All Lessons

(H) Graph the solution set of systems of two linear inequalities in two variables on the coordinate plane.

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

HSA.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

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

(Y9) Extend and apply the index laws to variables, using positive integer indices and the zero index (ACMNA212)

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.

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.

(C) Model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants

(Y10) Solve linear equations involving simple algebraic fractions (ACMNA240)

7.RP.2 Recognize and represent proportional relationships between quantities.

7.RP.2.C Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

HSA.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

(F) Solve quadratic and square root equations

(Y9) Extend and apply the index laws to variables, using positive integer indices and the zero index (ACMNA212)

HSA.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

(C) Model and solve one-variable equations with variables on both sides of the equal sign that represent mathematical and real-world problems using rational number coefficients and constants

(Y10) Solve linear equations involving simple algebraic fractions (ACMNA240)

8.EE.7 Solve linear equations in one variable.

8.EE.7.B Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

(E) Solve mathematic and scientific formulas, and other literal equations, for a specified variable.

(Y10) Solve problems involving linear equations, including those derived from formulas (ACMNA235)

HSA.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate

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

(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)

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

8.EE.7 Solve linear equations in one variable.

8.EE.7.A Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

(B) Represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0

(C) Contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation

(Y8) Plot linear relationships on the Cartesian plane with and without the use of digital technologies (ACMNA193)

HSA.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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).

(C) Contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation

(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

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

HSA.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

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).

(B) Write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y - y1 = m(x - x1), given one point and the slope and given two points

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

(Y10) Solve problems involving linear equations, including those derived from formulas (ACMNA235)

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

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.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

(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 Analyze and solve pairs of simultaneous linear equations.

8.EE.8.A Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

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.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).

HSA.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

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

(C) Solve systems of two linear equations with two variables for mathematical and real-world problems.

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

8.EE.8 Analyze and solve pairs of simultaneous linear equations.

8.EE.8.A Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

8.EE.8.C Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

HSA.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

(D) Graph the solution set of linear inequalities in two variables on the coordinate plane

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

HSA.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

(D) Graph the solution set of linear inequalities in two variables on the coordinate plane

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

HSA.REI.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

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

(Y9) Extend and apply the index laws to variables, using positive integer indices and the zero index (ACMNA212)

8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.

HSA.APR.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

(A) Represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity

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

6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

6.EE.3 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

(E) Describe the meaning of parentheses and brackets in a numeric expression

(F) Simplify numerical expressions that do not involve exponents, including up to two levels of grouping

(Y6) Explore the use of brackets and order of operations to write number sentences (ACMNA134)

5.OA.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.

5.OA.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2" as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.

(B) Represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity

(Y7) Introduce the concept of variables as a way of representing numbers using letters (ACMNA175)

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

6.EE.2.A Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5" as 5 - y.

6.EE.2.B Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.