Grade 8 Math South Dakota standards Standards

Bivariate Data

Statistics and Probability

Transformation, Similarity, and Congruency

Pythagorean Theorem

Triangles and Angles

Area, Volume, and Surface Area

Geometry

System of Equations

Equations and Inequalities

Algebra

Functions

Proportional and Linear Relationships

Relationships and Functions

Rational Number Operations

Rational and Irrational Numbers

Numbers Concepts and Computations

Identify linear equations in one variable with one solution, infinitely many solutions, or no solution.

Solve linear equations with rational number coefficients using the distributive property and combining like terms.

Analyze and solve one-variable linear inequalities with rational coefficients.

Understand a system of linear equations to be a set of two or more equations

Determine whether an ordered pair is a solution to a system of two linear equations by substituting the values into both equations and explaining why the pair satisfies both.

Analyze a system of two linear equations to determine whether it has one solution, infinitely many solutions, or no solution, and explain what each case means in terms of the graphs or equations.

Analyze and solve systems of linear equations algebraically and estimate solutions by graphing the equation.

Solve real-world and mathematical problems involving leading to two linear equations in one and/or two variables.

Explain the formulas for the volumes of cones, cylinders, and spheres and apply them to solve real-world and mathematical problems.

Understand a rigid transformation to be a change in location or orientation that generates a congruent shape by preserving distances between vertices.

Identify, draw, and describe the three types of rigid transformations on two-dimensional figures: rotations, reflections, and translations.

Identify, draw, and describe mathematical dilations on, two-dimensional figures, as non-rigid transformations that generate a similar shape

Describe a sequence of rigid transformations that moves and aligns one congruent shape onto another.

Understand and explain the angle-angle criterion for determining the similarity of triangles.

Understand properties of interior and exterior angles in triangles (triangle sum theorem, exterior angle theorem).

Understand and explain the relationships between angles when parallel lines are intersected by a transversal.

Identify, describe, and draw elements of triangles including base, height, leg, and hypotenuse.

Explain the Pythagorean theorem and apply it to find unknown side lengths in right triangles.

Explain the converse of the Pythagorean theorem and apply it to determine if a triangle is a right triangle.

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

Understand congruent figures to be geometric objects that have exactly the same size and shape.

Understand similar figures to be geometric objects that have the same shape (congruent angles), but different sizes (proportional) sizes.

Understand that a real number is any number that can be represented on a number line.

Write numbers in scientific notation using positive and negative exponents.

Perform operations (add, subtract, multiply, divide) with numbers in scientific notation

Classify numbers as rational or irrational and explain how both sets together form the set of real numbers

Understand an irrational number to be a non-terminating, non-repeating decimal that cannot be expressed as a fraction of two integers (rational number).

Compare the value of irrational numbers and locate them on a number line by finding the rational approximations.

Explain how the properties of integer exponents work by using patterns and reasoning to generate and justify equivalent expressions.

Understand the root index to be the number placed above the radical symbol indicating which root to find and recognize that if the root index is not specified, it is assumed to be 2

Understand a square root of a number to be the number that when multiplied by itself produces the original number.

Understand the cube root of a number to be a number that when cubed produces the original number.

Use square root and cube root symbols to represent solutions to equations of the form 𝑥^2 = 𝑝 and 𝑥^3 = 𝑝, where p is a positive rational number. Evaluate square roots of small perfect squares (up to 100) and cube roots of small perfect cubes (up to 125).

Graph proportional relationships, interpreting the rate of change as the slope of the graph.

Determine whether a relationship is linear or nonlinear given a table, graph, equation, or verbal description.

Understand a linear function to be a relationship between two variables with a constant rate of change whose graph is a straight line on the coordinate plane and to have the equation y = mx + b, give examples of functions that are not linear.

Explain how the rate of change and y-intercept describe the relationship between two quantities, using multiple representations to justify interpretations.

Explain how the rate of change (slope) and y-intercept (initial value) describe the relationship between two quantities, using multiple representations to justify interpretations

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear).

Construct a function to model a linear relationship between two quantities.

Compare two different proportional relationships represented in different ways.

Explain how the slope m of a line is the same between any two points on a line. Types of slope include: positive, negative, zero (horizontal), and undefined (vertical)

Define a function as a rule where each input has exactly one output.

Understand the domain as the set of inputs allowed by the function.

Understand the range as the set of outputs produced by the function

Determine whether a relationship is a function given a table, graph, equation, or verbal description.

Compare properties of two functions represented in different ways including tables, graphs, equations, and verbal descriptions.

Construct and interpret scatter plots using bivariate data; determine if the data displays a linear or nonlinear pattern and describe the patterns as clustering, outliers, positive, negative, or no association.

Construct a line of fit to approximately fit data displaying a linear association when presented in scatter plot.

Construct and interpret a relative frequency table.

Use functions to model relationships between quantities.

Define, evaluate and compare functions.

Functions

Investigate patterns of association in bivariate data.

Statistics and Probability

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

Understand the connections between proportional relationships, lines and linear equations.

Work with radicals and integer exponents.

Expressions and Equations

Know that there are numbers that are not rational, and approximate them by rational numbers.

The Number System

Solve real-world and mathematical problems involving volume of cylinders, cones and spheres.

Understand and apply the Pythagorean Theorem.

Understand congruence and similarity using physical models, transparencies or geometry software.

Geometry

Standards for Mathematical Practice

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

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.

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.

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.

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.

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.

Solve linear equations in one variable.

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).

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

Analyze and solve pairs of simultaneous linear equations.

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.

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

Solve real-world and mathematical problems involving leading to two linear equations in one and/or two variables.

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. (Function notation is not required in Grade 8).

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.

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.

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.

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

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

Angles are mapped to angles of the same measure.

Parallel lines are mapped to parallel lines.

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.

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

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.

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.

Explain a proof of the Pythagorean Theorem and its converse.

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

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

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

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 or terminates. Convert a decimal expansion which repeats or terminates into a rational number.

Use rational approximations of irrational numbers to compare the size of irrational numbers. Locate irrational numbers approximately on a number line diagram, and estimate the value of expressions such as (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.

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.

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 (i.e. line of fit), and informally assess the model fit by judging the closeness of the data points to the line.

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and the y-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.

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?

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.