CHAPTER TEN
PROBLEM SOLVING
10.1 Objectives
At the end of this chapter, you should be able to:
i. define and explain the term mathematical problem
ii. list the characteristics of mathematical problem
iii. explain the classification of mathematical problem
iv. define problem solving
v. enumerate attributes of problem solver
vi. discuss briefly problem-solving models
10.2 Introduction
In this chapter, the meaning of mathematical problem, characteristics of mathematical problem and problem solving are emphasized. Also, in this chapter classification of mathematical problem, qualities of good problem solver and importance of teaching problem solving were clearly explained.
10.3 What is Mathematical Problem?
Some mathematics students and individuals regard every mathematical question and textbook practice as a problem. However, this definition of a problem is not expected by many mathematics instructors since it does not take into account the nature of the individual confronting a particular difficulty and the situation or moment in which they find themselves.
Nonetheless, some mathematics instructors defined a mathematical problem as a circumstance for which there is no procedure that guarantees a solution. The relevant information of the individual must be recombined in a novel manner to address the challenge. This description is consistent with Godwin (2006), who defines a mathematical issue as "a circumstance in which an individual or group is required to undertake a job for which there is no easily available algorithm that totally dictates the solution approach."
This definition suggests that an issue is more than just a query or a circumstance; it must also be puzzling. Consider the query: what is the perimeter of a square that is 5cm long and 5cm wide? This question should not provide any difficulty for SSII students. A question or predicament is a sort of phrase that demands or seems to require a response. Consequently, a subject or circumstance can only be deemed confusing and thus an issue in respect to a person and a period. In the past, this question presented a challenge for every student in that class, but it no longer does.
10.4 Mathematical Problem Elements
The elements or qualities of a mathematical problem are:
§ Acceptance: the individual or student recognizes it as an issue.
§ Blockage: that the individual or student cannot quickly solve the problem with his or her first try.
The individual's drive and desire to address the problem by investigating other solutions.
As obvious as it may seem, a mathematical problem is a novel situation involving perplexing simple and compound questions about number, space, or relationship that can be answered by the essential utilization of necessary information given directly or indirectly in an analytical procedure that is purely interpretive, logically argumentative, computational, and otherwise dependent on mathematics. Numerous mathematics instructors have praised this concept, which distinguishes exercises from problems in a variety of ways. That exercises are utilized in mathematics class education to offer practice in previously taught learning abilities or as applications of comprehension. In contrast to exercises, problems demand the student or individual to use synthesis or insight. To solve a problem, a student must rely on previously acquired information, abilities, and understandings, but he synthesizes prior learning while applying them to a new context.
10.5 Classification of Mathematical Problems
Even in antiquity, it was believed that certain pupils possessed great intelligence, medium intelligence, and low intelligence. Previously, however, the term intelligence had little psychological value. Gall, Lavator, Benjamin, and others have identified six (6) degrees of intellectual load that a normal problem places on a human problem-solver. The challenges may consequently be divided into six categories requiring solutions: Knowledge is the ability to recall specific information, terminology, steps and sequence, categorization, and criteria. Understanding and interpreting information is comprehension. Application: the use of a principle or body of information to solve issues Analysis is the process of breaking down information into its component elements in order to identify a link or principle. Synthesis is the process of combining components and pieces to make a whole. Evaluation: a determination based on internal and external information as well as requirements for precision.
Polya saw assessment as looking back to assess the relevance of the problem, the adequacy of assumed facts, relations, and methods, the adequacy of all logical, mathematical, or algebraic processes, as well as the accuracy and reasonableness of the produced answer.
According to Akpan (1989) in Godwin (2006), mathematics issues are categorized as mixture problems, distance-rate and time problems, real-world questions, number theory difficulties, proof open-search challenges, and so forth.
This domain was initially published by Bloom et al. and focuses on problem-solving (1956).
10.6 What is the Definition of Problem-Solving?
According to Polya, problem-solving is finding a way around an impediment when none is immediately apparent. He also argued that to know mathematics is to solve difficulties. Problem-solving is the methodical application of previously learned information and abilities to the demands of a new circumstance. This method combines meticulous investigation of a particular circumstance with the synthesis of previously learned information in order to discover a solution to an issue. Critical thinking is a crucial skill for issue resolution.
Educators of mathematics stressed that knowledge is a collection of problem-solving techniques. The mathematics educators' definitions were congruent with the NCTM's definition. Particularly, each and every mathematics instructor emphasized the importance of non-repetitive issues. As long as the learner is unfamiliar with the solution route, everything the student undertakes to arrive at a solution qualifies as issue solving. The majority of responses ended with this demand. In addition to the non-routine aspect of the issue, some mathematics instructors specified other requirements. These extras increase adaptability and independence of thought. The solution path must be within reach of the learner, there must be a "desire or need to find a solution," logic and knowledge must be employed in the process of solving, and there must be a "want or need to find a solution." In addition to the ability to reach a solution, motivation, a process of reasoning, and the use of existing information, the additional requirements listed by the mathematics educators included the possibility of reaching a solution, motivation, a process of reasoning, and the use of existing information.
In conclusion, from the perspective of a mathematics educator, problem solving is the process by which a student resolves a situation for which the technique of approach is unknown or not clear (non-routine) to the learner.
10.7 Who is an Effective Problem Solver?
A proficient problem-solver possesses the following qualities:
§ They do not need to always be correct.
§ They surpass their own predispositions.
§ They search for opportunities inside the issue.
§ They can distinguish between complicated and simple thought.
§ They have a good understanding of the situation.
§ They consider their alternatives.
§ They possess realistic anticipations.
10.8 Importance of Problem-Solving Instruction
According to mathematics educators, the major goal of mathematical problem-solving teaching is not to provide pupils with a collection of skills and procedures, but rather to foster independent thought. The value of training in skills and processes should be determined by the extent to which the skills and processes foster flexible, autonomous thought. Aliyu provided the following as examples of the significance of teaching problem solving: Integration of prior knowledge: Through problem solving, students may apply and integrate previously taught concepts, generalizations, and abilities. The ability to solve problems is one of the most valuable benefits someone may receive from the study of mathematics. Students are able to solve common problems successfully and critically reason. Pupils' Development: problem solving enhances students' mathematical reasoning and communication skills. It offers the intellectual environment for mathematical growth and transforms the learner into a junior researcher in mathematics. Problem-solving provides pupils with the opportunity to discover mathematical formulas and techniques. It is considered to be the most fundamental mathematical activity. Problem-solving is the foundation for other mathematical processes such as generalization, abstraction, theory construction, and idea development. Class interaction: problem solving increases relationships between students and between students and the teacher. Consequently, performance and response in mathematics improved. Creativity of facts: problem-solving is a prerequisite for innovation in mathematics and encourages pupils to become more analytical in their decision-making abilities. It is an excellent preparation for advanced mathematical studies.
There are key parts of mathematical material that enhance or facilitate problem-solving in the discipline. Facts, concepts, definitions, symbols, axioms, principles, operations, algorithms, and theorems, according to Godwin (2006), are fundamental aspects that facilitate problem resolution.
It provides a solid natural foundation for verifying a broader problem-solving scenario. 2+6=8
This symbolizes the items or elements under consideration in the problem, such as a triangle with a right angle.
This gives appropriate meaning for the problem's ideas and facilitates problem understanding. The following are examples of Integer properties: Vacant property Community Property Inverse Quality Other cumulative properties
Symbols communicate distinct meanings and aid in issue interpretation; for instance, f(x)dx will evoke an integration reaction and not a differentiation response.
Axioms are the fundamental and defining characteristics or laws of a mathematical system.
Principles - There are general rules that aid in problem-solving by determining some essential steps and work patterns in advance and providing a standard approach. For example, the mathematical induction principle. If p (1) is true and p (k+1) is true, then whenever p (k) is true, p (n), a positive integer or natural number, is also true.
Operations- There is specific basic functions for changing one, two or more items of a set at once to a single element. It gives issue solution with legitimate procedural strategies for reducing complicated expressions to their basic versions. E.g. (24×3)/8=9 here 24, 3 and 8 have become 14.
Algorithm- There is steps to be followed to address a given problem.
Theorems- It constitutes the ultimate objective of all pure mathematical endeavor. In regular or non-routine application problem solving, they offer circumstances to hunt for or to develop, so that a beneficial truth may assumed or employed more directly.
10.9 The use of heuristics in problem solving
It is any strategy to problem-solving, learning, or discovery that involves a method that is not guaranteed to be ideal or faultless but is enough for achieving immediate objectives. Heuristic objectives are not formal problem-solving models, but they can be utilized as an approach to problem-solving when flawless or optimum solutions are not expected. Heuristics are often mental short-cuts that facilitate the problem-solving process.
It is a mental shortcut that enables individuals to solve issues and make decisions fast and effectively. It helps us uncover the most efficient and effective solutions to challenges. Mathematics encompasses much more than the study of numbers, spaces, and patterns.
While an algorithm is a series of steps designed to solve a certain issue. Heuristics play a significant part in the problem-solving process. Heuristics exist because, most of the time, they facilitate the discovery of a straightforward solution to difficult issues. In the case of insight difficulties, heuristics are not a poor problem-solving technique. Indeed, an innovative and creative approach is the only way to reach a solution.
10.10 Methodologies and heuristics for problem resolution
There are two (2) problem-solving methods: the pure discovery method and the guided discovery method. The pure discovery method relies on the student's prior knowledge to approach the problem without the teacher's guidance. This strategy has benefits and drawbacks. The majority of the time, the student wastes a great deal of time, but he may discover more concepts than the teacher anticipated. Oftentimes, the concept discovered is not the one anticipated, and the concept discovered may be incorrect. It is a very challenging approach and an incorrectly discovered concept. Therefore, this practice is discouraged. The following are examples of pure discovery approaches falling under this category
a. Intelligent Guessing Discovery Approach
The following are suggested simple steps to approach a problem, intelligent by the students without involving the teacher:
Step 1: Search for entry cues at every level
Step 2: Apply simple rules of arithmetic such as if A or B is digit, then
i. A+A is always even
ii. A+B cannot be more than 18
Step 3: Do not be afraid to guess an answer
Step 4: Test your result. This process is very useful in the solution of cry arithmetic problem such as:
Example: If ZOO + Z =BEE what number do these letters represent
Working Apply the steps suggested above,
For a cue, notice that in the ten’s digit, no number is added to 0 to get another letter E, this means that there must be a “carryover”,from the unit column.
Again, since the letters O+Z can never be more than 18 the number carried must be 1.
Then 1+0=E the only digit to be added to 1 to have a carryover is 9.
0=9 if 0=9 then E=0(zero) since 1+9=10
Then Z, must be 1
The solution is 199+1=200
Z=1, 0=9, E=0, B=2.
ii) Solve the following problems using alpha metric(cryptarithmetic) approach in which each letter represents a unique digit.
HAVE
+ SOME
HONEY
You can adopt the steps provided to verify this solution.
Sol: 1486+9076=10562 thus, H=1, E=6, N=5, M=7, S=9, O=0
iii) SPEND
ˉ MORE
MONEY
Sol: S=6, P=4, E=7, N=9, D=0, M=5, O=8, R=1, Y=3 which gives: 64790-5817=58973
Iv) SEND
+ MORE
MONEY
Sol: S=9, E=5, N=6, D=7, M=1, O=0, Y=2, R=8, which give: 9567+1085=10652
b. Solution by Intelligent Elimination
Consider the following problem WAEC, June, 1999 question objective
Example 2:
It is observed that
1+3 = 22
1+3+5 = 32
1+3+5+7 =42
1+3+5+7+9 =52
If 1+3+5+7+9+11+13+15 = p2 find p
(A) 6 (B) 7 (C) 8 (D) 9
An intelligent solution by elimination goes like this. Sum is always a perfect square of all the options only C is a perfect square of number of terms, 8. Answer is C.
Group Two
Guided discovery Approach is a process of instruction in which the student discovers the concept by himself under the special directives given by the teacher. The directive may be in form of:
a) Socratic question
b) Teacher initiated activities
c) Discovery chart
d) Inductive reasoning
e) Deductive reasoning
a. Socratic Questioning Approach
Example 3: how would you discover that for any number, no=1
Working: consider the following dialogue between the teacher (T) and the student (S1)
T What is the meaning of 30?
S1 It is zero, sir
T Do you agree with the answer, S2?
S2 I don’t know, sir
T What is the value of a number divided by it?
S2 It gives 1
T Good, then 32 Divides 32: gives what?
S2 1, sir
T Very Good, what do you think 80 will be?
S2 It will be 1 also
T Very Good, what about any number, n0
S2 1 sir
T Good, what is your conclusion about the value of any number raised to power?
S Any number raised to power zero is 1
b) Inductive Reasoning:
Is a process examining particular cases for a pattern and coming to generalization from that pattern.
Example 4: consider the following and answer the question that follows: 142-132 = 14+13 272-262 = 27+26 392-382 = 39+38 1992-1982 = _ _ _ _
Do you agree with this, check this it is true?
What pattern do you discover? Can you generalize with any number n, the formula?
If you are able to come to generalization from the pattern, then this is an example of inductive reasoning. The generalization is that for any number n: n2-(n-1)2= n+(n-1). Inductive reasoning is very important in prediction, in planning, in budgeting etc.
Student Activity
1. Given that p=8, find values for these letters so that AT+PT=FEE (All letters are one-Digit numbers)
2. Examine the sequence of number; 1,1,2,3,5,8,x,y,z. find the values of x,y,z.
3. Apply Socratic questioning approach to set a student discover that a-1= 1/a
4. It is observed that
2+4 = 22+2
2+4+6 =33+3
2+4+6+8 = 44+4
If 2+4+6+8+10+12+14+16= p2+p, find p.
10.11 Deductive Reasoning Approach
It is a process of giving deductive logical argument in which all the reasons call premises are accepted as true and the conclusions necessarily follow from the reasons. This leads to mathematical proof in geometry or other branches of mathematics. In geometry proof, the format is made up of four parts: the givens, required to prove, the construction (If any) and the proof. All these techniques are contained in the senior secondary mathematics and further mathematics curricula. Each of these parts must be learnt separately in one full lesson or more.
a) Givens: Given include explicit and implicit givens. Explicit givens are data stated and easily seen contained in the problem implicit givens are made up of pieces of knowledge, information, theorems definitions and previous knowledge implied in the problem, not stated but are necessary to unable you solve the problem.
b) Required to Prove: In identifying the givens, you also required to identify the unknown or what you are required to prove in the problem.
c) Construction: A combination of explicit and implicit givens usually leads to the required construction and statements, which are premises to the conclusion which we call.
d) The proof.
10.12 Definition of Mathematical Proof
A mathematical proof is a valid argument using axioms or already proved true statements to establish the truth of a proposition. A proof is therefore valid argument in which the conclusion is necessary derived from a set of true statements known as Premises.
An argument is a mathematical proof if and only if it satisfies the following two conditions.
i. All the premises are true, and
ii. The argument is valid and leads to a valid conclusion.
10.13 Kinds of mathematical proof
i. Deductive proof: this is the usual formal proof used in geometric proof. e.g. prove that the sum of angles of a triangle is 180
ii. Conditional proof: the argument is of the form” if p…then q”. E.g. if a line is perpendicular to a radius of circle at it point of extremity, and then the line is a tangent.
iii. Bi-conditional proof: this type of proof contains two conditional statements. Take example of this statement “quadrilateral is a square if and only if all the four sides are equal and one of the interior angles is a right angle”
iv. Proof by counter example: the general statement is usually used to prove or disapprove. If one example can be found which obeys all the premises but makes the general statement wrong, then that example is a counter example, and it proves the general statement wrong.
Example: the general statement “any 4 points in a plane form quadrilateral when joined in order is proved wrong if any 3 of the points be in a straight line.
10.14 Problem solution in mathematics
Some discussions would be made to guide you in finding solutions to mathematical problems, especially in the areas of proofs in geometry, variations, proportion.
Example 1: Prove that the sum of angles of triangle is 780°. Write down all the explicit and implicit given, required to prove and construction in the problem.
Working
Explicit given include:
i. Any triangle
ii. The three sides of triangle
iii. The sum of the three angles
iv. The angle 180°
Implicit given include:
i. A specific triangle ABC representing any triangle
ii. The sum of the angles ABC+ABC+BCA
iii. The angle on a straight line is 180°
iv. When lines are parallel (a) Corresponding angles are equal
(b)Alternative angles are equal
The unknown is the required to prove that: BAC+ABC+BCA= 180°.
Construction: the explicit and implicit given suggested drawing any triangle. In this case, we have chosen to draw triangle ABC. You are free to draw another triangle PQR or XYZ. But once you have drawn a specifically named triangle, restrict your references to that triangle.
Based on given, we draw straight line at by producingBC to D to set the straight line.
_ _ _ _ _ _ _ _ _ _
Proof: Now after considering the explicit and implicit given, the proof becomes very clear.
Variations
Direct variation: A relationship between any two quantities in such a way that is constant and called a direct variation OR we can also write where is a constant. A short form of writing varies directly as x is “ ’’
Characteristics of Direct Variation
i. OR where is a constant.
ii. If is increased, y is also increased in the same ratio.
iii. If is decreased, y is also decreased in the same ratio.
iv. When the graph of direct variation is drawn, it is a straight line.
Working Examples
Consider the relationship between two quantities as in the discovery chart
2 | 3 | 5 | 4 | 7 | 6 | 1 | 10 | 11 | m | X | |
6 | 9 | 15 | 12 | - | - | - | - | - | - | - |
a) Fill the blank spaces and answer the following questions
i. Discover the relationship between Ans: Y=3times x
ii. Increased what happens to y? Ans: Y is also increased
iii. is multiplied by 3, what happens to y? Ans: Y is also multiplied by 3
iv. Consider the ratio of for each pair of values. What do you notice? Ans: always.
v. What is the equation connecting Ans: Y=
b) Fill the following blank spaces
i. Y=----of x ii. X=----of y Ans: i. 3 ii.
c) Is it true that what is the constant Ans: constant =3
2. The cost (c) of feeding students per day is directly proportional to the number (n) of students. The cost of feeding 20 students a day is N800:00
a) What is the cost of feeding 36 students at the same rate?
b) What is the cost of feeding 80 students at the same rate?
Indirect or Inverse Variation
It is a relationship between x and y in which xy= constant, is called an inverse variation. The equation of an inverse variation is xy=k or where k is a constant.
Characteristics of an Inverse Variation
i. In an inverse variation or where k is a constant.
ii. If x is increased, y is decreased by the same ratio
iii. If x is decreased, y is increased by the same ratio
iv. If the graph of inverse variation is drawn, it forms a straight line bending backwards.
Working Examples
2. The number of days certain of quantity food lasts is inversely proportional to the number of students. The food lasts 2oo days for 100 students.
a) How long will the food last for 250 students.
b) How many students shall be retained to that the food lasts for 550 days?
Working: Let d stand for number of days and s stands for number of students now let d varies inversely as S when d=20, S=100,
Relationship between d and s is
a. When s=250, then d= food will last 8 days for 250 students.
b. When 0=50 then
Number of students will be 40.
Discovery Chart
It is a pattern chart in which the teacher has supplied some of the solutions and left some columns empty. The students are expected to follow the teacher’s model to complete the missing columns and generalization.
Consider the discovery chart and answer the questions that follow.
12 | 6 | 2 | 1 | 3 | 16 | 48 | X | |
4 | 8 | 24 | 48 | - | - | - | - |
i. If x is increased, what happens to y? Ans: y decreases
ii. If x is decreased, what happens to y? Ans: y increases
iii. If x is multiplied by 2, what changes occur in y? Ans: y is divided by 2
iv. Is it true that if x is divided by 2, y is multiplied by 2? Ans: yes
v. Calculate the product xy in each pair what do you notice
vi. Is it true that xy=constant? What is the value of the constant?
Conclusion: xy=constant =48 in each case.
Proportion
Consider this example
In Aliyu’s family, there are 2 boys and 3 girls while in Ndagara’s family there are 4 boys and 6 girls. Calculate the ratio of boys to girls in both families.
Working: Aliyu’s family, boys is to girls as 2 is to 3 in short in Aliyu’s family, boys:girls =2:3
We say in Aliyu’s family the ratio of boys to girls is 2/3 (read 2 is to 3) similarly in Ndagara's family the ratio of boys to girls is 4:6 or in fraction 4/6
Now when written in a lowest term
4:6=2:3 0r we say that have the same proportion.
Definition: A proportion is a statement that two ratios are equal. In general is a proportion which can be written as a:b=c read “a” is to b as c is to d.
Terms of Proportion: In the proportion or a: b=c: d a is the 1st term, b is the 2nd term, c is the 3rd term and d is the 4th term. Thus, b and c are the mean term, a and d are the extreme terms. So the mean product is the product of the mean terms = b×c.
The extreme product is the product of the extreme terms =a×d.
In ordinary language, we discover that bc=ad, that is mean product= product of extremes. We call this rule cross multiplication. Thus, in the proportion we have that 2×6 = 4×3 i.e. product of means = product of extremes. Now, identify (a) the 1st, 2nd, 3rd and 4th terms (b) the mean product (c) the extreme product of the proportion .
Example 1: If find the value of m.
Working: product of mean = 14×m = 14m
Product of extremes = 18×21 Equating, we have 14m = 18×21 m=
Students Activity
1. Write down at least 2 characteristics of mathematical proof
2. Give one example of a conditional statement
3. Construct one example of bi-conditional statement
4. What type of statement is this? A rectangle is a square if and only if all the sides are equal
5. The distance, 5km. travelled by a motorist at a constant speed varies directly as the time, t hours. In time 5hours the motorist covered 400km, find
(a) The time he will cover at distance 960km
(b) The distance he covered in 8 hours. Give two reasons for saying that, this is a direct variation
6. Given that
(a) Identify the 1st, 2nd, 3rd and the fourth terms
(b) Write down the mean product
(c) Write an equation connecting the mean product and the extreme product
7. If p:q:r = 2:3:4 evaluate(i) (ii)
8. Write down main differences between direct and indirect variation
9. Write down all the explicit and implicit given and use them to prove that the angles on the same segment of a circle are equal
10. Examine the sequence of numbers and find the missing numbers: 0, 3, 8, 15, 24, X, Y, 63, 80. Find the values of X and Y.
11. State two principles necessary for solving cryptarithmetic problem
12. Apply Socartic questioning approach to discover that = given that = and logaM= x, logaN= y and M=ax, N=ay use a named discovery approach to discover that logaMN= logaM+logaN.
10.15 Problem Solving Models
The following models in problem solving are as follows:
Polya’s model
Scope’s model
Krolik’s model
Rubenstin’s model
Polya’s Model
Polya. G(1957) defined four phases for the solution of any problem.
Phase 1: Understanding the problem
Phase 2: Planning for the solution
Phase 3: Computation skills
Phase 4: Looking back.
Understanding the problem:Ability to identify the givens and the structure of the problem, and also ability to grasp the relationships among the given.
Planning for the solution: This involves translation the givens and their relations to form symbols, diagrams, equations or mixed models then using the givens and their relations to develop the algorithm necessary for the solution of the problem.
Computation skills: This involves a variety of ways of carrying out computations in the algorithm.
Competence in the skill is acquired through repetitive practice. Many students learn by solving similar and identical kinds of problems.
Looking back: This involves checking through the processes of understanding, planning and computation
Scope (1973): He proposed six steps for the solution of any problem.
Step 1: Rephrase
Step 2: Trial and error
Step 3: Search for pattern
Step 4: Insight
Step 5: Justify insight
Step 6: Generalization
Rephrase: Here the problem solver needs to restate the problem in a precise language, so that if there are ambiguities, there are eliminate.
Trial and Error:Applying trial and error to find way to the clue, by trying several approaches. That is if one’s effort fails, let him proceeds to next.
Search for Pattern:The problem solver should begin to look for patterns which may link the present problem to others that he might have come across in the past.
Insight: Problem solver at particular times may not have solution to a problem at his finger tip. But at later time, the problem solver discovers the clue that will lead him to solve the problem at hand.
Justify insight: The problem solver must now utilize the approaches or logical reasons to back up his intentions.
Generalization: The problem solver now looks back on the completion of work to see if the results or the methods used to be generalized.
Robenstein (1975): defined four stages in problem solving. Which include: Preparation, Incubation, Inspiration and Verification?
Preparation:At this stage, the problem solver examines the elements of the problem and studies their relationships.
Incubation: At this stage, if the problem solvers search for a solution could no yields, no positive result. Let him put the problem aside and sleep on it for a while.
Inspiration: After sleeping on the problem for sometimes, the problem solver may suddenly get an idea or insight that will enable him to solve the problem at hand.
Verification:Here at this stage, the problem solvers look back to his solution to verify if it makes meaning.
Krulik and Rudnick (1980): the authors proposed five stages in problem solving. This includes: Read, Explore, Select a strategy, Solve and Review & Extend.
Read: The problem solvers read the problem, note keywords and know the unknown.
Explore: Here the problem solver draw the diagrams, making a chart a search for a pattern.
Select a strategy:At this stage, the problem solver experiment, think of simpler problem and make a guess to a solution.
Solve:The problem solver carryout the plan.
Review &Extend: At this stage, you verify your answer and look for interesting variation of the original problem.
10.16 Summary
§ Problem in mathematics and its characteristics are discussed.
§ Problem solving and qualities of good problem solver were outlined.
§ A mathematical proof is a valid argument applying axioms or already proved true statements to establish the truth of the proposition.
§ A direct variation is any relationship in the form A=KB where K is the constant. Inverse variation is any relationship in the form AB=K where K is the constant
§ Problem solving models were explained and
§ Phases involved in each of the model are well described
References
Channon, J. B. (1991). New General Mathematics for West Africa, SSS2, Longman Group UK Limited.
National Teachers’ Institute (2000). Nigeria Certificate of Education Course Book on Mathematics Cycle 2