PROBLEM (MATHEMATICAL PROBLEM)
Definition of Mathematical Problem
Mathematical problem can be defined as a task for which:
· The person confronting it wants or needs to find a solution.
· The person has no readily available procedure for finding the solution.
· The person must take action to find a solution.
Types of Problems
Problems can be classified as routine or non-routine.
a Routine Problem
Routine problem defined as one of the problem which the problem solver already possesses a ready-made solution procedure. It involves using one of the four arithmetic operations to solve problems in nature. Children are able to practice routine problem as early as they are in the age of 5 or 6. They practice using routine problem such as combining, or separating toys. This shows one of the basic activities of using routine problem every day. Apart from that, students may also practice routine problems in schools. For example, if a student has learned the procedure for long division of whole numbers, then a long- division problem represents a routine problem.
bNon-routine Problem
Non-routine problem is one of which the problem solver does not have a previously learned solution procedure. In non-routine math problems, it is also a problem that requires certain skills, formulas and some arithmetic operations such as addition, subtraction, division and multiplication to solve the problem. For example, students cannot simply solve a two unknown type of question. They have to learn basic formula or skills that are related to the question and then solve the problem. Through this way, students have invented a solution method that is new for the students.
Problem Solving
Definition of Problem Solving
In the early 1990, problem solving was viewed as a mechanical, systematic, and often abstract (decontextualized) set of skills, such as those use to solve riddles or mathematical equations. It is also defined as a process which used to define the correct answer for the question or statement.
FOUR STEPS IN PROBLEM SOLVING
In Polya strategy, it is stated that one must apply a routine procedure to arrive at an answer when it comes to solving an exercise. To solve a problem, one has to pause, reflect and perhaps take some original step never taken before to arrive at a solution.
The four steps of problem solving are as follows:
A) Understanding the problem
B) Device a plan
C) Carry out the plan
D) Look back
1. UNDERSTANDING THE PROBLEM
Before you solve a problem, you must read and try to understand what exactly the statement is written about. Make sure you are able to find the main point or clue from the statement. You should know how to practice these few steps to enable you to understand more clearly about the statement given.
· Do you understand all the words?
· Can you restate the problem in your own words?
· Do you know what is given?
· Do you know what the goal is?
· Is there enough information?
· Is there extraneous information?
· Is the problem similar to another problem you have solved?
2. DEVICE A PLAN
There are many ways to solve problems. We should choose the appropriate strategy to solve different kinds of questions. A partial of strategies are included:
· Guess and check
· Use a variable
· Draw a picture
· Look for a pattern
· Make a list
· Solve a simpler problem
· Draw a diagram
· Use direct reasoning
· Use properties of numbers
· Solve an equivalent problem
· Work backward
· Use cases
· Solve an equation
· Look for a formula
· Do a simulation
· Use a model
· Use dimensional analysis
· Identify sub goals
· Use coordinates
· Use symmetry
3. CARRY OUT THE PLAN
· After deciding on what strategy we are going to use, then we shall proceed to carry out the plan. Implement the strategy or strategies that you have chosen until the problem is solved.
· You may need a reasonable amount of time to solve the problem by using the strategy or strategies that you have chosen. If you failed to solve the problem, seek hints from the others or put aside for a while.
· Always try to use new strategies to solve the problem if you have failed by using the previous strategy that you had chosen before.
4. LOOK BACK
Once you have done or solve the problem, try to look back and make sure that the strategy you used is correct and satisfy the statement of the problem. If you missed steps for the problem solving, make sure you redo the problem again.
EXAMPLE
The houses on Main Street are numbered consecutively from 1 to 150. How many house numbers contain at least one digit 7?
STEP 1: UNDERSTAND THE PROBLEM
Teachers should ask students to read the text and try to understand the question. Then, teachers should explain briefly about the statement and ask students to underline the main point. In this question, the main point is to find the house number that contains at least one digit 7.
STEP 2: DEVICE A PLAN
Strategy used 1: Make a table
Strategy used 2: Identify the pattern
STEP 3: CARRY OUT THE PLAN
Strategy used 1: Make a table
Make a table for the arrangement of numbers from 1 to 150. Highlight and circle all the numbers that is under number 7. The numbers highlighted are the numbers that contain at least one digit number 7. Count the quantity of the numbers that are highlighted and circled
1
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2
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3
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4
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5
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6
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7
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8
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9
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15
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32
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150
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v The quantity numbers of the numbers that are highlighted and circled is 15.
Therefore, the Main Street contains 15 houses which contain at least one digit 7.
Strategy used 2: Identify the pattern
Identify the pattern by making a list. List all the numbers that contain at least one digit 7.
7, 17, 27, 37, 47, 57, 67, 77, 87, 97, 107, 117, 127, 137, 147
Count the quantity of the numbers that are listed out.
v The quantity numbers of the numbers that are listed is 15.
Therefore, the Main Street contains 15 houses which contain at least one digit 7.
STEP 4: LOOK BACK
This question is suitable to use a look for pattern strategy and also make a table. It is easier for students to identify the numbers when they highlight and circle the numbers. Moreover, they can understand clearer about the strategy used.
The figure below shows twelve toothpicks arranged to form three squares. How can you form five squares by moving only three toothpicks?
STEP 1: UNDERSTAND THE PROBLEM
Twelve toothpicks are given and are arranged. Students should understand that they have to form five squares by move three toothpicks only.
STEP 2: DEVICE A PLAN
Strategy used 1: Draw a diagram
Strategy used 2: Analogy (Act out)
STEP 3: CARRY OUT THE PLAN
Strategy used 1: Draw a diagram
Teachers should teach students to draw all possible diagrams that can form by moving three toothpicks only. Blank paper, pencils and erasers are given to each of the student. Then, ask students to try to come out with a diagram that forms by move only three toothpicks.
1. Draw the diagram in a piece of paper using pencil.
2. Erase three lines that you want to remove.
Strategy used 2: Analogy
If students failed to draw a simple diagram, then teachers can use analogy strategy to make a clear cut explanation about this statement. First, teachers should understand the statement well and try to come out with real toothpicks as model to solve this problem. Twelve toothpicks are needed for this question. Therefore, teachers have to act out the process on how to move the toothpick correctly.
Twelve toothpicks are used to build the model. We rearrange only three toothpicks to form five squares:
1. Build model by using real toothpicks.
2. Remove three toothpicks.
3. Try to rearrange three toothpicks to form five squares.
4. Finally, five squares are formed. (4 small squares and 1 big square)
STEP 4: LOOK BACK
Draw a diagram is a simple strategy because it gives a clear view about the problem. For those who are using visualizing intelligence, this strategy is the best strategy to solve problems. Besides that, this strategy is also easy to carry out. We got the answer for this problem by using draw a diagram strategy. Besides draw a diagram strategy, build a model also a best strategy. It also can give a clear view to solve the problem. We managed to get answer by using this strategy.
EXAMPLE
Three apples and two pears cost 78 cents. But two apples and three pears cost 82 cents. What is the total cost of one apple and one pear?
STEP 1: UNDERSTAND THE PROBLEM
First of all, ask students to read and try to understand the text. Students should able to highlight the main point and what is it the statement wanted them to find. If students failed to explain the meaning, then teachers have to guide and explain more to the students by using certain methods such as:
v Do apples and pears have the same cost?
v Do we need to find the cost for an apple and a pear?
v What is the total cost for an apple and a pear?
STEP 2: DEVICE A PLAN
Strategy used 1: Simplify the problem
Strategy used 2: Identify sub goals
STEP 3: CARRY OUT THE PLAN
Strategy 1: Simplify the problem
Create a table which consists of apple, pear and cost to be seen more clearly.
EVENT
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APPLE
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PEAR
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COST (CENTS)
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1.
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78
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2.
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82
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TOTAL
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160
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From the table, we can conclude that:
v Total apples = 5
v Total pears = 5
v Total cost = 160 cents
To find the total cost of an apple and a pear, divide 160 cents with all the amount of the apple or the pear, 5.
Now, the answer of 160 cents divide by 5 is 32.
Therefore, the total cost of an apple and a pear is 32 cents.
Strategy 2: Identify sub goals
The statement have state that the total of 3 apples and 2 pears equal to 78 cents; 2 apples and 3 pears equal to 82 cents. From that, we can total up the amount of apples and pears from the two events and equal to 160 cents (total cost). We can simply get the cost for an apple and a pear by divide the total cost by 5.
The answer of 160 cents divide by 5 is 32.
Therefore, we can conclude that the cost for an apple and a pear is 32 cents.
STEP 4: LOOK BACK
There are many methods that can be used in this kind of question. However, simplify the problems and identify sub goals are among the easiest and quickest way or strategy to solve this problem. Make sure the students understand the steps and are able to follow the steps that the teachers given. Teachers can also use analogy strategy to solve this problem if necessary.
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