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Maths Extension 1 Topics

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1. Adding and Subtracting Fractions

1.1

Using models to add and subtract fractions

1.2

Adding fractions with like denominators

1.3

Subtracting fractions with like denominators

1.4

Adding and subtracting fractions with unlike denominators

1.5

Adding and subtracting mixed numbers

2. Multiplying and Dividing Fractions

2.1

Multiplying fractions and whole numbers

2.2

Dividing fractions with whole numbers

2.3

Multiplying proper fractions

2.4

Multiplying improper fractions and mixed numbers

2.5

Dividing fractions and mixed numbers

2.6

Applications of fraction operations

3. Pythagorean Theorem

3.1

Squares and square roots

3.2

Pythagorean theorem

3.3

Estimating square roots

3.4

Using the pythagorean relationship

3.5

Applications of pythagorean theorem

4. Percents

4.1

Representing percents

4.2

Percents, fractions, and decimals

4.3

Percent of a number

4.4

Adding and multiplying percents

4.5

Taxes, discounts, tips and more

4.6

Simple interest

5. Surds

5.1

Square and square roots

5.2

Cubic and cube roots

5.3

Evaluating and simplifying surds

5.4

Converting radicals to mixed radicals

5.5

Converting radicals to entire radicals

5.6

Adding and subtracting surds

5.7

Multiplying and dividing surds

5.8

Rationalize the denominator

6. Algebraic Expressions

6.1

Equivalent algebraic expressions

6.2

Adding and subtracting algebraic expressions

6.3

Multiplying monomial by monomial

6.4

Multiplying monomial by binomial

6.5

Multiplying binomial by binomial

6.6

Multiplying polynomial by polynomial

6.7

Applications of polynomials

7. Introduction to Relations and Functions

7.1

Relationship between two variables

7.2

Understand relations between x- and y-intercepts

7.3

Domain and range of a function

7.4

Identifying functions

7.5

Function notation

8. Linear Functions

8.1

Distance formula:

d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}

8.2

Midpoint formula:

M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)

8.3

Gradient equation:

m = \frac{y_2-y_1}{x_2- x_1}

8.4

Gradient intercept form: y = mx + b

8.5

General form: Ax + By + C = 0

8.6

Gradient-point form:

y - y_1 = m (x - x_1)

8.7

8.8

Graphing linear functions using table of values

8.9

Graphing linear functions using x- and y-intercepts

8.10

Graphing from slope-intercept form y=mx+b

8.11

Graphing linear functions using a single point and gradient

8.12

Word problems of graphing linear functions

8.13

Parallel and perpendicular lines in linear functions

8.14

Applications of linear relations

8.15

Perpendicular line equation

9. Simultaneous Equations

9.1

Determining number of solutions to linear equations

9.2

Solving simultaneous linear equations by graphing

9.3

Solving simultaneous linear equations by elimination

9.4

Solving simultaneous linear equations by substitution

9.5

Money related questions in linear equations

9.6

Unknown number related questions in linear equations

9.7

Distance and time related questions in linear equations

9.8

Rectangular shape related questions in linear equations

9.9

Simultaneous linear-quadratic equations

9.10

Simultaneous quadratic-quadratic equations

9.11

Solving 3 variable simultaneous equations by substitution

9.12

Solving 3 variable simultaneous equations by elimination

9.13

Solving 3 variable simultaneous equations with no or infinite solutions

9.14

Word problems relating 3 variable simultaneous equations

10. Factorisation

10.1

Factorise by taking out the greatest common factor

10.2

Factorise by grouping

10.3

Factorising difference of squares: x^2 - y^2

10.4

Factorising trinomials

10.5

Factoring difference of cubes

10.6

Factoring sum of cubes

11. Quadratic Functions

11.1

Introduction to quadratic functions

11.2

Transformations of quadratic functions

11.3

Quadratic function in general form: y = ax^2 + bx + c

11.4

Quadratic function in vertex form: y = a(x-p)^2 + q

11.5

Completing the square

11.6

Converting from general to vertex form by completing the square

11.7

Shortcut: Vertex formula

11.8

Graphing quadratic functions: General form VS. Vertex form

11.9

Finding the quadratic functions for given parabolas

11.10

Applications of quadratic functions

12. Solving Quadratic Equations

12.1

Solving quadratic equations by factorising

12.2

Solving quadratic equations by completing the square

12.3

Using quadratic formula to solve quadratic equations

12.4

Nature of roots of quadratic equations: The discriminant

12.5

Applications of quadratic equations

13. Inequalities

13.1

Express linear inequalities graphically and algebraically

13.2

Solving one-step linear inequalities

13.3

Solving multi-step linear inequalities

13.4

Compound inequalities

13.5

Solving absolute value inequalities

13.6

Graphing linear inequalities in two variables

13.7

Graphing simultaneous inequalities

13.8

Solving quadratic inequalities in one variable

14. Algebraic Fractions

14.1

Simplifying algebraic fractions and restrictions

14.2

Adding and subtracting algebraic fractions

14.3

Multiplying algebraic fractions

14.4

Dividing algebraic fractions

14.5

Solving equations with algebraic fractions

14.6

Applications of equations with algebraic fractions

14.7

Simplifying complex fractions

14.8

Partial fraction decomposition

15. Reciprocal Functions

15.1

Graphing reciprocals of linear functions

15.2

Graphing reciprocals of quadratic functions

16. Angles, Lines, and Transversals

16.1

Pairs of lines and angles

16.2

Parallel lines and transversals

16.3

Parallel line proofs

16.4

Perpendicular line proofs

17. Scale Factors and Similarity

17.1

Enlargements and reductions with scale factors

17.2

17.3

Similar triangles

17.4

Similar polygons

18. Properties of Triangles

18.1

Classifying Triangles

18.2

Isosceles and Equilateral Triangles

19. Congruent Triangles

19.1

Congruence and Congruent Triangles

19.2

Triangles Congruent by SSS Proofs

19.3

Triangles Congruent by SAS and HL Proofs

19.4

Triangles Congruent by ASA and AAS Proofs

20. Circles

20.1

Angles in a circle

20.2

Chord properties

20.3

Tangent properties

20.4

Circle and circumference

20.5

Arcs of a circle

20.6

Areas and sectors of circles

20.7

Inscribed Angles and Proofs

20.8

Central and inscribed angles in circles

21. Functions

21.1

Function notation

21.2

Operations with functions

21.3

Adding functions

21.4

Subtracting functions

21.5

Multiplying functions

21.6

Dividing functions

21.7

Composite functions

21.8

Even and odd functions

22. Polynomial Functions

22.1

What is a polynomial function?

22.2

Polynomial long division

22.3

Polynomial synthetic division

22.4

Remainder theorem

22.5

22.6

Rational zero theorem

22.7

Characteristics of polynomial graphs

22.8

Multiplicities of polynomials

22.9

Imaginary zeros of polynomials

22.10

Determining the equation of a polynomial function

22.11

Applications of polynomial functions

22.12

Solving polynomial inequalities

22.13

Fundamental theorem of algebra

22.14

Descartes' rule of signs

23. Rational Functions

23.1

What is a rational function?

23.2

Point of discontinuity

23.3

Vertical asymptote

23.4

Horizontal asymptote

23.5

Slant asymptote

23.6

Graphs of rational functions

23.7

Solving rational inequalities

24. Indices

24.1

Indices: Product rule (a^x)(a^y) = a^(x+y)

24.2

Indices: Division rule (a^x / a^y) = a^(x-y)

24.3

Indices: Power rule (a^x)^y = a^(x * y)

24.4

Indices: Negative exponents

24.5

Indices: Zero exponent: a^0 = 1

24.6

Indices: Rational exponents

24.7

Combining laws of indices

24.8

Scientific notation

24.9

Convert between radicals and rational exponents

24.10

Solving for indices

25. Exponential Functions

25.1

Solving exponential equations using exponent rules

25.2

Graphing exponential functions

25.3

Graphing transformations of exponential functions

25.4

Finding an exponential function given its graph

26. Logarithmic Functions

26.1

What is a logarithm?

26.2

Converting from logarithmic form to exponential form

26.3

Evaluating logarithms without a calculator

26.4

Common logarithms

26.5

Natural log: ln

26.6

Evaluating logarithms using change-of-base formula

26.7

Converting from exponential form to logarithmic form

26.8

Solving exponential equations with logarithms

26.9

Product rule of logarithms

26.10

Quotient rule of logarithms

26.11

Combining product rule and quotient rule in logarithms

26.12

Evaluating logarithms using logarithm rules

26.13

Solving logarithmic equations

26.14

Graphing logarithmic functions

26.15

Finding a logarithmic function given its graph

27. Trigonometric Ratios and Angle Measure

27.1

Angle in standard position

27.2

Coterminal angles

27.3

Reference angle

27.4

Find the exact value of trigonometric ratios

27.5

ASTC rule in trigonometry (All Students Take Calculus)

27.6

27.7

Converting between degrees and radians

27.8

Trigonometric ratios of angles in radians

27.9

Radian measure and arc length

28. Sine Rule and Cosine Rule

28.1

28.2

28.3

Sine rule and cosine rule word problems

29. Trigonometric Graphs

29.1

Sine graph: y = sin x

29.2

Cosine graph: y = cos x

29.3

Tangent graph: y = tan x

30. Trigonometric Identities

30.1

Quotient identities and reciprocal identities

30.2

Pythagorean identities

30.3

Sum and difference identities

30.4

Cofunction identities

30.5

Double-angle identities

31. Solving Trigonometric Equations

31.1

Solving first degree trigonometric equations

31.2

Determining non-permissible values for trig expressions

31.3

Solving second degree trigonometric equations

31.4

Solving trigonometric equations involving multiple angles

31.5

Solving trigonometric equations using pythagorean identities

31.6

Solving trigonometric equations using sum and difference identities

31.7

Solving trigonometric equations using double-angle identities

32. Inverse Trigonometric Functions

32.1

Finding inverse trigonometric function from its graph

32.2

Evaluating inverse trigonometric functions

32.3

Finding inverse reciprocal trigonometric function from its graph

32.4

Finding exact value of inverse reciprocal trig functions

33. Bearings

33.1

Introduction to bearings

33.2

Bearings and direction word problems

33.3

Angle of elevation and depression

34. Series

34.1

Arithmetic series

34.2

Geometric series

34.3

Convergence and divergence of normal infinite series

34.4

Convergence & divergence of geometric series

35. Growth and Decay

35.1

Exponential growth and decay by a factor

35.2

Exponential growth and decay by percentage

35.3

Finance: Compound interest

35.4

Finance: Future value and present value

36. Conics

36.1

Conics - Parabola

36.2

Conics - Ellipse

36.3

Conics - Circle

36.4

Conics - Hyperbola

37. Permutations and Combinations

37.1

Fundamental counting principle

37.2

Factorial notation

37.3

Path counting problems

37.4

Permutation vs. Combination

37.5

37.6

37.7

Problems involving both permutations and combinations

37.8

Pascal's triangle

37.9

Binomial theorem

38. Discrete Probabilities

38.1

Probability distribution - histogram, mean, variance & standard deviation

38.2

Binomial distribution

38.3

Mean and standard deviation of binomial distribution

39. Probability

39.1

Addition rule for "OR"

39.2

Multiplication rule for "AND"

40. Limits

40.1

Finding limits from graphs

40.2

40.3

Finding limits algebraically - direct substitution

40.4

Finding limits algebraically - when direct substitution is not possible

40.5

41. Derivatives

41.1

Definition of derivative

41.2

41.3

Slope and equation of tangent line

41.4

41.5

Derivative of trigonometric functions

41.6

Derivative of exponential functions

41.7

41.8

41.9

Derivative of inverse trigonometric functions

41.10

Derivative of logarithmic functions

41.11

Higher order derivatives

42. Derivative Applications

42.1

Rectilinear Motion: Derivative

42.2

Critical number & maximum and minimum values

42.3

l'Hospital's rule

42.4

Curve sketching

42.5

43. Integrals

43.1

Antiderivatives

43.2

43.3

Definite integral

43.4

Fundamental theorem of calculus

43.5

43.6

Integration using trigonometric identities

44. Integral Applications

44.1

Areas between curves

44.2

Volumes of solids of revolution - Disc method

44.3

Volumes of solids of revolution - Shell method

45. Parametric Equations

45.1

Defining curves with parametric equations

45.2

Tangent and concavity of parametric equations

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