All Lessons

(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

(Y10) Solve simple quadratic equations using a range of strategies (ACMNA241)

HSA.REI.4 Solve quadratic equations in one variable.

HSA.REI.4.B Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

(Y10A) Factorise monic and non-monic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts (ACMNA269)

HSA.REI.4 Solve quadratic equations in one variable.

HSA.REI.4.A Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.

HSA.REI.4.B Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

(A) Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry

(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

(Y10) Solve simple quadratic equations using a range of strategies (ACMNA241)

HSA.REI.4 Solve quadratic equations in one variable.

HSA.REI.4.B Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

(F) Graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist

(Y10) Solve linear simultaneous equations, using algebraic and graphical techniques, including using digital technology (ACMNA237)

8.EE.8.B Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

HSA.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

HSA.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

(A) Add and subtract polynomials of degree one and degree two

(B) Multiply polynomials of degree one and degree two

(D) Rewrite polynomial expressions of degree one and degree two in equivalent forms using the distributive property

(Y10A) Investigate the concept of a polynomial and apply the factor and remainder theorems to solve problems (ACMNA266)

HSA.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

(A) Simplify numerical radical expressions involving square roots

(Y7) Investigate and use square roots of perfect square numbers (ACMNA150)

8.EE.2 Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

HSN.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^3 = (5^(1/3))^3 to hold, so (5^(1/3))^3 must equal 5.

HSN.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

(B) Solve linear inequalities in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides

(Y10) Solve linear inequalities and graph their solutions on a number line (ACMNA236)

HSA.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

(8) Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.

(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

(Y10) Expand binomial products and factorise monic quadratic expressions using a variety of strategies (ACMNA233)

HSA.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*

HSA.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

(8) Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.

(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

HSA.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*

HSA.SSE.3.A Factor a quadratic expression to reveal the zeros of the function it defines.

(C) Write exponential functions in the form f(x) = abx (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay

(D) Solve exponential equations of the form y = abx where a is a nonzero real number and b is greater than zero and not equal to one and single logarithmic equations having real solutions

(I) Generate and solve exponential equations in mathematical and real-world problems

(Y10A) Solve simple exponential equations (ACMNA270)

HSA.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

(B) Simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents.

(Y7) Extend and apply the laws and properties of arithmetic to algebraic terms and expressions (ACMNA177)

HSN.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

(Y10) Solve simple quadratic equations using a range of strategies (ACMNA241)

HSA.REI.4.A Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Derive the quadratic formula from this form.

HSA.REI.4.B Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

(A) Solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides

(Y8) Solve linear equations using algebraic and graphical techniques. Verify solutions by substitution (ACMNA194)

HSA.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

(7) Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations.

(A) Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry

(B) Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions

(C) Determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.

(Y10A) Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (ACMNA267)

HSF.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

(A) Determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities

HSF.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

(7) Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations.

(B) Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions

(Y10A) Factorise monic and non-monic quadratic expressions and solve a wide range of quadratic equations derived from a variety of contexts (ACMNA269)

HSF.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HSF.IF.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

(7) Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations.

(A) Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry

(B) Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions

(C) Determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.

(8) Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data.

(A) Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula

(Y10A) Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (ACMNA267)

HSF.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

HSF.IF.8.A Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

HSF.IF.8.B Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)ᵗ, y = (0.97)ᵗ, y = (1.01)12ᵗ, y = (1.2)ᵗ/10, and classify them as representing exponential growth or decay.

(B) Write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f(x) = a(x - h)2+ k), and rewrite the equation from vertex form to standard form (f(x) = ax2+ bx + c)

(B) Describe the relationship between the linear factors of quadratic expressions and the zeros of their associated quadratic functions

HSF.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

(D) Graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems

(Y10A) Describe, interpret and sketch parabolas, hyperbolas, circles and exponential functions and their transformations (ACMNA267)

HSF.IF.7.E Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

(C) Identify terms of arithmetic and geometric sequences when the sequences are given in function form using recursive processes

(Y6) Continue and create sequences involving whole numbers, fractions and decimals. Describe the rule used to create the sequence (ACMNA133)

HSF.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.