NEW ZEALAND MATH YEAR 13 – EDmama New Zealand

A. Introduction to Probability

A.01. Define probability and describe its basic principles.
A.02. Calculate the probability of simple events.
A.03. Use two-way frequency tables to determine probabilities.
A.04. Apply the addition rule to find probabilities of mutually exclusive and non-mutually exclusive events.
A.05. Differentiate between independent and dependent events.
A.06. Calculate conditional probabilities using formulas and two-way tables.
A.07. Apply combinations and permutations to calculate probabilities.
A.08. Solve complex probability problems involving multiple events and conditional probabilities.
A.09. Apply probability concepts to real-world scenarios and decision-making.

B. Introduction to Statistics

B.01. Define basic statistical terms such as population, sample, parameter, and statistic.
B.02. Calculate measures of central tendency (mean, median, mode) and dispersion (range, variance, standard deviation).
B.03. Identify and interpret outliers in a data set.
B.04. Describe the effect of removing outliers on measures of central tendency and dispersion.
B.05. Create and interpret scatter plots to identify correlation.
B.06. Match correlation coefficients to scatter plots.
B.07. Calculate and interpret correlation coefficients.
B.08. Find the equation of a regression line and interpret its slope and intercept.
B.09. Analyse a regression line of a data set.
B.10. Analyse a regression line using statistics of a data set.
B.11. Design experiments to minimise bias.
B.12. Identify biased samples and discuss potential sources of bias.
B.13. Find confidence intervals for population means and proportions.
B.14. Interpret confidence intervals for population means and proportions.
B.15. Apply statistical methods to analyse real-world data and draw conclusions.

C. Sequences and Series

C.01. Define a sequence and series, distinguishing between arithmetic and geometric types.
C.02. Find terms of a sequence given an explicit formula.
C.03. Find terms of a recursive sequence.
C.04. Identify a sequence as explicit or recursive.
C.05. Find a recursive formula.
C.06. Find recursive and explicit formulas.
C.07. Convert a recursive formula to an explicit formula.
C.08. Convert an explicit formula to a recursive formula.
C.09. Convert between explicit and recursive formulas.
C.10. Use sigma notation to represent a series.
C.11. Identify arithmetic and geometric series.
C.12. Find the sum of a finite arithmetic or geometric series.
C.13. Introduction to partial sums.
C.14. Calculate partial sums of arithmetic series.
C.15. Calculate partial sums of geometric series.
C.16. Partial sums: mixed review.
C.17. Determine if a geometric series is convergent or divergent.
C.18. Find the value of an infinite geometric series.
C.19. Write a repeating decimal as a fraction using the concept of infinite geometric series.
C.20. Apply sequences and series to solve financial problems (e.g., annuities, compound interest).

D. Functions: Foundations

D.01. Define a function and its key components: domain, range, and mapping.
D.02. Identify functions represented graphically, algebraically, and through mappings.
D.03. Evaluate functions for given input values.
D.04. Find values using function graphs.
D.05. Complete a table for a function graph.
D.06. Determine the domain and range of a function from its graph or equation.
D.07. Find the gradient of a linear function.
D.08. Graph a linear function.
D.09. Write the equation of a linear function.
D.10. Analyse the rate of change of linear functions over unit intervals.
D.11. Explore real-world applications of functions and their graphical representations.

E. Families of Functions and Transformations

E.01. State and apply the basic function transformation rules.
E.02. Perform translations of functions and describe the effect on the graph.
E.03. Perform reflections of functions over the x-axis and y-axis, and describe the effect on the graph.
E.04. Perform dilations (stretches and compressions) of functions and describe the effect on the graph.
E.05. Apply combinations of transformations to functions.
E.06. Describe a sequence of transformations applied to a function.
E.07. Compose functions and evaluate the resulting composite function.
E.08. Determine if two functions are inverses of each other.
E.09. Find values of inverse functions from tables.
E.10. Find values of inverse functions from graphs.
E.11. Find inverse functions and relations.
E.12. Understand the relationship between a function and its inverse graphically and algebraically.
E.13. Analyse the effect of transformations on the domain and range of a function.

F. Quadratic Functions

F.01. Identify the key characteristics of quadratic functions, including vertex, axis of symmetry, and intercepts.
F.02. Graph a quadratic function in vertex form, standard form, and factored form.
F.03. Match quadratic functions and graphs.
F.04. Find the maximum or minimum value of a quadratic function using various methods.
F.05. Solve a quadratic equation using square roots.
F.06. Solve a quadratic equation by factorising.
F.07. Solve a quadratic equation by completing the square.
F.08. Solve a quadratic equation using the quadratic formula.
F.09. Apply the discriminant to determine the number and nature of solutions to a quadratic equation.
F.10. Model and solve real-world problems using quadratic functions.
F.11. Analyse the relationship between the coefficients of a quadratic equation and the shape and position of its graph.

G. Exponential and Logarithmic Functions

G.01. Convert between exponential and logarithmic forms.
G.02. Evaluate simple logarithms.
G.03. Determine the domain and range of exponential and logarithmic functions.
G.04. Apply the change of base formula to evaluate logarithms with different bases.
G.05. Apply the product property of logarithms to simplify expressions.
G.06. Apply the quotient property of logarithms to simplify expressions.
G.07. Apply the power property of logarithms to simplify expressions.
G.08. Evaluate logarithms using properties.
G.09. Solve exponential equations by rewriting the base.
G.10. Solve exponential equations using logarithms.
G.11. Solve logarithmic equations with one logarithm.
G.12. Solve logarithmic equations with multiple logarithms.
G.13. Distinguish between linear and exponential functions.
G.14. Analyse the rate of change of exponential functions over unit intervals.
G.15. Describe and model linear and exponential growth and decay.
G.16. Solve exponential growth and decay word problems.
G.17. Solve compound interest word problems.
G.18. Analyse the relationship between exponential and logarithmic functions as inverses.
G.19. Model and solve real-world problems involving exponential growth and decay, including financial applications and population dynamics.

H. Trigonometry: Angles, Ratios, and Functions

H.01. Convert between radians and degrees.
H.02. Calculate arc length using radians.
H.03. Identify angles in different quadrants.
H.04. Determine coterminal and reference angles.
H.05. Find trigonometric ratios (sine, cosine, tangent, cosecant, secant, cotangent) using right triangles.
H.06. Find trigonometric ratios using the unit circle.
H.07. Find trigonometric ratios using reference angles.
H.08. Determine the inverses of trigonometric functions.
H.09. Solve trigonometric equations.
H.10. Use trigonometric ratios to find a side length in a right triangle.
H.11. Use trigonometric ratios to find an angle measure in a right triangle.
H.12. Solve a right triangle given sufficient information.
H.13. Apply the Law of Sines to solve triangles.
H.14. Apply the Law of Cosines to solve triangles.
H.15. Solve a triangle using appropriate trigonometric laws.
H.16. Calculate the area of a triangle using the sine formula.
H.17. Calculate the area of a triangle using Heron’s formula.
H.18. Apply trigonometric concepts to solve real-world problems involving angles, distances, and heights.

I. Trigonometric Functions: Graphs and Properties

I.01. Find the properties of sine functions (amplitude, period, phase shift, vertical shift).
I.02. Write equations of sine functions from graphs.
I.03. Write equations of sine functions using properties.
I.04. Graph sine functions, identifying key features.
I.05. Find the properties of cosine functions (amplitude, period, phase shift, vertical shift).
I.06. Write equations of cosine functions from graphs.
I.07. Write equations of cosine functions using properties.
I.08. Graph cosine functions, identifying key features.
I.09. Graph sine and cosine functions.
I.10. Analyse the relationship between the unit circle and the graphs of trigonometric functions.
I.11. Model periodic phenomena using trigonometric functions, such as sound waves and alternating current.

J. Trigonometric Identities

K. Polynomials

K.01. Divide polynomials using long division.
K.02. Divide polynomials using synthetic division.
K.03. Evaluate polynomials using synthetic division.
K.04. Write a polynomial from its roots.
K.05. Find the roots of factorised polynomials.
K.06. Apply the rational root theorem to find possible rational roots of a polynomial.
K.07. Apply the complex conjugate theorem to identify complex roots of a polynomial.
K.08. Apply conjugate root theorems.
K.09. Apply Descartes’ Rule of Signs to determine the possible number of positive and negative real roots of a polynomial.
K.10. State and apply the Fundamental Theorem of Algebra.
K.11. Match polynomials and graphs.
K.12. Factorise sums and differences of cubes.
K.13. Solve equations with sums and differences of cubes.
K.14. Factorise using a quadratic pattern.
K.15. Solve equations using a quadratic pattern.
K.16. Use Pascal’s triangle to determine binomial coefficients.
K.17. Relate Pascal’s triangle to the Binomial Theorem.
K.18. Apply the Binomial Theorem I.
K.19. Apply the Binomial Theorem II.
K.20. Analyse the relationship between the roots and coefficients of a polynomial.

L. Inequalities and Linear Programming

M. Simultaneous Equations

M.01. Solve simultaneous equations by graphing.
M.02. Solve simultaneous equations by graphing: word problems.
M.03. Classify simultaneous equations as independent, dependent, or inconsistent.
M.04. Solve simultaneous equations using substitution.
M.05. Solve simultaneous equations using substitution: word problems.
M.06. Solve simultaneous equations using elimination.
M.07. Solve simultaneous equations using elimination: word problems.
M.08. Solve simultaneous equations in three variables using substitution.
M.09. Solve simultaneous equations in three variables using elimination.
M.10. Determine the number of solutions to simultaneous equations in three variables.
M.11. Apply simultaneous equations to model and solve real-world problems with multiple variables and constraints.
M.12. Solve simultaneous equations using matrices.

N. Conic Sections

N.01. Identify the key properties of parabolas (vertex, focus, directrix, axis of symmetry).
N.02. Write equations of parabolas in vertex form.
N.03. Graph parabolas given their equations.
N.04. Identify the key properties of circles (centre, radius).
N.05. Write equations of circles in standard form.
N.06. Graph circles given their equations.
N.07. Identify the key properties of ellipses (centre, vertices, foci, major axis, minor axis).
N.08. Find the eccentricity of an ellipse and interpret its meaning.
N.09. Write equations of ellipses in standard form.
N.10. Identify the key properties of hyperbolas (centre, vertices, foci, transverse axis, conjugate axis, asymptotes).
N.11. Find the eccentricity of a hyperbola and interpret its meaning.
N.12. Write equations of hyperbolas in standard form.
N.13. Convert equations of conic sections from general to standard form.
N.14. Graph ellipses and hyperbolas given their equations.
N.15. Apply conic sections to model real-world phenomena, such as planetary orbits and satellite dishes.

O. Complex Numbers

O.01. Define the imaginary unit i and understand its properties.
O.02. Add and subtract complex numbers.
O.03. Identify and find complex conjugates.
O.04. Multiply and divide complex numbers.
O.05. Simplify complex expressions involving complex conjugates.
O.06. Find absolute values of complex numbers.
O.07. Simplify powers of i.
O.08. Perform arithmetic operations with complex numbers, including addition, subtraction, multiplication, and division.
O.09. Solve quadratic equations with complex solutions.

P. Complex Plane

P.01. Introduce the complex plane and its axes (real and imaginary).
P.02. Graph complex numbers in the complex plane.
P.03. Perform addition in the complex plane graphically.
P.04. Perform subtraction in the complex plane graphically.
P.05. Graph complex conjugates in the complex plane.
P.06. Find the absolute value of a complex number in the complex plane.
P.07. Find midpoints in the complex plane.
P.08. Calculate the distance between two complex numbers in the complex plane.
P.09. Explore geometric transformations of complex numbers in the complex plane.

Q. Polar Form of Complex Numbers

Q.01. Find the modulus (magnitude) and argument (angle) of a complex number.
Q.02. Convert complex numbers from rectangular form to polar form.
Q.03. Convert complex numbers from polar form to rectangular form.
Q.04. Convert complex numbers between rectangular and polar form.
Q.05. Match polar equations and graphs.
Q.06. Perform multiplication and division of complex numbers in polar form.
Q.07. Apply De Moivre’s Theorem to find powers and roots of complex numbers in polar form.
Q.08. Solve equations involving complex numbers in polar form.

R. Introduction to Limits

S. Calculate Limits

S.01. Find limits using addition, subtraction and multiplication laws.
S.02. Find limits using the division law.
S.03. Find limits using power and root laws.
S.04. Apply limit laws to evaluate limits of functions.
S.05. Find limits of polynomials and rational functions.
S.06. Find limits involving factorisation and rationalisation.
S.07. Find limits involving absolute value functions.
S.08. Find limits involving trigonometric functions.
S.09. Evaluate indeterminate forms (0/0, ∞/∞) using algebraic techniques and L’Hôpital’s Rule.
S.10. Apply limits to analyse the behaviour of functions near specific points.

T. Limits Involving Infinity

U. Rational Functions: Asymptotes and Continuity

V. Continuity: Definition and Properties

V.01. Identify graphs of continuous functions.
V.02. Determine continuity using graphs.
V.03. Determine one-sided continuity using graphs.
V.04. Find and analyse points of discontinuity using graphs.
V.05. Determine continuity on an interval using graphs.
V.06. Determine the continuity of a piecewise function at a point.
V.07. Make a piecewise function continuous.
V.08. State and apply the Intermediate Value Theorem.
V.09. Analyse the continuity of functions defined by limits.

W. Introduction to Derivatives: Rates of Change

W.01. Calculate the average rate of change of a function over an interval I.
W.02. Calculate the average rate of change of a function over an interval II.
W.03. Find instantaneous rates of change.
W.04. Understand velocity as a rate of change.
W.05. Find values of derivatives using limits.
W.06. Find the gradient of a tangent line using limits.
W.07. Find equations of tangent lines using limits.
W.08. Relate the derivative to the slope of the tangent line and the instantaneous rate of change.

X. Derivative Rules

Y. Calculate Derivatives: Specific Functions

Y.01. Find derivatives of polynomials.
Y.02. Find derivatives of rational functions.
Y.03. Find derivatives of trigonometric functions I.
Y.04. Find derivatives of trigonometric functions II.
Y.05. Find derivatives of exponential functions.
Y.06. Find derivatives of logarithmic functions.
Y.07. Find derivatives of inverse trigonometric functions.
Y.08. Find derivatives of radical functions.
Y.09. Find derivatives using the product rule I.
Y.10. Find derivatives using the product rule II.
Y.11. Find derivatives using the quotient rule I.
Y.12. Find derivatives using the quotient rule II.
Y.13. Find derivatives using the chain rule I.
Y.14. Find derivatives using the chain rule II.
Y.15. Apply derivative rules to solve real-world problems involving rates of change and optimisation.

Course Content

A. Introduction to Probability

A.01. Define probability and describe its basic principles.

A.02. Calculate the probability of simple events.

A.03. Use two-way frequency tables to determine probabilities.

A.04. Apply the addition rule to find probabilities of mutually exclusive and non-mutually exclusive events.

A.05. Differentiate between independent and dependent events.

A.06. Calculate conditional probabilities using formulas and two-way tables.

A.07. Apply combinations and permutations to calculate probabilities.

A.08. Solve complex probability problems involving multiple events and conditional probabilities.

A.09. Apply probability concepts to real-world scenarios and decision-making.

B. Introduction to Statistics

B.01. Define basic statistical terms such as population, sample, parameter, and statistic.

B.02. Calculate measures of central tendency (mean, median, mode) and dispersion (range, variance, standard deviation).

B.03. Identify and interpret outliers in a data set.

B.04. Describe the effect of removing outliers on measures of central tendency and dispersion.

B.05. Create and interpret scatter plots to identify correlation.

B.06. Match correlation coefficients to scatter plots.

B.07. Calculate and interpret correlation coefficients.

B.08. Find the equation of a regression line and interpret its slope and intercept.

B.09. Analyse a regression line of a data set.

B.10. Analyse a regression line using statistics of a data set.

B.11. Design experiments to minimise bias.

B.12. Identify biased samples and discuss potential sources of bias.

B.13. Find confidence intervals for population means and proportions.

B.14. Interpret confidence intervals for population means and proportions.

B.15. Apply statistical methods to analyse real-world data and draw conclusions.

C. Sequences and Series

C.01. Define a sequence and series, distinguishing between arithmetic and geometric types.

C.02. Find terms of a sequence given an explicit formula.

C.03. Find terms of a recursive sequence.

C.04. Identify a sequence as explicit or recursive.

C.05. Find a recursive formula.

C.06. Find recursive and explicit formulas.

C.07. Convert a recursive formula to an explicit formula.

C.08. Convert an explicit formula to a recursive formula.

C.09. Convert between explicit and recursive formulas.

C.10. Use sigma notation to represent a series.

C.11. Identify arithmetic and geometric series.

C.12. Find the sum of a finite arithmetic or geometric series.

C.13. Introduction to partial sums.

C.14. Calculate partial sums of arithmetic series.

C.15. Calculate partial sums of geometric series.

C.16. Partial sums: mixed review.

C.17. Determine if a geometric series is convergent or divergent.

C.18. Find the value of an infinite geometric series.

C.19. Write a repeating decimal as a fraction using the concept of infinite geometric series.

C.20. Apply sequences and series to solve financial problems (e.g., annuities, compound interest).

D. Functions: Foundations

D.01. Define a function and its key components: domain, range, and mapping.

D.02. Identify functions represented graphically, algebraically, and through mappings.

D.03. Evaluate functions for given input values.

D.04. Find values using function graphs.

D.05. Complete a table for a function graph.

D.06. Determine the domain and range of a function from its graph or equation.

D.07. Find the gradient of a linear function.

D.08. Graph a linear function.

D.09. Write the equation of a linear function.

D.10. Analyse the rate of change of linear functions over unit intervals.

D.11. Explore real-world applications of functions and their graphical representations.

E. Families of Functions and Transformations

E.01. State and apply the basic function transformation rules.

E.02. Perform translations of functions and describe the effect on the graph.

E.03. Perform reflections of functions over the x-axis and y-axis, and describe the effect on the graph.

E.04. Perform dilations (stretches and compressions) of functions and describe the effect on the graph.

E.05. Apply combinations of transformations to functions.

E.06. Describe a sequence of transformations applied to a function.

E.07. Compose functions and evaluate the resulting composite function.

E.08. Determine if two functions are inverses of each other.

E.09. Find values of inverse functions from tables.

E.10. Find values of inverse functions from graphs.

E.11. Find inverse functions and relations.

E.12. Understand the relationship between a function and its inverse graphically and algebraically.

E.13. Analyse the effect of transformations on the domain and range of a function.

F. Quadratic Functions

F.01. Identify the key characteristics of quadratic functions, including vertex, axis of symmetry, and intercepts.

F.02. Graph a quadratic function in vertex form, standard form, and factored form.

F.03. Match quadratic functions and graphs.

F.04. Find the maximum or minimum value of a quadratic function using various methods.

F.05. Solve a quadratic equation using square roots.