IF YOU THINK WAEC MATHS IS HELL,HERE ARE 84 STEPS TO MATHS HEAVEN (7)

BASIC QUICK REVISION MATHS FORMULAS & HINTS FOR WAEC EXAM

PREAMBLE

Most of the material in this write-up are basic indeed.However,those still struggling with foundation maths or maths clinics will find them very useful.Going through the write-up it will be very obvious that the formulas and hints were identified and listed using the 7 branches of maths as earlier listed by us…


We have listed 7 class activities or private study uses for the content of this write up and we shall be willing to implement them in any school in Nigeria. Please note, however, that another list made up of  slightly different formulas/ hints  of a slightly improved standard will come on line soonest to complement our efforts in these areas.

ARITHMETIC (STRUCTURE EXCLUDED)

 THE BINARY SYSTEM

 A number in Base 2 is called a BINARY NUMBER

A binary number is a sum of multiples of powers of 2

In base 2 we have only two numbers 0 & 1 to work with

To express a binary number as a decimal number write the binary number as a sum of multiples of 1, 2, 4, 8, 16 etc

WORD PROBLEMS AND FRACTIONS

You should be able to express word problems in numerical terms

You should be able to  simplify expressions having brackets and fractions

NON – RATIONAL NUMBERS AND APPROXIMATIONS

Rational numbers consist of all counting numbers, integers and fractions (positive and negative)

A class of non – rational numbers include square roots of natural numbers which are not perfect squares

An important example of non – rational number is π (PI)

Approximations of square roots which are non – rational can be obtained by using trial and error method.

 ALGEBRA

FACTORIZATION

You should be able to factorize expressions of the form a² – b². Same for

 a² + 2ab + b²

You should be able to factorize quadratic expressions by completing the square.

SIMPLE EQUATIONS INVOLVING FRACTIONS

Find the L.C.M of all denominators.

Then multiply each term in the equation by this L.C.M.

 Simplify and solve the resulting equation.

 SIMULTANEOUS LINEAR EQUATIONS

You should be able to solve a given system of linear equations by graph, by elimination or by substitution methods

To solve using the graph, we will first draw two lines or graphs for the equations given. The point of intersection of the two lines  is the solution. If the two graphs coincide, then there is an infinite number of solutions.

The graphical method is the slowest method.

VARIATION

Direct Variation: A varies directly as B can be expressed as A α B means A α B means A=K/B  where K is known as the constant of the variation.

 Inverse Variation: A varies inversely as B can be expressed as A α 1/B i.e. A=K/B where K is also the constant

 Joint Variation: A jointly varies directly as B and inversely as the square of C= A α B/ C²
or A = K/ C²   

Partial Variation: Suppose A varies directly as B and partly as the square of C then A can be broken into parts A₁ α A₂ such that A =  A₁+A₂ where A₁ = k₁ B and A₂ = k₂ C^2. Then A = K₁ B + K₂ C²

    CHANGE OF SUBJECT IN FORMULA

 A formula is an equation containing two or more variables and it describes how the variables are related.

We solve many problems in maths by the use of formulae But we often need to simplify by rearranging it to make one of the variables the subject of the formula. Making a specified variable the subject of the formula simply means expressing the specified variable in terms of the other variables. When values are given for other variables in a formula then we can find the value of any of the other variables.

 GEOMETRY/ MENSURATION

 VIEWS AND PLANS

 You should be able to draw common solids with your hands (free hand drawing).

 One way of drawing a solid shape is known as the method of Parallel Projection.

 You should also be able to draw views and plans of common solids. We have the Top View, Side View, Front View and Back View.

 In Parallel Projection, Vertical Lines are always drawn vertical, parallel edges are always drawn parallel, perpendicular lines are not always drawn perpendicular and lines which have equal lengths in the solids may not have equal lengths in the drawing.

 When edges and faces of the solid that are equal in size are drawn to have equal sizes we call this Proportional Drawing. If we combine Parallel Projection with Proportional Drawing then we call this method Orthogonal Projection. Orthogonal projection therefore means drawing to scale.

 SIMPLE CONSTRUCTIONS

 Use sharp pencil and ruler with straight edges and compasses that are not loose.

 Make, clear, thin points of intersection as opposed to an “area of intersection”

 You can copy angles, bisect angles, construct perpendicular bisector of a line segment, construct a perpendicular line to a given line segment from a point either on it or outside it. You can also construct medians and altitudes of triangles, construct circumscribed and inscribed circles of triangles or construct special angles of 60, 120, 30, 15, 90, 45 or any other combination.

 SIMILAR FIGURES AND ENLARGEMENT

 Shapes of the same sizes are similar

 All squares are similar

 All equilateral triangles are similar

 All cubes are similar

 All n-sided regular polygons are similar

 Triangles having equal corresponding angles (i.e. equiangular triangles) are similar.

 Triangles whose corresponding sides are in a constant proportion are similar. What is important about similar figures is the shape and not the size

 In scale drawing the drawing may be smaller (reduction) or bigger (enlargement)

 If the ratio of corresponding sizes of two shapes is a constant and the corresponding angles are equal then the two shapes are similar.

 Any two circles or spheres are “similar”.

 The corresponding arcs subtending equal angles at the centre of a sphere are similar and they are in constant ratio of their radii. (This is the idea behind π)

 CONCLUSION

 If two figures are similar with scale factors 1: a then the length in the second figures is “a” times the corresponding figure and the volumes in the second figure is “a” times the corresponding volume in the first figure.

 FURTHER MENSURATION

 TRIANGLE, PARALLELOGRAM, RHOMBUS(KITE) AND TRAPEZIUM

 Any side of a triangle can serve as it base with a chosen corresponding height when calculating its area.

 We can calculate the area of a triangle if we know its base and its height or two of its sides and the included angle

 We can calculate the area of a parallelogram if we know its base and its height orits sides and angles or

the diagonals in the case of a rhombus. (Note that the diagonals of a rhombus divides it into equal triangles. Note also that a kite is a rhombus)

 The area of a rhombus = ½ of the product of its diagonals.

 The area of a trapezium is equal to half of all the product of the sum of the parallel sides and the perpendicular distances between them. Or A = ½ (sum of parallel sides) × perpendicular distance between them

CIRCLES

 Area of circle= πr²

 Circles having the same centre are called concentric circles and the area between two concentric circles is called an annulus

 A semi-circle is half of a circle and a quadrant is one quarter of a circle.

 The area of an annulus is equal to the difference between the areas of the two circles.

 TRIGONOMETRICAL RATIOS







c= HYPOTENUSE(H),    a= OPPOSITE(O),    b= ADJACENT(A)

The sine of an angle included in a right- angled triangle is the ratio of the length of the opposite side to the   hypotenuse i.e. Sin X^O = O/H (SOH)


 The sine of an angle is also equal to the cosine of its complement.

 The cosine of an acute angle in a given right-angled triangle is the ratio of the length of the adjacent side to the hypotenuse i.e Cos X^O = A/H (CAH)

The tangent of an acute angle in a given right-angled triangle is the ratio of the length of the opposite side to the adjacent side of the triangle i.e. Tan X^O = O/A (TOA)

Note also that Tan X^O = sine x^o / cos x^o   = O/H x A/H = O/H × H/A = O/A

The first 3 formulas  under Trigonometry are also labeled as SOHCAHTOA

 PYTHAGORAS THEOREM

 Hypothenuse² = (opposite)² + (Adjacent)²

 PROBABILITY

 Usually expressed as a fraction.  If required, this fraction can be converted to a % or decimal.

 Probability = No of required outcomes/No of possible outcomes

 For two events A and B that are mutually exclusive, the probability that event A or B  will occur = P(A) + P(B). And the probability that event A and B occurs = P(A) × P(B).

 STATISTICS

 You should be able to present information as a frequency table, bar chart, line graph or pie chart or present it as a pictogram.

 Measures of average include the mode, median and mean.

 Where the number of data is even the median is calculated by dividing the sum of the two middle scores or calculated as ½ (nth)score/2  + (n+2)th score/2

Where n is odd then the median is the (n + 1)th score/2

 MODE

 Where the observation have been arranged in order of magnitude the mode can be determined from

the frequencies in a frequency table.

 the lengths of the line in a line graph.

 the lengths of the bars in a bar chart

 the sizes of the sectors in a pie chart

 the numbers of pictures in a pictogram

 MEDIAN

 The median can be determined from

 the data arranged in order of magnitude

 the frequency distribution

 the cumulative frequency distribution

MEAN

 If the mean is given and the number of scores in the data is known, then the total sum of all the data can be determined by multiplying the mean by number of scores.

 The sum of deviation from the mean is always zero.

 The range indicates the amount of Speed in the distribution of only given set of data. ( It is also the difference between the highest value and the lowest value)