MATHS SUMMARIES FOR JSS
(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 π (See 3 C under part 1 of this write-up)
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.
(11) TRIGONOMETRICAL RATIOS (see diagram here JSCE MATHS MEMORY JOGGERS)
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
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 (n) of data is even 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 observations have been arranged in order 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)
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).
d The probability that event A AND B occurs = p (A) × p (B)