(B) Add up to four two-digit numbers and subtract two-digit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations

(Y2) Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

(Y3) Recall addition facts for single-digit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation (ACMNA055)

1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

1.NBT.4 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

(B) Add up to four two-digit numbers and subtract two-digit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations

(Y2) Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

(Y3) Recall addition facts for single-digit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation (ACMNA055)

1.OA.5 Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

(E) Represent multiplication facts by using a variety of approaches such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting

(F) Recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts

(G) Use strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties

(D) Use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties

(Y2) Recognise and represent multiplication as repeated addition, groups and arrays (ACMNA031)

(Y4) Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)

(Y4) Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (ACMNA076)

3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

(F) Recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts

(J) Determine a quotient using the relationship between multiplication and division

(B) Represent and solve one- and two-step multiplication and division problems within 100 using arrays, strip diagrams, and equations

(C) Solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm

(Y2) Recognise and represent division as grouping into equal sets and solve simple problems using these representations (ACMNA032)

(Y4) Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)

(Y4) Develop efficient mental and written strategies and use appropriate digital technologies for multiplication and for division where there is no remainder (ACMNA076)

(Y5) Solve problems involving division by a one digit number, including those that result in a remainder (ACMNA101)

3.OA.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.

3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

5.NBT.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

(G) Apply properties of operations to add and subtract two or three numbers.

(A) Recall basic facts to add and subtract within 20 with automaticity

(Y2) Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

(Y3) Recall addition facts for single-digit numbers and related subtraction facts to develop increasingly efficient mental strategies for computation (ACMNA055)

2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

(F) Recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts

(B) Determine products of a number and 10 or 100 using properties of operations and place value understandings

(Y3) Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)

(Y4) Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)

3.NBT.3 Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

4.NBT.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

(F) Recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts

(J) Determine a quotient using the relationship between multiplication and division

(F) Use strategies and algorithms, including the standard algorithm, to divide up to a four-digit dividend by a one-digit divisor

(Y3) Recall multiplication facts of two, three, five and ten and related division facts (ACMNA056)

(Y4) Recall multiplication facts up to 10 × 10 and related division facts (ACMNA075)

(Y5) Find unknown quantities in number sentences involving multiplication and division and identify equivalent number sentences involving multiplication and division (ACMNA121)

4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

(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.

(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

(D) Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

(Y8) Simplify algebraic expressions involving the four operations (ACMNA192)

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.

6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

(D) Generate equivalent expressions using the properties of operations: inverse, identity, commutative, associative, and distributive properties.

(Y8) Extend and apply the distributive law to the expansion of algebraic expressions (ACMNA190)

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.

6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

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

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

(Y7) Create algebraic expressions and evaluate them by substituting a given value for each variable (ACMNA176)

6.EE.2.C Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2.

(B) Distinguish between expressions and equations verbally, numerically, and algebraically

(Y7) Create algebraic expressions and evaluate them by substituting a given value for each variable (ACMNA176)

6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

(B) Identify a number, its opposite, and its absolute value

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

(Y7) Solve simple linear equations (ACMNA179)

7.NS.1.B Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

(A) Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts

(Y7) Solve simple linear equations (ACMNA179)

6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

(B) Represent solutions for one-variable, one-step equations and inequalities on number lines

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

6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

(A) Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts

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

7.EE.4.B Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

(A) Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts

(Y7) Solve simple linear equations (ACMNA179)

6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

(A) Recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values

(A) Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts

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

(Y7) Solve simple linear equations (ACMNA179)

7.NS.2.A Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

(A) Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts

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

7.EE.4.B Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

(A) Model and solve one-variable, two-step equations and inequalities

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

7.EE.4.A Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

(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

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

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

(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.

(C) Represent integer operations with concrete models and connect the actions with the models to standardized algorithms

(D) Add, subtract, multiply, and divide integers fluently

(Y7) Compare, order, add and subtract integers (ACMNA280)

7.NS.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

7.NS.1.D Apply properties of operations as strategies to add and subtract rational numbers.

(D) Add, subtract, multiply, and divide integers fluently

(Y8) Carry out the four operations with rational numbers and integers, using efficient mental and written strategies and appropriate digital technologies (ACMNA183)

7.NS.2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

7.NS.2.A Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

7.NS.2.B Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.

7.NS.2.C Apply properties of operations as strategies to multiply and divide rational numbers.

(B) Determine if the given value(s) make(s) one-variable, one-step equations or inequalities true.

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

6.EE.5 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

(A) Model and solve one-variable, two-step equations and inequalities

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

7.EE.4.A Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

(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

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

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

(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.

(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)

(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.

(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).

(A) Generate equivalent numerical expressions using order of operations, including whole number exponents and prime factorization

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

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

(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).

(C) Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems

(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

(Y9) Sketch linear graphs using the coordinates of two points and solve linear equations (ACMNA215)

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

HSF.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

HSF.IF.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima.

(G) Identify functions using sets of ordered pairs, tables, mappings, and graphs

(Y9) Graph simple non-linear relations with and without the use of digital technologies and solve simple related equations (ACMNA296)

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

8.F.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

(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

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

HSF.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

(A) Determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and y - y1 = m(x - x1)

(C) Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems

(Y9) Sketch linear graphs using the coordinates of two points and solve linear equations (ACMNA215)

HSF.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

HSF.IF.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

(C) Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems

(Y9) Sketch linear graphs using the coordinates of two points and solve linear equations (ACMNA215)

8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

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).

HSF.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

HSF.IF.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima.

(C) Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems

(Y9) Sketch linear graphs using the coordinates of two points and solve linear equations (ACMNA215)

(Y10) Substitute values into formulas to determine an unknown (ACMNA234)

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

HSF.IF.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima.

(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

(C) Write linear equations in two variables given a table of values, a graph, and a verbal description

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

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).

(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.

(A) Model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts

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

7.EE.4.B Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

(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) 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

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

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.

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

(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.

(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.

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

(B) Add, subtract, and multiply polynomials

(Y9) Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213)

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.

(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.

(B) Multiply polynomials of degree one and degree two

(Y9) Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213)

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.

(B) Add, subtract, and multiply polynomials

(Y9) Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213)

L.2.4 Determine or clarify the meaning of unknown and multiple-meaning words and phrases based on grade 2 reading and content, choosing flexibly from an array of strategies.

L.2.4.A Use sentence-level context as a clue to the meaning of a word or phrase.

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.

(F) Decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial.

(E) Determine linear and quadratic factors of a polynomial expression of degree three and of degree four, including factoring the sum and difference of two cubes and factoring by grouping

(Y9) Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate (ACMNA213)

HSA.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

(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) 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)

HSF.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

HSF.IF.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima.

(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

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

HSF.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

HSF.IF.7.A Graph linear and quadratic functions and show intercepts, maxima, and minima.

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

(E) Factor, if possible, trinomials with real factors in the form ax2 + bx + c, including perfect square trinomials of degree two

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

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

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

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

L.2.4 Determine or clarify the meaning of unknown and multiple-meaning words and phrases based on grade 2 reading and content, choosing flexibly from an array of strategies.

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.

(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)

L.1.2.C Use commas in dates and to separate single words in a series.

HSA.SSE.3.B Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

(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)

(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.B Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

(F) Solve quadratic and square root equations

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

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

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

(A) Distinguish between categorical and quantitative data

(Y11-12) Identify examples of categorical data (ACMEM043)

(Y11-12) Identify examples of numerical data (ACMEM044)

HSS.IDA.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

(F) Analyze categorical data, including determining marginal and conditional distributions, using two-way tables.

(Y11-12) Interpret information presented in two-way tables (ACMEM038)

HSS.IDA.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

(F) Analyze categorical data, including determining marginal and conditional distributions, using two-way tables.

(Y11-12) Interpret information presented in two-way tables (ACMEM038)

HSS.IDA.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

(A) Construct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data

(Y11-12) Describe the patterns and features of bivariate data (ACMEM138)

(Y11-12) Describe the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak) (ACMEM139)

HSS.IDA.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

(A) Calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association

(Y11-12) Describe the patterns and features of bivariate data (ACMEM138)

(Y11-12) Describe the association between two numerical variables in terms of direction (positive/negative), form (linear/non-linear) and strength (strong/moderate/weak) (ACMEM139)

HSS.IDA.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

(B) Use regression methods available through technology to write a linear function, a quadratic function, and an exponential function from a given set of data

(Y11-12) Use technology to find the line of best fit (ACMEM142)

HSS.IDA.6.C Fit a linear function for a scatter plot that suggests a linear association.

(A) Distinguish between undefined terms, definitions, postulates, conjectures, and theorems

(A) Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools

(Y11-12) Use symbols and conventions for the representation of geometric information; for example, point, line, ray, angle, diagonal, edge, curve, face and vertex (ACMEM107)

HSG.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

(A) Describe and perform transformations of figures in a plane using coordinate notation

(Y8) Define congruence of plane shapes using transformations (ACMMG200)

HSG.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

(A) Describe and perform transformations of figures in a plane using coordinate notation

(Y8) Define congruence of plane shapes using transformations (ACMMG200)

HSG.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

(A) Describe and perform transformations of figures in a plane using coordinate notation

(Y8) Define congruence of plane shapes using transformations (ACMMG200)

HSG.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

(A) Describe and perform transformations of figures in a plane using coordinate notation

(Y8) Define congruence of plane shapes using transformations (ACMMG200)

HSG.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

(B) Determine the image or pre-image of a given two-dimensional figure under a composition of rigid transformations, a composition of non-rigid transformations, and a composition of both, including dilations where the center can be any point in the plane

(C) Apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles

(Y8) Define congruence of plane shapes using transformations (ACMMG200)

8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

HSG.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

(B) Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions

(Y8) Develop the conditions for congruence of triangles (ACMMG201)

HSG.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.