Course Overview
A - Algebra
• interpret parts of an expression such as terms, factors, and coefficients, in context • interpret expressions that represent a quantity in terms of its context • interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors • create equations and inequalities in one variable and use them to solve problems; (Include equations arising from linear, quadratic, and exponential functions - integer inputs only) • create linear or exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. • justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers. Justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties. • represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as possible (a solution) or not possible (non-solution) under the established constraints • 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; rearrange the formula for the area of a circle to highlight the radius) • solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents) • 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 (elimination method) • demonstrate that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane • solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables • 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 values, or find successive approximations. Include cases where f(x) and/or g(x) are linear and exponential functions) • solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions) • 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 ax² + bx + c = 0 • 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 • add, subtract, and multiply polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations • complete the square in a quadratic expression to reveal the maximum and minimum value of the function defined by the expression • factor any quadratic expression to reveal the zeroes of the function it defines • choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression • use the structure of an expression to rewrite it in different equivalent forms.
B - Statistics and Probability
• 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 • use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range) of two or more different data sets • represent data with plots on the real number line (including dot plots, histograms, and box plots) • compute (using technology) and interpret the correlation coefficient of a linear fit. For instance, by looking at a scatterplot, students should be able to tell if the correlation coefficient is positive or negative and give a reasonable estimate of the value. After calculating the line of best fit using technology, students should be able to describe how strong the goodness of fit of the regression is, using "r" • distinguish between correlation and causation • represent data on two quantitative variables on a scatter plot and describe how the variables are related • determine and interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data • fit a linear function for a scatter plot that suggests a linear association using given or collected bivariate data • interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers) • fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models
C - Geometry
• represent transformations in the plane using 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 angle to those that do not (e.g., translation versus horizontal stretch) • verify experimentally the properties of dilations given by a center and a scale factor • recognize that 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 • recognize that the dilation of a line segment is longer or shorter in the ratio given by the scale factor • 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 and the proportionality of all corresponding pairs of sides • apply the properties of similarity transformations to establish the AA criterion for two triangles to be similar • prove and apply theorems about triangles including a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity • apply congruence and similarity criteria to solve problems and prove relationships in geometric figures • demonstrate that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles • explain and apply relationships between the sine and cosine of complementary angles • use 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 • employ properties of rectangles, parallelograms, trapezoids, and regular polygons to describe rotations and reflections that map a polygon onto itself • explain, apply and experimentally verify definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments • 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 • solve application problems using the trigonometric ratios and the Pythagorean Theorem • given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using graph paper, tracing paper, or geometry software; specify a sequence of transformations that will carry a given figure onto another • use coordinates to prove simple geometric theorems algebraically • 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 segment are exactly those equidistant from the segment's endpoints • explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions • prove the slope criteria for parallel and perpendicular lines and 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) • prove theorems about parallelograms (Theorems include opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals) • prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point • use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent • make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line • construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle
D - Functions
• 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 n1 (n is greater than or equal to 1); draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences) • relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function) (Focus on linear and exponential functions.) • interpret key features of graphs and tables for a function that models a relationship between two quantities in terms of the quantities for a function that models a relationship between two 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) • understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range; each input value maps to exactly one output value. [(e.g., 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).)] • relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function) • interpret the parameters in a linear or exponential function in terms of a context. In context, students should describe what these parameters mean in terms of change and starting value. • show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function • 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 • recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another • graph functions expressed algebraically and show key features of the graph both by hand and by using technology • prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. This can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals. • write a function that describes a relationship between two quantities • compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) (e.g., given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum) • determine an explicit expression, a recursive process, or steps for calculation from a context • write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions • distinguish between situations that can be modeled with linear functions and with exponential functions • graph exponential functions showing intercepts and end behavior • 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) • recognize situations in which one quantity changes at a constant rate per unit interval relative to another • graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context) • 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; experiment with cases and illustrate an explanation of the effects on the graph using technology (Include recognizing even and odd functions from their graphs and algebraic expressions for them. • 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 context. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms. • write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function
E - Numbers and Quantity
• use the 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 • use units of measure (linear, area, capacity, rates, and time) as a way to understand problems. identify, use, and record appropriate units of measure within context, within data displays, and on graphs. Convert units and rates using dimensional analysis (English to English and Metric to Metric without conversion factor provided and between English and Metric with conversion factor). Use units within multistep problems and formulas; interpret units of input and resulting units of output • choose a level of accuracy appropriate to limitations on measurement when reporting quantities. For example, money situations are generally reported to the nearest cent (hundredth). Also, an answer's precision is limited to the precision of the data given. • determine appropriate quantities for the purpose of descriptive modeling. Given a situation, context, or problem, students will determine, identify, and use appropriate quantities for representing the situation. • Rewrite expressions involving radicals and rational exponents using the properties of exponents • Explain why the sum or the product of rational numbers is rational and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational
A - Algebra
• interpret parts of an expression such as terms, factors, and coefficients, in context • interpret expressions that represent a quantity in terms of its context • interpret the meaning of given formulas or expressions in context of individual terms or factors when given in situations which utilize the formulas or expressions with multiple terms and/or factors • create equations and inequalities in one variable and use them to solve problems; (Include equations arising from linear, quadratic, and exponential functions - integer inputs only) • create linear or exponential equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. • justify the steps of a simple one-solution equation using algebraic properties and the properties of real numbers. Justify each step, or if given two or more steps of an equation, explain the progression from one step to the next using properties. • represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret data points as possible (a solution) or not possible (non-solution) under the established constraints • 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; rearrange the formula for the area of a circle to highlight the radius) • solve linear equations and inequalities in one variable, including equations with coefficients represented by letters (Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential equations that rely only on application of the laws of exponents) • 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 (elimination method) • demonstrate that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane • solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables • 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 values, or find successive approximations. Include cases where f(x) and/or g(x) are linear and exponential functions) • solve quadratic equations by inspection, taking square roots, factoring, completing the square, and the quadratic formula, as appropriate to the initial form of the equation (limit to real number solutions) • 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 ax² + bx + c = 0 • 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 • add, subtract, and multiply polynomials; understand that polynomials form a system analogous to the integers in that they are closed under these operations • complete the square in a quadratic expression to reveal the maximum and minimum value of the function defined by the expression • factor any quadratic expression to reveal the zeroes of the function it defines • choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression • use the structure of an expression to rewrite it in different equivalent forms.
B - Statistics and Probability
• 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 • use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range) of two or more different data sets • represent data with plots on the real number line (including dot plots, histograms, and box plots) • compute (using technology) and interpret the correlation coefficient of a linear fit. For instance, by looking at a scatterplot, students should be able to tell if the correlation coefficient is positive or negative and give a reasonable estimate of the value. After calculating the line of best fit using technology, students should be able to describe how strong the goodness of fit of the regression is, using "r" • distinguish between correlation and causation • represent data on two quantitative variables on a scatter plot and describe how the variables are related • determine and interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data • fit a linear function for a scatter plot that suggests a linear association using given or collected bivariate data • interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers) • fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models
C - Geometry
• represent transformations in the plane using 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 angle to those that do not (e.g., translation versus horizontal stretch) • verify experimentally the properties of dilations given by a center and a scale factor • recognize that 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 • recognize that the dilation of a line segment is longer or shorter in the ratio given by the scale factor • 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 and the proportionality of all corresponding pairs of sides • apply the properties of similarity transformations to establish the AA criterion for two triangles to be similar • prove and apply theorems about triangles including a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity • apply congruence and similarity criteria to solve problems and prove relationships in geometric figures • demonstrate that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles • explain and apply relationships between the sine and cosine of complementary angles • use 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 • employ properties of rectangles, parallelograms, trapezoids, and regular polygons to describe rotations and reflections that map a polygon onto itself • explain, apply and experimentally verify definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments • 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 • solve application problems using the trigonometric ratios and the Pythagorean Theorem • given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using graph paper, tracing paper, or geometry software; specify a sequence of transformations that will carry a given figure onto another • use coordinates to prove simple geometric theorems algebraically • 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 segment are exactly those equidistant from the segment's endpoints • explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions • prove the slope criteria for parallel and perpendicular lines and 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) • prove theorems about parallelograms (Theorems include opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals) • prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180 degrees; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point • use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent • make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line • construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle
D - Functions
• 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 n1 (n is greater than or equal to 1); draw connection to F.BF.2, which requires students to write arithmetic and geometric sequences) • relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function) (Focus on linear and exponential functions.) • interpret key features of graphs and tables for a function that models a relationship between two quantities in terms of the quantities for a function that models a relationship between two 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) • understand that a function from one set (the input, called the domain) to another set (the output, called the range) assigns to each element of the domain exactly one element of the range; each input value maps to exactly one output value. [(e.g., 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).)] • relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes (e.g., if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function) • interpret the parameters in a linear or exponential function in terms of a context. In context, students should describe what these parameters mean in terms of change and starting value. • show using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or more generally as a polynomial function • 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 • recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another • graph functions expressed algebraically and show key features of the graph both by hand and by using technology • prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. This can be shown by algebraic proof, with a table showing differences, or by calculating average rates of change over equal intervals. • write a function that describes a relationship between two quantities • compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) (e.g., given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum) • determine an explicit expression, a recursive process, or steps for calculation from a context • write arithmetic and geometric sequences both recursively and with an explicit formula; use them to model situations, and translate between the two forms. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions • distinguish between situations that can be modeled with linear functions and with exponential functions • graph exponential functions showing intercepts and end behavior • 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) • recognize situations in which one quantity changes at a constant rate per unit interval relative to another • graph linear and quadratic functions and show intercepts, maxima, and minima (as determined by the function or by context) • 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; experiment with cases and illustrate an explanation of the effects on the graph using technology (Include recognizing even and odd functions from their graphs and algebraic expressions for them. • 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 context. For example, compare and contrast quadratic functions in standard, vertex, and intercept forms. • write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function
E - Numbers and Quantity
• use the 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 • use units of measure (linear, area, capacity, rates, and time) as a way to understand problems. identify, use, and record appropriate units of measure within context, within data displays, and on graphs. Convert units and rates using dimensional analysis (English to English and Metric to Metric without conversion factor provided and between English and Metric with conversion factor). Use units within multistep problems and formulas; interpret units of input and resulting units of output • choose a level of accuracy appropriate to limitations on measurement when reporting quantities. For example, money situations are generally reported to the nearest cent (hundredth). Also, an answer's precision is limited to the precision of the data given. • determine appropriate quantities for the purpose of descriptive modeling. Given a situation, context, or problem, students will determine, identify, and use appropriate quantities for representing the situation. • Rewrite expressions involving radicals and rational exponents using the properties of exponents • Explain why the sum or the product of rational numbers is rational and an irrational number is irrational; and why the product of a nonzero rational number and an irrational number is irrational
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