IF YOU THINK WAEC MATHS IS HELL,HERE ARE 84 STEPS TO MATHS HEAVEN (4)

[caption id="attachment_2628" align="alignleft" width="500"] WAEC EXAM HALL[/caption]

100 EXAMPLES OF QUESTION-PHRASES USED BY WAEC IN MATHS EXAMS

QUESTION  TYPES, VISUALIZATION & CRYSTALLIZATION

46. Learning maths through concepts and rules will be difficult without visualization of concepts in the mind. Learning maths vocabulary makes little practical sense to a student when learnt by itself. For example “numerators” or “denominators” are famously important yet they serve no practical purpose for most learners except when they provoke thoughtfulness and program understanding. Most vocabulary and rules lead to what is called inert knowledge .But students can make sense with a slight shift of emphasis to VISUALIZATION as a central concept.

47. The difference between a ”good” and “bad” maths  student is the amount of visualization each brings to bear on the subject. Non-performing students need careful help to visualize. Apart from the activities listed under 44 above, lots of instructional objects/educational aids need to be used in the classroom. Visualization crystallizes what can be called “PICTURES IN THE MIND”. Class activities directing students on how to visualize questions and concepts IN THE MIND should be emphasized with many instructional materials.

48. The problem that students face when they are confronted with maths questions are mostly in 3 areas:

              a. Inability  understanding the questions asked.

              b. Lack of knowledge of how to start or how to proceed in finding a solution to questions.

              c. Ignorance of what is really expected by the examiner.

49. To overcome these problems there is also a need for students to get familiar with mathematical question phrases Since maths is a science  of precision and exactness, the words used in forming the  questions or the concepts combined within each questions show or require specific meanings  or visualizations which must be  understood by students. The terms usually as follows:

 50. CALCULATE: Ascertain the solution by maths methods you know

51. CONSTRUCT: Form by putting together the part, build up, fit together, frame according to certain conditions.

52. DETERMINE: Settle, decide what is to be done, ascertain or calculate.

53. DRAW: Make a neat pencil figure, clearly or neatly.

54. EVALUATE:  Determine the value or the amount.

55. EXPRESS: State definitely (or show clearly)

56. FACTORIZE: turn into factors or break the multiples into smaller units.

57. FIND: Discover, seek out or obtain by search.

58. LOCATE: Find the place or discover the exact place or determine the where about.

59. MAKE A SUBJECT: Separate out, make to stand  alone or express in terms of others.

60. MEASURE: State the size or weight or ascertain the amount by quantifying.

61. PROVE: Ascertain as a truth by argument or otherwise i.e. show that something is true or not.

62. SHOW: Present a better view, allow or cause to be seen.

63. SIMPLIFY: Make less difficult or easier to understand.

64.  SKETCH: Give only a rough line of a figure of it. Make out a rough finished drawing. 

65. SOLVE: find the answer to the problem.

66. In the exam hall the problem of students and candidate principally is how to understand the meaning of the questions asked, determine applicable concepts and  visualize what principles are been tested. There is also a general lack of ability to answer questions orderly or meaningfully.

67. Please find below a list of 100 exam question-phrases usually used by examiners for class or private study exercises on imagination and visualization (numerically, graphically and pictorially).

 CAN YOU VISUALIZE THESE  WAEC MATHS QUESTION-PHRASES/CONCEPTS IN YOUR MIND?

MATHEMATICS IS ALL ABOUT PSYCHOLOGY.BUT YOUR UNDERSTANDING IS ALSO USUALLY AFFECTED BY HOW VAST YOUR ENGLISH VOCABULARY IS.IT IS NOT JUST WHETHER YOU ARE BRILLIANT OR NOT. IT IS IMPORTANT THAT YOU CAN INTERPRETE FACTS, FIGURES & CONCEPTS CONTAINED IN THE PHRASES BELOW.

1. AREA OF THE CURVED SURFACE OF A CYLINDER


2. LENGTH OF THE BASE RADIUS OF A CYLINDER


3. MODAL CLASS


4. VOLUME OF A RIGHT-SIDED CONE


5. CYLINDRICAL CONTAINER


6. CYCLIC QUADRILATERAL


7. MINOR SECTOR OF A CIRCLE


8. VERTEX OF A CONE


9. HEMISPHERICAL BASE


10. ARC WHICH SUBTENDS ANGLE AT CENTER OF A CIRCLE


11. HOLLOW CYLINDER/SPHERE


12. ARC OF RADIUS


13. RADIUS OF A TUBE


14. PYTHAGORAS RULE


15. PARALLEL OF LATITUDE


16. ELEMENTS OF A SET/SUBSET


17. COMPLEMENTS OF THE SET


18. SECTOR OF A CIRCLE


19. CENTER OF PARALLEL OF LATITUDE


20. SHADED PORTION OF TWO CIRCLES OR OF A QUADRILATERAL


21. LINEAR GRAPH


22. PERCENTAGE ERROR


23. VERTICAL CROSS SECTION


24. HORIZONTAL CROSS SECTION


25. FREQUENCY OF SEQUENCE


26. SUM OF DEGREES


27. ANGLE SUBTENDED AT THE CENTER OF THE EARTH


28. ARC OF THE EQUATION


29. SECANT VALUE


30. RIGHT-ANGLED/ISOSCELES/SCALENE/TRIANGLE


31. RATIONALIZE/SIMPLIFY


32. OPPOSITE AND ADJACENT SIDES


33. LOCUS OF POINTS


34. CYCLIC QUADRILATERAL


35. RIGHT CIRCULAR CYLINDER


36. WALLS OF A ROOM


37. CUBOID/RECTANGULAR BLOCK


38. LINEAR SEQUENCE


39. NAUTICAL MILE


40. STANDARD ZERO


41. BY SUBSTITUTION


42. CORRECT TO 2 SIGNIFICANT FIGURES


43. SHORTEST DISTANCE BETWEEN TWO POINTS


44. THE NTH TERM


45. CORRECT TO I DECIMAL PLACE


46. DIFFERENCE IN LONGITUDE


47. SATISFY INEQUALITIES


48. AREA OF CURVED SURFACE


49. ANGLE OF SECTOR OF AN ANGLE


50. PERPENDICULAR TO/FROM


51. AREA OF THE BASE OF THE CONE


52. ILLUSTRATE ON GRAPH PAPER AND SHADE THE REGION


53. DRAW THE GRAPH OF THE RELATION -2 × 2


54. USING THE GRAPH TO SOLVE AN EQUATION


55. ANSWER TO THE NEAREST 100KM


56. DISTANCE BETWEEN TWO POINTS OF LATITUDE USING A RULER AND COMPASS


57. CONSTRUCTING A TRIANGLE OR TRAPEZIUM OR A RECTANGLE


58. THE LOCUS (LOCI) OF POINTS EQUIVALENT FROM 2 POINTS OR FROM 2 LINES WHICH PASS THROUGH A TRIANGLE


59. LABELING POINTS OF INTERSECTION


60. USING THE OGIVE  50TH PERCENTILE


61. CALCULATING THE ANGULAR DISTANCE


62. RANGE OF DISTRIBUTION


63. LENGTH OF CORD


64. SURFACE AREA OF A HOLLOW CYLINDER OR OF A CYLINDER CLOSED AT ONE END


65. VOLUME OF A CONE


66. CUTTING A SECTOR OF A CIRCLE TO FORM A CONE


67. CALCULATING THE LENGTH OF ARC WHICH SUBTENDS AN ANGLE OF 700


68. THE SCOPE OF A GRAPH


69. DERIVATION OF EQUATIONS WHOSE COEFFICIENT ARE INTEGERS AND WHICH HAS ROOTS OF 1/2 & -7 (AS EXAMPLE)


70. ARC OF A RADIUS


71. SOLUTION SET OF INEQUALITIES


72. AREA OF A SPHERE OF RADIUS CM


73. AREA OF A CYLINDRICAL CONE CLOSED AT BOTH ENDS


74. PRODUCT OF TWO % ERRORS


75. VOLUME OF A RIGHT-ANGLED CONE


76. VOLUME OF A HEMISPHERICAL TANK


77. CONSTRUCTION OF A GROUPED DATA


78. UNDEFINED EXPRESSION FOR VALUES OF Y


79. A CYLINDER OF BASE RADIUS 4 GIVEN AT ONE END


80. RATIO OF THE BASE OF A CYLINDER TO THAT OF ITS CURVED SURFACE


81. EVALUATION WITH OR WITHOUT MATHEMATICAL TABLES


82. CONVERSION OF 80KM/HR TO METERS/SECOND


83. FINDING THE AREA OF THE ENCLOSED REGION OF 2 SEMI CIRCLES


84. PERIMETER OF THE REGION


85. FINDING THE COORDINATES OF A POINT


86. COMMON RATIO OF A GP


87. CUMULATIVE FREQUENCY CURVE OF DATA


88. CLASS BOUNDARIES OF DISTRIBUTION


89. SUBJECT OF THE FORMULA


90. GRADIENT OF A LINE


91. DIRECT/INVERSE VARIATION


92. CALCULATING IPQI


93. MEANING OF “TOSSING A FAIR DICE”


94. MEANING OF “VERTICAL WALL OR HORIZONTAL GROUND”


95. MEANING OF “CAPACITY OF” (VOLUME)


96. MEANING OF “LINE OF SIGHT”


97. MEANING OF “FLIGHT OF A BIRD OR OF AN EAGLE OR AN AEROPLANE”


98. MEANING OF “TWO SHIPS LEFT PORT”


99. MEANING OF FLIGHT DUE NORTH (ALONG THE MERIDIAN)


100. STANDARD DEVIATION OF A GROUPED DATA          

CAN YOU IMAGINE, EXPLAIN OR DRAW YOUR INTERPRETATION OF THESE QUESTION-PHRASES TO A STUDY GROUP OF YOUR CLASSMATES?


GOOD LUCK!

IF YOU THINK WAEC MATHS IS HELL,HERE ARE 84 STEPS TO MATHS HEAVEN (2)

THE SEVEN BRANCHES OF WAEC MATHS

 OPENING VEILS TO REVEAL NATURE OF WAEC EXAM QUESTIONS  

[caption id="attachment_4275" align="alignleft" width="300"] maths symbols[/caption]

14.The branches of maths are Arithmetic, Algebra,Trigonometry, Geometry, Probability, Statistics and Structure. These categorizations are important because when you are asked a question in maths you should ask yourself two questions.

a.What branch of maths is this question from?

b.What topic in that branch of maths is the question?

  Now let us look at each branch briefly.

ARITHMETIC

15.The operations of arithmetic revolve around playing with numbers by changing them from one form to another. However some people define arithmetic as the laws of numbers on business arithmetic. This may be so because arithmetic is the basis of all commercial transactions be it computations, bargaining, negotiating or counting.

16.Arithmetic is also the basis of other branches of mathematics and other subjects such as Business Studies, Economics, Accounting, Law, Engineering etc. This means that Arithmetic holds the key of operation to these branches to understand them too.

17. 2 important laws under arithmetic, which should be explained by the tutor, are:

          The rule of BODMAS

          The law of PRECEDENCE

The two are interrelated in the sense that they specify the order for calculations in maths.

18. 6 key terms commonly used for questioning under arithmetic are: EVALUATE, SIMPLIFY, FACTORIZE, CALCULATE, FIND and EXPRESS. These should also be explained by the tutor or see relevant paragraphs later  for more explanation.

ALGEBRA

19. Unlike arithmetic which deals with known numbers, this is the branch of mathematics, which deals with what are grouped as “UNKNOWNS”. This is because not everything in life is known. The interest of algebra is to seek out the unknown number with what is known. To do this, the first step is that the unknown number which is sometimes called a variable is denoted in either English or Greek alphabets. These alphabets are then developed as algebraic expressions or equations. In summary, the point to note is that unlike arithmetic that relates to calculating with numbers known or definite in values, Algebra on the other hand relates to calculating with equations containing unknown numbers But the same arithmetic rules and steps of  BODMAS and PRECEDENCE are still used. Some people see algebra as a twin brother of arithmetic or as equations, which rely on rules of arithmetic for their solutions.

20. There are 4 key terms normally used for questioning under algebra. These are: SOLVE, SIMPLIFY, PLOT/CHART and FIND.Please note that key terms for the remaining branches of maths shall be specified under another write-up titled QUESTION TYPES,VISUALIZATION AND CRYSTALLIZATION…

TRIGONOMETRY

21. Trigonometry developed in the earlier days because of the need for measurements of days and distances between two points on the surface of the earth by human beings traveling from place to place. This was done by triangles. However, it was also extended to measurements of chords for music and arcs of a circle.

22.Trigonometry tests the ability of students on the use of triangles through TRIGONOMETRIC RATIOS and IDENTITIES using the acronym known as SOHCAHTOA.

23.Trigonometry covers the calculation of trigonometric functions of a certain angle from those of other angles using such typical relationship of formulae as: SIN 2A = 2 SIN A COS A.

24.Trigonometry tests the ability to understand the relationship between ratios of angles e.g. Tan Q = Sin Q/Cos Q.

25.Trigonometry also tests the knowledge of trigonometry tables and the ability to determine the form of angle ratios in various quadrants. It also shows how to plot, sketch or calculate values and measurements on Sine or Cosine curves.

GEOMETRY

26.This is the earliest developed branch of maths. It is the study of different shapes of figures and proportions of space and object. It developed to address the problems of shapes and beauty.

27.Geometry covers three major areas –plane, solid and analytical.

28.“Plane” refers to 2—dimensional flat surface e.g triangles and squares.

29.“Solid” refers to 3—dimensional objects i.e those with length, breadth and height e.g. Cuboids, Cylinder, Cone, Pyramid and Sphere.

30. The usual question on “Plane” geometry relates to length, breadth and areas of physical objects whereas, “solid” geometry include volume as an additional measure. 

31.“Analytical” usually refers to the addition of algebra to plain or solid geometry.

32.Under geometry there are many theories which student might be asked to “prove” they might also be asked to “trace” the shape of figures based on the principles of “Locus” of a point using mathematical instruments.

  PROBABILITY

33. We live in a world of uncertainties. A world in which no one knows what will happen the next minute. Our lives are governed by chances, which are unknown to us though they are clearly known to God. The study of probability developed in relations to things that are surround us. Probability is likened to a game of likelihood and revolves around the extent to which an event is likely to occur. It is measured by the ratio of the number of the favorable/likely cases to the number of the possible cases.

34. Probability is therefore a ratio of measurement and has a value that lies between 0(that is, the event not occurring at all) expected) and 1(tha is,the event occurring as expected). Re-stated in another way when the probability is zero it means that the event will not happen or take place, but when the probability is one it follows that the event will  certainly happen. For example, the probability that the sun will rise tomorrow is 1.

35. To measure probability, data must be collected, arranged and processed e.g the number of times rain falls in a month. Secondly, the data can be presented in various ways such as charts, figures; tables etc for explanatory purposes. Even graphs can be used. After presentation; those who want to use the data will then study and analyze them in terms of locations within the total set and according to dispersal from one another. Thereafter the data is interpreted for decision – making purposes. However when the data is used for prediction purposes if is said to be used for “Regression” and “Extrapolation”. The process then moves from collecting data to predicting future events. Sometimes people say probability is used for prophesy or for foretelling  future events so that human beings can take measures of safety before unfortunate events come upon them.

36.The different types of probability are: Experimental, Theoretical, Mutually exclusive probability and Independent events. (go over this again in your text books or seek  for assistance from your tutors). However take note of the following expressions and know their implications clearly.

               Mutually exclusive events (one event excludes the other e.g. positions in an exam)

               Mutually independent event (one event is not affected by another e.g. tossing a coin many times)

               Dependent event (when the outcome of an event depends on another. E.g.

              passing Jamb or not will affect your admission into a university.)

 STATISTICS

37. To prophesy under probability, we have to gather data which is then processed to predict an outcome. The data we gather is called statistics. It is therefore obvious that probability and statistics are related to one another.  

38. Data gathered as said earlier can be used to describe the habits shown by a set of information gathered. The principal measure is called a MEAN and others are the MODE, the MEDIAN, CENTRAL TENDENCY, VARIANCE and STANDARD DEVIATION. (your textbooks or tutors again please). These measurements are called descriptive statistics. Another type is called theoretical statistics, which depends on an  advanced form of mathematics called Probability and Games theory. In theoretical statistics the laws of probability are merged with statistical methods, to infer the nature or characteristic or ways of behavior of the composition of the population from the simple data. You may ask what statistics are used for, there are 3 major uses, and these are

             Scientific research, commercial and industrial activities.

             Social uses like elections and population census

             Inferences and predictions using estimation from arithmetic.

 STRUCTURE

39. This is the last and newest branch of mathematics relating to set theory and logic.

TO BE CONTINUED. WE STILL HAVE 45 STEPS TO GO.

IF YOU THINK WAEC MATHS IS HELL,HERE ARE 84 STEPS TO MATHS HEAVEN (1)

MAKING  WAEC/NECO  MATHS MAKE SENSE  – 

COMPILED BY OLUKAYODE  ODUMOSU, EX-PROPRIETOR, MASON COLLEGE, FESTAC,LAGOS

                               maths can be heaven but you need to be born again about it!...

PREAMBLE

84 STEPS TO MATHS HEAVEN IS A BOLD ATTEMPT TO PORTRAY MATHS IN A MORE REALISTIC AND FRIENDLIER MANNER THAN MANY TUTORS BOTHER TO MAKE IT APPEAR.IT  IS A COMPILATION OF MATHS METHODS AND NOT A WRITE-UP ON  PARTICULAR TOPICS UNDER MATHEMATICS.THOUGH IT MIGHT BE USEFUL FOR REGULAR SENIOR SCHOOL MATHS CLASSES IT IS MORE USEFUL AS A PSYCHOLOGICAL BOOSTER FOR MATHS CLINICS OR MATHS REMEDIAL CLASSES.KINDLY TAKE NOTE OF THIS

1.In this write-up “maths” means mathematics.

2.So you think maths is a difficult subject? Yes many other students think so too. But have you asked yourself why?

3.Experiences over the years have pointed to many reasons why and this include family stories that "maths is difficult".

4.Terror-producing teaching methods by teachers who don’t really know how to teach the subject is another reason. Many students have been victims of their fuzzy or meaningless explanations. Many of them forget that maths teaching should be seen as "assisted performance" to students. How do student make any progress in maths when they cannot make sense of any key topic or mathematical operations? For instance, what does 20 ÷ 5 or 30 ÷ ¹/₂  + 10 mean in real life? Here I am not referring to the answer but that the teacher needs to explain what the operations mean in practical terms.

5.Similarly, many teachers do not believe that a student can work above what they think is the competence of the student. Hence they make very little effort at what is called remedial work, maths clinics, maths continuous practice and maths special programs as in Mason College.

6.Emphatically ,it is obvious that many teachers fail to link maths to real life. Many teach maths without physical or pictorial examples. They expect students to be "mathematically correct" by assuming that each one should be able to think ABSTRACTLY. Not even the commonest and cheapest educational aids are used. They assume  that every student must have a "mathematical mind" leading to a rather austere portrayal of the subject.

7.Yes, we will not leave teachers alone! We cannot but refer  to many of them using robotic, imitative, memorized styles of teaching with little conceptual guiding. It is a fact that no one can make maths easy. But it should be taught with a soul.

8.Naturally, many students are afraid of maths because of its difficult words. Sometimes, the problem comes from maths questions too. These can be confusing. Add to these a lot of formulae which students are not asked to prove nowadays as in the past .In which case many cannot be remembered easily. In summary, many students cannot even understand the purpose of maths.

"Poor teaching leads to the inevitable idea that the subject (mathematics) is only adapted to peculiar minds, when it is the one universal science and the one whose ground rules are taught us almost in infancy and reappear in the motions of the universe." Henry J.S. Smith (above)

GOOD NEWS!!

9.But there is good news for you. There are actually other tutors who care for your feelings about maths. We believe we can make you  make sense out of maths.  You can become friendlier with maths. You can “walk the walk and talk the talk” of maths.

10.To do this, you must understand that maths is a language and not a "spirit". Yes the teaching or learning of mathematics should be regarded initially as the extension of the teaching of a foreign language from which you acquire some proficiency through certain rules and practices.Those rules must be observed otherwise practice will be difficult.On the other hand those rules will be not be useful without practice.


11.It must be noted that the rules are based on logic and reasoning before maths can be made  intelligible. Fortunately all of us can reason whether we are in JSS 1 or SS 3. In fact, we do reason all the time or don’t we? When we have our bath early in the morning, we do not bite the soap. When we eat we do not put the fork or spoon in our eyes or on our heads. We know what is good and what can be harmful to us. When our parents speak to us or when we listen to our church pastor or the imam in the mosque we “reason out what they tell us. We query what appears to make no sense at all. Therefore the first basic ingredient for our maths soup which is ability for logic and reasoning is possessed by all of us.


12.Next maths is a living subject which requires thinking (mental effort) along with reasoning. There must be a deliberate thought process that we want to reason things out  in a maths situation because many of them are abstract (the tutor must explain this clearly to the student).

Under paragraph 10 above we referred to “rules and practice” as the basis for making sense out of mathematics. So what are these rules? They are as follows:

CATEGORIZE the seven branches of maths (maths can be categorized into 7 main branches)


EMPHASIZE your understanding of the basic concepts and principles of maths.

VISUALIZE & CRYSTALLIZE maths through question phrases.

RECOGNIZE useful hints, formulas, signs, symbols and notations.

MEMORIZE the procedural steps for maths solutions.

Now let us consider additional notes on these in detail

NEXT:

DETAILED NOTES ON THE SEVEN BRANCHES OF MATHEMATICS AND HOW KNOWING THEM CAN HELP YOU MAKE SENSE OUT OF MATHS