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WV.M.1HS8.8th Grade High School Mathematics I
8th Grade High School Mathematics I
Congruence, Proof, and Constructions Understand and apply the Pythagorean theorem.M.1HS8.60. Apply the Pythagorean theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
M.1HS8.61. Apply the Pythagorean theorem to find the distance between two points in a coordinate system.
Experiment with transformations in the plane.M.1HS8.49. Know precise definitions of angle, circle, perpendicular line, parallel line and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
M.1HS8.50. Represent transformations in the plane using, example, transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and
M.1HS8.51. Given a rectangle, parallelogram, trapezoid or regular polygon, describe the rotations and reflections that carry it onto itself.
M.1HS8.52. Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.
M.1HS8.53. Given a geometric figure and a rotation, reflection or translation draw the transformed figure using, e.g., graph paper, tracing paper or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Understand congruence in terms of rigid motions.M.1HS8.54. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
Reasoning with Equations Analyze and solve linear equations and pairs of simultaneous linear equations.M.1HS8.34. Analyze and solve pairs of simultaneous linear equations.M.1HS8.34.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.
M.1HS8.34.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
Understand solving equations as a process of reasoning and explain the reasoning.M.1HS8.32. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Solve systems of equations.M.1HS8.35. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
M.1HS8.36. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Solve equations and inequalities in one variable.M.1HS8.33. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Relationships between Quantities Reason quantitatively and use units to solve problems.M.1HS8.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Create equations that describe numbers or relationships.M.1HS8.5. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions and simple rational and exponential functions.
M.1HS8.6. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.1HS8.7. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. (e.g., Represent inequalities describing nutritional and cost constraints o
M.1HS8.8. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.)
Connecting Algebra and Geometry through Coordinates Use coordinates to prove simple geometric theorems algebraically.M.1HS8.63. Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems. (e.g., Find the equation of a line parallel or perpendicular to a given line that passes through a given point.)
M.1HS8.64. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula).
Descriptive Statistics Summarize, represent, and interpret data on a single count or measurement variable.M.1HS8.37. Represent data with plots on the real number line (dot plots, histograms, and box plots).
M.1HS8.40. 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 associat
M.1HS8.41. 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 dat
Summarize, represent, and interpret data on two categorical and quantitative variables.M.1HS8.44. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data.
M.1HS8.45. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.M.1HS8.45.b. Informally assess the fit of a function by plotting and analyzing residuals. (Focus should be on situations for which linear models are appropriate.)
M.1HS8.45.c. Fit a linear function for scatter plots that suggest a linear association.
Linear and Exponential Relationships Construct and compare linear, quadratic, and exponential models and solve problems.M.1HS8.28. Distinguish between situations that can be modeled with linear functions and with exponential functions.M.1HS8.28.a. Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.Quiz, Flash Cards, Worksheet, Game & Study Guide Functions
Analyze functions using different representations.M.1HS8.23. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.1HS8.23.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Understand the concept of a function and use function notation.M.1HS8.15. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f cQuiz, Flash Cards, Worksheet, Game & Study Guide Functions
M.1HS8.16. Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.
M.1HS8.17. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. (e.g., The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for n ≥ 1.)Quiz, Flash Cards, Worksheet, Game & Study Guide Sequences
M.1HS8.18. 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
Represent and solve equations and inequalities graphically.M.1HS8.10. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, (e.g., using technology to graph the functions, make tables of v
M.1HS8.11. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality) and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding
Interpret expressions for functions in terms of the situation they model.M.1HS8.31. Interpret the parameters in a linear or exponential function in terms of a context.
Build a function that models a relationship between two quantities.M.1HS8.25. Write a function that describes a relationship between two quantities.M.1HS8.25.a. Determine an explicit expression, a recursive process or steps for calculation from a context.
Interpret functions that arise in applications in terms of a context.M.1HS8.20. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercept
M.1HS8.22. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Define, evaluate, and compare functions.M.1HS8.12. 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.Quiz, Flash Cards, Worksheet, Game & Study Guide Functions
M.1HS8.14. 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. (e.g., The function A = s^2 giving the area of a square as a function of its side length is not linear because
WV.M.8.EE.Expressions and Equations
Expressions and Equations
Analyze and solve linear equations and pairs of simultaneous linear equations.M.8.10. Analyze and solve pairs of simultaneous linear equations.M.8.10.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.
M.8.10.b. Solve systems of two linear equations in two variables algebraically and estimate solutions by graphing the equations. Solve simple cases by inspection. (e.g., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.)
M.8.9. Solve linear equations in one variable.M.8.9.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
M.8.9.b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.
Work with radicals and integer exponents.M.8.3. Know and apply the properties of integer exponents to generate equivalent numerical expressions. (e.g., 3^2 × 3^–5 = 3^–3 = 1/3^3 = 1/27.)
M.8.4. 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 and cube roots of small perfect cubes. Know that √2 is irratioQuiz, Flash Cards, Worksheet, Game & Study Guide Real numbers
M.8.5. 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. (e.g., Estimate the population of the United States as 3 × 10^8
M.8.6. 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.
Understand the connections between proportional relationships, lines, and linear equations.M.8.7. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. (e.g., Compare a distance-time graph to a distance-time equation to determine which of
M.8.8. 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 v
WV.M.8.F.Functions
Define, evaluate, and compare functions.M.8.11. 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.Quiz, Flash Cards, Worksheet, Game & Study Guide Functions
M.8.13. 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. (e.g., The function A = s^2 giving the area of a square as a function of its side length is not linear becaus
Use functions to model relationships between quantities.M.8.14. 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
WV.M.8.G.Geometry
Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.M.8.24. Know the formulas for the volumes of cones, cylinders and spheres and use them to solve real-world and mathematical problems.
Understand congruence and similarity using physical models, transparencies, or geometry software.M.8.16. Verify experimentally the properties of rotations, reflections and translations:M.8.16.a. Lines are taken to lines, and line segments to line segments of the same length.
M.8.16.b. Angles are taken to angles of the same measure.
M.8.16.c. Parallel lines are taken to parallel lines.
M.8.17. 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 t
M.8.20. 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. (e.g., Arrange three copies of
Understand and apply the Pythagorean Theorem.M.8.22. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
M.8.23. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
WV.M.8.NS.The Number System
Know that there are numbers that are not rational, and approximate them by rational numbers.M.8.1. 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 eventuall
WV.M.8.SP.Statistics and Probability
Statistics and Probability
Investigate patterns of association in bivariate data.M.8.25. 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 associat
M.8.26. 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 dat
WV.M.A18.High School Algebra I for 8th Grade
High School Algebra I for 8th Grade
Quadratic Functions and Modeling Analyze functions using different representations.M.A18.67. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.A18.67.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
M.A18.68. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.M.A18.68.b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^12t, y = (1.2)^t/10, and classify them as representing exponential
Understand and apply the Pythagorean theorem.M.A18.62. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
M.A18.63. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Build a function that models a relationship between two quantities.M.A18.70. Write a function that describes a relationship between two quantities.M.A18.70.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Interpret functions that arise in applications in terms of a context.M.A18.64. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercep
M.A18.66. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Descriptive Statistics Summarize, represent, and interpret data on two categorical and quantitative variables.M.A18.46. Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal and conditional relative frequencies). Recognize possible associations and trends in the data.
M.A18.47. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.M.A18.47.b. Informally assess the fit of a function by plotting and analyzing residuals.
M.A18.47.c. Fit a linear function for scatter plots that suggest a linear association.
Summarize, represent, and interpret data on a single count or measurement variable.M.A18.39. Represent data with plots on the real number line (dot plots, histograms, and box plots).
Investigate patterns of association in bivariate data.M.A18.42. 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 associa
M.A18.43. 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 da
Relationships between Quantities and Reasoning with Equations Understand solving equations as a process of reasoning and explain the reasoning.M.A18.9. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Reason quantitatively and use units to solve problems.M.A18.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Solve equations and inequalities in one variable.M.A18.10. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Create equations that describe numbers or relationships.M.A18.5. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
M.A18.6. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.A18.7. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. (e.g., Represent inequalities describing nutritional and cost constraints o
M.A18.8. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.)
Expressions and Equations Create equations that describe numbers or relationships.M.A18.55. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
M.A18.56. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.A18.57. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.)
Write expressions in equivalent forms to solve problems.M.A18.53. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.M.A18.53.c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as (1.15^1/12)^12t ≈ 1.012^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
Perform arithmetic operations on polynomials.M.A18.54. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Linear and Exponential Relationships Interpret functions that arise in applications in terms of a context.M.A18.27. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercep
M.A18.29. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
Understand the concept of a function and use function notation.M.A18.22. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f cQuiz, Flash Cards, Worksheet, Game & Study Guide Functions
M.A18.23. Use function notation, evaluate functions for inputs in their domains and interpret statements that use function notation in terms of a context.
M.A18.24. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)+ f(n-1) for n ≥ 1.Quiz, Flash Cards, Worksheet, Game & Study Guide Sequences
Represent and solve equations and inequalities graphically.M.A18.17. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately (e.g., using technology to graph the functions, make tables of va
M.A18.18. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the correspondin
Construct and compare linear, quadratic, and exponential models and solve problems.M.A18.35. Distinguish between situations that can be modeled with linear functions and with exponential functions.M.A18.35.a. Prove that linear functions grow by equal differences over equal intervals; exponential functions grow by equal factors over equal intervals.Quiz, Flash Cards, Worksheet, Game & Study Guide Functions
Analyze and solve linear equations and pairs of simultaneous linear equations.M.A18.13. Analyze and solve pairs of simultaneous linear equations.M.A18.13.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.
M.A18.13.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
Analyze functions using different representations.M.A18.30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.A18.30.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Use functions to model relationships between quantities.M.A18.25. 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
Define, evaluate and compare functions.M.A18.19. 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.Quiz, Flash Cards, Worksheet, Game & Study Guide Functions
M.A18.21. 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. (e.g., The function A = s^2 giving the area of a square as a function of its side length is not linear because
Interpret expressions for functions in terms of the situation they model.M.A18.38. Interpret the parameters in a linear or exponential function in terms of a context.
Solve systems of equations.M.A18.14. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
M.A18.15. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Build a function that models a relationship between two quantities.M.A18.32. Write a function that describes a relationship between two quantities.M.A18.32.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
WV.M.MHM.Mathematical Habits of Mind
Mathematical Habits of Mind
MHM1. Make sense of problems and persevere in solving them.
MHM2. Reason abstractly and quantitatively.