MATHS SUMMARIES FOR JSS
(REFERENCE SUMMARIES FOR TEACHERS AND STUDENTS)
(1) THE BINARY SYSTEM
a. A number in base 2 is called a BINARY NUMBER
b. A binary number is a sum of powers of 2
c. In base 2 we have only two numbers 0 & 1 to work with
d. To express a binary number as a decimal number write the given
binary number as a sum of multiples of 1, 2, 4,8,16 etc.
(2)
WORD PROBLEMS AND FRACTIONS
a. You should be able to express word problems in numerical terms
b. You should be able to simplify expressions having brackets and
fractions.
(3)
NON-RATIONAL NUMBERS AND APPROXIMATIONS
a. Rational numbers consist of all counting numbers, integers, and
fractions (positive and negative).
b. A class of non-rational numbers includes square roots, of natural
numbers which are not perfect squares.
c. An important example of a non-rational is π.
d. Approximations of square roots which are non-rational can be
obtained by using trial and error method.
(4)
FACTORIZATION
a. You should be able to factorize expressions of the form a2-b2. Same
for a2 ± 2ab + b2.
b. You should be able to factorize quadratic expressions by
completing the squares.
(5)
SIMPLE EQUATIONS INVOLVING FRACTION
a. Find the L.C.M of all denominations.
b. Multiply each term in the equation by this L.C.M
c. Simplify and solve the resulting equation.
(6)
SIMULTANEOUS LINEAR EQUATIONS
a. You should be able to solve a given system of linear equations by
graphs, by elimination method or substitution method.
b. To solve by graph, we will draw the two lines of the two equations.
The point of intersection of the two lines or two graphs is the
solution of the simultaneous equation. When the graphs coincide,
then there is an infinite number of solutions.
c. The graphical method is the slowest method.
(7)
VARIATION
a. Direct variation: A varies directly as B can be expressed as A∞B:
A∞B means A = KB Where K is constant known as the constant of
the variation.
b. Inverse variation: A varies directly as B can be expressed as A ∞ 1/B
i.e. A = K/B Where K is constant.
c. Joint variation: A jointly varies directly as B and inversely as the
square of C; = A ∞ B/C2 OR A = KB/C2
d. Partial variation: Suppose A varies directly as B and partly as the
square of C then A can be broken into two parts A1 ∞ A2 such as
that A = A1 + A2 where A1 = K1B2 and A2 = K2C2. Then A = K1B1 + K2C2.
(8)
CHANGE OF SUBJECT OF FORMULA
a. A formula is an equation containing two or more variables and it
describes how the variables are related.
b. We solve many problem in maths by the use of formulae
c. We often need to rearrange a formula in order to make any one of
the variables the subject of the formula.
d. Making a specified variable the subject of the formula simply
means expressing the specified variable in terms of the other
variables.
e. When values are given for other variables in a formula we can find
the value of specified variables.
(9)
VIEWS AND PLANS
a. You should be able to draw common solids with your hands (free
hand drawing).
b. One way of drawing solid shapes is known as the method of parallel
projection.
c. You should also be able to draw views and plans of common solids.
d. We have the top view, side view front view and back view.
e. In parallel projection, vertical lines are always drawn vertical,
parallel edges are always drawn parallel, perpendicular and lines
which have equal lengths in the solid may not have equal lengths in
the drawing.
f. When edges and faces of the solid that have equal sizes 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.
(10) SIMILAR FIGURES AND ENLARGEMENT
Shapes of the same sizes are similar
a. All squares are similar
b. All equilateral triangles are similar.
c. All cubes are similar
d. All n-sided regular polygons are similar
e. Triangles having equal corresponding angles (i.e. equilateral
triangles) are similar.
f. Triangles whose corresponding sides are in constant proportion are
similar.
g. What is important in similar figures is the shape and not the size.
h. In scale drawing the drawing may be smaller (reduction) or bigger
(enlargement).
i. If the ration of corresponding sizes of two shapes is a constant and
corresponding angles are equal then the two shapes are similar.
j. Any two circles or spheres are similar.
k. 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 the II (Sec 3 C)
l. If two figures are similar with scale factors 1: a then the lengths in
the second figure are “a” times the corresponding lengths in the
first figure. The areas in the second figure are “a” times the
corresponding volumes in the first figure.
H
(11) TRIGONOMETRICAL RATIOS
O
A
a. The sine of an acute angle included in a right-angles triangle is the
ratio of the length of the opposite side to the hypotenuse
sine XO = O/H
b. The cosine of an acute angle in a given right-angles triangle is the
ration of the length of the adjacent side to the hypotenuse
Cos XO/H = A/H
c. The tangent of an acute angle in a given right-angled is the ratio of
the length of opposite side to the adjacent side of the triangle tan
XO = O/A
d. Note also that Tan X = Sin XO/ Cos XO = O/H ÷ A/H = O/H × H/A = O/A
e. The sine of an angle is equal to the cosine of its complement.
(12) FURTHER MENSURATION
a. Any side of a triangle can serve as its base with its corresponding
height when calculating its area.
b. We can calculate the area of a triangle if we know
i.
Its base and its height or
ii.
Two of its side and the included angle.
c. We can calculate the area of a parallelogram if we know
(i) Its base and its height or (ii) its sides and angle or
(iii) The diagonals in the case of a rhombus. (Note that the
diagonals of a rhombus divides it into equal triangle. Note also that
a kite is rhombus). The area of a rhombus = ½ of the product of its
diagonals.
d. The area of a trapezium is equal to half of the product of the sum of
the parallel sides and the perpendicular distance between them.
e. Pythagoras theorem (hypotenuse)2 = (opposite)2 + (adjacent)2
f. To state (d) again, A = ½ (sum of parallel sides) × perpendicular
distance between them.
g. Area of a circle = πr2
h. Circles having the same center are called concentric circles and the
area between two concentric circles is called annulus.
i. The area of an annulus is equal to the difference between the areas
of the two circles.
j. A semi-circle is half of a circle and a quadrant is one quarter of a
circle.
(13) SIMPLE CONSTRUCTIONS
a. Use a sharp pencil and ruler with straight edges and compasses
that are not loose.
b. Make clear thin points of intersection as opposed to an “area of
intersection”
c. You can copy angles, bisect angles construct perpendicular bisector
of a line segment from a point either on it or outside it, construct
medians and attitudes of triangles, construct circumscribed,
inscribed and enscribed circles of triangles or construct special
angles of 60, 120, 30, 15, 90, 45 and any other combinations
(14) STATISTICS
a. You should be able to present information as a frequency table, bar
chart, line graph or pie chart or present it as a pictogram.
b. Measures of average include the mode, median and mean
c. Where the number of data is given the median is calculated by
dividing the sum of the two middle scores or calculated as ½ ( nth/2
score + (n + 2)th/ 2 score)
d. Where n is odd then the median is the (n + 1)th/2 score
Where the observation have been arranged in other of magnitude
under (c) and (d) above
e. The mode can be determined from
i.
The frequencies in a frequency table
ii.
The lengths of the line in a line graph
iii. The lengths of the bars in a bar chart
iv. The sizes of the sectors in a pie chart
v.
The numbers of pictures in a pictogram
f. The median van be determined from
i.
The data arranged in order of magnitude
ii.
The frequency distribution
iii. The cumulative frequency distribution
g. 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 the number of scores.
h. The sum of deviation from the mean is always zero
i. The range indicates the amount of speed in the distribution of any
given set of data. (it is also the difference between the highest
value and the lowest value)
15. PROBABILITY
a. Usually expressed as a fraction. If required his fraction can be
converted to % or decimal.
b. Probability = No of required outcomes
No of possible outcomes
c. For two events A and B that are mutually exclusive, the probability
that event A OR B occurs= p(A) + p(B).
The probability that event A AND B occurs = p (A) × p (B)