basic math ass

7/31/2019 Basic Math Ass

1/26

Unit : PPISMP SEM 2 PT/PJ/KS AMBILAN JULAI

2009

Group member s name : SASIKALA A/P ELANGOVAN

(911222-05-5048)

DIVYA A/P LOGANADAN

(901224-08- 5962)

J. PUNNIA MURRTY A/L JAYARAMAN

(910523-01 -6187)

Course/Code : Basic Mathematics- Semester 2

Lecturers Name : Puan Rafidah Binti Wahab

Date of Submission : 16 Mac 2010

TUGASAN KERJA KURSUS PENDEKBasic Mathematics


2/26

CONTENTNo. Subject Pages1. Acknowledgement2. Introduction3. Non-routine problems4. Strategies in problem solving5. summary6. Problems with solution7. Reflection

8. References


3/26

AcknowledgementWe would like to send a special thanks to Madam, lecturer of Mathematic class

for Basic Mathematics subject as she has help us a lot in completing this assignment. When

facing with problems, she guide me through it. She teach us a lot about problem solving

during class and asked question about how to solve a question by using all those problem

solving strategies.

This help us to understand more and know how to apply those strategies to answer a non-

routine problems. Some mistake that we had make, she will correct us and teach us the right

way to do it.

Next, a special thanks to my classmates who have helped me to made this

assignment better than it otherwise could be. They give us some information that we left out

about problem solving from website and some books too. We also share information that we

got from either books or website. Some question that we do not understand, we will discuss

with each other.

Last but not least, a special gratitude towards the best parents in the world, who

continue to encourage and support us in every way. We are who we are because of both of

them. They also gave me some idea on how to start this assignment.


4/26

IntroductionProblem is something that is difficult to deal with or to understand. It can also definedas question that can be answer by logical or mathematics. So people have to make an

attempt to find a solution. There are some problem solving strategies to solve problems.

Problem solving is an important approach in teaching and learning of mathematics in

primary schools. To select the best an appropriate strategy of solving a problem especially a

non-routine is one of the main aim of a good problem solver. In complete this task we are

required to solve 3 non-routine problems by using two or more types of problem solving

strategies. From this, we can know which type of problem solving strategies is the most

efficient.

There are 9 common problem solving strategies that is

I. Guess and check

II. Draw a diagram

III. Construct a table, chart and graph

IV. Do a simulation or experiment

V. Look for a pattern

VI. Simplify a problem

VII. Identify subgoal

VIII. Analog

IX. Work backwards

Guess and check means have to try an error and guess the answer. After that check

it to difference whether it is right or wrong. Guessing is done systematically and in a certain

order. From the diagram, we clearly know that what is happening from the question. We can

solve the problem easier. Simplify the problem can let us understand the problem easier. We

can solve the problem in a short time.

Do a simulation we can carry out an experiment to identify the following number. We

can arrange it in an order. Identifying sub-goal can make problem become clearer that


5/26

finding the answer might be made less difficult by finding the solution to an easier problem

embedded within the problem. Analogy is create a similar question .

So from all this problem solving strategies can help us to solve any problem. We can

used it at anytime or for non-routine problem.


6/26

Non routine

problems

The non-routine problem is a kind of unique problem solving which requires the

application of skills, concept or principle which have been learned and mastered. It is

also meant that whenever we are facing an unusual problem or situation which we

dont know the procedures to solve it, we are facing the non-routine problem.

Non-routine problem solving serves a different purpose than routine problem

solving. While routine problem solving concerns solving problems that are useful for

daily life (in the present or in the future), non-routine problem solving concerns that

only indirectly. Non-routine problem solving is mostly concerned with developing

students mathematical reasoning power and fostering the understanding that

mathematics is a creative endeavor.

Non-routine problem solving can be seen as evoking an I tried this and I tried

that, and eureka, I finally figured it out. reaction. That involves a search for heuristics

(strategies seeking to discover). There is no convenient model or solution path that is

readily available to apply to solving a problem. That is in sharp contrast to routine

problem solving where there are readily identifiable models (the meanings of the

arithmetic operations and the associated templates) to apply to problem situations.


7/26

Strategies in

problem solvingEIGHT PROBLEM SOLVING STRATEGIES.

1. USING SYMMETRY

It helps us to reduce the difficulty level of a problem. Playing Noughts and

crosses, for instance, you will have realized that there are three and not nine ways to

put the first symbol down. This immediately reduces the number of possibilities for

the game and makes it easier to analyze. This sort of argument comes up all the

time and should be grabbed with glee when you see it.

2.ACT IT OUT

Meaning that two strategies combine together because they are closely

related.Young children especially, enjoy using Act it Out. Children themselves take

the role of things in the problem. In the FARMYARD problem, the children might take

the role of the animals though it is unlikely that you would have 87 children in your

class. But if there are not enough children you might be able to press gang the odd

teddy or two.

There are pros and cons for this strategy. It is an effective strategy for

demonstration purposes in front of the whole class. On the other hand, it can also be

cumbersome when used by groups, especially if a largish number of students are

involved. We have, however, found it a useful strategy when students have had

trouble coming to grips with a problem.

The on-looking children may be more interested in acting it out because

other children are involved. Sometimes, though, the children acting out the problem


8/26

may get less out of the exercise than the children watching. This is because the

participants are so engrossed in the mechanics of what they are doing that they dont

see through to the underlying mathematics. However, because these children are

concentrating on what they are doing, they may in fact get more out of it and

remember it longer than the others, so there are pros and cons here2. WORK BACKWARDS

To solve some problems, you may need to undo the key actions in the problem. This

strategy is called Work Backward.Working backwards is a standard strategy that

only seems to have restricted use. However, its a powerful tool when it can be used.

In the kind of problems we will be using in this web-site, it will be most often of value

when we are looking at games. It frequently turns out to be worth looking at what

happens at the end of a game and then work backward to the beginning, in order to

see what moves are best.

3. LOOK FOR A PATTERN

In many ways looking for patterns is what mathematics is all about. We want to know

how things are connected and how things work and this is made easier if we can find

patterns. Patterns make things easier because they tell us how a group of objects

acts in the same way. Once we see a pattern we have much more control over what

we are doing.

5. MAKE A TABLE

There are a number of ways of using Make a Table. These range from tables of

numbers to help solve problems like the Farmyard, to the sort of tables with ticks and

crosses that are often used in logic problems. Tables can also be an efficient way of

finding number patterns.


9/26

6. DRAW A PICTURE

It is fairly clear that a picture has to be used in the strategy Draw a Picture. But the

picture need not be too elaborate. It should only contain enough detail to solve the

problem. Hence a rough circle with two marks is quite sufficient for chickens and a

blob plus four marks will do for pigs. There is no need for elaborate drawings

showing beak, feathers, curly tails, etc., in full colour. Children should be encouraged

to use this strategy at some point because it helps children see the problem and it

can develop into quite a sophisticated strategy later.

7. MAKE A LIST

Making Organised Lists and Tables are two aspects of working systematically. Most

children start off recording their problem solving efforts in a very haphazard way.

Often there is a little calculation or whatever in this corner, and another one over

there, and another one just here. It helps children to bring a logical and systematic

development to their mathematics if they begin to organize things systematically as

they go. This even applies to their explorations.

When an Organised List is being used, it should be arranged in such a way that

there is some natural order implicit in its construction. For example, shopping lists

are generally not organised. They usually grow haphazardly as you think of each

item. A little thought might make them organised. Putting all the meat together, all

the vegetables together, and all the drinks together, could do this for you. Even more

organisation could be forced by putting all the meat items in alphabetical order, and

so on.

.


10/26

8.GUESS AND CHECK

Some problems cannot be solved directly. You need to use a strategy called Guess

and Check.Guess and check is one of the simplest strategies. Anyone can guess an

answer. If they can also check that the guess fits the conditions of the problem, then

they have mastered guess and check. This is a strategy that would certainly work on

the Farmyard problem but it could take a lot of time and a lot of computation.

Guess and improve is slightly more sophisticated than guess and check. The idea is

that you use your first incorrect guess to make an improved next guess. You can see

it in action in the Farmyard problem. In relatively straightforward problems like that, it

is often fairly easy to see how to improve the last guess. In some problems though,where there are more variables, it may not be clear at first which way to change the

guessing.


11/26

Summary

Require application of skill orprinciple that we have masteredbefore

Non- routine problem areproblem that unusual orwe cantrecognise the way to solve it

Routine problems are problemuseful for daily life whereasnon-routine problem solvingconcerns that only indirectly

Non routine problems need us tothink and use our critical thinking skilloptimally in order to solve thequestions.

Non- routine problem


12/26

Strategiesin

Problem

Consider Special Cases-Find out the main point make thebest choice, solve it.

Work BackworkDo the work from beginning in orderto get the most accuracy

Use Direct Reasoning-Select the points and elaborate it.

Solve an Equation

-Determine the variables and simplify.

Use a Formula

-By using formula that is given ,recognise the variable and solve it

Be IngeniousChange the example with anotherobjects,


13/26

Strategiesin

Problem

Solving

Use SymmetryMake the question easier byrecognise the possibilities ofways to solve the question

Use a ModelMake an example by using

another situationMake and Orderly listArranged the objects by itscharacteristics

Eliminate PossibilitiesRemove all the possibility that

repeated.

Look for a patternDraw the pattern and look for

the similarities and how itconnected. Then we can

predict the possible outcome

Draw a PictureDraw pictures to easilyunderstand the questions.Pictures do not need anyfurther explanations because

we can see it

Solve a Simpler ProblemMake a question similar to the

problem but an easier one

Guess and CheckGuess the answer of thequestion and check the

answer whether it is possibleor not. If not redo the guess.


14/26

Problem and solution

Attachment 1

Problem 1

Three darts hit this dart board and each scores a 1 , 5 , or 10 . The total score is the

sum of the scores for the three darts . There could be 1s , two 1s and 5 and one 5

and two 10s , and so on . How many different possible total scores could a person

get with three darts ?

Answer:

Using the four basic principles for problem solving or heuristic polyas model, firstly

we must understanding the problem.

1.Understanding the problem:

i. Three darts hits this dart board and each scores a 1,5 or 10.

ii. There could be three 1s, two 1s and 5, one 5 and two 10s and so on.

iii. we have to find the different possible total scores could a person get with

darts.

2. Devising a plan :

i. we devising a plan to solve this problem by using a correct strategies, and we

choose to make a table to solve this question.

1

5

10


15/26

ii. Then, we list out the possible total scores could a person get with darts by

using multiplying and addition.

3. Carrying out the problem:

i. Transfer the informations into the table.

Scores Total

scores1 5 10

3 0 0 3

2 1 0 7

1 1 1 16

1 2 0 11

0 3 0 15

0 2 1 20

0 0 3 30

0 1 2 25

1 0 2 21

2 0 1 12

The different possible total score could be a person get with darts is 3 , 7 , 11 , 12 ,15 , 16 , 20 , 25 ,21 and 30. . So, the total different possible total scores could aperson get with three darts is 10.

4. Examine and check back.i. The possible outcome scores tables is checked several times to avoid

repeatation of possible outcome scores or miss to write the possible outcome

scores.

ii. the total scores is calculate several times to avoid miscalculation.

iii. So, All the possible outcome scores is write without repeatation or left behind

and the total scores is count correctly. Therefore, the answer is verify.

For example :

(0x1)+(2x5)+(1x10)=20

(2x1)+(1x5)+(0x10)=7

(3x1)+(0x5)+(0x10)=3

(1x1)+(1x5)+(1x10)=16


16/26

(0x1)+(0x5)+(3x10)=30

(0x1)+(3x5)+(0x10)=15

(0x1)+(1x5)+(2x10)=25


17/26

Problem 2

List the 4-digit numbers that can be written using each of 1 , 3 , 5 and 7 once and

only once . Which strategy did you use ?

Answer:

1.Understanding the problem

i. List the 4-digit numbers that can be written using each of 1 ,3 ,5 and 7

once and only once.

2.Devising a plan

i. Find out the best strategies to get the answer.

ii. List down the 4-digit numbersiii. List down each of 1, 3 , 5 and 7 once and only once.

3.Carrying out the problem

i. List down all the 4-digit numbers 1 , 3 , 5 and 7 once and only once.

1 , 3 , 5 ,7 3 , 1 , 5 ,7 5 ,7 ,3 ,1 3 , 5 ,7 ,1

1 , 5 , 7 , 3 3 , 5 ,7 ,1 7 ,1 ,3 ,5 3 ,7 ,5 ,1

1 , 7 , 5 , 3 3 ,5 ,1 ,7 7 ,3 ,1 ,5 5 , 1 ,7 ,31 , 5 , 3 , 7 3 ,7 ,1 ,5 7 ,5 ,3, 1 5 ,3 ,7 ,1

1 , 3 , 7 , 5 5 ,1 ,3 ,7 7 ,3 , 5 ,1 5 ,7 ,1 ,3

1 , 7 ,3 , 5 5 ,3 ,1 ,7 7 ,5 ,1 ,3 7 ,1 ,5 ,3

4.Looking back

i. Determine whether the list is relevant.

ii. The list number of 1 , 3 ,5 ,7 above are used once and only one.

iii. We can also used another strategies to solve and get the answer such

as guess and check.


18/26

Problem 3Pedar Soint has a special package for large groups to attend their amusement park :a flat fee of $20 and $6 per person. If a club has $100 to spend on admission. Whatis the most number of people who can attend?

a). Understand the problem.

Students may need to discuss this a little before attempting to tackle the problem.

b). Devise a plan.

Make a table. But develop what should be in the table with the students. Let them

assist how you make this table.

c). Carry out the plan.

# of people Cost X $6 +$20 Total free Result

10 60 20 80 Too low

15 90 20 110 Too high

13 78 20 98 Too low

14 84 20 104 Too high

Answer : At most, 13 people can attend for $100 and they will have $2 left over.

d). Look back.

Is there another way this could be done? Yes, guess and check (which is part ofWhat we did). The difference is that we tried to do this in an orderly fashion notjust guess randomly. We tried to surround the solution.


19/26

Problem 4

The houses o Main Street are numbered consecutively from 1 to 150. How many

house numbers contain at least one digit 7?

Heuristics of Polyas Model

1.Explore and understand the question.

The problem want we to count the number of houses that contain at least one digit 7.

It means, whether the digit 7 is in hundred places, tens places or in digit places it still

be count. But the number of houses cannot be excced 150. So, we must find how

many houses on Main Street that the house number contain at least one digit 7.

2. Plan out strategy to solve the problem.

Strategy used : Make and orderly list by using a table and Look for a pattern.

Since the problem involve a series of numbers, the easier way to solve this problem

is by make and orderly list of number by using a table and look for a pattern which is

numbers that contain digit 7. List carefully to avoid missing number.

3. Solve and carry out the strategy.

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

100 101 102 103 104 105 106 107 108 109

110 111 112 113 114 115 116 117 118 119


20/26

120 121 122 123 124 125 126 127 128 129

130 131 132 133 134 135 136 137 138 139

140 141 142 143 144 145 146 147 148 149

We colour all the numbers which contain digit 7 in red in colour.

Try look at the 8th column. The digit 7 in red in colour is appears 15 times. So,

there will be 15 number ended with digit 7 for 7th column.

(pattern : number ended with digit 7)

Now,take a look at 8th row. The digit 7 in red in colour is appears 10 times. So,

there will be 7 number started with digit 7 for 8th row.

(pattern : number started with digit 7)

Hence,we add the number of appears times of digit 7 for the column and the row to

get the total number of houses on Main Street that the house number contain at

least one digit 7.

15 + 10 = 25

Since the number 77 is already count in the first pattern (number ended with digit 7),

we deduct the answer by one.

25 1 = 24

Hence, the total number of houses on Main Street that the house number

contain at least one digit 7 is 24.

4. Examine and check back.

i. The consecutive numbers is write carefully and checked several times to

avoid missing number.

ii. The number with digit 7 is seek saveral times to avoid wrong calculation.

iii. Llist back all the number contain digit 7 without repeatation and calculate the

numbers.

.7,17,27,37,47,57,67,70,71,72,73,74,75,76,77,78,79,87,97,107,127,137,147.

iv. The total number of houses that the house number contain at least one digit

7 is 24.


21/26

v. So, All the numbers is write without repeatation or left behind and the

numbers of digit 7 is calculate correctly. Therefore, the answer is verify.


22/26

Problem 5

The figure below shows twelve toothpicks arranged to form three squares. How can

you form five squares by moving only three toothpicks ?

Answer:

1.Understanding the problem

i. Form five squares by moving only three toothpicks from the twelve

toothpicks that arranged to form three squares.

2.Devising a plan.

i. Find out the best strategies to get the answer.ii. Draw a diagram to form the square by removing the three toothpicks..

3.Carrying out a plan.

= 5 squares

4.Looking back

Determine the 5 squares that formed after moving the three of the toothpic


23/26

ReflectionWe have to be sure that you are clear about what the problem really is before you areready to take any steps to solve the problem. Problem is something that is difficult to dealwith or to understand. It can also defined as question that can be answer by logical or

mathematics. So people have to make an attempt to solve it.

At times, however, situations arise which we cannot solve automatically. In those

situations the use of problem-solving skills becomes an invaluable asset that allows we tomake the best choices and decisions available This is a problem-solving strategy that can be

used with difficult concepts such as manipulating ratios or fractions. If a problem is

confusing, the numbers can be rounded, or simpler numbers can be used to help make a

plan to solve it.

This is why there are some problem solving strategies to solve problems that we

learnt during class lesson. Problem-solving is a tool, a skill, and a process. As a tool is helps

you solve a problem or achieve a goal. As a skill you can use it repeatedly throughout yourlife.

The main objective of problem solving strategies is to make our life easier by solving

those difficult problem that we are facing whether is non-routine problem or routine problem.

There are nine common problem solving strategies that is guess and check, draw a diagram,

construct a table, chart and graph, do a simulation or experiment, look for a pattern, simplify

a problem, identify subgoal, analogy and lastly work backwards.

Before solve the problems we need to follow some steps to understand more on the

problems. First, we have to understand the definition of the problems and then analysis it.

Once you have looked at the problem from different perspectives, you can decide what you

want to achieve and establish your goals. You need to answer the very specific question.

After that generate he possible solutions. Before the last steps, have to eanalyse the

solution. The last step is to implement the solution you have chosen.


24/26

It is not unusual for problems to arise when you are working towards a goal and

encounter obstacles along the way. Students usually have many and varied goals, both

related to school and to other areas of their lives, and it is likely that you will encounter

barriers to your success at times. As these barriers are encountered, problem-solving

strategies can be utilized to help you overcome the obstacle and achieve your goal. With

each use of problem-solving strategies, these skills become more refined and integrated so

that eventually their use becomes second nature.

It can be very helpful to write down the answers to these questions so that you are

forced to clarify that the problem you are defining is the actual one you want to solve. Just

thinking about things in your head can cause confusion and end up distracting you from theactual problem at hand.If you are dealing with more than one problem at a time, it may be

helpful to prioritize them. That way you can focus on each one individually, and give them all

the attention they require.

It can be very helpful to ask yourself what you have done in the past when faced with

similar problems, and how other people you know have dealt with similar situations. In

addition, you can also approach friends, family, a counselor, teachers, books, or the internet

to obtain ideas for solutions. Be sure to write down all the possibilities you generate so thatyou can approach this task systematically.

If the problem is successfully solved, then we will feel very satisfied with our efforts to

solve it. As a conclusion all those problem solving strategies are very helpful in our life to

solve any problem no matter is routine or non-routine problem.


25/26

ReferenceInternet

CHARLES, R. I. AND MASON, R. P. AND NOFSINGER, J. M.AND WHITE, C. A.

(1985). Problem-solving in Mathematics. Retrieved Febuary 19, 2010, from Learning

Resources: http://library.thinkquest.org/25459/learning/problem/

Ellington, D. (n.d.). Problem Solving Strategies. Retrieved Febuary 19, 2010, from

http://pred.boun.edu.tr/ps/

Word Problem Solving Strategies. (n.d.). Retrieved Febuary 19, 2010, from

MathStories: http://www.mathstories.com/strategies.htm

Long, C. T., & DeTemple, D. W., Mathematical reasoning for elementary

teachers. (1996). Reading MA: Addison-Wesley

Reimer, L., & Reimer, W. Mathematicians are people too. (Volume 2). (1995)

Dale Seymour Publications

Polya, G. How to solve it. (1957) Garden City, NY: Doubleday and Co., Inc.

Polya, G. (1973). How to solve it. Princeton, NJ: Princeton University Press.

Krulik, S. & Reys, R.E. (1980). Problem solving in school mathematics 1980

Yearbook. Reston, Virginia: NCTM.

Henderson, K. B. & Pingry, R. E. (1953). Problem solving in mathematics.

In the learning of mathematics: Its theory and practice (21st Yearbook of the

National Council of Teachers of Mathematics), pp. 228-270. Washington,

DC: NCTM.


26/26

Suydam, M. (1987). Indications from research on problem solving. In

teaching and learning: A problem solving focus. Reston, VA: NCTM.

http://www.math.wichita.edu/history/men/polya.html

http://www.mathstories.com

http://math.tntech.edu/techreports/TR_1999_5.pdf
http://www.math.wichita.edu/history/men/polya.htmlhttp://www.math.wichita.edu/history/men/polya.htmlhttp://www.mathstories.com/http://www.mathstories.com/http://math.tntech.edu/techreports/TR_1999_5.pdfhttp://math.tntech.edu/techreports/TR_1999_5.pdfhttp://math.tntech.edu/techreports/TR_1999_5.pdfhttp://www.mathstories.com/http://www.math.wichita.edu/history/men/polya.html

basic math ass

Documents