IRELAND MATH YEAR 6 – EDmama Ireland

A. Roots and Rational Exponents

A.01. Find roots of integers, including perfect squares and perfect cubes.
A.02. Find roots of rational numbers.
A.03. Evaluate rational exponents.
A.04. Find roots using a calculator.
A.05. Simplify radical expressions with variables.
A.06. Simplify expressions involving rational exponents.
A.07. Perform operations with rational exponents, including multiplication, division, and exponentiation.
A.08. Define and apply the concept of nth roots.
A.09. Convert between radical expressions and expressions with rational exponents.

B. Complex Numbers

B.01. Define the imaginary unit ‘i’ and its properties.
B.02. Add and subtract complex numbers.
B.03. Multiply and divide complex numbers.
B.04. Understand and apply the concept of complex conjugates.
B.05. Simplify expressions involving powers of i.
B.06. Calculate absolute values of complex numbers.
B.07. Perform all four arithmetic operations with complex numbers.
B.08. Represent complex numbers geometrically on the Argand plane.
B.09. Apply complex numbers to solve quadratic equations with negative discriminants.

C. Functions: Domains, Ranges, and Representations

D. Linear Functions

E. Transformations of Functions

F. Exponential and Logarithmic Functions

F.01. Convert between exponential and logarithmic forms.
F.02. Evaluate logarithmic expressions using basic properties.
F.03. Identify the domain and range of exponential and logarithmic functions.
F.04. Apply the change of base formula for logarithms.
F.05. Apply the product, quotient, and power properties of logarithms.
F.06. Solve exponential equations by rewriting the base.
F.07. Solve exponential equations using logarithms.
F.08. Solve logarithmic equations with one or multiple logarithms.
F.09. Identify linear and exponential functions from data or graphs.
F.10. Model exponential growth and decay scenarios.
F.11. Solve word problems involving exponential growth, decay, and compound interest.
F.12. Graph exponential and logarithmic functions, identifying key features such as asymptotes.

G. Quadratic Functions

G.02. Find the maximum or minimum value of a quadratic function.
G.03. Graph a quadratic function.
G.04. Match quadratic functions and their graphs.
G.05. Solve a quadratic equation using square roots.
G.06. Solve a quadratic equation by factoring.
G.07. Solve a quadratic equation by completing the square.
G.08. Solve a quadratic equation using the quadratic formula.
G.09. Use the discriminant to determine the nature of the roots.
G.10. Apply quadratic functions to model projectile motion and optimization problems.
G.01. Identify characteristics of quadratic functions (vertex, axis of symmetry, intercepts).

H. Polynomial Functions

H.01. Divide polynomials using long division.
H.02. Divide polynomials using synthetic division.
H.03. Evaluate polynomials using synthetic division.
H.04. Write a polynomial from its roots.
H.05. Find the roots of factored polynomials.
H.06. Apply the Rational Root Theorem to find potential rational roots.
H.07. Apply the Complex Conjugate Theorem.
H.08. Apply Descartes’ Rule of Signs to predict the number of positive and negative real roots.
H.09. State and apply the Fundamental Theorem of Algebra.
H.10. Factorize sums and differences of cubes.
H.11. Solve equations involving sums and differences of cubes.
H.12. Factorize polynomials using a quadratic pattern.
H.13. Solve equations using a quadratic pattern.
H.14. Match polynomials with their graphs.
H.15. Apply Pascal’s triangle to expand binomials using the Binomial Theorem.
H.16. Apply the Binomial Theorem to find specific terms in a binomial expansion.

I. Rational Functions

J. Radical Functions

K. Systems of Equations

K.01. Solve simultaneous equations by graphing.
K.02. Solve simultaneous equations by graphing: word problems.
K.03. Classify simultaneous equations (consistent, inconsistent, dependent, independent).
K.04. Solve simultaneous equations using substitution.
K.05. Solve simultaneous equations using substitution: word problems.
K.06. Solve simultaneous equations using elimination.
K.07. Solve simultaneous equations using elimination: word problems.
K.08. Solve simultaneous equations in three variables using substitution.
K.09. Solve simultaneous equations in three variables using elimination.
K.10. Determine the number of solutions to simultaneous equations in three variables.
K.11. Model and solve real-world problems using systems of equations.

L. Inequalities and Linear Programming

M. Nonlinear Inequalities

N. Trigonometry: Angles and Ratios

N.01. Convert between radians and degrees.
N.02. Calculate arc length using radians.
N.03. Identify quadrants on the unit circle.
N.04. Find coterminal and reference angles.
N.05. Find trigonometric ratios (sin, cos, tan, csc, sec, cot) using right triangles.
N.06. Find trigonometric ratios using the unit circle.
N.07. Find trigonometric ratios using reference angles.
N.08. Evaluate inverses of trigonometric functions.
N.09. Solve trigonometric equations within a specified interval.
N.10. Determine trigonometric ratios for special angles (30°, 45°, 60°).

O. Trigonometry: Triangle Solutions and Applications

O.01. Use trigonometric ratios to find a side length in a right triangle.
O.02. Use trigonometric ratios to find an angle measure in a right triangle.
O.03. Solve a right triangle completely.
O.04. Apply the Law of Sines to solve oblique triangles.
O.05. Apply the Law of Cosines to solve oblique triangles.
O.06. Determine if a triangle can be uniquely solved given specific information (SSA, SAS, SSS).
O.07. Calculate the area of a triangle using the sine formula.
O.08. Calculate the area of a triangle using Heron’s formula.
O.09. Apply trigonometric principles to solve real-world problems involving angles, distances, and heights.

P. Trigonometric Functions

P.01. Identify properties of sine functions (amplitude, period, phase shift, vertical shift).
P.02. Write equations of sine functions from graphs.
P.03. Write equations of sine functions using properties.
P.04. Graph sine functions.
P.05. Identify properties of cosine functions (amplitude, period, phase shift, vertical shift).
P.06. Write equations of cosine functions from graphs.
P.07. Write equations of cosine functions using properties.
P.08. Graph cosine functions.
P.09. Graph sine and cosine functions with transformations.
P.10. Apply trigonometric functions to model periodic phenomena, such as sound waves and oscillations.

Q. Trigonometric Identities

Q.01. Apply complementary angle identities.
Q.02. Use symmetry and periodicity properties of trigonometric functions.
Q.03. Find trigonometric ratios using Pythagorean or reciprocal identities.
Q.04. Find trigonometric ratios using multiple identities.
Q.05. Simplify trigonometric expressions using identities.
Q.06. Verify trigonometric identities.
Q.07. Apply sum and difference identities for sine, cosine, and tangent.
Q.08. Apply double-angle and half-angle identities.

R. Circles in the Coordinate Plane

R.01. Find the center of a circle given its equation or graph.
R.02. Find the radius or diameter of a circle given its equation or graph.
R.03. Write equations of circles in standard form from graphs.
R.04. Write equations of circles in standard form using properties (center and radius).
R.05. Convert equations of circles from general to standard form.
R.06. Find properties of circles from equations in general form.
R.07. Graph circles given their equations in standard or general form.
R.08. Solve problems involving tangent lines to circles.

S. Sequences and Series

S.01. Find terms of a sequence given an explicit formula.
S.02. Find terms of a sequence given a recursive formula.
S.03. Identify a sequence as explicit or recursive.
S.04. Find a recursive formula for a given sequence.
S.05. Find both recursive and explicit formulas for a given sequence.
S.06. Convert a recursive formula to an explicit formula.
S.07. Convert an explicit formula to a recursive formula.
S.08. Understand and use sigma notation to represent series.
S.09. Identify arithmetic and geometric series.
S.10. Find the sum of a finite arithmetic or geometric series.
S.11. Define and calculate partial sums of arithmetic and geometric series.
S.12. Determine whether a geometric series is convergent or divergent.
S.13. Find the value of an infinite convergent geometric series.
S.14. Write a repeating decimal as a fraction using the concept of infinite geometric series.
S.15. Apply sequences and series to model financial situations such as annuities and loans.

T. Probability

T.09. Find conditional probabilities using two-way frequency tables.
T.10. Find probabilities using the addition rule.
T.11. Apply probability concepts to analyze games of chance and make informed decisions.
T.01. Define and understand basic probability concepts.
T.02. Calculate probabilities of simple events.
T.03. Calculate combinations and permutations.
T.04. Find probabilities using combinations and permutations.
T.05. Find probabilities using two-way frequency tables.
T.06. Define and identify independent events.
T.07. Calculate conditional probabilities.
T.08. Determine independence using conditional probability.

U. Probability Distributions

U.01. Identify discrete and continuous random variables.
U.02. Write a discrete probability distribution.
U.03. Graph a discrete probability distribution.
U.04. Calculate expected values of random variables.
U.05. Calculate the variance of random variables.
U.06. Calculate the standard deviation of random variables.
U.07. Write the probability distribution for a game of chance.
U.08. Calculate expected values for a game of chance.
U.09. Determine which bet is the better one based on expected value.
U.10. Find probabilities using the binomial distribution.
U.11. Calculate the mean, variance, and standard deviation of binomial distributions.
U.12. Find probabilities using the normal distribution I.
U.13. Find probabilities using the normal distribution II.
U.14. Find z-values corresponding to specific probabilities.
U.15. Find values of normal variables given a probability.
U.16. Explain the distributions of sample means.
U.17. State and apply the Central Limit Theorem.
U.18. Use normal distributions to approximate binomial distributions.

V. Statistics

V.01. Identify biased samples.
V.02. Calculate variance and standard deviation.
V.03. Identify outliers in a data set.
V.04. Describe the effect of removing an outlier from a data set.
V.05. Identify outliers in scatter plots.
V.06. Match correlation coefficients to scatter plots.
V.07. Calculate correlation coefficients.
V.08. Find the equation of a regression line.
V.09. Interpret regression lines.
V.10. Analyze a regression line of a data set.
V.11. Analyze a regression line using statistics of a data set.
V.12. Find confidence intervals for population means.
V.13. Find confidence intervals for population proportions.
V.14. Interpret confidence intervals for population means.
V.15. Understand and apply principles of experiment design.
V.16. Conduct hypothesis tests for population means and proportions.

W. Introduction to Limits

X. Calculating Limits

X.01. Find limits using addition, subtraction, and multiplication laws.
X.02. Find limits using the division law.
X.03. Find limits using power and root laws.
X.04. Find limits using all limit laws.
X.05. Find limits of polynomials and rational functions.
X.06. Find limits involving factorization and rationalization.
X.07. Find limits involving absolute value functions.
X.08. Find limits involving trigonometric functions.
X.09. Evaluate indeterminate forms (0/0) using algebraic techniques.
X.10. Apply L’Hôpital’s Rule to evaluate limits of indeterminate forms.

Y. Limits Involving Infinity

Z. Limits and Rational Functions

AA. Continuity

AA.01. Identify graphs of continuous functions.
AA.02. Determine continuity using graphs.
AA.03. Determine one-sided continuity using graphs.
AA.04. Find and analyze points of discontinuity using graphs.
AA.05. Determine continuity on an interval using graphs.
AA.06. Determine the continuity of a piecewise function at a point.
AA.07. Make a piecewise function continuous by finding appropriate parameter values.
AA.08. Apply the Intermediate Value Theorem to prove the existence of a root.
AA.09. Distinguish between different types of discontinuities (removable, jump, infinite).

BB. Introduction to Derivatives

BB.01. Calculate average rate of change over an interval I.
BB.02. Calculate average rate of change over an interval II.
BB.03. Find instantaneous rates of change.
BB.04. Interpret velocity as a rate of change.
BB.05. Find values of derivatives using limits.
BB.06. Find the slope of a tangent line using limits.
BB.07. Find equations of tangent lines using limits.
BB.08. Relate the derivative to the slope of a curve at a given point.
BB.09. Apply the concept of the derivative to solve real-world problems involving rates of change.

CC. Derivative Rules

CC.01. Apply the sum and difference rules of differentiation.
CC.02. Apply the product rule of differentiation.
CC.03. Apply the quotient rule of differentiation.
CC.04. Apply the power rule of differentiation I.
CC.05. Apply the power rule of differentiation II.
CC.06. Apply the chain rule of differentiation.
CC.07. Apply the inverse function rule of differentiation.
CC.08. Differentiate composite functions using multiple rules.
CC.09. Determine higher-order derivatives.

DD. Calculating Derivatives

DD.01. Find derivatives of polynomials I.
DD.02. Find derivatives of polynomials II.
DD.03. Find second derivatives of polynomials.
DD.04. Find derivatives of rational functions.
DD.05. Find derivatives of trigonometric functions I.
DD.06. Find derivatives of trigonometric functions II.
DD.07. Find derivatives of exponential functions.
DD.08. Find derivatives of logarithmic functions.
DD.09. Find derivatives of inverse trigonometric functions.
DD.10. Find derivatives of radical functions.
DD.11. Find derivatives using the product rule I.
DD.12. Find derivatives using the product rule II.
DD.13. Find derivatives using the quotient rule I.
DD.14. Find derivatives using the quotient rule II.
DD.15. Find derivatives using the chain rule I.
DD.16. Find derivatives using the chain rule II.

Course Content

A. Roots and Rational Exponents

A.01. Find roots of integers, including perfect squares and perfect cubes.

A.02. Find roots of rational numbers.

A.03. Evaluate rational exponents.

A.04. Find roots using a calculator.

A.05. Simplify radical expressions with variables.

A.06. Simplify expressions involving rational exponents.

A.07. Perform operations with rational exponents, including multiplication, division, and exponentiation.

A.08. Define and apply the concept of nth roots.

A.09. Convert between radical expressions and expressions with rational exponents.

B. Complex Numbers

B.01. Define the imaginary unit ‘i’ and its properties.

B.02. Add and subtract complex numbers.

B.03. Multiply and divide complex numbers.

B.04. Understand and apply the concept of complex conjugates.

B.05. Simplify expressions involving powers of i.

B.06. Calculate absolute values of complex numbers.

B.07. Perform all four arithmetic operations with complex numbers.

B.08. Represent complex numbers geometrically on the Argand plane.

B.09. Apply complex numbers to solve quadratic equations with negative discriminants.

C. Functions: Domains, Ranges, and Representations

C.01. Identify the domain and range of a function from its graph.

C.02. Evaluate functions for given inputs.

C.03. Determine whether a relation is a function.

C.04. Find values using function graphs.

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

C.06. Explore and apply piecewise-defined functions.

C.07. Understand and apply function notation in various contexts.

D. Linear Functions

D.01. Calculate the slope of a linear function.

D.02. Graph a linear function using slope-intercept form.

D.03. Write the equation of a linear function in slope-intercept form.

D.04. Determine the rate of change of linear functions over unit intervals.

D.05. Write equations of parallel and perpendicular lines.

D.06. Solve linear equations and inequalities.

D.07. Model real-world situations using linear functions.

E. Transformations of Functions

E.01. Identify translations of functions (horizontal and vertical shifts).

E.02. Identify reflections of functions (over x-axis and y-axis).

E.03. Identify dilations of functions (horizontal and vertical stretches and compressions).

E.04. Apply multiple transformations to a function.

E.05. Describe function transformations in words based on their equations.

E.06. Write the transformation rules for a given function.

E.07. Analyze and predict the effect of transformations on key function characteristics (e.g., intercepts, extrema).

F. Exponential and Logarithmic Functions

F.01. Convert between exponential and logarithmic forms.

F.02. Evaluate logarithmic expressions using basic properties.

F.03. Identify the domain and range of exponential and logarithmic functions.

F.04. Apply the change of base formula for logarithms.

F.05. Apply the product, quotient, and power properties of logarithms.

F.06. Solve exponential equations by rewriting the base.

F.07. Solve exponential equations using logarithms.

F.08. Solve logarithmic equations with one or multiple logarithms.

F.09. Identify linear and exponential functions from data or graphs.

F.10. Model exponential growth and decay scenarios.

F.11. Solve word problems involving exponential growth, decay, and compound interest.

F.12. Graph exponential and logarithmic functions, identifying key features such as asymptotes.

G. Quadratic Functions

G.02. Find the maximum or minimum value of a quadratic function.

G.03. Graph a quadratic function.

G.04. Match quadratic functions and their graphs.

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

G.06. Solve a quadratic equation by factoring.

G.07. Solve a quadratic equation by completing the square.

G.08. Solve a quadratic equation using the quadratic formula.

G.09. Use the discriminant to determine the nature of the roots.

G.10. Apply quadratic functions to model projectile motion and optimization problems.

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

H. Polynomial Functions

H.01. Divide polynomials using long division.

H.02. Divide polynomials using synthetic division.

H.03. Evaluate polynomials using synthetic division.

H.04. Write a polynomial from its roots.

H.05. Find the roots of factored polynomials.

H.06. Apply the Rational Root Theorem to find potential rational roots.

H.07. Apply the Complex Conjugate Theorem.

H.08. Apply Descartes’ Rule of Signs to predict the number of positive and negative real roots.

H.09. State and apply the Fundamental Theorem of Algebra.

H.10. Factorize sums and differences of cubes.