH_PROBLEM SOLVING] 1 + 1 + 1 = 3 pounds

Compare the amount. 5 > 3, so there is more meat.

Now solve the problem the same way, using the fractions. Add to find how much meat. 1/4 + 1/2 + 1/4 + 1/4 = 1 ¼ pounds

Add to find how much cheese. 1/3 + 1/3 + 1/3 = 1 pounds

Compare the amounts. 1 ¼ > 1 so there is more meat than cheese.

(vii)

Logical reasoning

A carnival worker needs to fill an animal’s drinking tank with 6 gallons of water. He has a 5-gallon pail and 8-gallon pail. How can he use these pails to get exactly 6 gallons of water into the tank?

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April 5, 2007 [MATH_PROBLEM SOLVING]

Solution:

How much water in each:

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Fill 8-gallon pail.

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Pour water from 8-gallon pail to fill 5-gallon pail.

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Pour 3 gallons left in the 8-gallon pail into the tank. 13

April 5, 2007 [MATH_PROBLEM SOLVING]

Now we can empty the water from the 5-gallon pail and repeat steps 1 through 3 to get 6 gallons in the tank. . (viii) Act a situation

We may find it helpful to act out a problem. This will help us create a visual image of the problem. By taking an active role in finding a solution, we are more likely to remember the process we used and be able to use it again for solving similar problems. Sample problem:

The Marble Collectors’ Club luncheon is today! The club president wants to seat 24 members so that every table is filled. Each round table seats 5 people. Each rectangular table seats 6. Which shape tables should she use? How many tables will she need?

Solution:

Model using round tables. Four tables are filled and 4 people are left over.

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April 5, 2007 [MATH_PROBLEM SOLVING]

Model using rectangular tables. The president of the rectangular tables. She will need 4 tables. Zero people will be left over.

(viiii) Working backwards

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April 5, 2007 [MATH_PROBLEM SOLVING] The strategy of working backwards entails starting with the end results and reversing the steps you need to get those results, in order to figure out the answer to the problem.

When do we use this strategy? What are real life examples?

There are at least two different types of problems which can best be solved by this strategy:

(1) When the goal is singular and there are a variety of alternative routes to take. In this situation, the strategy of working backwards allows us to ascertain which of the alternative routes was optimal. An example of this is when you are trying to figure out the best route to take to get from your house to a store. You would first look at what neighborhood the store is in and trace the optimal route backwards on a map to your home. (2) When end results are given or known in the problem and you're asked for the initial conditions. An example of this is when we are trying to figure out how much money we started with at the beginning of the day, if we know how much money we have at the end of the day and all of the transactions we made during the day.

Sample problem:

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April 5, 2007 [MATH_PROBLEM SOLVING] Ali and his father are going to the Athletic Club to play in a basketball tournament. The game starts at 7:30 p.m. It takes those about 35 minutes to walk to the Market Street bus stop. Which bus should they leave home?

Bus Route p.m. Schedule Bus Stop Time Market Street 3:00 Mini Mall 3:10 Athletic Club 3:32

Time 6:30 6:40 7:02

Time 7:00 7:10 7:32

Solution:

We can work backward to solve this problem. We can use the bus schedule given. Start by finding the last bus that can arrive at the Athletic Club before 7:30 p.m.

Think: A bus arrives at the Athletic Club at 7:02 p.m. It leaves Market Street at 6:30 p.m.

Then find the time Ali and his father need to leave home. Think: End time

: 6:30 p.m.

Elapsed time : 35 minutes ------ Time to walk to bus stop Start time

: 5:55 p.m.

They should take the 6:30 p.m. bus and leave home by 5:55 p.m. 17

April 5, 2007 [MATH_PROBLEM SOLVING]

(x)

Make a graph

Each year, San Francisco, California, hosts the Aquatic Beach Sandcastle Competition. How could a newspaper display this data to show the age group that had the most people in the contest?

Solution:

Make the pictograph. Then compare the number of symbols for each age group.

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April 5, 2007 [MATH_PROBLEM SOLVING]

The age group 16 and up has the most symbols. So more participants were 16 years and older than any other age group.

Question 2

Three non-routine and the solution using the Polya’s strategies and they have an alternative method to solve them. Polya spelled out the thought processes of a problem solver step by step. The bold face segment following each step is a summary for the reference process. Each part of Polya's reflections have great bearing on the daily job of information problem solving. As such, they are thought processes that need to be modeled for, and inculcated in, library users. Problem solving process has common elements across domains:

UNDERSTANDING THE PROBLEM •

First. You have to understand the problem.

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What is the unknown? What are the data? What is the condition?

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Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory? 19

April 5, 2007 [MATH_PROBLEM SOLVING] •

Draw a figure. Introduce suitable notation.

•

Separate the various parts of the condition. Can you write them down?

DEVISING A PLAN •

Second. Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a plan of the solution.

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Have you seen it before? Or have you seen the same problem in a slightly different form?

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Do you know a related problem? Do you know a theorem that could be useful?

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Look at the unknown! And try to think of a familiar problem having the same or a similar unknown.

•

Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?

•

Could you restate the problem? Could you restate it still differently? Go back to definitions.

•

If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? 20

April 5, 2007 [MATH_PROBLEM SOLVING] Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other? •

Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?

CARRYING OUT THE PLAN •

Third. Carry out your plan.

•

Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?

LOOKING BACK •

Fourth. Examine the solution obtained.

•

Can you check the result? Can you check the argument?

•

Can you derive the solution differently? Can you see it at a glance?

•

Can you use the result, or the method, for some other problem?

Sample question 1 Strategy: Draw a diagram

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April 5, 2007 [MATH_PROBLEM SOLVING] It has often been said that a picture is worth a thousand words. This is particularly true in problem solving. In geometry, drawing often provides the insight necessary to solve a problem. The following problem is a non geometric problem that can be solved by drawing a diagram or could be done using a piece of paper; means by using act it out strategy.

Problem:

Amin has a single sheet of 8 ½-by-11in. paper. He needs to measure exactly 6 in. Can he do it using the sheet paper?

Understanding the problem The problem is to use only an 8 ½-by-11 in. paper to measure something of exactly 6 in. There are the two edge lengths with which to work.

Devising a plan A natural thing to ask is what combinations of lengths can be made from the two given lengths? We could fold to halve the lengths, but this would lead to such fractional lengths as 4 ¼ in., 5 ½ in., 2 1/8 in., and so on, that appear unhelpful. Another idea is to consider other folds that could be used to combine 8 ½ in. and 11 in. to get 6 in. For example, consider how to obtain 6 from 8 ½ and 11.

11 – 8 ½ = 2 ½ 8½-2½=6 22

April 5, 2007 [MATH_PROBLEM SOLVING]

If we could fold the paper to obtain 2 ½ in. from the two given lengths and the ‘subtract’ the 2 ½ from 8 ½, then we could obtain the desired result of 6. One strategy to investigate how to obtain 2 ½ in. from the 11 in. and 8 ½ in. lengths is to draw a diagram.

Carrying out the plan By folding the paper as shown by the arrows in figure below, we can obtain a length of 6 in.

With the fold in figure, we obtain a length of 11 – 8 1/2, or 2 ½ in. With the fold figure (c), we obtain 8 ½ - 2 ½, or 6 in. Looking back Though there are other ways to solve this problem, drawing a diagram showing the folds of the paper combines notions of geometry and gives a way to fold a square from a rectangle. An entirely different way to solve the problem is possible if have more than one sheet of paper. Other than that, we also can act it out and fold the paper. 23

April 5, 2007 [MATH_PROBLEM SOLVING]

Sample question 2

Strategy: Use a variable

Problem: In a small town, three children deliver all the newspapers. Abu delivers three times as many papers as Busu, and Chuan delivers 13 more than Abu. If the 24

April 5, 2007 [MATH_PROBLEM SOLVING] three children delivered a total of 496 papers, how many papers does each deliver?

Understanding the problem The problem asks for the number of papers that each child delivers. It gives information that compares the number of papers that each child delivers as well as the total numbers of papers delivered in the town.

Devising a plan Let a, b, and c is the number of papers delivered by Abu, Busu and Chuan, respectively. We translate the given information into equations as follows:

Abu delivers three times as many papers as Busu

: a = 3b

Chuan delivers 13 more papers than Abu

: c = a + 13

Total delivery is 496

: a + b + c =496

To reduce the number of variable, substitute 3b for a in the second and third equations:

c = a + 13

becomes c = 3b +13

a + b + c = 496

becomes 3b + b + c = 496

Next, make an equation in one variable, b, by substituting 3b + 13 for c in the equation 3b + b + c = 496, solve for b, and then find a and c. 25

April 5, 2007 [MATH_PROBLEM SOLVING]

Carrying out the plan

3b + b + 3b + 13 = 496 7b + 13 = 496 7b = 483 b = 69

Thus a = 3b = (3 x 69) = 207. Also, c = a + 13 = 207 + 13 = 220. So, Abu delivers 207 papers, Busu delivers 69 papers, and Chuan delivers 220 papers.

Looking back To check the answers, follow the original information, by using make a sentence, start by using a = 207, b = 69, and c = 220.

The information in the first sentence, “Abu delivers three times as many papers as Busu” checks,

Since 207 = 3 x 69. 26

April 5, 2007 [MATH_PROBLEM SOLVING]

The second sentence, “Chuan delivers 13 more papers than Abu” is true because

220 = 207 + 13

The information on the total delivery checks, since

207 + 69 + 220 = 496.

Sample question 3

Strategy: Guess, check and improve

Problem:

A farmer had some cows and ducks. One day he counted 20 heads and 56 legs. How many cows and ducks did he have?

Step 1: Understanding the problem 27

April 5, 2007 [MATH_PROBLEM SOLVING]

How many the cows and the ducks that the farmer’s have? •

Cows have four legs

•

Ducks have two legs

Step 2: Devising a plan

We make a plan to shows various combination of 20 ducks and cows and how many legs they have altogether. We use guess and checks strategy.

Step 3: Carrying out the plan

Numbers of cows

Numbers of ducks

Numbers of legs

12 11 10 9 8

12 9 10 11 12

80 + 24 = 104 44 + 18 = 62 40 + 20 = 60 36 + 22 = 58 32 + 24 = 56

Step 4: Looking back

Checks for the answers by using a variable Lets a = duck, b = cow 28

April 5, 2007 [MATH_PROBLEM SOLVING]

2a + 4b = 56 --------------------------------(i) a + b = 20 -----------------------------------(ii)

From (ii), a = 20 – b ----------------------------------(iii)

Replace (iii) into (i), 2(20-b) + 4b = 56 40 - 2b + 4b = 56 2b = 16

b=8

Replace b = 8 into (iii), a = 20 - 8 =12

Therefore, the answer 8 cows and 12 ducks are correct.

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April 5, 2007 [MATH_PROBLEM SOLVING]

REFERENCES http://rhlschool.com Max A. Sobel, Evan M. Malestsky (1970), Teaching Mathematics A Sourcebook of Aids, Activities and Strategies, Second Education, Prentice – Hall Rick Billstein, Shlomo Libeskind (2004), A Problem Solving Approach to Mathematics, Pearson Education McGraw-Hill (2002), Mathematics Level 9, McGraw-Hill School Division McGraw-Hill (2002), Mathematics Level 5, McGraw-Hill School Division

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