All West Virginia teachers are responsible for classroom instruction that integrates content standards and mathematical habits of mind. Transition Mathematics for Seniors prepares students for their entry-level credit-bearing liberal studies mathematics course at the post-secondary level. Students will solidify their quantitative literacy by enhancing numeracy and problem solving skills as they investigate and use the fundamental concepts of algebra, geometry, and introductory trigonometry. Mathematical habits of mind, which should be integrated in these content areas, include: making sense of problems and persevering in solving them, reasoning abstractly and quantitatively; constructing viable arguments and critiquing the reasoning of others; modeling with mathematics; using appropriate tools strategically; attending to precision, looking for and making use of structure; and looking for and expressing regularity in repeated reasoning. Students will continue developing mathematical proficiency in a developmentally-appropriate progressions of standards. Continuing the skill progressions from previous courses, the following chart represents the mathematical understandings that will be developed:
Number and Quantity – The Real Number System
M.TMS.1
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
M.TMS.2
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Transition Mathematics for Seniors
M.TMS.1
Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
M.TMS.2
Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Algebra – Seeing Structure in Expressions
M.TMS.4
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2).
Transition Mathematics for Seniors
M.TMS.4
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2) (x2 + y2).
M.TMS.5
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
- Factor a quadratic expression to reveal the zeros of the function it defines.
- Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
M.TMS.6
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
Transition Mathematics for Seniors
M.TMS.5
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
- Factor a quadratic expression to reveal the zeros of the function it defines.
- Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
M.TMS.6
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
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 which of two moving objects has greater speed.
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 vertical axis at b.
M.TMS.9
Solve linear equations in one variable.
Transition Mathematics for Seniors
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 which of two moving objects has greater speed.
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 vertical axis at b.
M.TMS.9
Solve linear equations in one variable.
Algebra – Arithmetic with Polynomials and Rational Expressions
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.
Transition Mathematics for Seniors
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.
Algebra – Creating Equations
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 constraints on combinations of different foods.
M.TMS.14
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Transition Mathematics for Seniors
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 constraints on combinations of different foods.
M.TMS.14
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Algebra – Reasoning with Equations and Inequalities
M.TMS.15
Solve simple rational and radical equations in one variable and give examples showing how extraneous solutions may arise.
Transition Mathematics for Seniors
M.TMS.15
Solve simple rational and radical equations in one variable and give examples showing how extraneous solutions may arise.
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.
M.TMS.18
Solve quadratic equations in one variable. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
Transition Mathematics for Seniors
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.
M.TMS.18
Solve quadratic equations in one variable. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
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.20
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
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 values or find successive approximations).
Transition Mathematics for Seniors
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.20
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
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 values or find successive approximations).
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 half-planes.
Transition Mathematics for Seniors
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 half-planes.
Functions – Interpreting Functions
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 corresponding to the input x. The graph of f is the graph of the equation y = f(x).
Transition Mathematics for Seniors
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 corresponding to the input x. The graph of f is the graph of the equation y = f(x).
M.TMS.25
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
M.TMS.26
Interpret the parameters in a linear or exponential function in terms of a 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: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
M.TMS.28
Distinguish between situations that can be modeled with linear functions and with exponential functions.
Transition Mathematics for Seniors
M.TMS.25
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
M.TMS.26
Interpret the parameters in a linear or exponential function in terms of a 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: intercepts; intervals where the function is increasing, decreasing, positive or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
M.TMS.28
Distinguish between situations that can be modeled with linear functions and with exponential functions.
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.30
Describe qualitatively the functional relationship between two quantities by analyzing a graph.
M.TMS.31
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.
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.
- Graph linear and quadratic functions and show intercepts, maxima, and minima.
- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
M.TMS.33
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasingly linearly, quadratically, or (more generally) as a polynomial function.
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 symmetry of the graph, and interpret these in terms of a context.
M.TMS.35
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Transition Mathematics for Seniors
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.30
Describe qualitatively the functional relationship between two quantities by analyzing a graph.
M.TMS.31
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.
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.
- Graph linear and quadratic functions and show intercepts, maxima, and minima.
- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
M.TMS.33
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasingly linearly, quadratically, or (more generally) as a polynomial function.
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 symmetry of the graph, and interpret these in terms of a context.
M.TMS.35
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
Functions – Building Functions
M.TMS.36
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
M.TMS.37
Write a function that describes a relationship between two quantities.
- Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
- Compose 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, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Transition Mathematics for Seniors
M.TMS.36
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
M.TMS.37
Write a function that describes a relationship between two quantities.
- Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
- Compose 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, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Geometry – Geometric Measuring and Dimension
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.
M.TMS.39
Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
Transition Mathematics for Seniors
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.
M.TMS.39
Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
Geometry – Expressing Geometric Properties with Equations
M.TMS.40
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
M.TMS.41
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula).
Transition Mathematics for Seniors
M.TMS.40
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
M.TMS.41
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, (e.g., using the distance formula).
Geometry – Modeling with Geometry
M.TMS.42
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with topographic grid systems based on ratios).
Transition Mathematics for Seniors
M.TMS.42
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with topographic grid systems based on ratios).
Statistics and Probability – Interpreting Categorical & Quantitative Data
M.TMS.43
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Interpret linear models.
M.TMS.44
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
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 data points to the line.
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 data.
Transition Mathematics for Seniors
M.TMS.43
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Interpret linear models.
M.TMS.44
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
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 data points to the line.
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 data.
M.TMS.47
Represent data with plots on the real number line (dot plots, histograms, and box plots).
M.TMS.48
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
M.TMS.49
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
M.TMS.50
Computer (using technology) and interpret the correlation coefficient of a linear fit.
M.TMS.51
Distinguish between correlation and causation.
Transition Mathematics for Seniors
M.TMS.47
Represent data with plots on the real number line (dot plots, histograms, and box plots).
M.TMS.48
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
M.TMS.49
Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
M.TMS.50
Computer (using technology) and interpret the correlation coefficient of a linear fit.
M.TMS.51
Distinguish between correlation and causation.
Statistics and Probability – Interpreting Categorical & Quantitative Data
M.TMS.52
Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
Transition Mathematics for Seniors
M.TMS.52
Understand statistics as a process for making inferences about population parameters based on a random sample from that population.