Standards
USE PROPORTIONAL RELATIONSHIPS TO REPRESENT AND REASON ABOUT LINEAR EQUATIONS.
Generate resourceGraph proportional relationships and interpret the unit rate as the slope of the graph.
Generate resourceCompare proportional relationships represented in different forms (e.g., graphs, equations, and tables) by analyzing their slopes.
Generate resourceUse proportional reasoning to identify slope and represent linear relationships.
Generate resourceUse proportional reasoning to explain why the slope m is the same between any two points on a non-vertical line by showing that the ratios of vertical change to horizontal change between points are equivalent.
Generate resourceUse this relationship to write equations of lines in the form for lines that pass through the origin, and in the form for lines that include a vertical shift from the origin, where b represents the y-intercept.
Generate resourceGive examples of linear equations in one variable with one solution, infinitely many solutions, or no solution. Justify each example by showing how it can be rewritten in an equivalent equation in the form x=a, a=a, or a=b (where a and b are different rational numbers) and explaining what this form reveals about the solution.
Generate resourceSolve linear equations in one variable with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combing like terms.
Generate resourceSolve inequalities including those requiring the use of the distributive property and combining like terms.
Generate resourceRepresent the solution set on a number line and describe the set of possible values in the context of the problem.
Generate resourceInterpret the solution to a system of two linear equations as the point where their graphs intersect and verify that the coordinates of the intersection satisfy both equations.
Generate resourceSolve systems of two linear equations by graphing and substitution. When using graphing, find an approximate or exact solution depending on the precision of the graph. Select a method strategically based on the structure of the system or the context of the problem and explain why the method is appropriate.
Generate resourceUse structure to determine when a system has no solution, one solution, or infinitely many solutions and justify the conclusion.
Generate resourceModel and solve contextual problems using systems of two linear equations.
Generate resourceDetermine whether a relationship is a function by analyzing whether each input is assigned to exactly one output.
Generate resourceGiven a clearly defined set of input-output pairs in a table, mapping, or real-world description, determine whether the relationship is a function and explain why.
Generate resourceGiven a graph, identify input-output pairs and determine whether the graph represents a function.
Generate resourceCompare the properties (i.e., rate of change, intercepts) of two linear functions each represented in a different way (algebraically, graphically, numerically in tables, or by narrative descriptions).
Generate resourceIdentify whether a function is linear or non-linear given equations, graphs, or input-output pairs, and provide a justification using reasoning about the constant rate of change and its relationship to proportional reasoning.
Generate resourceConstruct and interpret a linear function to model a relationship between two quantities.
Generate resourceGiven two input-output pairs, a graph, a table, or a description of a relationship, determine the rate of change and initial value and write the linear equation that models the situation.
Generate resourceInterpret the rate of change and initial value in the context of the problem.
Generate resourceDescribe 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 based on a narrative description.
Generate resourceGiven the graph of a linear function, identify the slope and y-intercept.
Generate resourceDetermine whether observed associations in sample data justify generalizations about a broader population.
Generate resourceIdentify limitations or potential bias in data collection or interpretation when analyzing trends in scatter plots.
Generate resourceDESCRIBE, ANALYZE, AND COMPARE DATA USING VISUAL AND NUMERICAL REPRESENTATIONS TO MODEL SITUATIONS AND DRAW INFERENCES.
Generate resourceConstruct 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.
Generate resourceCompare the fit of different linear models shown on scatter plots of the same data by describing the overall pattern and the closeness of data points to each line.
Generate resourceUse a provided linear model to make predictions based on bivariate measurement data. Interpret the meaning of the slope and y-intercept in context.
Generate resourceMODEL AND ANALYZE PROBABILITY TO INTERPRET CHANCE EVENTS AND MAKE PREDICTIONS.
Generate resourceConstruct a two-way table to organize data on two categorical variables collected from the same subjects. Calculate and interpret relative frequencies for rows and/or columns and use them to describe possible associations between the variables.
Generate resourceFind probabilities of compound events by representing or simulating the sample space or simulating events.
Generate resourceRepresent the sample space for a compound event using organized lists, tables, or tree diagrams. Given an event described in everyday language, identify the outcomes in the sample space that make up the event.
Generate resourceDetermine the probability of a compound event as a fraction of outcomes in the sample space that result in the event.
Generate resourceKnow and apply the properties of integer exponents to generate equivalent numerical expressions.
Generate resourceUse square root and cube root symbols to represent solutions to equations of the form <em>x² = p and x³ = p</em>, where <em>p</em> is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Generate resourceUse 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.
Generate resourcePerform 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.
Generate resourceGraph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
Generate resourceUse similar triangles to explain why the slope <em>m</em> is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation <em>y=mx</em> for a line through the origin, and the equation <em>y = mx + b</em> for a line intercepting the vertical axis at <em>b</em>.
Generate resourceGive 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 <em>x = a, a = a</em>, or <em>a = b</em> results (<em>a</em> and <em>b</em> are different numbers).
Generate resourceSolve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Generate resourceUnderstand 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.
Generate resourceSolve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
Generate resourceUnderstand the connections between proportional relationships, lines, and linear equations.
Generate resourceAnalyze and solve linear equations and pairs of simultaneous linear equations.
Generate resourceUnderstand 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).
Generate resourceCompare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Generate resourceInterpret the equation <em>y = mx + b</em> as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.
Generate resourceConstruct 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 (<em>x, y</em>) 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.
Generate resourceDescribe 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.
Generate resourceVerify experimentally the properties of rotations, reflections, and translations.
Generate resourceLines are taken to lines, and line segments to line segments of the same length.
Generate resourceUnderstand 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.
Generate resourceDescribe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
Generate resourceUnderstand 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.
Generate resourceUse 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.
Generate resourceApply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
Generate resourceApply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Generate resourceKnow the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Generate resourceUnderstand congruence and similarity using physical models, transparencies, or geometry software.
Generate resourceSolve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
Generate resourceUse the relationships between supplementary, complementary, vertical, and adjacent angles to write and solve equations for an unknown angle.
Generate resourceDraw or build triangles using tools (ruler, protractor, technology, or physical manipulatives) given three measures of angles or sides. Determine whether the given conditions result in a unique triangle, more than one triangle, or no triangle.
Generate resourceUse informal arguments to verify that the sum of the interior angles of a triangle is 180 degrees and why an exterior angle equals the sum of the two remote interior angles.
Generate resourceUse diagrams, examples, or informal reasoning to explain the Pythagorean Theorem and why it applies to all right triangles.
Generate resourceUse the converse of the Pythagorean Theorem to determine whether a set of three side lengths forms a right triangle, and extend this reasoning to identify whether a triangle is acute or obtuse using Pythagorean inequalities.
Generate resourceFind the distance between two points in a coordinate system using the Pythagorean Theorem.
Generate resourceIdentify a number as irrational when its decimal expansion neither terminates nor repeats, and explain that irrational numbers cannot be expressed as the ratio of two integers.
Generate resourceUse rational approximations of irrational numbers to estimate their locations on a number line, compare their size, and apply this understanding to estimate the value of expressions (e.g., ) in context.
Generate resourceKnow and apply the properties of integer exponents to generate equivalent numerical expressions.
Generate resourceUse square and cube roots to represent solutions to equations and describe numbers.
Generate resourceUse square and cube roots symbols to represent solutions to equations of the form and , where p is a positive rational number.
Generate resourceEvaluate square roots of perfect squares from 1 to 100 and cube roots of perfect cubes from 1 to 125 by inspection.
Generate resourceApply scientific notation to represent and compare very large and very small quantities in context, and use technology to compute with numbers in scientific notation.
Generate resourceExpress numbers as a single digit times an integer power of 10 to estimate and compare magnitudes.
Generate resourceUse scientific notation to model and compute with real-world quantities, choose appropriate units, and interpret their meaning based on the context.
Generate resourceKnow that there are numbers that are not rational, and approximate them by rational numbers.
Generate resourceKnow 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.
Generate resourceUse 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).
Generate resourceConstruct 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.
Generate resourceKnow 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.
Generate resourceUse the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
Generate resourceUnderstand 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.
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