Grade 8 Math Montana Standards

Work with radicals and integer exponents.

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

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.

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.

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.

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

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

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.

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

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 collecting 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.

Solve real-world and mathematical problems leading to two linear equations in two variables.

Define, evaluate, and compare functions.

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.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

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.

Use functions to model relationships between quantities.

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.

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.

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

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

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

Angles are taken to angles of the same measure.

Parallel lines are taken 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.

Understand and apply the Pythagorean Theorem.

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.

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

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

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

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.

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).

Investigate patterns of association in bivariate data.

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, 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 intercept.

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.

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.

Expressions and Equations

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

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 and evaluate square roots of small perfect squares and cube roots of small perfect cubes.

Represent very large or very small quantities using scientific notation, limited to a single digit times an integer power of ten.

Perform operations with numbers expressed in scientific notation.

Graph proportional relationships, interpreting the unit rate as the slope of the graph, and compare two different proportional relationships as tables, graphs, and equations.

Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane and 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 by:

*Solve linear equations in one variable by* giving examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions and showing 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 in one variable by* solving linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Analyze and solve pairs of simultaneous linear equations by:

*Analyze and solve pairs of simultaneous linear equations by* understanding 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.

*Analyze and solve pairs of simultaneous linear equations by* solving systems of two linear equations in two variables algebraically, estimating solutions by graphing the equations, and solving simple cases by inspection.

*Analyze and solve pairs of simultaneous linear equations by* solving problems in context that lead to two linear equations in two variables. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Functions

Understand that a function is a rule that assigns to each input exactly one output and the graph of a function is the set of ordered pairs (x,y) each consisting of an input, x, and the corresponding output, y.

Compare properties of two functions using tables, graphs, and equations.

Interpret the equation y = mx + b as defining a linear function whose graph is a straight line with slope m passing through the point (0, b).

Given linear data relating two quantities, construct a linear function that models the data and interpret the rate of change and initial value of a linear function in terms of the situation it models.

Given the graph of a function, describe qualitatively the functional relationship between quantities, and, given a verbal description of a functional relationship, sketch a graph that exhibits the qualitative features of a function.

Geometry

Verify experimentally the properties of rotations, reflections, and translations and understand that these are rigid transformations, lines are taken to lines, line segments to line segments of the same length, angles are taken to angles of the same measure, and parallel lines are taken to parallel lines. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

Understand that a two-dimensional figure is congruent to another if the second can be obtained by a sequence of rigid transformations, and, 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. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

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, and, 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.

Explain a proof of the Pythagorean Theorem and its converse.

Apply the Pythagorean Theorem to determine unknown side lengths in right triangle problems, including problems in context in two and three dimensions. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

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

Know, use, and apply the formulas for the volumes of cones, cylinders, and spheres to solve problems, including problems in context.

The Number System

Know real numbers are made up of rational and irrational numbers, understand informally that every number has a decimal expansion, and convert a decimal expansion which repeats eventually into a rational number.

Use rational approximations of irrational numbers to compare the value of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.

Statistics and Probability

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities and 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 and 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.

Use the equation of a linear model to solve problems in the context of bivariate measurement data, and interpret the slope and intercept.

Construct and interpret frequencies and relative frequencies for bivariate categorical data in a two-way table to investigate patterns of association. This standard should incorporate cultural context relating to Montana Indigenous Peoples and local communities.

The Number System

Standards for Mathematical Practice

Expressions and Equations

Functions

Geometry

Statistics and Probability

Mathematical practice standard 1 is to problem-solve and persevere. Mathematically proficient students: make conjectures, plan, and follow solution strategies; evaluate their progress and accuracy; engage in sense-making and self-monitoring; and persevere in seeking solutions, and value alternative approaches.

Mathematical practice standard 2 is to abstract and generalize. Mathematically proficient students are able to decontextualize and symbolically represent both mathematical and non-mathematical situations to search for and analyze regularities, patterns, and structures.

Mathematical practice standard 3 is to justify and prove. Mathematically proficient students create, evaluate, justify, and refute mathematical claims in developmentally and mathematically appropriate ways.

Mathematical practice standard 4 is to model with mathematics. Mathematically proficient students: make sense of a scenario; identify a problem to be solved, and mathematize it; and apply a mathematical model to reach a solution and verify its viability.

Mathematical practice standard 5 is to represent. Mathematically proficient students recognize, use, create, interpret, and translate representations using appropriate methods and tools; and understand multiple ways of representing mathematical ideas and how they are related.

Mathematical practice standard 6 is to collaborate mathematically. Mathematically proficient students engage in mathematics as a social enterprise through discussion and collaborative inquiry where ideas are offered, debated, connected, and built upon toward solutions, shared understanding, and appreciation of other perspectives.

Mathematical practice standard 7 is to culturally connect. Mathematically proficient students recognize cultural connections and contributions to mathematics; and appreciate the role of mathematics in various cultural contexts, including those of tribally-specific Montana Indigenous Peoples.