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BASIC MATHEMATICS
CONTENTS..
ACKNOWLEDGEMENTIn the name of Allah, the most gracious and most merciful.
A special Thanks to:Mrs. Afza Binti Abdollah, our mathematic lecturer who teach and give us a lot of information in this project, to our group member, Amera Suriyati Binti Saad, Nur Aina Sofia Binti Mohmad Soffi , Nur Zaihayu Binti Zainoddin and Nurunnajwa Binti Ani for their co-operation and dedication, and all of our friends and classmates for their guidance and support to make sure our coursework finish with fully success. A token of appreciation and thanks to our family, and everyone who is involved in our coursework.
Thank youDECLARATION FORMWe hereby declare that all our assignment coursework for Basic Mathematics is an original place of work prepared by ourself in our own words and we did not copy or plagiarise any part of the paper that we have submitted.We hereby acknowledge, that we:Please check
i. Have acknowledged all work and ideas in our assignment taken from printed and electronically published resources;/
ii.Did not copy from any other resources or another student;/
iii.Did not allow another student copy our assignment; and/
iv.Have faithfully represented the views of authors cited in our paper/
Therefore, we Amera Suriyati Bt Saad,Nor Aina Sofia Bt Mohmad Soffi,Nur Zaihayu Bt Zainoddin, and Nurunnajwa Bt Ani (students name) understand that if any the above is found to be untrue, Institut Pendidikan Guru Malaysia, Kampus Perlis has the full right to take any disciplinary action that the Institut deems fit as denoted under the Coursework For Basic Mathemathic...
. (signature) (signature)
Name: AMERA SURIYATI BT SAAD Name :NOR AINA SOFIA BT MOHMAD SOFFII / C. No:901114-02-5766 I/C No. :900713-02-5448 Date: Date :
. (signature) (signature) Name : NUR ZAIHAYU BT ZAINODDIN Name :NURUNNAJWA BT ANI I/C No. :890905-02-5788 I/C No :900706-09-5018 Date : Date :1.0 INTRODUCTION OF PROBLEM SOLVING
Problem solving is an integral part of all mathematics learning. In everyday life and in the workplace, being able to solve problems can lead to great advantages. However, solving problems is not only a goal of learning mathematics but also a major means of doing so. Problem solving should not be an isolated part of the curriculum but should involve all Content Standards.
Problem solving means engaging in a task for which the solution is not known in advance. Good problem solvers have a "mathematical disposition"--they analyze situations carefully in mathematical terms and naturally come to pose problems based on situations they see.
Good problems give students the chance to solidify and extend their knowledge and to stimulate new learning. Most mathematical concepts can be introduced through problems based on familiar experiences coming from students' lives or from mathematical contexts. For example, middle-grades students might investigate which of several recipes for punch giving various amounts of water and juice is "fruitier." As students try different ideas, the teacher can help them to converge on using proportions, thus providing a meaningful introduction to a difficult concept.
Students need to develop a range of strategies for solving problems, such as using diagrams, looking for patterns, or trying special values or cases. These strategies need instructional attention if students are to learn them. However, exposure to problem-solving strategies should be applied across the curriculum. Students also need to learn to monitor and adjust the strategies they are using as they solve a problem.
Teachers play an important role in developing students' problem-solving dispositions. They must choose problems that engage students. They need to create an environment that encourages students to explore, take risks, share failures and successes, and question one another. In such supportive environments, students develop the confidence they need to explore problems and the ability to make adjustments in their problem-solving strategies.1.1 TIPS ON PROBLEM SOLVING
Apply Plyas four-step process such as:1. The first and most important step in solving a problem is to understand the problem, that is, identity exactly which quantity the problem is asking you to find or solve for (make sure you read the whole problem.
2. Then devise a plan, that is, identify which skills and techniques you have learned can be applied to solve the problem at hand.
3. Carry out the plan. By using the chosen strategy, the problem is solved step by step in this stage. If this solution cannot be found, the strategies are change. All you need is care and patience, given that you have the necessary skills. 4. Look back to check whether reasonable or not and whether there still have another way for the solution.. PLYAS FOUR PRINCIPLES
Understand the Problem
This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they dont understand it fully, or even in part. Plya taught teachers to ask students questions such as:
Do you understand all the words used in stating the problem?
What are you asked to find or show?
Can you think of a picture or a diagram that might help you understand the problem?
Is there enough information to enable you to find a solution?
Devise a plan
Plya mentions (1957) that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
Guess and check
Make an orderly list
Eliminate possibilities
Use symmetry
Consider special cases
Use direct reasoning
Solve an equation
Also suggested:
Look for a pattern
Draw a picture
Solve a simpler problem
Use a model
Work backward
Use a formula
Carry out the plan
This step is usually easier than devising the plan. In general (1957), all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Dont be misled; this is how mathematics is done, even by professionals.
Review or Extend
Plya mentions (1957) that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didnt doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.
1.2 PROBLEM SOLVING STRATEGIESThere are many strategies and lots of practice with these strategies can improve your chances for finding correct solutions to some of the most difficult problems. Problem solving strategies include:
Making a table, chart or graph Making a list Guess and check Finding a pattern Drawing pictures or diagrams Making and using models Choosing an operation Writing a number sentence or equation Using estimation Acting out the problem Working backward Solving multi-step problems Interpreting remainders Using formulas Solving a simpler problem Try and error Eliminate all possibility Solve a simpler but related to problems. Logical deduction Find a general rules
Simulation or experimental
Explain why
Try some simpler cases
Divide and conquer
Pattern reconnection
1.3What do good problem solvers do?
Good problem solvers use a variety of processes and strategies as they read and represent the problem before they make a plan to solve it.
As they read, they use comprehension strategies to translate the linguistic and numerical information in the problem into mathematical notations. For example, good problem solvers may read the problem more than once and may reread parts of the problem as they progress and think through the problem. They use self-regulation strategies by asking themselves if they understood the problem.
Identify the important information and may even underline parts of the problem. Good problem solvers ask themselves what the question is and what they are looking for.
Developing a schematic representation of the problem so that the picture or image reflects the relationships among all the important problem parts. Using both verbal translation and visual representation, good problem solvers not only are guided toward understanding the problem, but are also guided toward developing a plan to solve the problem. This is the point at which students decide what to do to solve the problem. They have represented the problem and they are now ready to develop a solution path.
Think logical solutions and the types of operations and number of steps needed to solve the problem. They may write the operations symbols as they decide on the most appropriate solution path and the algorithms they need to carry out the plan. They ask themselves if the plan makes sense given the information they have.
Using mental calculations or may even quickly use paper and pencil as they round the numbers up and down to get a ballpark idea.
Do the arithmetic and then compare their answer with their estimate. They also ask themselves if the answer makes sense and if they have used all the necessary symbols and labels such as dollar signs and decimals.
To make sure they used the correct procedures and that their answer is correct.
NON-ROUTINE PROBLEM
What is 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 living (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 endeavour.
From the point of view of students, non-routine problem solving can be challenging and interesting. From the point of view of planning classroom instruction, teachers can use non-routine problem solving to introduce ideas (EXPLORATORY stage of teaching); to deepen and extend understandings of algorithms, skills, and concepts (MAINTENANCE stage of teaching); and to motivate and challenge students (EXPLORATORY and MAINTENANCE stages of teaching).
There are other uses as well. Having students do non-routine problem solving can encourage the move from specific to general thinking; in other words, encourage the ability to think in more abstract ways. From the point of view of students growing to adulthood, that ability is becoming more important in todays technological, complex, and demanding world.
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.
The following is an example of a problem that concerns non-routine problem solving.
Consider what happens when 35 is multiplied by 41. The result is 1435. Notice that all four digits of the two multipliers reappear in the product of 1435 (but they are rearranged). One could call numbers such as 35 and 41 as pairs of stubborn numbers because their digits reappear in the product when the two numbers are multiplied together. Find as many pairs of 2-digit stubborn numbers as you can. There are 6 pairs in all (not including 35 & 41).
Solving problems like the one above normally requires a search for a strategy that seeks to discover a solution (a heuristic). There are many strategies that can be used for solving unfamiliar or unusual problems. The strategies suggested below are teachable to the extent that teachers can encourage and help students to identify, to understand, and to use them. However, non-routine problem solving cannot be approached in an automatized way as can routine problem solving. To say that another way, we cannot find nice, tidy methods of solution for all problems. Inevitably, we will be confronted with a situation that evokes the response; I haven't got much of a clue how to do this; let me see what I can try.
The list below does not contain strategies like: read the question carefully, draw a diagram, or make a table. Those kinds of strategies are not the essence of what it takes to be successful at non-routine problem solving. They are only preliminary steps that help in getting organized. The hard part still remains - to actually solve the problem - and that takes more powerful strategies than drawing a diagram, reading the question carefully, or making a table. The following list of strategies is appropriate for Early and Middle Years students in that the strategies involve ways of thinking that are likely to be comfortable for these students.
Look for a pattern
Guess and check
Make and solve a simpler problem.
Work backwards.
Act it out/make a model.
Break up the problem into smaller ones and try to solve these first.
It is important that students share how they solved problems so that their classmates are exposed to a variety of strategies as well as the idea that there may be more than one way to reach a solution. It is unwise to force students to use one particular strategy for two important reasons. First, often more than one strategy can be applied to solving a problem. Second, the goal is for students to search for and apply useful strategies, not to train students to make use of a particular strategy.
Finally, non-routine problem solving should not be reserved for special students such as those who finish the regular work early. All students should participate in and be encouraged to succeed at non-routine problem solving. All students can benefit from the kinds of thinking that is involved in non-routine problem solving.
Examples of non-routine problems and its solutions
Problem 1:Which car was fastest?
CARLAP 01
Saga02:3.50
Waja02:1.41
Understand the problemWhat are the facts?
What do you need to know?Car Saga: 2 min 3.50 s
Car Waja: 2 min 1.41 s
Which car was faster?
2Make a planWhat do you do to solve the problems?
Compare the times.
3Do the workShow the work.2 min 3.50 s > 2 min 1.41 s
4Interpret the answerIs the answer reasonable?
Does it answer the question?
Car Waja is faster.
Yes, it answers the question and is reasonable.
SOLUTIONS2.0 NON-ROUTINE PROBLEMS2.1 QUESTION 1
PROBLEM 1MKB Project Kelvin Grove Primary school + Huron Street Public School.
Spaceship Shuffle Problem
The big spaceships and small spacepods are lined up touching each other as shown. Your mission is to move the spacecraft so that all the big spaceships are at one end of the line and all the small pods at the other end of the line. BUT you can only move two touching spacecraft in each move (so they must move together in pairs).
The minimum number of moves you can make for this problem is 3. Find as many ways of solving this problem in three moves as you can.Now try the same problem with 4 big spaceships and 3 small spacepods. When you have worked out the minimum number of moves for this problem, record it in the table.Now try the same problem with 5 big spaceships and 4 small spacepods. When you have worked out the minimum number of moves for this problem, record it in the table.Look at the table. Can you see a pattern that can help you predict the minimum number of moves for 6 big spaceships and 5 small spacepods and the following problems? Test your pattern.Table Number of big spaceshipsNumber of small spacepodsMinimum number of moves
323
436
5410
6515
7621
8728
9836
10945
STRATEGIES
Draw a table
Drawing sample
SOLUTION
STRATEGY 1
Draw a table
From the table I found a pattern that the minimum number of moves for any number of big spaceships and any number of small spacepods. I found a new formula to count the minimum number of moves for the spaceships and spacepods. I use the formula and list out in the table below.
Number of big spaceships
( x )Number of small spacepods
( y )Minimum number of moves
(xy / 2)
323 x 2 / 2 = 3
434 x 3 / 2 = 6
545 x 4 / 2 = 10
656 x 5 / 2 = 15
767 x 6 / 2 = 21
878 x 7 / 2 = 28
989 x 8 / 2 = 36
10910 x 9 / 2 = 45
The formula that I make is xy / 2. x is to be the number of big spaceships and y is to be the number of small spacepods. We must multiply x and y and then divide it by 2. After that we can get the minimum number of moves. We can use this formula for all numbers of spaceships and spacepods. For example,
Number of big spaceships
= 100
Number of small spacepods = 99
Minimum number of moves= xy / 2
= 100(99) / 2
= 4950
So, we must not list out all the numbers of big spaceships and the numbers of small spacepods more. We only put it into the formula.STRATEGY 2 Drawing sample
When we use 3 big spaceships and 2 small spacepods we worked it out with two type of movement. First type is move, twist and moves and moves again the spaceships and the small spacepods. The second type is twist, twist and twist.
First type
Move to 1 movement
Twist and move to 2 movement
Move to 3 movement
Second type
Twist
Twist
Twist
Try with 4 big spaceships and 3 small spacepods use the second type.
Twist
Twist
Twist
Twist
Twist
Twist
Try with 5 big spaceships and 4 small spacepods use second type.
Twist
Twist
Twist
Twist
Twist
Twist
Twist
Twist
Twist
Twist
Number of big spaceships
( x )Number of small spacepods
( y )Minimum number of moves
323
433 + 3 = 6
546 + 4 = 10
6510 + 5 = 15
7615 + 6 = 21
8721 + 7 = 28
9828 + 8 = 36
10936 + 9 = 45
The conclusion that I can make to predict the minimum number of moves for 6 big spaceships and 5 small spacepods is 15 moves. My pattern is shown at the bottom that is twisting the spacecraft. So that my mission to move the spacecraft with the minimum number of moves until all the big spaceships are at one end of the line and all the small pods at the other end of the line.
JUSTIFICATION
I have two strategies to solve this question. I think the best pattern that most suitable in this problem is the second type of movement. That is twist, twist and twist. We can use this type of movement for any number of big spaceships and any number of small spacepods. It is interesting and makes us be creative thinking. It also makes us easy to understand the movement. We can use a coin for the sample.
2.3 QUESTION 3
I was exploring a puzzle in which matchsticks had to be moved to make a different number of triangles.
I made 1 triangle with 3 matchsticks.
I made it into 4 triangles by a row of two triangles (i.e. 6 matches)
I added another row and counted the triangles and counted the matches.
I made a table of my result and continue adding rows. I found many patterns.
Have a go and see what patterns you find.
See if you can find a rule to predict the total number of triangles if you have 5, 6, 7, 8, 9, 10 rows.
What if you had 100 rows? What would the total number of triangles be?
See if you can find a rule to predict the number of matchsticks needed if you have 5, 6, 7, 8, 9, and 10 rows.
Number of rowsNumber of trianglesTotal number of matchsticks
113
249
3918
41630
52545
63663
74984
864108
981135
10100165
STRATEGIES
Arithmetic progression
Drawing
Use a formulaSOLUTION
STRATEGY 1
Arithmetic progression
The formulaTn = a + (n 1) d
Sn = n / 2
T1 = 1 + (1 1) 3
S1 = 1 / 2
= 1
= 3
T2 = 1 + (2 1) 3
S2 = 2 / 2
= 4
= 9
T3 = 1 + (3 1) 3
S3 = 3 / 2
= 7
= 18
T4 = 1 + (4 1) 3
= 10
T5 = 1 + (5 1) 3
= 13
T6 = 1 + (6 1) 3
= 16
T7 = 1 + (7 1) 3
= 19
T8 = 1 + (8 1) 3
= 22
T9 = 1 + (9 1) 3
= 25
T10= 1 + (10 1) 3
S10=10 / 2
= 28
= 165
Therefore,
T100= 1 + (100 1)3
S100=100/2
= 298
= 15150 Matchsticks
STRATEGY 2
Drawing
I had drawn until 10 number or rows. I get 100 number of triangles when I drawn 10 number of rows. After that I count the number of matchsticks for 10 rows are 165. So, the number of matchsticks for each row is we must plus 3 for all rows. From that I will get a patterns and shown in the table above.
Number of rowsNumber of trianglesNumber of matchsticks for each row
1 x 113
2 x 243 + 3 = 6
3 x 396 + 3 = 9
4 x 4169 + 3 = 12
5 x 52512 + 3 = 15
6 x 63615 + 3 = 18
7 x 74918 + 3 = 21
8 x 86421 + 3 = 24
9 x 98124 + 3 = 27
10 x 1010027 + 3 = 30
100 x 10010000270 + 30 = 300
STRATEGY 3
Use a formulam = t + 2
m = t + 2 +3
m = t + 2 + 3 + 4
m = t + 2 + 3 + 4 + 5
m = t + 2 + 3 + 4 + 5 + 6
m = t + 2 + 3 + 4 + 5 + 6 + 7
m = t + 2 + 3 + 4 + 5 + 6 + 7 + 8
m = t + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9
m = t + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
m = t + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11
So the formula is:
m = t + r [(r + 3) / 2]
To prove,
E.g. for the third row
m = t + r [(r + 3) / 2]
m = t + 2 [(2 + 3) / 2]
m = t + 5
If we want to get the number of triangles we need to square the number of rows. That mean we must multiply the number of each row by itself. So, the formula is t = r.
Number of triangles for 100 rows.
t = r
t = (100)
t = 10000
So, by using the formula m = t + r [(r + 3) / 2] we can get the number matchsticks for 100 rows.
Let m be the matchsticks
Let r be the rows
Let t be the trianglesm = t + r [(r + 3) / 2]
m = 10000 + 100 [(100 + 3) / 2]
m = 10000 + 50 (103)
m = 10000 + 5150
m = 15150 matchsticks
JUSTIFICATION
I have three strategies to solve this problem. I get many patterns in this problem. I have made a new formula that shown in the third strategy. So that I think the third strategy is most suitable to solve this problem. It is because:
i. I will more understand the steps of solution in this strategy.
ii. It also makes me be more creative thinking to solve it.
iii. This strategy shows us step by step until we get the answer.
iv. It gives us clear and easy to solve this problem.
v. To make the formula true I have proof it.
The other strategies are not too suitable than the third strategy. But it can be use for primary students.
JUSTIFICATION OF ALL STRATEGIESAfter doing all the strategies.I think the best strategy is constructing a table because:i. When we had construct a table it can make us clear and more understand about this problem.ii. Besides that it also will save our time because we just tick at the table when we choose all the player sports.iii. This strategy also make students more interesting to solve the question
iv. It also suitable for all level.PROBLEM WITH SELECTED STRATEGIES
MAKE A TABLE
Question: You save $3 on Monday. Each day after that you save twice as much as you saved the day before. If this pattern continues, how much would you save on Friday?
Strategy:
1) UNDERSTAND:
You need to know that you save $3 on Monday. Then you need to know that you always save twice as much as you find the day before.
2) PLAN:
How can you solve the problem?
You can make a table like the one below. List the amount of money you save each day. Remember to double the number each day.DayAmount of Money Saved
Monday$3
Tuesday$6
Wednesday$12
Thursday$24
Friday$48
You save $48 on Friday
REFLECTION
Problem solving has two method, problem representation and problem execution. In this task, we have to search for non-routine mathematical problem. To solve the problem, we need to understand the problem based on Polya four principles.
Before this, we never know what the meaning of non-routine problem is. After doing some research from others resources, we finally understood.
In this task, we also had been asked to build a same problem with selected strategy. We decided to use draw a table or diagram strategy for the solution of the problem. We choose that kind of strategy because we find it was the easiest method to solve the problem. It was also the most simple and easier solution. We think that it was the most efficiently strategy for the children to do and understand.
Discussion:
We decided to use a table method in solving the problem. Based on the Polyas (1957) four phrases of problem solving have become the framework often recommended for teaching problem solving; understanding the problem, devising the plan to solve the problem, implement a solution plan and reflecting on problem, we successfully got the reasonable answer.
3.0 REFERENCES
5.1 Website
1. http://facmathforum.org/library/topics/problem_solving2. ulty.goucher.edu/jcampf/webquest_resources.htm3. http://mathforum.org/library/topics/problem_solving/4. www.rhlschool.com/math.htm5. standards.nctm.org/document/chapter3/prob.htm 5.2 Book1. Vickie K. Doris, 1996. Problem Solving Experiences In Mathematics. New Jersey. Dale Seymour Publications.2.Billstein.(2003).A Problem Solving Approach to Mathematics for Elementary School Teachers, United States of America, Pearson Addison Wesley
3.Ernest R. Duncan.(1985).Mathematics, United States of America, Houghton Mifflin Co.
4.Sybilla Beckmann.(2005).Mathematics for Elementary School Teacher, United States of America, Pearson Addison Wesley
5.W.George Cathcart.(2003).Learning Mathematics in Elementary and Middle Schools, Canada Inc., Pearson Education
READ THE PROBLEM FOR UNDERSTANDING
PARAPHRASE THE PROBLEM BY PUTTING IT INTO THEIR OWN WORDS
VISUALIZING OR DRAWING A PICTURE OR DIAGRAM
HYPOTHESIZE
ESTIMATE OR PREDICT THE ANSWER
COMPUTE
CHECK
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