Key Concept Chart - Pearson

Mathematics
Key Concept Chart
Form 2
Properties of triangles
r
A
C
F
r r
All the angles of an equilateral
triangle are equal to 60°.
p
q
r
The sides of a scalene triangle are
different lengths and the angles
are therefore all unequal.
x
e e
The base angles of an isosceles
triangle are equal.
y
b
a
The sum of the interior angles
of a triangle is equal to 180°
(a b c = 180°).
z e
The exterior angle of a triangle is equal to the sum
of the opposite two interior angles (e = x y).
c
B
D
P
60° Q
55°
E 55° 65° R
F
D
Congruent triangles are exactly the
same shape and size. Conditions of
congruency are side, side, side (SSS);
side, angle, side (SAS); angle, angle,
side (AAS); right-angle, hypotenuse,
side (RHS).
Similar triangles are exactly the same
shape but aren’t necessarily the same
size. The condition of similarity is
angle, angle, angle (AAA).
E
Polygons
Quadrilaterals
The sum of the interior angles of a polygon:
(n 2) 180º (n is the number of sides)
Rectangle
5 cm
a
Parallelogram
The size of an interior angle of a regular polygon: A quadrilateral is a polygon with four sides.
a ____________
​ (n 2) 180°
n ​ The sum of the interior angles of a quadrilateral
equals 360º.
in Action Mathematics in Action Mathematics in Action Mathematics in Action Mathematics in Action Mathematics

Algebra and numbers
Definitions
Factor: a number that divides exactly into another number without leaving a remainder (1, 2, 3 and 6 are
factors of 6).
Prime factor: a factor of a number that is also a prime number (2 and 3 are prime factors of 6).
Prime number: a number that only has two factors: itself and one (2, 3, 5, 7…).
Square number: the product of a number multiplied by itself (4 is the square number of 2: 2 2 = 4).
Square root: a number that when multiplied by itself gives a square number (4 is the square root of 16).
Cube number: the product of a number multiplied by itself twice (8 is the cube number of 2: 2 2 2 = 8).
Cube root: a number that when multiplied by itself twice gives a cubed number (2 is the cube root of 8).
A monomial expression has only one term, for example 5 x.
A polynomial expression has more than one term, for example 3 x 5 6a.
A binomial expression has two terms, for example 3 x 5.
A trinomial expression has three terms, for example 5 x 2y 7.
In a quadratic equation, the highest power of the variable is 2, for example 5 x 2 x 2 5.
Equations
Linear equation: 5 x 7 = 3
5 x = 3 7 [Group like terms]
5 x = 10 [Simplify]
x = 2 [Solve for x]
Simultaneous linear equation:
Solve for p and q if 3p 5q = 29 and 4p q = 8.
3p 5q = 29 (1)
4p q = 8 (2)
(2) 5: 20p 5q = 40 (3) [Multiply each term of (2) by 5]
(1) (3): 23p 0 = 69 [Add equations (1) and (3)]
\ p = 3 [Divide both sides by 23]
Substitute p = 3 into (1): 3(3) 5q = 29
\ 5q = 20 [Subtract 9 from both sides]
\ q = 4 [Solve for q]
\ p = 3 and q = 4
Square of binomial: (a b) 2 = (first term) 2 2(product of two terms) (last term) 2 = a 2 2ab b 2 .
Indices
Useful infomation
Order of Operations
B: Brackets
O: Of
D: Division
M: Multiplication
A: Addition
S: Subtraction
a 2 index
base
Multiplying a 3 a 2 = (a a a) (a a) = a 3 2 = a 5
Add the indices
Dividing a 3 a 2 = _______ ​ a a a a
a ​ = a 3 2 2 = a 1
Subtract the indices
Raising to (a 3 ) 2 = (a a a) (a a a) = a 3 2 = a 6
a power Multiply the indices
Zero as
index
A negative
index
3 0 = 1
All numbers with the index zero = 1, so a 0 = 1
2 1 = ​ 1_
2 ​
A negative index becomes positive when written below the fraction line a n = __ ​
a 1 n ​
in Action Mathematics in Action Mathematics in Action Mathematics in Action Mathematics in Action Mathematics

Graphs
–5
–4
–3
–2
y
5
4
3
2
1
–1 0
–1
1 2 3 4 5
–2
–3
–4
–5
x
Graphs are visual representations of figures.
A Cartesian plane is a plane with coordinate grids that is used to
plot coordinates.
Every point on the Cartesian plane represents an ordered pair
and an ordered pair represents the coordinates of the point.
The first number the x-coordinate is the horizontal position
of the point with respect to the origin. The second number the
y-coordinate is the vertical position of the point with respect to
the origin. Always write the x-coordinate first and the y-coordinate
second, e.g. (3; 2) represents the point x = 3 and y = 2.
y = 2x – 3
y
8
6
4
2
Straight-line graphs are also known as linear graphs.
Straight-line graph equations: y = mx c where m is the gradient
and c is the y-intercept.
Alternative equations: y = mx, y = c and x = c.
–5
–4
–3
–2 –1 0
–2
1 2 3 4 5 6
–4
–6
–8
–10
–12
x
The gradient (m) is the slope of the straight-line graph. If m is
positive, the straight line slopes upwards from the bottom left to
the top right of the Cartesian plane. If m is negative, the straight
line slopes downwards from the top left to the bottom right of the
Cartesian plane.
m = ________
change in y
​
change in x​ (represented as a fraction).
Another way to write this equation is m = __ ​ Δy
Δx ​.
–6
–5
y = x 2 + 4x – 4
y
2
1
–4 –3 –2 –1 0
–1
–2
–3
–4
–5
–6
–7
–8
–9
1
2
x
Parabola graphs are curved graphs.
Parabola graph equations:
y = ax 2 bx c; y = ax 2 bx; y = ax 2 c; y = ax 2 .
Remember: The sign of the coefficient of x 2 indicates whether
the arms of the parabola point upwards or downwards. If the
coefficient is positive the arms of the graph point upward. If it
is negative the arms of the graph point downward.
in Action Mathematics in Action Mathematics in Action Mathematics in Action Mathematics in Action Mathematics

Perimeter (P)
Perimeter (P) is the total distance around a
2D shape.
Perimeter formulae
Triangle = a b c
Square = 4s
Rectangle = 2l 2b or 2(l b)
Circle = C = π d = 2 π r
Volume (V ) and total surface area (TSA)
Volume (V ) is the amount of space that a 3D object
occupies in cubic metres (m 3 ), cubic centimetres
(cm 3 ), or cubic millimetres (mm 3 ).
TSA cylinder
= 2 area of base area of curved side
= 2pr 2 2prh
V prism
= area of base perpendicular height
Example:
TSA = 2 area of the circular base area of the
rectangle
= 2 π r 2 2π r h
= 2 π (5 cm) 2 2 π 5 cm × 8 cm
= 157.08 cm 2 251.33 cm 2
Measurements
Area (A)
5 cm
Solution:
= 408.41 cm 2
5 cm
8 cm
Data
Area (A) is the number of square units that a 2D
shape covers.
Area formulae
Triangle = ​ 1_
2 ​bh
Square = s 2
Rectangle = lb
Parallelogram = bh
Trapezium = ​ 1_
2 ​(s 1 s 2 ) h
Kite = ​ 1_
2 ​(d 1 d 2 )
Circle = π r 2 or π ​ ( ​ d__
2 ​ ) ​2 ​ = ​ 1_
4 ​ π d 2
Speed (v)
Speed (v) is the measure of how fast an object is
moving given by the ratio of distance versus time.
distance (s)
time (t)
Speed (v) = ________ ​ ​
Example: A lion starts from 0 m/s to chase a buck
for 4 min over 1.5 km. The buck outruns the lion and
the lion gives up and stops. Calculate the average
speed of the lion in km/h.
Speed = ______ ​ distance
time
​
= _______ ​
1.5 km
(4 60) h ​ [4 min = __ ​
60 4 ​ h]
= 22.5 km/h
Central tendency: the middle values
of a data set i.e. the mean, median
and mode.
Frequency polygon: a graph based
on a histogram that you create by
joining the midpoints of the top of
each bar of the histogram.
Frequency table: a table in which
you record data as tallies.
Scatter graph: a graph on which you
plot two variables, usually x and y,
that form a set of points on a set
of axes.
Scatter graph of Language marks
versus Mathematics marks
100
80
60
40
20
0
20 40 60 80 100
Mathematics mark
Language mark
Probability: the likelihood of an event
occurring expressed as a number between
0 and 1 in the form of a normal fraction,
decimal or percentage.
Mean = _______________
​
total of all values
total number of values ​
Theoretical probability
__________________________________
the number of ways in which the outcome can occur
= ​
the total number of possible equally likely outcomes ​
Experimental probability
= _______________________________
​
the number of times that the outcome occurred​
the total number of trials
Contact details
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Plot 14386, New Lobatse Road, G-West Industrial Site,
Gaborone, Botswana. Website: www.longmanafrica.co.za