All Lessons

8(10)(D)(ii) Edit drafts using standard English conventions, including consistent, appropriate use of verb tenses and active and passive voice.

(C) Use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

(A) Extend previous knowledge of sets and subsets using a visual representation to describe relationships between sets of real numbers

(Y8) Investigate the concept of irrational numbers, including π (ACMNA186)

(Y10A) Define rational and irrational numbers and perform operations with surds and fractional indices (ACMNA264)

8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

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

(Y7) Investigate index notation and represent whole numbers as products of powers of prime numbers (ACMNA149)

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.

(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) Approximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number approximation on a number line

(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.

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.

(A) Represent linear proportional situations with tables, graphs, and equations in the form of y = kx

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

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

(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

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

(9) Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.

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).

(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

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.

(9) Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.

(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.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.

(C) Use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

(C) Use data from a table or graph to determine the rate of change or slope and y-intercept in mathematical and real-world problems.

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

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

(A) Represent linear proportional situations with tables, graphs, and equations in the form of y = kx

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

8.EE.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

(C) Convert between standard decimal notation and scientific notation

(Y8) Use index notation with numbers to establish the index laws with positive integral indices and the zero index (ACMNA182)

8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.

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

(C) Convert between standard decimal notation and scientific notation

(Y9) Express numbers in scientific notation (ACMNA210)

8.EE.4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology

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

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.