Grade 8 Common Core Standards For Math

A1. 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.1

Skill check: Computing derivatives, part 2

Intercepts from a graph

Intercepts from an equation

Slope-intercept intro

Evaluate functions

Evaluate functions from their graph

Function inputs & outputs: graph

Compare linear functions

Domain and range from graph

Comparing linear functions word problem

Recognize functions from graphs

Graphing linear functions word problems

Function domain word problems

Writing linear functions word problems

Recognize functions from tables

Evaluate piecewise functions

Evaluate step functions

Piecewise functions graphs

Linear & nonlinear functions

Factoring quadratics intro

Finite geometric series

Graphing exponential growth & decay

Graph quadratics in standard form

Evaluate composite functions

Graph quadratics in factored form

Evaluate composite functions: graphs & tables

Model with function combination

Interpret change in exponential models

Amplitude of sinusoidal functions from graph

Model with composite functions

Interpret time in exponential models

Evaluate inverse functions

Interpret change in exponential models: changing units

Interpret change in exponential models: with manipulation

Determine if a function is invertible

Restrict domains of functions to make them invertible

Radical functions & their graphs

Graphs of logarithmic functions

Fitting quadratic and exponential functions to scatter plots

Domain of advanced piecewise functions

Periodicity of algebraic models

Compare features of functions

Removable discontinuities

Derivative as slope of tangent line

Polynomial functions differentiation

Exponential functions differentiation intro

Composite exponential function differentiation

Absolute minima & maxima (entire domain)

Points of inflection (algebraic)

Linear approximation

Indefinite integrals: sin & cos

Functions defined by integrals graphically

Differential equations intro

A2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

Converting recursive & explicit forms of arithmetic sequences

Evaluate composite functions

Graphs of nonlinear piecewise functions

A3. 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.

Absolute minima & maxima (closed intervals)

B4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Estimating slope of line of best fit

DEPRECATED Domain & range of piecewise functions

Converting recursive & explicit forms of geometric sequences

B5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Linear equations word problems

Solutions of inequalities: graphical

Recursive formulas for geometric sequences

Construct exponential models

A1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.

Extend geometric sequences: negatives & fractions

Powers of products & quotients (integer exponents)

Powers of products & quotients (structured practice)

Solving quadratics by taking square roots: with steps

Quiz: Looking Forward

Solve for t

Tangents to polar curves

Powers of fractions

Simplify hairy fractions

Number of solutions to equations

Exponents with negative fractional bases

Negative exponents

Multiply & divide powers (integer exponents)

Square and cube challenge

Properties of exponents challenge (integer exponents)

Permutations

Factors & divisibility

Multiply monomials

Factor monomials

Graph interpretation word problems

Multiply monomials challenge

Multiply binomials intro

Unit-fraction exponents

Fractional exponents

Special products of binomials intro

Simplify square-roots (variables)

Simplify square-root expressions

Properties of exponents (rational exponents)

Special products of binomials

Rational exponents challenge

Finite geometric series in sigma notation

Powers of the imaginary unit

Write exponential functions

Interpret basic exponential functions

Interpret basic exponential functions: graphs & tables

Equivalent forms of exponential expressions

Expand binomials

Rewrite exponential expressions

Prove polynomial identities

Evaluate logarithms

Simplify rational expressions: common monomial factors

Solve square-root equations (basic)

Add & subtract rational expressions

Equations with two rational expressions

Partial fraction expansion

Trapezoidal rule

Area between two curves

Net change (graphical)

Ratio test

A2. 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.

Quiz: Looking Forward

Tangents to polar curves

Powers of fractions

Square roots

Cube roots

Factors & divisibility

Evaluate radical expressions challenge

A3. 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.

Tangents to polar curves

Scientific notation

A4. 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

Tangents to polar curves

Adding & subtracting in scientific notation

Multiplying & dividing in scientific notation

Scientific notation challenge

B5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

Horizontal & vertical lines

Tangents to polar curves

Number of solutions to equations challenge

Graphing proportional relationships

Intercepts from a table

Slope from equation

Parallel & perpendicular lines from graph

Two-variable inequalities from their graphs

Graphs of inequalities

Interpreting graphs of functions

Use geometric sequence formulas

Sequences word problems

Domain of advanced functions

Related rates (Pythagorean theorem)

B6. 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.

Slope from graph

Slope from two points

Graph from slope-intercept form

Point-slope form

Linear equations word problems: tables

Estimating equations of lines of best fit, and using them to make predictions

Parallel & perpendicular lines from equation

Linear vs. exponential growth: from data

Equations & inequalities word problems

C7. Solve linear equations in one variable.

Commutative and non-commutative transformations

Exponential model equations

Series basics challenge

Distributive property with variables

Equations with variables on both sides: decimals & fractions

Equations with parentheses: decimals & fractions

Ordered pair solutions to linear equations

Linear equations with unknown coefficients

Solutions to 2-variable equations

Complete solutions to 2-variable equations

Equation practice with angle addition

Equation practice with angles

Equation practice with segment addition

Function inputs & outputs: equation

Determine the domain of functions

Add & subtract polynomials: two variables

Solutions of inequalities: algebraic

Linear systems of equations capstone

Constraint solutions of two-variable inequalities

Multiply monomials by polynomials challenge

Multiply binomials

Solve absolute value equations

Arithmetic series in sigma notation

Convergent & divergent geometric series

Positive & negative intervals of polynomials

Simplify rational expressions: common monomial factors

Multiply & divide rational expressions (basic)

Equations with one rational expression

Determinant of a 2x2 matrix

Limits by factoring

Differentiate quotients

Exponential functions differentiation

Vector-valued functions differentiation

Antiderivatives

Evaluating definite integrals

C7a. 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).

C7b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

C8. Analyze and solve pairs of simultaneous linear equations.

Ray intersection with plane

Systems of equations word problems capstone

Systems of linear equations word problems: Level 2

Convert linear equations to standard form

Graph from linear standard form

Linear equations in any form

Solutions of systems of equations

Systems of equations with graphing

Number of solutions to a system of equations graphically

Systems of equations with elimination

Systems of equations with substitution

Equivalent systems of equations

Number of solutions to a system of equations algebraically

Systems of equations word problems

Age word problems

Constraint solutions of systems of inequalities

Systems of inequalities graphs

Systems of inequalities word problems

Recognize direct & inverse variation

Second derivatives (vector-valued functions)

Riemann sums with sigma notation

Function as a geometric series

C8a. 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.

Related rates (multiple rates)

C8b. 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.

C8c. 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.

A1. Verify experimentally the properties of rotations, reflections, and translations:

Angular Measure 2

Calculating the touching point

Pythagorean theorem word problems

Ray intersection with line

Ray tracing

Rotating a point around the origin

Transformation puzzles 3

Use Pythagorean theorem to find area and perimeter

Area & perimeter on the coordinate plane

Dividing by zero

Perform translations

Determine translations

Reflect shapes

Rotate shapes: center ≠ (0,0)

Determine dilations

Congruence & transformations

Points inside/outside/on a circle

Similarity & transformations

Determine similar triangles: AA

Solve triangles: angle bisector theorem

Equivalent vectors

Graph a circle from its features

Features of a circle from its graph

Analyze scalar multiplication

Add & subtract vectors

Direction of vectors

Foci of an ellipse from radii

Transform vectors using matrices

Determine invertibile matrices

Matrices as transformations

Related rates (advanced)

A1a. Lines are taken to lines, and line segments to line segments of the same length.

A1b. Angles are taken to angles of the same measure.

A1c. Parallel lines are taken to parallel lines.

A2. 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.

Transformation puzzles 3

Use Pythagorean theorem to find right triangle side lengths

Tangents of circles problems

A3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Transformation puzzles 1

Transformation puzzles 2

Transformation puzzles 3

Understanding rotation of arbitrary points

Transforming polygons

Distance between two points

Distance between point & line

Perform reflections

Rotate shapes

Dilate shapes

Determine mappings

Unit vectors

A4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Composite transformations

Advanced reflections

Determine rotations

Perform dilations

Determine reflections

Identify transformations

Determine similar triangles: SSS

Solve similar triangles (basic)

Solve similar triangles (advanced)

Use similar & congruent triangles

A5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Constructing triangles

Finding angle measures between intersecting lines

Pythagorean theorem challenge

Find angles in congruent triangles

Period of sinusoidal functions from equation

B6. Explain a proof of the Pythagorean Theorem and its converse.

Pythagorean theorem in 3D

Infinite geometric series

B7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

Coordinate plane word problems: polygons

B8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

C9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Volume of cones

Volume of spheres

Solid geometry

Volumes of cylinders

Solid geometry word problems

A1. 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.

Converting repeating decimals to fractions 1

Approximating square roots

Classify numbers: rational & irrational

Classify numbers

Roots of decimals & fractions

Approximating square roots

Comparing irrational numbers with a calculator

Comparing irrational numbers

Converting multi-digit repeating decimals to fractions

Significant figures

Plot numbers on the complex plane

Multiply complex numbers (basic)

Matrix equations: scalar multiplication

A2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Approximating square roots

Comparing irrational numbers

Simplify square roots

A1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

p-series

Scatterplots & correlation

Constructing scatter plots

Making good scatter plots

Positive and negative linear correlations from scatter plots

Describing trends in scatter plots

Basic set notation

Direct comparison test

A2. Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

A3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

A4. Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?