Answer ALL the questions.
For items 1 to 18, each stem is followed by four options lettered A to D. Read each item carefully and circle the letter of the correct or best option.
1. Which of these statement(s) about odd numbers should a B5 learner indicate as true about odd numbers?
I. Odd numbers are numbers which are not divisible by 2.
II. The sum of consecutive odd numbers gives square numbers.
III. Successive odd numbers have differences of 2.
A. I and II only
B. I and III only
C. II and III only
D. I, II, and III
2. Which of the statement(s) should a B5 learner discover as not true about even numbers?
I. Even numbers are natural numbers which has 2 as one of it factors.
II. The sum of consecutive even numbers is an even number.
III. Successive even numbers differ by 1.
A. I only
B. II only
C. III only
D. II and III only
3. The Sieve of Eratosthenes is a teaching activity a teacher teaching B6 learners can use to generate which of these numbers? ………………… numbers.
A. Even
B. Figurative
C. Odd
D. Prime
4. Which of these statement(s) should a teacher assist a B5 student to discover as false statement(s) about prime numbers?
I. All prime numbers are odd numbers.
II. One (1) is the least prime number.
III. Prime numbers are natural numbers with one (1) and the number itself as the only factors.
A. I only
B. II only
C. III only
D. All of the above.
5. Which of these numbers would you assist a B6 learner to identify as not a composite number?
A. 111
B. 246
C. 517
D. 563
6. Which of the resource materials would you recommend a B5 teacher to use in helping his learners to generate all the factors of 36?
A. Cuisenaire rods.
B. Dienes Multi-base ten materials.
C. Geoboard.
D. Place value chat.
7. A B5 learner should be led by his/her mathematics teacher to indicate which of the options as all the factors of 60?
A. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20 30, 60.
B. 1, 5, 10, 20, 30, 60.
C. 2, 3, 4, 5, 6, 10, 30, 20.
D. 2, 3, 5, 6, 12, 15, 20.
8. All except one of the options is not a multiple of 15. Which of the options should a B5 learner choose as the correct answer to the question?
A. 225
B. 270
C. 330
D. 445
9. A floor of length, 225 cm and width, 100 cm is to be tessellated with the least number of unit square tiles of side x cm. A teacher should lead a learner to discover that the required tile should have what value of x?
A. 10 cm
B. 15cm
C. 20cm
D. 25 cm
10. Which of the options should a B5 mathematics teacher guide his/her learners to state as the prime factorization (product of prime factors) of 600?
A. 21×33×52
B. 22×33×51
C. 23×31×52
D. 23×32×51
Use the Venn diagram shown below to answer questions 11 to 13.
11. Which of the options should a B5 teacher should assist his/her learners to state as the Least Common Multiple (L.C.M) of X and Y?
A. 22×31×50
B. 23×32×51
C. 23×33×51
D. 26×33×51
12. Which of these should a mathematics teacher in B5 lead his/her learners to indicate as the Highest Common Factor of X and Y?
A. 22×31×50
B. 23×32×51
C. 23×33×51
D. 26×33×51
13. The alarms on Mamamudu’s and Zenabu’s hand phones snooze at every 12 hours and 18 hours respectively. Which of these concepts should a B6 teacher employ to help a B6 pupil to determine the hours the two alarms will snooze at the time? ……………….. concept.
A. Divisibility
B. Highest Common Factor
C. Least Common Multiple
D. Prime Factorisation
14. A B3 learner should be guided by his/her class teacher to interpret the common fraction two-fifth as which of these options?
A. Three equal parts of a whole.
B. Three over five.
C. Three parts of a whole divided into five equal parts.
D. Three whole number divided by five whole number.
15. A B3 mathematics teacher should assist B3 learners to discover that a red Cuisenaire rod represents what fraction of a dark green Cuisenaire rod?
A. One-eighth
B. One-ninth
C. One-sixth
D. One-third
16. In teaching B5 learners about decimal fractions in the hundredth using Multi-base Ten Materials, what decimal fraction should a teacher use the long to represent?
A. A Hundredth.
B. A tenth.
C. A whole.
D. None of the above.
17. Which of these should a B6 mathematics teacher lead his class to discover about decimal numbers?
I. The sum of two or more decimal fractions will have the number of decimal places equal to the number of decimal places of the addend with the largest number of decimal places.
II. The number of decimal place of the product of two or more decimal fractions is equal to the sum of the number of decimal places of the factors.
III. The decimal point of a decimal fraction shifts to the right a number of times equal to the index of 10 when the decimal fraction is multiplied by a power of 10.
A. I only
B. II only
C. III only
D. All the three above.
Items 18 to 20 are statements followed by True and False options. Read each item carefully and indicate whether it is True or False by circling the letter of the correct option.
18. A B4 learner must be led to discover that to add or subtract two common unlike proper fractions, he/she has to find the L.C.M of the two fractions and use the LCM to express the two fractions as liked equivalent fractions before using the skill of adding or subtracting liked fractions to find the sum or difference respectively.
A. True
B. False
19. Simplification of a proper common fraction required dividing the numerator and denominator of the fraction by the H.C.F of the numerator and the denomination of the fraction.
A. True
B. False
20. A B4 learner must be led by his/her mathematics teacher to interpret division sentence: as ‘How many times can two-thirds be subtracted from 6?’
A. True
B. False
SECTION B
Q1. What is mathematics?
Ans: Mathematics like other subjects is difficult to define. It is pregnant with many definitions. Some mathematics considers mathematics as a science of number and space. Others think it is a way of finding answers to a problem. Many also think, it is the ability to think critically in order to establish relationship and use them.
Q2. Give THREE reasons why it is important to learn mathematics?
Ans: (a) To help the children to develop problem solving strategies aimed at dealing with specific problems.
(b) Equips children in finding solution to problem in everyday life and other situations such as the teaching profession, industry, law and economics.
(c) To be able to count and make simple calculation with numbers.
(d) To be able to read and understand graphs in their various forms.
(e) To know about money and be able to make simple calculations involving money.
(f) To be able to measure and do simple computations involving measurement.
(g) To be able to recognize shapes and know their properties.
Q3. Distinguish between a problem and a question.
Ans: Every problem is a question to be answered but not all questions are problems. For a question to be a problem, it must be challenging or perplexing. For example
15-7 may be a problem to a basic 2 child but not a basic 5 child. This means that what may be a problem to one child may not necessarily be a problem to another child. A problem therefore is normally considered to be a challenging or perplexing question or situation.
Q4. Distinguish between problem solving and mathematical investigation.
Answer: Problem solving is a process of finding solution to a problem or puzzling, or perplexing situation which involves the use of a known strategy to arrive at a solution. On the other hand, mathematical investigation is the process of trying things out, identifying strategies, using different approaches and methods for solving a problem one has not met before and for which there is no readily available means of solution. In other words, there is NO KNOWN strategy to arrive at solution to the problem.
Q5. State FOUR possible benefits pupils derive in carrying out mathematical investigation.
Answer: Mathematical investigation promotes logical reasoning in pupils. Pupils learn a lot through practical activities.
♦ Pupils understanding of mathematical processes are broadened and are able to think mathematically.
♦ Pupils interest is sustained and the ideas that mathematics is difficult and abstract are removed.
♦ Interest derived through mathematical investigations enable pupils to carry on more investigations thereby enabling them to increase their learning abilities.
♦ Mathematical investigations allow for more adaptability, flexibility and has positive transfer of learning in pupils.
Q6. State George Polya's model for problem solving in their correct order
Answer: V Understanding the problem
V Devising a plan
V Carrying out the plan
V Looking back.
Q7. The second step in George Polya's model for problem solving is 'Devising a plan' State what a child should do before devising a plan.
Answer: • The child should consider a wide range of approaches and make appropriate selection.
• Plan all strategies and consider what might go wrong.
Q8. The third step in George Polya's model for problem solving is "carrying out the plan"
What should a child consider before carrying out the plan?
Ans: • The child must be aware of the implications and make appropriate decisions in the light of the results obtained at various stages.
• Making generalizations and justifying them.
• Simplifying test and attempting to prove results where necessary.
• Completing the giving task.
Q9. The fourth step in George Polya's model for problem solving is "looking back" Explain what this involves.
Answer: V Checking your result and finding out whether it is reasonable.
V Amend the result in the light of your checks on its accuracy.
Q10. (a) State George Polya's four steps for problem solving.
(b) Using a suitable example, explain each step in (a).
Answer: • Understanding the problem.
• Devising a plan.
• Carrying out the plan.
• Looking back.
(b) given that there are 15 people and that each person should shake hands once with all other person in the group, how many handshakes shall we have altogether?
Step 1: Understanding the problem:
►This involves reading the question about three times as a means of trying to understand the demands of the question.
► This question involves shaking hands with all other people.
►Everyone will have a turn once only to shake hands with one person.
►The number of total handshake will be far more than 15.
►A person cannot shake his/her own hand. It is just impossible.
Step2: Devising a plan:
V You arrange the people in a straight line.
V Begin with a smaller number and then move to a large number. For example. 3 people will give 2 handshakes. If one person shakes hands with the remaining 2 persons and the remaining 2 also have 1 handshake each. This means for 3 people, there are 3 handshakes that is 2 + 1 =3.
Step 3: Carrying out the plan.
♦ 2 people give 1 handshake.
♦ 3 people give 2 + 1=3 handshakes.
♦ 4 people give 1 +2 + 3 = 6 handshakes.
♦ 5 people give 1 + 2 + 3 + 4 = 10 handshakes.
♦ Children can now generalize that for 15 people, the number of handshakes will be 14 + 13 + 12 + 11+10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1=105 handshakes.
Step 4: Looking back:
Checking to see whether the answer is correct and or can be applied in a new situation.
Is the answer reasonable or is the answer correct? Check:
Let solve back.
(1) :S = 1 + 2 + 3 + 4+5 + 6 + 7 + 8 + 9 + 10 + 11+12+13 + 14
(2) :S = 14 + 13 + 12 + 11 +10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1
(1) + (2) gives 2S = 15 x 14 (Note that when we reverse 14+13 + 12 + 11 +10 to 1 + 2 + 3 + 4 respectively and adding both figures vertically gives 15.e.g. 1 +14 = 2 +
2 q 15 x14
13 = 3 + 12 etc.) thus, = 2 = 15 x 7 = 105. Therefore the number of
handshakes is 105.
Thus for n number of people, the number of handshake is (n -1) + (n - 2) + (n - 3) + (n - 4) + (n - 5) + (n - 6) + can be used to determine the total number of handshakes if each person should shake his/her friend once.
Q11. Give and explain John Mason's model for problem solving processes.
Ans: (a) Entry phase: At this phase, the problem solver asks himself several questions to determine what he knows about the question. Example, what do I know about this problem. What do I want from this problem. This is followed by the attack phase.
(b) Attack phase: At this stage, the problem solver may get stuck and through deep thinking see relationships and step on clues and exclaim - AHA!, I see what to do now. These are some of the likely feeling of the problem solver.
(c) Review phase: At this stage the problem solver checks his solution, reflects on the key ideas and the key moments and sees if the solution can be extended to a wider context.
Q12. Briefly describe a game which you could use to teach a mathematical concept of your choice. Name the basic mathematics ideas involved in the game.
Ans: A game which can be used to teach the concept of probability is ludo. It is a game played by two, three or four people. A fair die numbered one to six (1 - 6) on its faces is tossed. It is fair since each face has an equal chance of appearing when tossed. Each player uses four buttons which are of different colours usually blue, red, green and yellow. Each player tosses the die in turn and moves the buttons depending on the number that appears on each toss. A button is brought out of home when the number 6 appears after each toss. A player is first to win the game when he/she is able to remove all his buttons from home and moves round through the paths back home first.
Basic mathematical ideas involved in this game:
i) Counting
ii) Idea of possible numbers to appear (sample space)
iii) The number which appears out of the lot (event)
iv) The idea of wanting a particular number to appear out of the lot (probability)
Q13. Give THREE general objective of the mathematics syllabus. Ans: ♦ Socialize and co-operate with other pupils effectively.
♦ Adjust and handle number words.
♦ Perform number operation.
♦ Collect, process and interpret data.
♦ Make use of appropriate strategies of calculations.
♦ Recognize and use functions, formulae, equation and inequalities.
♦ Recognize and use patterns, relationship and sequence and make generalization.
♦ Identify or recognize the arbitrary/standard units of measure.
♦ Use the arbitrary units to measure various quantities.
Q14. Give THREE general aims of mathematics syllabus for primary school in Ghana.
Ans: i) Develop basic idea of quantity and space.
ii) Develop the skill of selecting and applying criteria for classification and generalization.
iii) Communicate effectively using mathematical terms, symbols and explanation through logical reasoning.
iv) Use mathematics in daily life by recognizing and applying appropriate mathematical problem solving strategies.
v) Understanding the processes of measurement and acquire skills in using appropriate measuring instrument.
vi) Develop the ability and willingness to perform investigations using various mathematical ideas and operations.
vii) Work co-operatively with other pupils to carry out activities and project in mathematics.
viii) See the study of mathematics as a means for developing human values and personal qualities such as diligence, perseverance, confidence and tolerance, and as a preparation for profession and careers in science, technology, commerce, industry and variety of work areas.
ix) Develop interest in studying mathematics to a higher level.
Q15. Write down THREE points indicating the rationale for teaching mathematics in primary school in Ghana.
Ans: ► To enable pupils to reason logically.
► To function effectively in society.
► To help pupils develop interest in mathematics as an essential tool for science and research development.
► To enable pupils read and use numbers competently.
► To help pupils solve problem.
► To acquire knowledge and skills that will help pupils to develop the foundation of numeracy.
► To enable pupils communicate mathematics ideas effectively.
Q16. List SIX major areas of content covered in the primary school mathematics syllabus in Ghana.
Ans: 1. Number
2. Shape and Space
3. Measurement
4. Collecting and handing data
5. Problem solving
6. Investigation with numbers
Q17. Distinguish between syllabus and curriculum.
Ans: Syllabus is a list of mathematics topics to be studied.
Curriculum on the other hand is the whole educational experiences of the pupils inside and outside the classroom and the school environment. It includes what is taught and how it is taught. Thus, curriculum directs the teacher on what to teach, teaching and learning materials to select and how to use them.
Q18. What are the important pieces of information the syllabus contain?
Ans: ♦ General objectives
♦ Year and units
♦ Specific objectives
♦ Content
♦ Teaching and learning activities
♦ Evaluation
♦ Profile dimension
Q19. List THREE curriculum materials that are used in primary schools in Ghana. Ans: 1) Syllabus
2) Teacher's handbook or manual
3) Pupil's textbook
4) Other books or sources of reference relevant to the subject or topic to be treated by the teachers.
Q20. What do you understand by the term profile dimension? Ans: Profile Dimension is a concept in the syllabus which describes the underlying behavioral change in learning during teaching, learning and assessment.
N.B: The dimensions considered are knowledge, understanding and application. Knowledge and understanding together constitute lower profile dimension whilst application constitute higher profile dimension. The specified profile dimensions in Ghana are:
Knowledge and understanding (pry 1 - 3) 40% pry 4 - 6 (30%)
Applications of knowledge pry (1 - 3) 60% (pry 4-5) 70%.
Q21. Briefly explain the following terms and give example: (a) Gender (b) Gender bias (c) Gender balance.
Ans: Gender: gender refers to the different characteristics attributed to boys and girls or woman and man. It also means the different roles which society thinks boys or girls, males or females, men or woman should be assigned.
Example: Girls tend to have low interest in science and mathematics whilst men have high interest in mathematics and science as well as subject which demands higher thinking.
Gender bias: This means treating pupils unfairly based on their social roles. Example, where boys are made to study on returning from school, girls are instructed to prepare food in the kitchen or go to the market and sell. Whilst boys are given scientific toys to play with to help develop their problem solving skills, girls are given dull toys to play with.
Gender balance: (Gender equity or Gender equality) refers to a situation where pupils are given equal access to participate in any activity. Example: in the classroom situation, both boys and girls should be given equal opportunities to participate in class discussions.
Q22. Give FOUR reasons that account for less participation of girls in the study of mathematics.
Ans: 1) The perception of mathematics as a male dominated activity.
2) Boys are encouraged to be more independent than girls and this also encourage experiment and problem solving among boys.
3) Peer group pressure causes girls to fear that high attainment in mathematics will hurt the development of their relationships with boys.
4) Activities in most textbook reflect activities associated with males more than those associated with female. The same situation repeats itself in examination questions set for pupils and student.
5) In child care practice, boys are given significantly more scientific and constructional toys which encourage the development of every important mathematical concept and problem solving.
6) Schools, employers and industry appear to show little interest in encouraging or attracting girls with ability in mathematics into some of the fields and professions where there are good applicants. Boys are usually counseled to work towards a higher qualification in mathematics as a career but this is not always the case with girls.
7) In classroom, teachers tend to interact more with boys than with girls. They also give more serious consideration to boys than girls and tend to give opportunities to boys to answer more difficult higher level questions.
8) In school, more men than woman teach mathematics.
Q23. Write down THREE ways by which you would encourage girls in your class to develop interest in mathematics.
♦ Encourage group work and make girls leaders of their group, thus giving them the opportunity to talk
♦ Encourage girls to develop confidence in their mathematical abilities by motivating them through praise or complementing their good performances.
♦ Teaching learning materials used should not put girls at a disadvantage. They should be girl friendly.
♦ Plan appropriate career guidance for girls, so that they will be aware that lack of an appropriate mathematical qualification can bar them from entry into many fields of employment.
♦ Discourage boys from dominating or hijacking group and class discussions, but provides equal opportunities for both boys and girls in the discussion.
Q24. Describe briefly the Behaviorist's theory of learning.
Ans: behaviorist view all form of learning in terms of development of connections between stimuli received and responses displayed by organism or learners. Behaviorist concludes that learning is a process by which stimuli and responses bonds are established when a successful response immediately and frequently follows a stimulus. Some of the early behaviorists include Skinner, Pavlov and Thondike.
B.F Skinner's theory relates to stimulus response relationship and reinforcement. In his view learning is a change in behavior. Thus, as learners learn, their responses in terms of changed behavior increases.
Q25. Describe briefly the developmentalist's theory of learning.
Ans: Developmentalist holds the view that to understand learning; we must not confine our self to observable behavior but must be concerned with the learner's ability to mentally recognize his or her inner world of concept and memories. They believe that in learning, the mental processes of the child must be taken into account. This means children cannot learn the same content as adult and individual children will even differ in the way they learn. In other words, learning is a personal experience and the job of the teacher is to facilitate this process.
They believe that there are developmental stages in the ability of the child to think logically and that children have to reach a certain point of development before they are ready or able to understand a particular mathematical concept or the other.
Piaget, a developmentalist proposed 4 stages. These are Sensory motor stage, Pre- operational stage, concrete operational stage and formal operational stage.
Q26. Give THREE characteristics of pupil in the Sensory motor stage.
Ans 1) Children at this stage are mostly without language.
2) They learn to physically control objects in their environment.
3) At the latter part of this stage, children become aware of others and language begins to develop.
Q27. Give THREE characteristics of a child at the pre-operational stage.
Ans: 1) Children cannot carry out mentally simple operations such as addition, subtraction, multiplication and division because they are not able to relate to abstract stage of things.
2) They have difficulty with height, quantity, time and understanding space and idea of cardinal numbers.
3) They cannot classify things i.e. sort object into groups.
4) They have no reversible thought process and so cannot think logically.
5) They are good at imitating though without understanding.
6) They are self-centered and easily influenced by visual data.
Q28. Give THREE characteristics of children at the concrete operational stage.
Ans: a) Children begin to develop logical system of thought.
b) They achieve understanding of attribute one at a time.
c) They cannot combine a series of operations to perform complex task.
d) They begin to group things into classes and form one to one correspondence
e) They begin to understand time, quantity, space and become capable of operating independently.
f) They can reverse thought processes and order elements or events in time.
g) Cannot formulate all of the possible alternatives when they are given problem.
h) They are able to form mental actions rather than relying on the physical need to perform them.
i) They are also more comfortable with concrete materials during teaching and learning.
Q29. Give THREE characteristics of a child in the formal operational stage.
Ans: a) They can reverse thought processes.
b) They understand time, space, height, distance etc.
c) They can perform mathematical operations based on abstract thought.
d) They have the ability to deal with the abstract in both verbal and mathematical situations.
Q30. Describe briefly Bruner's theory of development.
Ans: Bruner's theory of development suggests that children and adults go through three main stages of development when they are learning some piece of mathematics. He states that any subject can be taught effectively in some intellectually honest form to any child at any stage of development. He proposed three levels. Enactive, laconic and symbolic stages. In simple terms it implies that whenever a child learns new concept, he/she will pass through the concept at each of these levels in turn.
Q31. What is the implication of Bruner's theory of development for teaching a new mathematical concept?
Ans: The implication is that, the presentation of new concept should be made first using concrete materials and followed later with a gradual introduction of abstract symbols after having used picture or diagrams (learning proceeds from concrete - semi concrete - abstract). It is to be noted that the proportion of time spent on each of these stages will vary according to age, experience and ability i.e. younger children will require lots of concrete activities to understand new concepts which may not be the case with older children though some models may still be useful to them.
Q32. Explain briefly Jerome Bruner's Enactive stage of development in learning.
Ans: At this stage the child uses motor responses to past events. The child at this level manipulates concrete materials directly. The child learning involves direct experience and manipulative skills. Example: In performing addition sums: three plus four, a child has to form a set of four objects and another three objects and put them together before determining that there are seven objects in all.
Q33. Explain briefly Jerome Bruner's "laconic" stage of development in mathematics learning giving example.
Ans: It is the stage where the child uses mental pictures of concrete objects. At this stage the child uses mental image to represent a concept. This is based on visual data such as pictures, diagrams and illustrations. Example, a child is able to picture mentally that a collection of two pens added to five pens give seven pens.
Q34. What do you understand by the term CONCEPT?
Ans: concept refers to the knowledge gained through experience or activities. Concept is formed when the meaning of a topic understudy is fully understood so as to apply the idea gained in a new situation.
Q35. How does Richard Skemp describe concept?
Ans: According to Richard Skemp, concept is the mental object which results when we abstracts from something which they all have in common a number of times.
For instance, young children learn the concept of "dog" from repeated contact with many types of dogs. Also children learn the meaning of number by experiencing number in many varied situations or through suitable collection of examples. The same holds for addition, subtraction, multiplication, division and measurement.
Q36.. In the teaching and learning of mathematics, the child learn "concept" and "skills" Explain these two terms.
Ans: concept refers to the knowledge or mental image or idea gained through experience or activities mathematically concept is formed when the meaning of a topic under study is fully
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