MATHEMATICS BASICS
1. FOUR BASIC OPERATIONS
1.1 Addition (Inverse Operation to Subtraction)
1.2 Subtraction (Inverse Operation to Addition)
1.3 Multiplication (Inverse Operation to Division)
1.4 Division (Inverse Operation to Multiplication)
https://www.universalclass.com/articles/math/pre-algebra/the-four-basic-mathematical-
operations-.htm
http://www.mathsteacher.com.au/year8/ch01_arithmetic/01_basic/oper.htm
https://www.slideshare.net/J_A_R_E/basic-operations-in-mathematics
https://www.aplustopper.com/four-basic-operations-in-mathematics/
2. B O D M A S RULE
B BRACKETS ( ) or [ ] or { }
O OF OF
D DIVISION
M MULTIPLICATION
A ADDITION
S SUBTRACTION
https://www.mathsisfun.com/operation-order-bodmas.html
http://www.math-only-math.com/bodmas-rule.html
https://www.skillsyouneed.com/num/bodmas.html
https://byjus.com/maths/bodmas-rule/
http://planetmath.org/bodmasruleapplication
https://en.wikipedia.org/wiki/Order_of_operations
http://www.bbc.co.uk/bitesize/ks3/maths/number/order_operation/revision/1/
https://www.youtube.com/watch?v=KBfI0-Kmj1g
3. PROPERTIES OF EQUAL RELATION
3.1 a = a (reflexive property; a quantity is equal to itself)
3.2 If a = b, and b = c, then a = c (Transitivity; transitive property; equal third)
3.3 If a = b, then a + c = b + c (Addition Property; equal addition)
3.4 If a = b, then a – c = b – c (Subtraction Property; equal subtraction)
3.5 If a = b, then a x c = b x c (Multiplication Property; equal multiplication)
3.6 If a = b, and c ≠ 0 , then a/c = b/c (Division Property; equal division)
https://en.wikipedia.org/wiki/Equality_(mathematics)
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4. COMMUTATIVE, ASSOCIATIVE AND DISTRIBUTIVE LAWS
https://www.mathsisfun.com/associative-commutative-distributive.html
4.1 COMMUTATIVE LAW
https://en.wikipedia.org/wiki/Commutative_property
https://www.britannica.com/topic/commutative-law
4.1.1 a+b=b+a (Commutative Law of Addition)
https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-arithmetic-
properties/v/commutative-law-of-addition
4.1.2 axb=bx a (Commutative Law of Multiplication)
https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-arithmetic-
properties/v/commutative-law-of-multiplication
4.2 ASSOCIATIVE LAW
4.2.1 a + (b + c) = (a + b) + c (Associative Law of Addition)
https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-arithmetic-
properties/v/associative-law-of-addition
4.2.2 a x(b x c) = (a x b) x c (Associative Law of Multiplication)
https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-arithmetic-
properties/v/associative-law-of-multiplication
4.3 DISTRIBUTIVE LAW
4.3.1 a(b + c) = a x b + a x c (Distributive Law of Multiplication over Addition)
4.3.2 a(b - c) = a x b - a x c (Distributive Law of Multiplication over Subtraction)
http://ictedusrv.cumbria.ac.uk/maths/pgdl/unit8/unit8/page_62.htm
https://en.wikipedia.org/wiki/Distributive_property
https://www.khanacademy.org/math/pre-algebra/pre-algebra-arith-prop/pre-algebra-ditributive-
property/v/the-distributive-property
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5. FACTORING (REVERSE PROCESS OF SIMPLIFICATION)
5.1 (Taking out a common factor)
http://www.mathsteacher.com.au/year8/ch04_algebra/14_taking/comfac.htm
5.2 x2 + (a+b)x + ab = (x + a) (x + b)
https://www.khanacademy.org/math/algebra/polynomial-factorization/factoring-quadratics-
1/v/factoring-quadratic-expressions
5.3 a2 + 2ab + b2 = (a + b)(a + b) = (a + b)2
a2 - 2ab + b2 = (a - b)(a - b) = (a - b)2
https://www.khanacademy.org/math/algebra/polynomial-factorization/factoring-polynomials-3-
special-product-forms/v/factoring-special-products-2
5.4 (Difference of two squares)
https://www.khanacademy.org/math/algebra/polynomial-factorization/factoring-
polynomials-3-special-product-forms/v/factoring-differ
5.5 (Sum of two cubes)
https://www.khanacademy.org/math/algebra2/polynomial-functions/advanced-polynomial-
factorization-methods/v/factoring-sum-
5.6 (Difference of two cubes)
https://www.khanacademy.org/math/algebra2/polynomial-functions/advanced-polynomial-
factorization-methods/v/difference-of-cubes-factoring
6. SIMPLIFICATION (REVERSE PROCESS OF FACTORING)
6.1 (Distributive Law)
http://www.mathsteacher.com.au/year8/ch04_algebra/09_dist/law.htm#distlaw
6.2 (Difference of two squares)
https://www.youtube.com/watch?v=esVcl7gJoIM
6.3 (Sum of two cubes)
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6.4 (Difference of two cubes)
7. SIMILAR FIGURES
7.1 (Invertendo)
7.2 (Alternando)
7.3 (Componendo)
7.4 (Dividendo)
6.5 (Componendo et dividendo)
http://academic.sun.ac.za/mathed/shoma/MATUNIT24_05.htm
http://academic.sun.ac.za/mathed/shoma/Index25.htm
http://academic.sun.ac.za/mathed/shoma/MATUNIT24_06.htm
http://academic.sun.ac.za/mathed/shoma/MATUNIT25_03.htm
http://academic.sun.ac.za/mathed/shoma/MATUNIT25_04.htm
http://academic.sun.ac.za/mathed/shoma/MATUNIT24_08.htm
http://academic.sun.ac.za/mathed/shoma/MATUNIT24_09.htm
http://academic.sun.ac.za/mathed/shoma/MATUNIT24_10.htm
http://academic.sun.ac.za/mathed/shoma/MATUNIT25_10.htm
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8. CONCEPT OF A GRADIENT
https://www.mathsisfun.com/gradient.html
http://www.mathsteacher.com.au/year10/ch03_linear_graphs/02_gradient/line.htm
http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/graphsrev4.shtml
https://www.youtube.com/watch?v=ykeCKm9JqSA
http://www.teacherschoice.com.au/Maths_Library/Gradient/gradient_-_two_fixed_points.htm
http://www.explainingmaths.com/what-is-the-gradient-of-a-line.html
https://www.intmath.com/plane-analytic-geometry/1b-gradient-slope-line.php
http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-gradstlnseg-2009-1.pdf
https://www.khanacademy.org/math/algebra2/intro-to-conics-alg2/standard-equation-circle-
alg2/e/graphing_circles
9. ALGEBRAIC GRAPHS
9.1 Linear Graphs:
9.1.1 Given gradient m and y-intercept
y= mx + c
9.1.2 Given a point P and gradient m:
9.1.3 Given two points P and Q :
9.1.4 Given dual intercepts (a;0) and (0;b)
http://www.staff.vu.edu.au/mcaonline/units/graphs/linear.html
http://www.mathsteacher.com.au/year8/ch15_graphs/04_plot/graphs.htm
http://www.analyzemath.com/Graphing/GraphingLinearFunction.html
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https://www.khanacademy.org/math/8th-engage-ny/engage-8th-module-4/8th-module-4-
topic-b/v/graphs-of-linear-equations
9.2 Parabola:
Given
92.1 Shape: and
9.2.2 Y-intercept:
9.2.3 X-intercepts/Roots:
Solve for x (factorisation; completing a square; formula)
9.2.4 Co-ordinates of the turning point:
Write in the form
Turning point will be (-p;q)
Alternatively,
Substitute into to obtain
y-coordinate of turning point
9.2.5 Use facts from 7.2.1 to 7.2.4 to plot the graph
https://www.mathsisfun.com/algebra/quadratic-equation-graphing.html
https://www.desmos.com/calculator/pfdwlq5qht
http://jwilson.coe.uga.edu/EMT668/EMAT6680.Folders/Barron/unit/Lesson%206/6.html
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https://www.khanacademy.org/math/algebra/quadratics/quad-standard-form-
alg1/v/graphing-a-parabola-using-roots-and-vertex
10. SIMULATNEOUS EQUATIONS
http://mathsfirst.massey.ac.nz/Algebra/SystemsofLinEq/EMeth.htm
http://www.analyzemath.com/Tutorial-System-Equations/Tutorial-System-Equations.html
http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/quadequationshirev4.shtml
https://en.wikipedia.org/wiki/Simultaneous_equations
https://www.youtube.com/watch?v=NPzlCNDEJqA
https://www.youtube.com/watch?v=YtyveL7SrEE
http://www.mathsmutt.co.uk/files/simgraph2.htm
11. TRIGONOMETRY
https://en.wikipedia.org/wiki/Trigonometry
https://www.skillsyouneed.com/num/trigonometry.html
https://www.mathsisfun.com/algebra/trigonometry.html
https://www.khanacademy.org/math/trigonometry
https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-trig-ratios-intro/v/basic-
trigonometry
https://betterexplained.com/articles/intuitive-trigonometry/
https://www.germanna.edu/documents/Howtographtrigfunctions.pdf
http://www.purplemath.com/modules/triggrph.htm
https://www.khanacademy.org/math/algebra2/trig-functions/graphs-of-sine-cosine-tangent-
alg2/v/we-graph-domain-and-range-of-sine-function
https://www.youtube.com/watch?v=ijTIr-aykUk
https://www.youtube.com/watch?v=vHYI93UV5Kg
http://www.themathpage.com/atrig/graphs-trig.htm
http://www.sparknotes.com/math/trigonometry/graphs/section3.rhtml
12. CALCULUS
https://revisionmaths.com/gcse-maths-revision/algebra/gradients-and-graphs
https://www.siyavula.com/read/maths/grade-12/differential-calculus
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http://www.ecdoe.gov.za/documents/learners/learner-support/maths-911/gr-12-calculus.pdf
https://www.youtube.com/watch?v=nYQ08qEZxQY
https://www.google.co.za/search?
dcr=0&source=hp&ei=SakjWpXVK4GPU8G5pGg&q=calculus+grade+12&oq=calculus&gs_l=
psy-ab.1.1.0l2j0i20i263k1j0l7.2625.8031.0.14438.9.6.0.2.2.0.283.1579.2-6.6.0....0...1c.1.64.psy-
ab..1.8.1596.0..35i39k1j0i131k1j0i67k1.0.46L0wbHngn0
https://cnx.org/contents/lKwDs6cA@1/Differential-Calculus-Introduc
https://www.youtube.com/watch?v=Nxa6IQPqlas
http://learn.mindset.co.za/resources/mathematics/grade-12/differential-calculus
13. SETS
https://www.mathsisfun.com/sets/sets-introduction.html
https://en.wikipedia.org/wiki/Set_(mathematics)
https://www.khanacademy.org/math/statistics-probability/probability-library/basic-set-
ops/v/intersection-and-union-of-sets
http://www.onlinemathlearning.com/math-sets.html
https://study.com/academy/lesson/mathematical-sets-elements-intersections-unions.html
http://mathworld.wolfram.com/Set.html
https://www.google.co.za/search?
rlz=1C2CAFA_enZA732ZA732&dcr=0&source=hp&ei=8MYjWoSCKMr5UOf7iuAO&q=sets&oq
=se&gs_l=psy-
ab.1.0.35i39k1l2j0i67k1l2j0i131k1l3j0j0i131k1l2.2130.2485.0.5446.4.3.0.0.0.0.396.668.2-
1j1.2.0....0...1c.1.64.psy-ab..2.2.667.0...0.tBH6jPZvzho
14. PROBABILITY AND STATISTICS
http://www.purplemath.com/modules/meanmode.htm
https://www.thoughtco.com/the-mean-median-and-mode-2312604
https://www.universalclass.com/articles/math/statistics/how-to-calculate-mean-median-
mode.htm
https://www.thoughtco.com/variance-and-standard-deviation-3026711
https://en.wikipedia.org/wiki/Standard_deviation
https://en.wikipedia.org/wiki/Variance
https://www.mathsisfun.com/data/probability.html
https://brilliant.org/courses/probability/
https://www.mathgoodies.com/lessons/vol6/intro_probability
https://en.wikipedia.org/wiki/Probability
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http://mathworld.wolfram.com/Probability.html
https://en.wikipedia.org/wiki/Statistics
https://en.wikipedia.org/wiki/Mathematical_statistics
15. SOLUTION OF QUADRATIC EQUATIONS
Using:
15.1 Factorisation
http://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/
U12_L3_T1_text_final.html
https://www.varsitytutors.com/hotmath/hotmath_help/topics/solving-quadratic-equations-
using-factoring
http://www.purplemath.com/modules/solvquad.htm
https://www.khanacademy.org/math/algebra/quadratics/solving-quadratic-equations-by-
factoring/v/example-1-solving-a-quadratic-equation-by-factoring
15.2 Completing a square
(Take the term that is independent of x to the
RHS)
( Divide throughout by the coefficient of )
(Add the square of half the coefficient of x to both
sides and simplify the RHS)
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(Express LHS as square of a binomial)
(Find the square root of each side)
(Solve for x)
https://www.mathsisfun.com/algebra/completing-square.html
http://www.purplemath.com/modules/solvquad3.htm
https://www.brainfuse.com/jsp/alc/resource.jsp?s=gre&c=35261&cc=108824
https://www.intmath.com/quadratic-equations/2-solving-quadratic-equations-completing-
square.php
15.3 a Formula:
https://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE
/U16_L5_T2_text_final.html
http://www.sosmath.com/algebra/quadraticeq/quadraformula/quadraformula.html
http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/quadequationshirev3.shtml
https://en.wikipedia.org/wiki/Quadratic_formula
16. QUADRATIC INEQUALITIES
https://www.siyavula.com/read/maths/grade-11/equations-and-inequalities/02-equations-
and-inequalities-07
https://www.mathsisfun.com/algebra/inequality-quadratic-solving.html
https://www.shmoop.com/quadratic-formula-function/quadratic-inequality-examples.html
https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/quadratics-in-one-
variable/solving-quadratic-inequalities
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17. NATURE OF ROOTS
17.1 Use of (Delta) to determine the nature of roots
17.1.1 If = 0, then the roots are real, rational and equal
17.1.2 If then the roots are real
17.1.3 If then the roots are non-real (imaginary)
17.1.4 If and a perfect square, then the roots are real, rational and unequal
17.1.5 If and not a perfect square, then the roots are real, irrational and
unequal
http://www.math-only-math.com/nature-of-the-roots-of-a-quadratic-equation.html
http://www.tutorsonnet.com/nature-of-roots-of-quadratic-equations-homework-help.php
http://www.ask-math.com/nature-of-roots-in-quadratic-equation.html
http://www.national5maths.co.uk/nature-of-roots/
http://www.mathwarehouse.com/quadratic/discriminant-in-quadratic-equation.php
17.2 Determine the value of an unknown in a quadratic equation if the nature of
roots
is given
https://www.siyavula.com/read/maths/grade-11/equations-and-inequalities/02-equations-
and-inequalities-06
http://mathinstructor.net/2012/05/sample-questions-nature-of-roots-of-quadratic-equation-
cbse-ncert-solutions-chapter-4-exercise-4-4-q2/
18. RELATIONS AND FUNCTIONS
http://www.sparknotes.com/math/algebra2/functions/section1.rhtml
http://www.purplemath.com/modules/fcns.htm
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https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-linear-equations-
functions/cc-8th-function-intro/v/relations-and-functions
http://www.mathwarehouse.com/algebra/relation/math-function.php
19. HYPERBOLA
https://www.google.co.za/search?
rlz=1C2CAFA_enZA732ZA732&dcr=0&source=hp&ei=8MYjWoSCKMr5UOf7iuAO&q=hyperbo
la&oq=HYPERBOLA&gs_l=psy-ab.1.0.0l10.3673.20601.0.23296.13.10.1.1.1.0.310.2528.2-
8j1.9.0....0...1c.1.64.psy-ab..2.11.2544.0..0i131k1j0i10k1.0.XiPzvh_dDyo
https://en.wikipedia.org/wiki/Hyperbola
https://www.mathsisfun.com/geometry/hyperbola.html
http://mathworld.wolfram.com/Hyperbola.html
http://www.purplemath.com/modules/hyperbola.htm
20. CIRCLE
https://www.mathsisfun.com/algebra/circle-equations.html
http://www.purplemath.com/modules/circle.htm
http://www.purplemath.com/modules/sqrcircle.htm
https://en.wikipedia.org/wiki/Circle
https://www.mathsisfun.com/geometry/circle.html
21. GEOMETRY
http://www.onlinemathlearning.com/geometry-help.html#CircleTheorems
http://study.com/academy/topic/high-school-geometry-triangles-theorems-and-proofs.html
https://www.mathsisfun.com/geometry/circle-theorems.html
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http://www.math.com/tables/geometry/circles.htm
22. WORD PROBLEMS
https://www.scribd.com/presentation/461055254/General-Procedure-in-the-Solution-of-Word-
Problems
https://www.scribd.com/presentation/461055511/Problems-Involving-Numbers
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