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WV.M.1HS.High School Mathematics I
High School Mathematics I
Congruence, Proof, and Constructions Experiment with transformations in the plane.M.1HS.39. 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.1HS.40. Represent transformations in the plane using, for 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
M.1HS.41. Given a rectangle, parallelogram, trapezoid or regular polygon, describe the rotations and reflections that carry it onto itself.
M.1HS.42. Develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.
M.1HS.43. 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.1HS.44. 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 Solve systems of equations.M.1HS.29. 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.1HS.30. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Understand solving equations as a process of reasoning and explain the reasoning.M.1HS.27. 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 equations and inequalities in one variable.M.1HS.28. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Relationships between Quantities Create equations that describe numbers or relationships.M.1HS.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.1HS.6. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.1HS.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.1HS.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.1HS.50. 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.1HS.51. 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.1HS.31. Represent data with plots on the real number line (dot plots, histograms, and box plots).
Summarize, represent, and interpret data on two categorical and quantitative variables.M.1HS.34. 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.1HS.35. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.M.1HS.35.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.1HS.35.c. Fit a linear function for scatter plots that suggest a linear association.
Linear and Exponential Relationships Build a function that models a relationship between two quantities.M.1HS.20. Write a function that describes a relationship between two quantities.M.1HS.20.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.1HS.15. 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.1HS.17. 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.
Represent and solve equations and inequalities graphically.M.1HS.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.1HS.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
Construct and compare linear, quadratic, and exponential models and solve problems.M.1HS.23. Distinguish between situations that can be modeled with linear functions and with exponential functions.M.1HS.23.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.1HS.18. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.1HS.18.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Understand the concept of a function and use function notation.M.1HS.12. 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.1HS.14. 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
WV.M.2HS.High School Mathematics II
High School Mathematics II
Expressions and Equations Write expressions in equivalent forms to solve problems.M.2HS.19. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.M.2HS.19.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%.
Create equations that describe numbers or relationships.M.2HS.20. Create equations and inequalities in one variable and use them to solve problems.
M.2HS.21. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.2HS.22. 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.)
Analyze functions using different representations. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.M.2HS.11.b. Use the properties of exponents to interpret expressions for exponential functions. (e.g., 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 growt
M.2HS.10. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.2HS.10.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Extending the Number System Perform arithmetic operations on polynomials.M.2HS.6. 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.
Circles With and Without Coordinates Explain volume formulas and use them to solve problems.M.2HS.60. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle and informal limit arguments.
M.2HS.61. Use volume formulas for cylinders, pyramids, cones and spheres to solve problems. Volumes of solid figures scale by k^3 under a similarity transformation with scale factor k.
Applications of Probability Understand independence and conditional probability and use them to interpret data.M.2HS.29. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities and use this characterization to determine if they are independent.
M.2HS.31. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. (e.g.
M.2HS.32. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. (e.g., Compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.)
Use the rules of probability to compute probabilities of compound events in a uniform probability model.M.2HS.36. Use permutations and combinations to compute probabilities of compound events and solve problems.
Build a function that models a relationship between two quantities.M.2HS.13. Write a function that describes a relationship between two quantities.M.2HS.13.a. Determine an explicit expression, a recursive process or steps for calculation from a context.
Quadratic Functions and Modeling Interpret functions that arise in applications in terms of a context.M.2HS.7. 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.2HS.9. 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.
Similarity, Right Triangle Trigonometry, and Proof Prove theorems involving similarity.M.2HS.46. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Understand similarity in terms of similarity transformations.M.2HS.39. Verify experimentally the properties of dilations given by a center and a scale factor.M.2HS.39.a. A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged.
M.2HS.39.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
M.2HS.40. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles
Use coordinates to prove simple geometric theorems algebraically.M.2HS.47. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Prove geometric theorems.M.2HS.42. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line
WV.M.3HSLA.High School Mathematics III LA
High School Mathematics III LA
Mathematical Modeling Analyze functions using different representations.M.3HSLA.39. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Create equations that describe numbers or relationships.M.3HSLA.31. 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.3HSLA.32. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.3HSLA.33. 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 on
M.3HSLA.34. 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.)
Interpret functions that arise in applications in terms of a context.M.3HSLA.35. 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.3HSLA.37. 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.
Inferences and Conclusions from Data Make inferences and justify conclusions from sample surveys, experiments, and observational studies.M.3HSLA.6. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
Polynomials, Rational, and Radical Relationships Perform arithmetic operations on polynomials.M.3HSLA.15. 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.
Represent and solve equations and inequalities graphically.M.3HSLA.23. 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
WV.M.3HSTR.High School Mathematics III TR (Technical Readiness)
High School Mathematics III TR (Technical Readiness)
Mathematical Modeling Analyze functions using different representations.M.3HSTR.39. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Create equations that describe numbers or relationships.M.3HSTR.31. 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.3HSTR.32. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.3HSTR.33. 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 on
M.3HSTR.34. 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.)
Interpret functions that arise in applications in terms of a context.M.3HSTR.35. 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.3HSTR.37. 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.
Inferences and Conclusions from Data Make inferences and justify conclusions from sample surveys, experiments, and observational studies.M.3HSTR.6. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
Polynomials, Rational, and Radical Relationships Perform arithmetic operations on polynomials.M.3HSTR.15. 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.
Represent and solve equations and inequalities graphically.M.3HSTR.23. 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
WV.M.4HSTP.High School Mathematics IV – Trigonometry/Pre-calculus
High School Mathematics IV – Trigonometry/Pre-calculus
Analysis and Synthesis of Functions Build a function that models a relationship between two quantities.M.4HSTP.20. Write a function that describes a relationship between two quantities, including composition of functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time
WV.M.4HSTR.High School Mathematics IV TR (Technical Readiness)
High School Mathematics IV TR (Technical Readiness)
Mathematical Modeling Create equations that describe numbers or relationships.M.4HSTR.31. 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.4HSTR.32. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.4HSTR.33. 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 on
M.4HSTR.34. 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.)
Analyze functions using different representations.M.4HSTR.39. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Inferences and Conclusions from Data Make inferences and justify conclusions from sample surveys, experiments, and observational studies.M.4HSTR.6. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
Polynomials, Rational, and Radical Relationships Represent and solve equations and inequalities graphically.M.4HSTR.23. 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
WV.M.A1HS.High School Algebra I
Quadratic Functions and Modeling Build a function that models a relationship between two quantities.M.A1HS.57. Write a function that describes a relationship between two quantities.M.A1HS.57.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.A1HS.51. 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.A1HS.53. 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.
Analyze functions using different representations.M.A1HS.54. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.A1HS.54.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
M.A1HS.55. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.M.A1HS.55.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
Descriptive Statistics Summarize, represent, and interpret data on a single count or measurement variable.M.A1HS.33. Represent data with plots on the real number line (dot plots, histograms, and box plots).
Summarize, represent, and interpret data on two categorical and quantitative variables.M.A1HS.36. 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.A1HS.37. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.M.A1HS.37.b. Informally assess the fit of a function by plotting and analyzing residuals.
M.A1HS.37.c. Fit a linear function for scatter plots that suggest a linear association.
Relationships between Quantities and Reasoning with Equations Solve equations and inequalities in one variable.M.A1HS.10. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
Create equations that describe numbers or relationships.M.A1HS.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.A1HS.6. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.A1HS.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.A1HS.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.)
Understand solving equations as a process of reasoning and explain the reasoning.M.A1HS.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.
Expressions and Equations Create equations that describe numbers or relationships.M.A1HS.45. 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.A1HS.46. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.A1HS.47. 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.A1HS.43. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.M.A1HS.43.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.A1HS.44. Recognize 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 Analyze functions using different representations.M.A1HS.24. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.A1HS.24.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Understand the concept of a function and use function notation.M.A1HS.18. Recognize 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 coQuiz, Flash Cards, Worksheet, Game & Study Guide Functions
M.A1HS.20. 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
Solve systems of equations.M.A1HS.13. 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.A1HS.14. 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.A1HS.26. Write a function that describes a relationship between two quantities.M.A1HS.26.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.A1HS.21. 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.A1HS.23. 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.
Represent and solve equations and inequalities graphically.M.A1HS.16. 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.A1HS.17. 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.A1HS.29. Distinguish between situations that can be modeled with linear functions and with exponential functions.M.A1HS.29.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
WV.M.A2HS.High School Algebra II
Inferences and Conclusions from Data Make inferences and justify conclusions from sample surveys, experiments, and observational studies.M.A2HS.42. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
Polynomial, Rational, and Radical Relationships Perform arithmetic operations on polynomials.M.A2HS.9. 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.
Represent and solve equations and inequalities graphically.M.A2HS.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
Modeling with Functions Analyze functions using different representations.M.A2HS.31. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Create equations that describe numbers or relationships.M.A2HS.23. Create equations and inequalities in one variable and use them to solve problems.
M.A2HS.24. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.A2HS.25. 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.A2HS.26. 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.) While functions will often be linear, exponential, or quadratic the types of problems
Interpret functions that arise in applications in terms of a context.M.A2HS.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.A2HS.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. Note: Emphasize the selection of a model function based on behavior of data and
WV.M.AMM.Advanced Mathematical Modeling
Advanced Mathematical Modeling
Geometry Concrete geometric representation (physical modeling).M.AMM.36. Solve geometric problems involving inaccessible distances.
Statistics Critiquing statistics.M.AMM.16. Describe strengths and weaknesses of sampling techniques, data and graphical displays and interpretations of summary statistics and other results appearing in a study, including reports published in the media.
Conducting statistical analysis.M.AMM.21. Determine possible sources of variability of data, both those that can be controlled and those that cannot be controlled.
Finance Understanding financial models.M.AMM.5. Determine, represent and analyze mathematical models for investments involving simple and compound interest with and without additional deposits. (e.g., Savings accounts, bonds, and/or certificates of deposit.)
Personal use of finance.M.AMM.8. Research and analyze taxes including payroll, sales, personal property, real estate and income tax returns.
Modeling Modeling data and change with functions.M.AMM.30. Collect numerical bivariate data; use the data to create a scatter plot; determine whether or not a relationship exists; if so, select a function to model the data, justify the selection and use the model to make predictions.
Probability Analyzing information using probability and counting.M.AMM.9. Use the Fundamental Counting Principle, Permutations and Combinations to determine all possible outcomes for an event; determine probability and odds of a simple event; explain the significance of the Law of Large Numbers.
WV.M.GHS.High School Geometry
Applications of Probability Understand independence and conditional probability and use them to interpret data.M.GHS.43. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
M.GHS.45. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For ex
M.GHS.46. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cance
Use the rules of probability to compute probabilities of compound events in a uniform probability model.M.GHS.50. Use permutations and combinations to compute probabilities of compound events and solve problems.
Extending to Three Dimensions Explain volume formulas and use them to solve problems.M.GHS.25. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
M.GHS.26. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Congruence, Proof, and Constructions Prove geometric theorems.M.GHS.9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line
Experiment with transformations in the plane.M.GHS.1. 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.GHS.2. Represent transformations in the plane using, for 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
M.GHS.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
M.GHS.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
M.GHS.5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, for example, 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.GHS.6. 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.
Connecting Algebra and Geometry Through Coordinates Use coordinates to prove simple geometric theorems algebraically.M.GHS.30. Prove the slope criteria for parallel and perpendicular lines and uses 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.GHS.31. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
M.GHS.32. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. This standard provides practice with the distance formula and its connection with the Pythagorean theorem.
Similarity, Proof, and Trigonometry Understand similarity in terms of similarity transformations.M.GHS.14. Verify experimentally the properties of dilations given by a center and a scale factor.M.GHS.14.a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
M.GHS.14.b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
M.GHS.15. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles
Prove theorems involving similarity.M.GHS.18. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
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.
WV.M.SRM.STEM Readiness
Functions and Modeling Building a function that models a relationship between two quantities.M.SRM.20. Write a function that describes a relationship between two quantities.
WV.M.TMS.Transition Mathematics for Seniors
Transition Mathematics for Seniors
Functions – Interpreting Functions Analyze functions using different representations.M.TMS.29. 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.
M.TMS.32. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.M.TMS.32.a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
M.TMS.34. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symm
Understand the concept of a function and use function notation.M.TMS.24. Understand 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 corresQuiz, Flash Cards, Worksheet, Game & Study Guide Functions
Interpret functions that arise in applications in terms of the context.M.TMS.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.TMS.28. Distinguish between situations that can be modeled with linear functions and with exponential functions.Quiz, Flash Cards, Worksheet, Game & Study Guide Functions
Algebra – Creating Equations Create equations that describe numbers or relationships.M.TMS.11. 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.TMS.12. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
M.TMS.13. Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraint
M.TMS.14. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Algebra – Seeing Structure in Expressions Understand the connections between proportional relationship, lines, and linear equations.M.MTS.9. Solve linear equations in one variable.
M.TMS.7. Graph proportional relationships, interpreting the unit rates 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
M.TMS.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 plan; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the ve
Statistics and Probability – Interpreting Categorical and Quantitative Data Summarize, represent, and interpret data on two categorical and quantitative variables.M.TMS.45. 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 d
M.TMS.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 dat
Summarize, represent, and interpret data on a single count or measurement variable.M.TMS.47. Represent data with plots on the real number line (dot plots, histograms, and box plots).
Geometry – Geometric Measuring and Dimension Explain volume formulas and use them to solve problems.M.TMS.38. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
Algebra – Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically.M.TMS.22. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
M.TMS.23. 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
Solve equations and inequalities in one variable.M.TMS.16. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
M.TMS.17. 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.TMS.19. 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.TMS.21. Explain why the x-coordinates of the points where the graphs of the equation y = f(x) and y = g(x) intersect are the solution of the equation f(x) = g (x); find the solution approximately (e.g., using technology to graph the functions, make tables of valu
Algebra – Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials.M.TMS.10. 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.
Geometry – Expressing Geometric Properties with Equations Use coordinates to prove simple geometric theorems algebraically.M.TMS.41. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula).