NEW ZEALAND MATH YEAR 12 – EDmama New Zealand

A. Numbers and Counting

A.01. Evaluate numerical expressions involving integers using order of operations.
A.02. Convert between decimals, fractions, and percentages with precision.
A.03. Understand and apply the properties of integers in addition, subtraction, multiplication, and division.
A.04. Perform operations with rational numbers, including addition, subtraction, multiplication, and division.
A.05. Simplify complex numerical expressions involving rational numbers.
A.06. Estimate and approximate values of irrational numbers in decimal form.
A.07. Explore the historical significance and real-world applications of rational and irrational numbers.

B. Coordinate Geometry

C. Measurement and Error

D. Solving Linear Equations and Inequalities

D.01. Solve linear equations in one variable using algebraic manipulation.
D.02. Apply linear equations to solve real-world word problems.
D.03. Graph a linear inequality in one variable on a number line.
D.04. Graph a linear inequality in two variables on the coordinate plane.
D.05. Write inequalities from given graphs on the coordinate plane.
D.06. Formulate linear inequalities to represent real-world constraints.
D.07. Solve linear inequalities in one variable.
D.08. Graph the solution set of linear inequalities in one variable on a number line.
D.09. Graph solutions to quadratic inequalities on a number line.
D.10. Solve quadratic inequalities algebraically.
D.11. Formulate and solve absolute value equations and inequalities.

E. Factorising

E.01. Factorise out the greatest common monomial factor from a polynomial.
E.02. Factorise quadratic expressions of the form ax² + bx + c.
E.03. Factorise quadratics using algebra tiles as a visual aid.
E.04. Factorise polynomials by recognising and applying quadratic patterns.
E.05. Factorise polynomials by grouping terms.
E.06. Factorise sums and differences of cubes.
E.07. Factorise general polynomials using a combination of techniques.
E.08. Apply factorisation to solve polynomial equations.

F. Functions: Identification, Evaluation, and Graphs

F.01. Determine the domain and range of a function from its graph or equation.
F.02. Identify whether a given relation represents a function.
F.03. Evaluate a function for a given input value using function notation.
F.04. Find function values using function graphs.
F.05. Complete a table of values for a given function graph.
F.06. Determine the gradient (slope) of a linear function.
F.07. Graph a linear function given its equation.
F.08. Write the equation of a linear function given its graph or other information.
F.09. Interpret linear functions over unit intervals in real-world contexts.
F.10. Identify linear, quadratic, and exponential functions from graphs.
F.11. Identify linear, quadratic, and exponential functions from tables of values.
F.12. Write linear, quadratic and exponential functions from verbal descriptions and data.

G. Quadratic Functions and Equations

G.03. Determine the equation of a quadratic function given specific points or properties.
G.04. Graph a quadratic function and identify its key features.
G.05. Match quadratic functions with their corresponding graphs.
G.06. Solve quadratic equations using the square root property.
G.07. Solve a quadratic equation using the zero product property after factorisation.
G.08. Solve a quadratic equation by factorising.
G.09. Complete the square to rewrite a quadratic equation in vertex form.
G.10. Solve a quadratic equation using the quadratic formula.
G.11. Use the discriminant to determine the nature and number of solutions to a quadratic equation.
G.12. Apply quadratic functions to model and solve real-world problems.
G.01. Identify key characteristics of quadratic functions (vertex, axis of symmetry, intercepts).
G.02. Complete a function table for quadratic functions.

H. Radical Functions and Expressions

H.01. Determine roots of perfect integer powers without a calculator.
H.02. Determine roots of perfect rational powers without a calculator.
H.03. Find roots of numbers using a calculator.
H.04. Understand and simplify expressions involving nth roots.
H.05. Simplify radical expressions with variables I (basic simplification).
H.06. Simplify radical expressions with variables II (more complex simplification).
H.07. Multiply radical expressions.
H.08. Divide radical expressions, including rationalizing the denominator.
H.09. Add and subtract radical expressions by combining like terms.
H.10. Simplify radical expressions using the distributive property.
H.11. Simplify radical expressions using conjugates.
H.12. Determine the domain and range of radical functions.
H.13. Solve radical equations, checking for extraneous solutions.

I. Rational Exponents

J. Simultaneous Equations

J.01. Determine whether an ordered pair (x, y) is a solution to a system of simultaneous equations.
J.02. Solve systems of simultaneous equations by graphing.
J.03. Solve simultaneous equations by graphing: word problems.
J.04. Determine the number of solutions to a system of simultaneous equations.
J.05. Classify systems of simultaneous equations as independent, dependent, or inconsistent.
J.06. Solve systems of simultaneous equations using the substitution method.
J.07. Solve simultaneous equations using substitution: word problems.
J.08. Solve systems of simultaneous equations using the elimination method.
J.09. Solve simultaneous equations using elimination: word problems.
J.10. Solve systems of simultaneous equations using any appropriate method.
J.11. Solve simultaneous equations using any method: word problems.
J.12. Solve systems of simultaneous equations in three variables using substitution.
J.13. Solve systems of simultaneous equations in three variables using elimination.
J.14. Determine the number of solutions to a system of equations in three variables.
J.15. Solve nonlinear systems of simultaneous equations.
J.16. Apply systems of equations to solve mixture and investment problems.

K. Rational Functions and Expressions

K.01. Identify asymptotes and excluded values of rational functions.
K.02. Evaluate rational expressions I (basic evaluation).
K.03. Evaluate rational expressions II (more complex evaluation).
K.04. Simplify rational expressions by cancelling common factors.
K.05. Multiply and divide rational expressions.
K.06. Add and subtract rational expressions, finding common denominators.
K.07. Solve rational equations, checking for extraneous solutions.
K.08. Graph rational functions, identifying key features.

L. Function Operations and Inverses

L.01. Add and subtract functions using function notation.
L.02. Multiply functions using function notation.
L.03. Divide functions using function notation.
L.04. Evaluate the composition of linear functions: find a value.
L.05. Determine the composition of linear functions: find an equation.
L.06. Evaluate the composition of linear and quadratic functions: find a value.
L.07. Determine the composition of linear and quadratic functions: find an equation.
L.08. Find values of inverse functions from tables.
L.09. Find values of inverse functions from graphs.
L.10. Find inverse functions and relations algebraically.
L.11. Verify inverse functions using composition.

M. Inequalities and Linear Programming

N. Polynomials

N.01. Understand and use polynomial vocabulary (degree, leading coefficient, etc.).
N.02. Add and subtract polynomials by combining like terms.
N.03. Multiply polynomials using the distributive property.
N.04. Divide polynomials using long division or synthetic division.
N.05. Write a polynomial from its roots.
N.06. Find the roots of factorised polynomials.
N.07. Apply the Complex Conjugate Theorem to find roots of polynomials.
N.08. Apply Conjugate Root Theorems to find roots of polynomials.
N.09. Use Descartes’ Rule of Signs to determine the possible number of positive and negative real roots of a polynomial.
N.10. Understand and apply the Fundamental Theorem of Algebra.
N.11. Match polynomials and graphs, identifying key features like end behaviour and intercepts.

O. Families of Functions and Transformations

P. Complex Numbers

P.01. Introduce the concept of complex numbers (real and imaginary parts).
P.02. Add and subtract complex numbers.
P.03. Understand and use complex conjugates.
P.04. Multiply complex numbers.
P.05. Divide complex numbers.
P.06. Perform all four basic arithmetic operations with complex numbers.
P.07. Calculate the absolute value (modulus) of complex numbers.
P.08. Simplify powers of the imaginary unit ‘i’.
P.09. Represent complex numbers graphically on the complex plane.
P.10. Convert between rectangular and polar forms of complex numbers.
P.11. Apply De Moivre’s Theorem to find powers and roots of complex numbers.

Q. Exponential and Logarithmic Functions

Q.01. Determine the domain and range of exponential and logarithmic functions.
Q.02. Evaluate exponential functions.
Q.03. Match exponential functions with their corresponding graphs.
Q.04. Solve exponential equations by rewriting the base.
Q.05. Solve exponential equations using common logarithms.
Q.06. Solve exponential equations using natural logarithms.
Q.07. Solve logarithmic equations I (basic equations).
Q.08. Solve logarithmic equations II (more complex equations).
Q.09. Identify linear and exponential functions from data or graphs.
Q.10. Interpret exponential functions over unit intervals in real-world contexts.
Q.11. Describe linear and exponential growth and decay.
Q.12. Solve exponential growth and decay word problems.
Q.13. Solve compound interest word problems.
Q.14. Solve continuously compounded interest word problems.
Q.15. Model real-world phenomena using exponential and logarithmic functions.

R. Logarithms: Properties and Evaluation

R.10. Apply the power property of logarithms to expand or condense expressions.
R.11. Apply properties of logarithms in mixed review problems.
R.12. Evaluate logarithms in mixed review problems.
R.01. Convert between exponential and logarithmic form with rational bases.
R.02. Convert between natural exponential and logarithmic form.
R.03. Convert between exponential and logarithmic form with all bases.
R.04. Evaluate logarithms.
R.05. Evaluate natural logarithms.
R.06. Apply the change of base formula to evaluate logarithms.
R.07. Identify and apply the properties of logarithms (product, quotient, power).
R.08. Apply the product property of logarithms to expand or condense expressions.
R.09. Apply the quotient property of logarithms to expand or condense expressions.

S. Conic Sections: Parabolas

S.01. Identify the direction a parabola opens.
S.02. Find the vertex of a parabola.
S.03. Find the focus and directrix of a parabola.
S.04. Find the axis of symmetry of a parabola.
S.05. Write equations of parabolas in vertex form from graphs.
S.06. Write equations of parabolas in vertex form using given properties.
S.07. Convert equations of parabolas from general to vertex form.
S.08. Find properties of a parabola from equations in general form.
S.09. Graph parabolas using their properties.

T. Conic Sections: Circles

U. Conic Sections: Ellipses

U.01. Find the centre, vertices, and co-vertices of an ellipse.
U.02. Find the length of the major and minor axes of an ellipse.
U.03. Find the foci of an ellipse.
U.04. Write equations of ellipses in standard form from graphs.
U.05. Write equations of ellipses in standard form using given properties.
U.06. Convert equations of ellipses from general to standard form.
U.07. Find properties of ellipses from equations in general form.
U.08. Graph ellipses using their properties.

V. Conic Sections: Hyperbolas

V.01. Find the centre of a hyperbola.
V.02. Find the vertices of a hyperbola.
V.03. Find the length of the transverse and conjugate axes of a hyperbola.
V.04. Find the equations for the asymptotes of a hyperbola.
V.05. Find the foci of a hyperbola.
V.06. Write equations of hyperbolas in standard form from graphs.
V.07. Write equations of hyperbolas in standard form using given properties.
V.08. Convert equations of hyperbolas from general to standard form.
V.09. Find properties of hyperbolas from equations in general form.
V.10. Graph hyperbolas using their properties.
V.11. Identify the type of conic section from its general equation.

W. Angle Measures and Trigonometry

W.01. Convert between radians and degrees.
W.02. Relate radians and arc length using the appropriate formula.
W.03. Identify the quadrant in which an angle lies.
W.04. Sketch graphs of angles in standard position.
W.05. Determine coterminal angles.
W.06. Find reference angles.
W.07. Apply Pythagoras’ Theorem and its converse to solve right triangles.
W.08. Apply the properties of special right triangles (30-60-90 and 45-45-90) to find side lengths.
W.09. Define and calculate the trigonometric ratios: sin, cos, and tan.
W.10. Define and calculate the trigonometric ratios: csc, sec, and cot.
W.11. Find trigonometric ratios in similar right triangles.
W.12. Find trigonometric ratios using the unit circle.
W.13. Determine the sin, cos, and tan of special angles (0, 30, 45, 60, 90 degrees).
W.14. Determine the csc, sec, and cot of special angles (0, 30, 45, 60, 90 degrees).
W.15. Find trigonometric function values using a calculator.
W.16. Find inverses of sin, cos and tan.
W.17. Find inverses of csc, sec and cot.
W.18. Solve trigonometric equations I (basic equations).
W.19. Solve trigonometric equations II (more complex equations).
W.20. Use trigonometric ratios to find a side length in a right triangle.
W.21. Use trigonometric ratios to find an angle measure in a right triangle.
W.22. Solve a right triangle (find all missing sides and angles).
W.23. Apply the Law of Sines to solve oblique triangles.
W.24. Apply the Law of Cosines to solve oblique triangles.
W.25. Solve a triangle given various combinations of sides and angles.
W.26. Calculate the area of a triangle using the sine formula.
W.27. Calculate the area of a triangle using Heron’s formula.
W.28. Explore applications of trigonometry in navigation and surveying.

X. Trigonometric Functions and Graphs

X.01. Identify properties of sine functions (amplitude, period, phase shift).
X.02. Write equations of sine functions from graphs.
X.03. Write equations of sine functions using given properties.
X.04. Graph sine functions.
X.05. Identify properties of cosine functions (amplitude, period, phase shift).
X.06. Write equations of cosine functions from graphs.
X.07. Write equations of cosine functions using given properties.
X.08. Graph cosine functions.
X.09. Graph sine and cosine functions together, observing their relationship.
X.10. Graph transformations of sine and cosine functions.
X.11. Model periodic phenomena using trigonometric functions.

Y. Trigonometric Identities

Z. Surface Area and Volume

Z.01. Understand the concepts of surface area and volume.
Z.02. Calculate the surface area of prisms and cylinders.
Z.03. Calculate the surface area of pyramids and cones.
Z.04. Calculate the volume of prisms and cylinders.
Z.05. Calculate the volume of pyramids and cones.
Z.06. Calculate the surface area and volume of spheres.
Z.07. Understand the concept of similar solids.
Z.08. Relate the surface area and volume of similar solids using the scale factor.
Z.09. Review surface area and volume concepts.
Z.10. Solve real-world problems involving surface area and volume.

AA. Sequences and Series

AA.01. Classify formulas and sequences as arithmetic, geometric, or neither.
AA.02. Find terms of an arithmetic sequence given a formula or initial terms.
AA.03. Find terms of a geometric sequence given a formula or initial terms.
AA.04. Find terms of a recursive sequence given a recursive formula.
AA.05. Evaluate formulas for sequences.
AA.06. Write a formula for an arithmetic sequence given information about its terms.
AA.07. Write a formula for a geometric sequence given information about its terms.
AA.08. Write a formula for a recursive sequence given information about its terms.
AA.09. Review sequences concepts.
AA.10. Understand and use sigma notation to represent series.
AA.11. Identify arithmetic and geometric series.
AA.12. Find the sum of a finite arithmetic or geometric series.
AA.13. Understand the concept of partial sums of a series.
AA.14. Calculate partial sums of arithmetic series.
AA.15. Calculate partial sums of geometric series.
AA.16. Review partial sums concepts.
AA.17. Determine whether a geometric series is convergent or divergent.
AA.18. Find the value of an infinite geometric series when it converges.
AA.19. Write a repeating decimal as a fraction using the formula for an infinite geometric series.

BB. Probability

BB.12. Calculate geometric probability.
BB.13. Find conditional probabilities.
BB.14. Determine independence using conditional probability.
BB.15. Find conditional probabilities using two-way frequency tables.
BB.16. Find probabilities using the addition rule.
BB.17. Find probabilities using the normal distribution.
BB.18. Apply probability concepts to analyze games of chance and real-world scenarios.
BB.01. Understand the basic concepts of probability (sample space, events, outcomes).
BB.02. Calculate theoretical and experimental probability.
BB.03. Calculate the number of outcomes for compound events.
BB.04. Calculate probabilities of events.
BB.05. Calculate permutations.
BB.06. Apply the counting principle to determine the number of possibilities.
BB.07. Understand and use permutation and combination notation.
BB.08. Calculate probabilities using permutations and combinations.
BB.09. Find probabilities using two-way frequency tables.
BB.10. Identify independent events.
BB.11. Calculate probabilities of independent and dependent events.

Course Content

A. Numbers and Counting

A.01. Evaluate numerical expressions involving integers using order of operations.

A.02. Convert between decimals, fractions, and percentages with precision.

A.03. Understand and apply the properties of integers in addition, subtraction, multiplication, and division.

A.04. Perform operations with rational numbers, including addition, subtraction, multiplication, and division.

A.05. Simplify complex numerical expressions involving rational numbers.

A.06. Estimate and approximate values of irrational numbers in decimal form.

A.07. Explore the historical significance and real-world applications of rational and irrational numbers.

B. Coordinate Geometry

B.01. Review the fundamental concepts of the coordinate plane (axes, quadrants, points).

B.02. Calculate the midpoint of a line segment given two endpoints.

B.03. Calculate the distance between two points using the distance formula.

B.04. Apply dilation transformations to geometric figures on the coordinate plane.

B.05. Determine the relationship between dilations and parallel lines.

B.06. Determine the equation of parallel and perpendicular lines.

B.07. Use coordinate geometry to solve geometric problems.

C. Measurement and Error

C.01. Convert between different units of measurement and rates.

C.02. Understand the concept of precision in measurement.

C.03. Calculate the greatest possible error in a measurement.

C.04. Determine the minimum and maximum possible area and volume based on measurement errors.

C.05. Calculate percent error in single measurements.

C.06. Calculate percent error in derived quantities like area and volume.

C.07. Analyze how measurement errors propagate through calculations.

D. Solving Linear Equations and Inequalities

D.01. Solve linear equations in one variable using algebraic manipulation.

D.02. Apply linear equations to solve real-world word problems.

D.03. Graph a linear inequality in one variable on a number line.

D.04. Graph a linear inequality in two variables on the coordinate plane.

D.05. Write inequalities from given graphs on the coordinate plane.

D.06. Formulate linear inequalities to represent real-world constraints.

D.07. Solve linear inequalities in one variable.

D.08. Graph the solution set of linear inequalities in one variable on a number line.

D.09. Graph solutions to quadratic inequalities on a number line.

D.10. Solve quadratic inequalities algebraically.

D.11. Formulate and solve absolute value equations and inequalities.

E. Factorising

E.01. Factorise out the greatest common monomial factor from a polynomial.

E.02. Factorise quadratic expressions of the form ax² + bx + c.

E.03. Factorise quadratics using algebra tiles as a visual aid.

E.04. Factorise polynomials by recognising and applying quadratic patterns.

E.05. Factorise polynomials by grouping terms.

E.06. Factorise sums and differences of cubes.

E.07. Factorise general polynomials using a combination of techniques.

E.08. Apply factorisation to solve polynomial equations.

F. Functions: Identification, Evaluation, and Graphs

F.01. Determine the domain and range of a function from its graph or equation.

F.02. Identify whether a given relation represents a function.

F.03. Evaluate a function for a given input value using function notation.

F.04. Find function values using function graphs.

F.05. Complete a table of values for a given function graph.

F.06. Determine the gradient (slope) of a linear function.

F.07. Graph a linear function given its equation.

F.08. Write the equation of a linear function given its graph or other information.

F.09. Interpret linear functions over unit intervals in real-world contexts.

F.10. Identify linear, quadratic, and exponential functions from graphs.

F.11. Identify linear, quadratic, and exponential functions from tables of values.

F.12. Write linear, quadratic and exponential functions from verbal descriptions and data.

G. Quadratic Functions and Equations

G.03. Determine the equation of a quadratic function given specific points or properties.

G.04. Graph a quadratic function and identify its key features.

G.05. Match quadratic functions with their corresponding graphs.

G.06. Solve quadratic equations using the square root property.

G.07. Solve a quadratic equation using the zero product property after factorisation.

G.08. Solve a quadratic equation by factorising.

G.09. Complete the square to rewrite a quadratic equation in vertex form.

G.10. Solve a quadratic equation using the quadratic formula.

G.11. Use the discriminant to determine the nature and number of solutions to a quadratic equation.

G.12. Apply quadratic functions to model and solve real-world problems.

G.01. Identify key characteristics of quadratic functions (vertex, axis of symmetry, intercepts).

G.02. Complete a function table for quadratic functions.

H. Radical Functions and Expressions

H.01. Determine roots of perfect integer powers without a calculator.

H.02. Determine roots of perfect rational powers without a calculator.

H.03. Find roots of numbers using a calculator.

H.04. Understand and simplify expressions involving nth roots.

H.05. Simplify radical expressions with variables I (basic simplification).

H.06. Simplify radical expressions with variables II (more complex simplification).

H.07. Multiply radical expressions.