Problem Solving in Mathematics

Content

1 Synopsis 2

1 Goals 3

2 Mathematical problem defined 4

3 What is problem solving? 7

4 Problem solving process 9

5 Teaching problem solving 11

6Problem solving at the primary level

12

7 Problem extension 14

8 Problem solving strategies 16

9Newman’s Error Analysis Procedure

51

10 References 57

      

0

SYNOPSIS

This topic enables the students to understand the definition of routine and non routine problems. Subsequently the students are guided to explore the various problem solving strategies. With regards to this the priority is given to the Polya Model of problem solving. Application of Newman Error Analysis method in identifying difficulties encountered in solving mathematical word problems is also discussed.

LEARNING OUTCOMES

To explain the meaning of problem and problem solving. To apply the Polya Model in solving mathematical problems. To make and examine conjectures. To explore and investigate alternative solutions. To evaluate and select apparopriate strategies in solving mathematical

problems. To apply the Newman Error Analysis procedure in identifying difficulties in

solving mathematical word problems.

1

GOALS

The ultimate goal of any problem-solving program is to improve students' performance at solving problems correctly. The specific goals of problem-solving in Mathematics are to:

1. Improve pupils' willingness to try problems and improve their perseverance when solving problems.

2. Improve pupils' self-concepts with respect to the abilities to solve problems.

3. Make pupils aware of the problem-solving strategies.

4. Make pupils aware of the value of approaching problems in a systematic manner.

5. Make pupils aware that many problems can be solved in more than one way.

6. Improve pupils' abilities to select appropriate solution strategies.

7. Improve pupils' abilities to implement solution strategies accurately.

8. Improve pupils' abilities to get more correct answers to problems.  

2

MATHEMATICAL PROBLEM DEFINED

A problem is a task for which the person confronting

it wants or needs to find a solution,

has no readily available procedure for finding a solution,

and must make an attempts to find a solution.

Charles & Lester (1982)

 

TYPES OF PROBLEMS

1. DRILL EXERCISEDrill exercise provide students with practice in using an algorithm and help maintain mastery of basic computational skills.

Example : 269 x 76

2. SIMPLE TRANSLATIONSimple translation problem provide students with experience in translating real world situations into mathematical models.

Example:Ahmad has 11 marbles and Cheah has 7 marbles. How many more marbles does Ahmad has than Cheah?

3. COMPLEX (OR MULTI-STEP) TRANSLATIONComplex translation problems provide students with the same experience as simple translation problems, except that more than one operation may be involved.

Example :Matches come in packs of 40. A carton holds 36 packs. If a shop owner ordered 4320 matchsticks, how many cartons did he order?

3

4. PROCESS (or NON-ROUTINE) PROBLEMSProcess problems lend themselves to exemplify the processes inherent in thinking through the solving of a problem. They serve to develop general strategies for understanding, planning and solving problems, as well as evaluating attempts at solutions.

Example :A tennis club held a tournament for its 25 members. If every member played one game against each other members, how many games were played?

5. APPLIED PROBLEMSApplied problem provide an opportunity for students to use a variety of mathematical skills, processes, concepts and facts to solve realistic problems. They make students aware of the value of usefulness of mathematics in everyday problem situations.

Example :How much paper of all kinds does your school uses in a fortnight?

6. PUZZLE PROBLEMSPuzzle problems allow students an opportunity to engage in potentially enriching recreational mathematics. They highlight the importance of flexibility in attaching a problem.

Example :A coin is in a “cup” formed by four matchsticks. Try to get the coin out of the cup by moving only two matchsticks to form a congruent “cup” but in a new position :

4

ROUTINE PROBLEM

Routine problems are those that merely involved an arithmetic operation with the characteristics:

Presents a question to be answered

Gives the facts or numbers to use

Can be solved by direct application of previously learned algorithms and the basic task is to identify the operation appropriate for solving the problem.

Example of problems :

What’s the area of a 100 meter x 1000 meter car lot?

An employee makes RM8.50 per hour. How much will the employee makes in 40 hours?

NON-ROUTINE PROBLEM

It occurs when an individual is confronted with an unusual problem situation, and is not aware of a standard procedure for solving it. The individual has to create a procedure. To do so, he or she must become familiar with the problem situation, collect appropriate information, identify an efficient strategy, and use the strategy to solve the problem.

Example of a non-routine problem :

Approximately how many hairs are there on your head?

Non-routine problems are those that call for the use of processes far more than those of routine problems with the characteristics:

Use of strategies involving some non-algorithmic approaches

Can be solved in many distinct ways requiring different thinking processes.

5

WHAT IS PROBLEM SOLVING?

Problem solving is the process of applying previously acquired knowledge, skills, and understanding to new and unfamiliar situations.Problem solving is the process used to find an answer to a statement or a question

(Hamada, R.Y. & Smith).

To solve a problem is to find a way where no way is known off-hand, to find a way out of a difficulty, to find a way around an obstacle, to attain a desired end, that is not immediately attainable, by appropriate means.

(G. Polya in Krulik and Reys 1980, p. 1)

WHY A SPECIAL EMPHASIS ON PROBLEM SOLVING?

In the classroom can lessen the gap between real world problem and the classroom world and thuis set a more positive mood in the classroom.

Problem solving shows an interaction between mathematical ideas.

Problem solving is an integral part of the larger area of critical thinking, which is universally accepted goal for all education.

Problem solving permits students to learn and practice heuristic thinking.

WHAT MAKES A GOOD PROBLEM SOLVER?

Characteristics exhibited by good problem solvers:Have a desire to solve a problem.Extremely perseverant when solving problems.Show an ability to skip some of the steps in the solution orocess.Not afraid to guess.

Good problem solvers are students who hold conversations with themselves. They know what questions to ask themselves, and what to do with the answers they receive as they think through the problem.

6

WHAT MAKES A GOOD PROBLEM?

A good problem contains some or all characteristics:The solution to the problem involves the understanding of distinct mathematical concepts or the use of mathematical skills.The solution of the problem leads to a generalization.The problem is open-ended in that it lead to extensions.The problem lends itself to a variety of solutions.The problem should be interesting and challenging to the students.

Example:There are 8 people in a room. Each person shakes hands with each of the other people once and only once. How many handshakes are there?

A farmer has some horses and some chickens. He finds that together they have 70 heads and 200 legs. How many horses and how many chickens does he have?

7

PROBLEM SOLVING PROCESS

Steps in problem solving process

George Polya identified four steps in the problem solving process :

Understanding the problem Devising a plan Carrying out the plan Looking back

Burton (1984) identified four phases in the problem solving process :

Entry Attack Review Extension

The Reality and Mathematics Education (RIME) program recommended the following steps for mathematical problem solving :

Introduce the problem Pose the problem Allow students to carry out initial investigations Encourage students to check their predictions Assist students to develop a summary and conclusion to what they have

been doing

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Problem Solving Process (George Polya)

1. Understanding the problem

Can you state the problem in your own words? What are you trying to find or do? What information do you obtain from the problem? What are the unknowns? What information, if any, is missing or not needed?

2. Devising a plan

Find the connection between the data and the unknown. Consider auxiliary problem if an immediate connection can be found What strategies do you know? Try a strategy that seems as if it will work.

3. Carrying out the plan

Use the strategy you selected and work the problem. Check each step of the plan as you proceed Ensure that the steps are correct

4. Looking Back

Reread the question Did you answer the question asked? Is your answer correct? Does your answer seems reasonable?

9

TEACHING PROBLEM SOLVING

How to establish a positive climate in the classroom for problem solving :

Be enthusiastic about the problem Have students bring in problems from their personal experiences Personalize problems whenever possible ( e.g. use students’ names) Recognize and reinforce willingness and preseverance Reward risk takers Encourage students to guess answers Accept unusual solutions Praise students for getting correct solutions Emphasize the selection and use of problem solving strategies Emphasize persistence rather than speed

Teaching problem solving

Before :

Read the problem to the class or have a student read the problem. Discuss words or phrases students may not understand.

Use a whole-class discussion about understanding the problems. Ask questions to help students understanding the problem.

Ask students which strategies might be helpful for finding a solution. Do not evaluate students’ suggestions. You can direct students’ attention to the list of strategies on problem solving when asking for suggestions.

During :

Observe and question students about their work. Give hints for solving the problems as needed. Require students who obtain a solution to check their work and answer the

problem. Give a problem extension to students who complete the original problem

much sooner than others.

After :

Show and discuss students’ solutions to the original problem. Have students name the strategies used.

Relate the problem to previous problems and solve an extension of the original problem.

Discuss special features of the original problem, if any.

PROBLEM SOLVING AT THE PRIMARY LEVEL

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Children begin solving problems long before they enter the primary grades. Some problems faced by pre-primary children require sophisticated problem solving skills. By the time children enter the primary level, children have developed their own personal approaches and strategies for solving these complex problems. It is the primary teacher’s responsibility to assist children in extending, refining and adding to their problem-solving approaches and strategies.

Problem solving at the primary level should be the focus of almost all mathematics instruction the child will encounter. Computational skills are taught so they can be used as tools in the problem-solving process.

It is not enough that teachers teach problem solving by exposing children to problems and hoping that somehow after working many problems the children would automatically develop into problem solvers. Children usually solve problems that used addition, division, fractions, etc.

Teachers should utilize the results of research on how children learn to solve problems. Students are systematically involved in examining various problem-solving strategies which can be used in a wide variety of problem-solving situations.

The strategies commonly delegated to the primary level are:

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Act it out. Teachers in kindergarten and first grade use this approach when developing models for addition and subtraction. Rebus problems, which use pictures to convey words, are used as basis for children physically “acting out” a problem.

Draw a picture. Children’s textbooks use drawings and pictures throughout the early development of mathematical concepts, as well as beginning problem-solving exercises. However, teachers must design activities that encourage children to use this skill.

Use a model. As early as kindergarten, teachers will be using representions on the flannel-board or magnetic board to illustrate addition and subtraction situations. Children also should have opportunities to use heir own sets of materials (at their desks) to “act out” the problem using counters or number lines.

Collect and organize data. Children start making simple, teacher-directed graphs as early as kindergartens. By second grade they should be constructing bar graphs and getting information from tables.

Look for a pattern. Bead-stringing, paper chains, and rhythm activities used in kindergarten lay a foundation for this strategy. Children start identifying patterns with pictures or symbols in the first grade. By the third grade, students will extend these skills to include numerical patterns (odd/even numbers, multiples of five etc)

Guess and check. Children begin “guessing” even before the teacher introduces this skill. Before a pupil can become very skillful with this strategy, some number and measurement skills must be developed. The teacher should be building estimations skills as children work with whole numbers, operations, and measurement so children will have a fairly good foundation in estimation skills before third grade.

PROBLEM EXTENSION

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Goals for Extension

1) Lead pupils to generalisation

2) Introduce or integrate other branches of mathematics

3) Provide opportunities for divergent thinking and making value judgements

Principles for Extending a Set of Problems

At a party I attended recently, I noticed that every person shook hands with each other person exactly one time. There were 12 people at the party. How many handshakes were there?

Principle for Problem Variation New Problem

A. Change the problem context/ setting (e.g., party to a Ping-Pong tournament).

A. Twelve students in Ms.Palmer's fifth-grade class decided to have a Ping-Pong tournament. They decided that each students would play one game against each other students. How many games were played?

B. Change the numbers (e.g., 12 becomes 20 or n).

B. At a party I attended recently, I noticed that every,person shook hands with each other person exactly one time. There were 20 people at the party. How many handshakes were there? That if there were n people at the party?

C. Change the number of conditions(e.g., instead of the single condition that "every person shook hands with each other person exactly one time," we add the condition "Tim shook hands with everyone twice."

C. At a party I attended recently I noticed that every person but the host,Tim, shook hands with each other person exactly one time. Tim shook hands with everyone twice (once when they arrived, once when they left). There were 12 people at the party. How many handshakes were there?

D. Reverse given and wanted D. At a party I attended recently I noticed

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Principle for Problem Variation New Problem

information (e.g., in the basic problem you are given the number of people at the party and you want to find how many handshakes there are; the reverse is true in the new problem).

that every person shook hands with each other person exactly one time. If I told you there were 66 handshakes, could you tell me how many people were at the party?

E. Change some combination of E. context, numbers' conditions, and given/wanted information (e.g., in problem E both the context and the numbers have been changed). Note: There are 11 combinations possible!

E. All 20 students in Ms. Palmer's fifth-grade class decided to have a Ping-Pong tournament. They decided that each student would play one game against each other student. How many games were played?

    

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PROBLEM-SOLVING STRATEGIES

1. Guess and check

2. Draw a diagram or graph

3. Work backwards

4. Make a table

5. Identify patterns

6. Simply problems

7. Make an organized list

The list of problem-solving strategies above is by no means exhaustive. You may like to read up on some other strategies such as

(i) Reading and restating problem.

(ii) Brainstorming

(iii) Looking in another way

(iv) Making a model

(v) Identifying cases

Note: Different strategies can be used to solve the same problem.

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PROBLEM SOLVING STRATEGY : GUESS AND CHECK

Process Problem 1

Marty did 2 of these activities. He paid for them with a RM10.00 bill. His change was RM3.75. What 2 activities did Marty do? (Hint: Make a guess. Then check your guess.)

Activity Cost

Movies RM3.50

Putt-Putt Golf RM3.00

Skating RM2.00

Go-Kart Rides RM2.75

Understanding the Problem

· How many activities did he do? (2)· How much money did he have? (RM10.00)    · What was his change? (RM3.75)

Planning a Solution

· How much money did he have? (RM10.00) What was the change? (RM3.75)· How much did he spend? (RM6.25)· If he saw the movies and golfed, how much money would he have spent? (RM6.50) Did he do these 2 activities? (No, they cost too much.)

Finding the Answer

Guess and Check· Try movies and skating—RM3.50 + RM2.00 = RM5.50. (too little)· Try movies and go-karts—RM3.50 + RM2.75 -- RM6.25. (correct)The activities that Marty did are the movies and the go-kart rides.

Problem Extension

Marty's friend Joe went also, but he was not limited to 2 activities. He took RM10.00 and brought backRM2.25. What activities did he do? (golf, skating, go-kart rides)

16

Process Problem 2

I wrote 5 different numbers on 5 cards. The sum of the numbers is 15. What numbers did I put on the cards? (Hint: Make a guess. Then check your guess.)

Understanding the Problem

· How many numbers did I write? (5)· What is the sum of the numbers? (15)· How many numbers on each card? (1)· Are any 2 numbers the same? (No, they are all different.)

Planning a Solution

· Could 1 of the numbers be 15? (No, because the rest would be 0 and we said that all the numbers were different.)· Select 5 numbers and check to see if their sum is 15.

Finding the Answer

Guess and Check· Try 0,1,2,3,4—0 + 1 + 2 + 3 + 4 = 10 (too little)· Try 1,2,3,4,5—1 + 2 + 3 + 4 + 5 - 15 (correct)The numbers are 1, 2, 3, 4, and 5.

Problem Extension

If the numbers are even and different and their sum is 30, what are the numbers? (2,4,6,8,10)

17

Process Problem 3

David's age this year is a multiple of 5. Next year, David's age will be a multiple of 7. How old is David now?

Understanding the Problem

· What do we know about David's age this year? (multiple of 5)· What do we know about David's age next year? (multiple of 7)

Planning a Solution

· List some multiples of 5. (5, 10, 15, . . .) Multiples of 7. (7, 14, 21, . . .)· Guess what David's age might be this year and add 1 to it to see if that number is a multiple of 7. (See solution.)

Finding the Answer

Guess and Check· Try 10 and 11. (No, 11 is not a multiple of 7.)and so on

Make an Organized List

Multiples of 5

5 10 15 20

Multiples of 7

7 14 21 28

David is 20 years old now.

Problem Extension

In how many more years will David's age be a multiple of both 5 and 7? (In 15 years, when he is 35)   

18

 Process Problem Worksheet: Guess and Check

1. Mary has 6 coins, which have a total value of 67 cents. What combinations of coins could she have? Use denominations of 1cents, 5cents, 10cents and 25cents.

2. Navigate your spaceship to the "Black Hole". The product of the numbers along your path must be 2592.

Start6

2 103 4

9 27 1

8 2

4 5Black Hole

  

3. The sums of numbers on each side of the magic triangle are all the same. Find two solutions for the magic triangle using a different number in each box. 

  

4. a) How many shots at this target are needed to make a score of 300?b) Find four different combinations where the value of the shots on the target totals 300.

19

10

24

16

5. If r is less than 10, what value of r makes r3749r0 divisible by 60? (there are two possibilities.)

6. Find a set of 3 consecutive even numbers whose sum is 294.

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SOLUTIONS

1. Mary has 6 coins, which have a total value of 67 cents. What combinations of coins could she have? Use denominations of 1cent, 5cents, 10cents and 20cents.

1cent 5cents 10cents 20cents Total

2 1 2 2 7

 

2. Navigate your spaceship to the "Black Hole". The product of the numbers along your path must be 2592.

Start6

2 103 4

9 27 1

8 2

4 5Black Hole

2 x 3 x 6 x 4 x 9 x 2 = 2592   

3. The sums of numbers on each side of the magic triangle are all the same. Find two solutions for the magic triangle using a different number in each box. 

  

21

10

24 A

16 C B

Sum = 10 + 24 + 16 = 50

A + B = 40 & B + C = 32

Fix a number for B and the  will difference from the above sumsyield A and C

4. a) How many shots at this target are needed to make a score of 300?

b) Find four different combinations where the value of the shots on the target totals 300.

Numbers 45 48 51 54 57 Total

No. of shots 0 3 2 1 0 300

No. of shots 2 1 1 1 1 300

No. of shots 2 2 0 0 2 300

No. of shots 2 1 0 3 0 300

 

5. If r is less than 10, what value of r makes r3749r0 divisible by 60? (there are two possibilities.)

r3749r0 ÷ 60 ====> r3749r ÷ 6

A number divisible by 6 is even and divisible by 3.So r must be even and r3749r is its divisibility by 3.

3 + 7 + 4 + 9 = 2323 + r + r  is divisible by 3 and r < 10 and even

===> r = 2 and 8 

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PROBLEM SOLVING STRATEGY : WORK BACKWARDS

Process Problem 1

Phil was given his allowance on Monday. On Tuesday he spent RM1.50 at the fruit stand. On Wednesday, Jed paid Phil the RM1.00 he owed him. If Phil now has RM2.00, how much is his allowance? (Hint: Using the facts given, start with the amount Phil has now and work backwards.)

Understanding the Problem

· How much money did Phil have after Wednesday? (RM2.00)· Do you know how much Phil's allowance is? (no)· How much did Phil spend at the fruit stand? (RM1.50)· Was Phil given any money after he got his allowance? (Yes, Jed gave him RM1.00.)

Planning a Solution

· Did Phil have Jed's RM1.00 on Tuesday night? (no)· How much money did Phil have at the end of Tuesday? (RM2.00 - RM1.00 = RM1 00)· Did Phil spend money on Tuesday? (yes, RM1.50) How much money did Phil have before he spent the RM1.50? (RM1.00 + RM1.50 = RM2.50)

Finding the Answer

Work Backwards   

Start with ?Subtract RM1.50Add RM1.00End with RM2.00     So--->

End with RM2.50Add RM1.50Subtract RM1.00Start with RM2.00

Phil's allowance is RM2.50.

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Problem Extension

On Phil's birthday his father increased his allowance. Phil was so happy he went to the store and bought 2 cans of spray paint for his model airplanes. The paint cost RM1.50. After Phil bought the paint, he had RM3.50 left. How much of an increase did Phil get in his allowance? (RM2.50, twice as much).   Process Problem 2

Jacob, Jesse, and James uncovered a strongbox containing some gold nuggets. They buried half of the nuggets in the Grand Canyon and divided the remaining nuggets evenly among themselves. Jesse received 2,000 gold nuggets. How many nuggets were in the strongbox? (Hint: Start with the number of nuggets Jesse received and work backwards.)

Understanding the Problem

· Do you know how many gold nuggets were in the strongbox? (no)· Do you know how many nuggets were left after they buried half in the Grand Canyon? (no) After they divided the remaining nuggets evenly among themselves? (yes, 2,000 each)

Planning a Solution

· If Jesse received 2,000 nuggets, how many did James and Jacob receive? (2,000 each) · Together, how many nuggets did the men have? (6,000)· The money the men had is how much of what was in the strongbox? (Half, because half of the money was buried in the Grand Canyon.)

Finding the Answer

Work Backwards

Start with 2,000 nuggets.Multiply by 3, the number of men—3 x 2,005 = 6,000.Multiply by 2 for the half they buried—2 x 6,000 = 12,000.

There were 12,000 gold nuggets in the strongbox.

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Problem Extension

The men decided to place half of the nuggets in the bank first, bury half of the remaining nuggets, and then divide the nuggets that were left evenly among themselves. Again, each man received 2,000 nuggets. How many nuggets were in the strongbox? (24,000)   

Process Problem 3

Leon wanted to know the age of a black bear at the zoo. The zoo keeper told Leon that if he added 10 years to the age of the bear and then doubled it, the bear would be 90 years old. How old is the bear? (Hint: Using the facts given, start with 90 years and work backwards.)

Understanding the Problem

· Did the zoo keeper tell Leon the bear's age? (no)· What was the last thing the zoo keeper did to the bear's age? (He doubled it.)· What was the first thing the zoo keeper did to the bear's age? (He added 10.)

Planning a Solution

· If you double a number and get 10, what number did you double? (5) What operation did you use to get 5? (division—10 ÷ 2 = 5)· The zoo keeper doubled a number and got 90. What operation could you use to get the number he doubled? (division—divide by 2)· Is the bear 45 years old? (no)· What did the zoo keeper do before he doubled the bear's age? (He added 10 to the bear's age.)· Which operation would you use to find out how old the bear is? (subtraction—45 - 10 = 35)

Finding the Answer

Work BackwardsStart with 90, the final number given by zoo keeper.Divide by 2 to get the number that was doubled—90 ÷ 2 = 45.Subtract 10 to get the age of the bear before 10 years was added—45 - 10 = 35The bear was 35 years old.

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Problem Extension

Leon also wanted to know the age of a turtle he saw at the zoo. The zoo keeper said that if he added 14 years to the age of the turtle and then doubled it, the turtle would be 200 years old. How old was the turtle? (86)   

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Process Worksheet: Work Backwards

1. Rabbits multiply at an amazing rate. In year 1 there are X rabbits. The rabbit population doubles each year. The forest is crowded in year 7 when there are 3200 rabbits. How many rabbits were there in year one if the population doubles each year?

2. I bought a bag of apples. I kept half of them for myself. I gave the rest to 3 friends. Each friend got 2 apples. How many apples did I buy?

3. What is the starting number?

?

Add 12Subtract

6

Multiply by 2 20

 

4.?

Divide by 7

Multiply by 2

Add 2

20

SOLUTIONS

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1. Make a table and work backward from year 7 when there are 3200 rabbits. Since population doubles each year, working backward means halving it.

Year No. of Rabbots

7 3200

6 1600

5 800

4 400

3 200

2 100

1 50

There were 50 rabbits in year one. 

2. Work backward from each of 3 friend having 2 apples.

Forward Step Backward Step

Each of 3 friends has 2 applesTotal : 3 x 2 = 6

apples

I kept half of them for myselfTotal : 6 x 2 = 12

apples

I bought 12 apples. 

3. Start with the end result 20.

28

Forward Step Backward Step Working

Final Result 20 20

Multiplt by 2 Divide by 2 20 / 2 = 10

Subtract 6 Add 610 + 6 =

16

Add 12 Subtract 12 16 - 12 = 4

The starting number is 4.  

4. Reverse the steps

Forward Step Backward Step Working

? ? 63

Divide by 7 Multiply by 7 9 x 7 = 63

Multiply by 2 Divide by 2 18 / 2 = 9

Add 2 Subtract 2 20 - 2 = 18

20 20 20

PROBLEM SOLVING STRATEGY : MAKE A TABLE

Process Problem 1

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Holly checked a book out of the library and read this notice about fines: It a book is 1 day overdue, the fine is 1cents, 2 days overdue, 2cents, 3 days overdue, 4cents, and so on. It Holly's book is 7 days overdue, how much is her fine? (Hint: Complete this table.)

Day 1 2 3 4 5 6 7

Fine 1cents 2cents 4cents 8cents      

Understanding the Problem

· How much is the fine for 1 day? (1 cents)· How much is the fine for 2 days? (2cents)· How much is the fine for 3 days? (4cents)

Planning a Solution

· How much would the fine be for 4 days if we double the previous day? (8cents)· How much is the fine for 5 days? (16cents)

Finding the Answer

Make a Table

Day

1 2 3 4 5 6 7

Fine

1cents

2cents

4cents

8cents

16cents

32cents

64cents

Holly's fine is 64cents.

Problem Extension

Holly had 2 books overdue. One book was 10 days overdue and the other was 5 days overdue. What was her total fine? (RM5.28)

Process Problem 2

30

Seth and Bob each began reading a Hardy Boys book today. If Seth reads 8 pages each day and Bob reads 5 pages each day, what page will Bob be reading when Seth is reading page 56? (Hint: Complete the table.)

DaySeth's Page

Bob's Page

1 8 5

2 16 10

3 24 15

Understanding the Problem

· How many pages does Seth read each day? (8) Bob? (5)· Did they start reading their books on the same day? (yes)

Planning a Solution

· How many pages had Seth read at the end of the first day? (8) Bob? (5) · When Seth has read 16 pages, how many pages will Bob have read?(10)· Find the number of pages Seth read for the first 5 days. (8, 16, 24, 32, 40)

Finding the Answer

Make a Table   

DaySeth's Page

Bob's Page

1 8 5

2 16 10

3 24 15

4 32 20

5 40 25

6 48 30

7 56 35

Bob will be reading page 35 when Seth is reading page 56.

Problem Extension

31

Mary reads 9 pages a day, Sue reads 10 pages a day, and Molly reads 8 pages a day. What page will Sue and Molly each be reading when Mary is reading page 72? (Sue--Page 80, Molly--Page 64) 

 Process Problem 3

Jerry was mowing his lawn when he noticed Christy was a so mowing her lawn next door. They stopped to talk and Jerry learned that Christy mows her lawn once every 8 days. Jerry mows his lawn once every 6 days. In how many days will they be mowing their lawns together again?

Understanding the Problem

· How often does Christy mow her lawn? (every 8 days)· How often does Jerry mow his lawn? (every 6 days)

Planning a Solution

· In how many days will Christy mow her lawn again? (8) Then the next time?(16)· When will Jerry mow his lawn again? (in 6 days) Then again? (12)· Try making a table. (See solution.)

Finding the Answer

Make a Table

Christy 8 16 24 32

Jerry 6 12 18 24

Jerry and Christy will be mowing their lawns together again in 24 days.

Problem Extension

Suppose Jerry's mom pays him RM1.50 each time he mows the lawn and Christy's mom pays herRM1.75 each time she mows the lawn. Who will have made the most money in 24 days? (Jerry)    

Process Problem Worksheet: Make a Table

32

1. Mr Green has a small farm near Steinbach, Manitoba. He has chickens and cows on his farm. If there are 32 legs altogether, what is the greatest number of cows possible?

2. List the different combinations of 5-cents coins and 10-cents coins that make 55cents.

3. Each time 2 dice are rolled, 2 numbers land flacing up.

(a) How many different combinations of numbers can there be?Hint: Only count different sets of numbers. Example: 1, 3 and 3, 1 are the same.

(b) How many different products can there be?

4. In a box there are twelve pieces of paper, each with a number. The first is numbered 1, the second 2, the third 3, and so on until 12. The box is shaken and the numbers drawn out in pairs. If the sums for each of the six pairs are 4, 6, 13, 14, 20, and 21, what are the numbers that make up the pairs?

5. Sally is having a early. The first time the doorbell rings, one person enters. If on each successive ring a group enters that has two more persons than the group thot entered on the previous wring, how many people enter on the sixth ring?

6. A kennel owner has the following dogs: a blonde collie, a brown terrier, a black poodle, a black collie, a blonde poodle, a while terrier, a brown collie, a black terrier, a while poodle, and a blonde terrier. If he wants to have one of each colour and breed combination, what types of dogs should he get?

SOLUTIONS

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1. Mr Green has a small farm near Steinbach, Manitoba. He has chickens and cows on his farm. If there are 32 legs altogether, what is the greatest number of cows possible?

Always maintain a constant total of 32 legs in total.If all the legs belong to cows only, then 8 cows (32÷4) are possible.But there must be at least 1 chicken on the farm.Exchange 1 cow for 2 chickens.

Chickens Cows Legs

0 8 32

2 7 32

There are at most 7 cows. 

2. List the different combinations of 5-cents coins and 10-cents coins that make 55cents.

5-cents coins

11 9 7 5 3 1

10-cents coins

0 1 2 3 4 5

As there must be at least 1 coin of each, only 5 different combinations exist.

 3. Each time 2 dice are rolled, 2 numbers land flacing up.

(a) How many different combinations of numbers can there be?Hint: Only count different sets of numbers. Example: 1, 3 and 3, 1 are the same.

(b) How many different products can there be?

DicePair

xDicePair

xDicePair

xDicePair

xDicePair

PROBLEM SOLVING STRATEGY : IDENTIFY PATTERNS

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Process Problem 1

A man was very overweight and his doctor told him to lose 36 kg. If he loses 11 kg the first week, 9 kg the second week, and 7 kg the third week, and he continues losing at this rate, how long will it take him to lose 36 kg? (Hint: Look for a pattern. Then complete the table.)

WeekTotal Kilograms

Lost

1 11

2 11 + 9 = 20

3 20 + 7 = 27

4

5

                    

Understanding the Problem            · How much does the man need to lose? (36 kg)             · How much did he lose the first week? (11 kg)             · How much did he lose the second week? (9 kg)             · How much did he lose the third week? (7 kg)

Planning a Solution

· How much less does he lose the second week than the first week? (2 kg)        · How much less does he lose the third week than the secornd? (2 kg)            

Finding the Answer

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Make a Table/Look for a Pattern

WeekTotal Kilograms

Lost

1 11

2 11 + 9 = 20

3 20 + 7 = 27

4 27 + 5 = 32

5 32 + 3 = 35

6 35 + 1 = 36

Pattern: The number of kilograms lost decreases by 2 each week. It will take the man 6 weeks to lose 36 kg.

Problem Extension

If the man gains his weight back at the rate of .2 kg the first week, 4 kg the second week, 6 kg the third week, and so on, in which week will he have gained back 36 kg? (the sixth)   

 

Process Problem 2

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Jose used 6 blocks to build this staircase with 3 steps. How many blocks will Jose need to make a 6-step staircase? (Hint: Make a table and look for a pattern.)

Understanding the Problem

· How many blocks are used to build a 3-step staircase? (6)· Do you know how many blocks are used to make a 6-step staircase? (No, that is what we are trying to find out.)

Planning a Solution

· How many blocks were used to build the first step? (1 )· How many new blocks were used for the second step? (2)· How many new blocks would be needed for the fourth step? (4) What would be the total number of blocks used to build a staircase with 4 steps? (10)

Finding the Answer

Make a TablelLook for a Pattern

Steps in Staircase

Blocks Needed to Build New Steps

Total Blocks Needed

1 1 1

2 2 1 + 2 = 3

3 3 1 + 2 + 3 = 6

4 4 1 + 2 + 3 + 4 = 10

5 5 1 + 2 + 3 + 4 + 5 = 15

6 61 + 2 + 3 + 4 + 5 + 6 = 21

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Pattern: The number of new blocks needed increases by 1 with eachnew step. The total number of blocks needed for nth step is the sum of the number 1 through n.It would take 21 blocks to build a 6-step staircase.

Problem Extension

How many steps would there be in a staircase using 78 blocks? (12)    

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Process Problem 3

Earl played a game using the figure below. First he covered the section numbered 1. Then he covered the sections numbered 1 and 2. Next he covered the sections numbered 1 and 4. What sections would he cover on his seventh round?

Understanding the Problem         · What numbers are in the circle? (1, 2, 4, 8)· What number(s) did he cover first? (C') Second? (1, 2) Next? (1, 4)        

Planning a Solution

· What is the sum of the numbers he covered first? (1)· What is the sum of the numbers he covered second? (3) Next? (5)· Make a table and look for a pattern. (See solution.)   

Finding the Answer

Make a Table/Look for a Pattern

Round Sum

First 1

Second 1 + 2 = 3

Third 1 + 4 = 5

Fourth 1 + 2 + 4 = 7

Fifth 1 + 8 = 9

Sixth 1 + 2 + 8 = 11

Seventh 1 + 4 + 8 = 13

Pattern: The sum of the numbers increases by 2 in each round.Earl would cover the 1, 4, and 8 on his seventh round.

Problem Extension

If he covered the 2 first, then the 4, then the 2 and the 4, what numbers would he cover on hisseventh round? (2, 4, 8)

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Process Worksheet: Identify Patterns

1. Find the next 3 numbers in the following sequence.    2, 5, 11, 23, ____, _____, ______.

2. If this pattern was formed to make a cube, what numbers would appear where the question marks are?

3. The number of line segments joining a set of points increases as the number of points increases. Find how many line segments there will be when there are 8 points; 10 points.

4. f the figure on the left is continued, how many letters will be in the J row?   Which row will contain 27 letters?

             A          BBB        CCCCC       DDDDDD

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SOLUTIONS

1. Find the next 3 numbers in the following sequence.    2, 5, 11, 23, ____, _____, ______.

Pattern : x 2 + 1

Answer : 47, 95, 191 

2. If this pattern was formed to make a cube, what numbers would appear where the question marks are?

?=1 ?=4 ?=2

 

3. The number of line segments joining a set of points increases as the number of points increases. Find how many line segments there will be when there are 8 points; 10 points.

Points 2 3 4 5 8 9 10 n

Lines 1 3 6 10 28 36 45 n(n-1)/2

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4. If the figure on the left is continued, how many letters will be in the J row?   Which row will contain 27 letters?           

A          BBB        CCCCC       DDDDDD

Letter A B C D E ...... J K L M N General

Numeral Order

1 2 3 4 5 ...... 10 11 12 13 14 n

Total No. 1 3 5 7 9 ...... 19 21 23 25 27 2n-1

 

PROBLEM SOLVING STRATEGY : SIMPLIFY PROBLEMS

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Process Problem 1

Sometimes a problem is too complex to solve in one step. When this happens, it is often useful to Simplify the problem by dividing it into cases and solving each one separately.

Use the hints provided to solve each of the following problems.

1. How many palindromes are there between 0 and 1000?(A palindrome is a number like 525 that reads the same backward or forward.)

Simplify the problem. Find the number of one, two, and three digit palindromes separately.Make a list. Use the list I've started, and look for a pattern!

a. How many of the numbers 1 through 9 are palindromes?All nine numbers are palindromes.

b. How many of the numbers 10 through 99 are palindromes?1122            nine::99

  Make a List

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c. Find the number of palindromes from 100 through 999.1 0 1 2 0 2 ......... 9 0 91 1 1 2 1 2 ......... 9 1 91 2 1 2 2 2 ......... 9 2 91 3 1 : :

::

::

::

1 9 1 2 9 2 ......... 9 9 9

9 columns x 10 palindromes/column = 9090 palindromes from 100- 999

Use the list I've started, and look for a pattern! 

d. What is the answer to this original question?There are 108 palindromes between 0 and 1000    

Process Problem 2

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Tony's restaurant has 30 small tables to be used for a banquet. Each table can seat only one person on each side. If the tables are pushed together to make one long table, how many people can sit at the table?

Strategies involved

Simplify the problem Draw a picture Make a table Look for a pattern

 

No. of tables No. of people

2 6

3 8

4 10

5 12

6 14

: :

: :

30 62

 

Pattern: 2 times the number of tables plus 2  ===>  2 x 30 = 60  ===> 60 + 2 = 6262 people can sit at the table.    

Process Worksheet: Solve a Simpler Problem

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1. How many squares are in this figure?

     

     

     

2. A total of 28 handshakes were exchanged at a party. Each person shook hands exactly once with each of the others. How many people were present at the party?

3. Find the thickness of one page in your mathematics text book.

4. Mike is paid for writing numbers on pages of a book. Since different pages require different numbers of digits, Mike is paid for writing each digit. In his last book, he wrote 642 digits. How many pages were in the book?

SOLUTIONS

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1. Make a table and work backward from year 7 when there are 3200 rabbits. Since population doubles each year, working backward means halving it.

Possible size of Square No. of each size

9

4

1

Total = 14 

2. Make a table and start with the least possible number of people and find a pattern.

No. of people 1 2 3 4 5 6 7 8

No. of handshakes

0 1 3 6 10 15 21 28

Pattern 0 +1 +2 +3 +4 +5 +6 +7

8 people were present at the party.  

3. Method: Measure the thickness of say, 100 pages of the text book. Then divide the result by 100 to obtain the thickness of one page.

 

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4. Divide the pages of the book into groups of 1-, 2- and 3-digit pages and count them separately in batches.

Page Number

No. of digits per page

No. of pages No. of digits

1 to 9 1 9 1x9=9

10 to 99 2 99 - 9 = 90 2x90=180

Total = 99 Total = 189

No. of digits left excluding the 1- and 2-digit pages = 642 - 189 = 453

No. of pages with 3-digit numbers = 453 / 3 = 151

Total no. of 1-, 2- and 3-digit pages totalling 642 digits = 99 + 151 = 250

There were 250 pages in the book.

PROBLEM SOLVING STRATEGY : EXPERIMENTATION AND SIMULATION/ACTING OUT

Acting out a problrm forces an understanding of the nature of the problem. If someone is capable of acting out the problem, we can almost best be certain that he or she understands it.

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Some manipulative such as bottles caps can be used in place of snowbalss, chips can be represent by people. We can simulate the action with pencil and paper, by making a drawing or a table.

PROCESS WORKSHEET : EXPERIMENTATION AND SIMULATION/ACTING OUT

1. A man bought a goat for RM 60, sells it for RM 70, buys it back for RM 80 and sells it for RM 90. How much does the man make profit or lose in the goat trading business?

2. A dog chasing a rabbit, which has a lead of 15 meters, jumps 3 meters for every time the rabbit jumps 2 meters. In how many lepas does the dog overtake the rabbit?

3. Determine the largest number of boxes of dimensions 2 X 2 X 3 that can be placed inside a box 3 X 4 X 5.

NEWMAN ERROR ANALYSIS

Reasearches :

WATSON (1980) CLARKSON (1983)

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FINDINGS : 50% OF ERRORS OCCURS IN THE READING, COMPREHENSION AND TRANSFORMATION STAGES.

MARINAS & CLEMENTS (1990) RESEARCH DONE ON FORM ONE STUDENTS IN PULAU PINANG FINDINGS : 90% OF ERRORS OCCURS IN THE COMPREHENSION AND

TRANSFORMATION STAGES

BACKGROUND

M. ANNE NEWMANAUSTRALIA1977

Analysis of students errors in written mathematical tasks

According to Newman, a problem solving item could be solved according to the hierarchy below :

o Able to read the problem giveno Able to comprehend the given problemo Able to transform the problem to specific mathematical statements o Able to solve the problem using correct algorithmo Able to transfer the answer to the space providedo Avoiding carelessnesso Have high motivation

Newman’s research

o 124 YEAR 6 low achievement students o Findings :

13% errors in the reading stage 22% errors in the comprehension stage 12% errors in the transformation stage total of 47% errors occurs before the process skill stage

DIRECTIONS SUGGESTED BY THE NEWMAN PROCEDURE IN ANALYSING STUDENTS’ ERROR IN PROBLEM SOLVING

1. Please read this question. (READING)

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2. Explain to me what is needed by the problem. (COMPREHENSION)

3. Suggest one way to solve the problem. (TRANSFORMATION)

4. Show and explain how you solve the problem in getting the answer.(PROCESS SKILL)

5. Now write your answer. (ENCODE)

HIERARCHY IN THE NEWMAN’S ERROR ANALYSIS PROCEDURE

NEWMAN PROCEDURE

1. Reading Recognition

Please read the question to me

2. Comprehension

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READING

COMPREHENSION

TRANSFORMATION

PROCESS SKILL

ENCODE

CARELESS MISTAKES

MOTIVATION

What does this word/sign mean? What is the question asking you to do?

3. Transformation

Tell me or show me how you start finding an answer to the question.

4. Process Skills

Show me how you work the answer out for this question.

5. Encoding Ability

Write down the answer to the question.

NEWMAN’S ERROR CATEGORIES

1. Reading Error

Words Symbols Numerals

2. Comprehension Error

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Specific terminology understanding General meaning

3 Transformation Error

Verbalization or showing how the item can be translated into a mathematical form which allow a correct answer to be worked out at the process skill stage.

4. Process Skill Error

Wrong operation Faulty algorithm Faulty computation Random response (guessing) Fail to attempt

5. Encoding Error

Can verbalize the answer to the task at (4) but write the answer incorrectly.

REMEDIATION

1. Teaching reading skills in mathematics

Word and symbol identification Interpreting terminology

2. Teaching comprehension skills

Gathering stated information Interpreting terminology Interpreting concepts and terminology

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Comprehension of main ideas literally stated Interpreting graphic materials Disregarding irrelevant information

3. Teaching transformation skills

Stepwise solution approach to solving word problems

o Read the problemo Write or underline the unknown wordo Show the problem in pictureo What is the main question being askedo Reread. Write given information.o Work out the answer.o Reread to check the answer.

Symbols for words

o Children translate mathematical word problems into symbolic form.

Example : eighty divided by four equals80 4 =

Creative word problem

o Allow children to write their own word problems which can then be acted out as real life situations and simultaneously recorded in symbolic form.

Reverse transformation

o Pupils are given symbolic sentences and are asked to write a story to illustrate the concepts.

EXAMPLE OF INTERPRETING ERRORS

Pupil 1 : Glenn

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Sam goes to bed at 10 minutes to 9.John goes to bed 15 minutes later than Sam.What time does John go to bed?

Initial error cause : Reading recognition : word recognition

From the pupil’s response column, it can be seen that Glenn answered the item incorrectly (No response) on the second attempt. He was then unable to read the item accurately – the words “minutes” and “later” being read incorrectly. Following this he was unable to give meaning to these terms and was unable to verbalise an understanding of the general meaning of the item. Glenn made no attempt to work the problem out. The pupil’s reading ability was not adequate to allow him to proceed to the comprehension stage.

Pupil 2 : MickyInitial error cause : Comprehension : Specific terminology understanding

On the second attempt at the item Micky answered the item incorrectly, giving 10 minutes to 9 as his answer. He read the item accurately. However, when questioned as to the meaning of “later than” Micky showed no understanding of the term though he knew the meaning of “minutes”. This pupil did appear to understand the general meaning of the item but he has no idea how to work out the item. In this example, even though the pupil appeared to know the general meaning of the item, his lack of understanding of an essential term caused him to guess the answer.

REFERENCES

Charles, R. I. & Barnet, C.S. (1992). Problem-Solving Experiences in Pre

Algebra. USA: Addison-Wesley.

Dolan, S. & Everton, T. (1994). Problem Solving. Cambridge : Cambridge

University Press.

Dolan, D.T. & Williamson, J. (1983). Teaching problem Solving Strategies.

USA : Addison-Wesley Publishing Company

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Hamada, R.Y. & Smith, B. (1989). Problem Solving in mathematics Skills and

Strategies. LA : Unified School District Office of Elementary Publications

No. EC-569

http://www.library.thinkquest.org/learning/problem

Krulik, S. & Rudnick, J.A. (1989). Problem Solving A handbook For Senior

High School Teachers. Massachusetts : Allyn and Bacon A Division of

Simon & Schuster.

Stein, B. & Lotf, L. (1990). A problem Solving Approach To Mathematics for

Elementary School Teachers. The Benjamin/Cummings Publishing

Companny.

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