Mathematics

Coventry High Schools Mathematics Program provides a rigorous standards based curriculum in a coherent program with a focus on developing students’ conceptual understanding, procedural skills and fluency, and application of mathematical knowledge to a variety of situations.

Courses

Algebra I

In Algebra 1, student learning focuses on five critical areas: (1) seeing structure in expressions to interpret and write expressions in equivalent forms to solve problems; (2) creating equations that describe numbers or relationships; (3) reasoning with equations and inequalities to solve equations, inequalities, and systems of equations; (4) interpreting and analyzing functions in and out of context using different representations; and (5) reasoning and using the properties of exponents, rational, and irrational numbers to solve problems.

Units

Linear Equations and Inequalities

Students interpret expressions and parts of expressions that represent a quantity in terms of its content.
Students create equations and inequalities in one variable and use them to solve problems, including those rising from linear functions.
Students 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.

Functions

Students create equations in two or more variables to represent relationships between quantities and graph equations on coordinate axes with labels and scales
Students use function notation; and evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
For a function that models a relationship between two quantities, students 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.

Linear Functions

Scatter Plots and Trend Lines

Students use statistics appropriate to the shape of the data distribution to compare center (mean, median) and spread (interquartile range, standard deviation).
Students interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
Students fit a function to the data and use functions fitted to the data to solve problems in the context of the data
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Systems of Linear Equations

Students solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
Students 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 functions.

Linear Inequalities and Systems of Linear Inequalities

Students understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Students 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.

Introduction to Exponential Functions

Students rewrite expressions involving radicals and rational exponents using the properties of exponents.
Students write geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
Students prove that exponential functions grow by equal factors over equal intervals.
Students construct exponential functions, given a graph, a description of a relationship, or two input-output pairs (including reading these from a table).
Students interpret the parameters in an exponential function in terms of a context.

Introduction to Quadratic Functions and Equations

Students 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.
Students graph quadratic functions and show intercepts, maxima, and minima.
Students 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.
Students identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative), and find the value of k given the graphs.

Geometry

In Geometry, student learning focuses on five critical areas: (1) experimenting with transformations on the coordinate plane to understand and prove theorems on congruence and similarity; (2) modeling situations with geometry; (3) defining and applying trigonometric ratios and Pythagorean Theorem to solve problems involving triangles (4) explaining volume formulas by visualizing the relationship between two-dimensional and three-dimensional objects and using those formulas to solve problems; and (5) understanding and applying theorems about circles to find arc lengths and area of sectors in circles.

Units

Pythagorean Theorem and Radicals

Building Blocks of Geometry

Polygons and Properties of Triangles

Students prove theorems about triangles.
Students use coordinates to prove simple geometric theorems algebraically.
Students 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).
Students use geometric shapes, their measures and their properties to describe objects (e.g., modeling a ladder against a building as a triangle).

Congruent Triangles

Similarity

Trigonometry

Properties of Quadrilaterals

Area and Volume

Students use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Students give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone (dissection arguments, Cavalieri's principle, and informal limit arguments).
Students use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk as a cylinder).

Circles and Other Conics

Transformations

Students develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Students represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs; and compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Given a geometric figure and a rotation, reflection, or translation, students will draw the transformed figure and/or specify a sequence of transformations that will carry a given figure onto another.

Algebra II

In Algebra II, student learning focuses on five critical areas: (1) summarizing, representing, and interpreting data on one or two variables; (2) building functions that model a relationship between two quantities or stem form an existing function; (3) constructing, interpreting, and comparing linear and exponential models; (4) using polynomial identities to rewrite or perform arithmetic operations on polynomials; and (5) using the unit circle to extend the domain of trigonometric functions and model phenomena with trigonometric functions.

Units

Probability and Statistics

Students represent data with plots on the real number line (dot plots, histograms, and box plots).
Students 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.
Students represent data on two quantitative variables on a scatter plot and describe how the variables are related (ex: fitting a function to the data to solve problems).
Students compute (using technology) and interpret the correlation coefficient of a linear fit.

Linear Functions, Scatter Plots, and Systems

Function Notation and Transformations

Students graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.; including square root, cube root, and piecewise-defined functions, step functions and absolute value functions.
Students write a function that describes a relationship between two quantities.
Students find inverse functions.

Quadratic Functions

For a function that models a relationship between two quantities, students interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features (intercepts, relative maxima and minima, symmetries, and interval descriptions) given a verbal description of the relationship.
Students graph quadratic functions and show intercepts, maxima, and minima.
Students 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.
Students identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

Exponential and Logarithmic Functions

Students write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Students use the properties of exponents to interpret expressions for exponential functions.
Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Polynomial Functions

Trigonometric Functions

Students understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Students explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Students use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

AP Statistics/ECE Statistics

In AP Statistics/ ECE Statistics, student learning focuses on four critical areas: (1) summarizing, representing, and interpreting data on one or two variables; (2) making inferences and justifying conclusions from sample surveys, experiments, and observational studies; (3) using independence and conditional probability to interpret data; and (4) using probability to calculate expected values and evaluate outcomes of decisions.

Units

Exploring Data

Students represent data with plots on the real number line (dot plots, histograms, and box plots).
Students 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.
Students interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Density Curves and Normal Distributions

Linear Regression and Two Variable Data

Students 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.
Students represent data on two quantitative variables on a scatter plot and describe how the variables are related.
Students interpret the slope (rate of change) and the intercept (constant term) of a linear fit in the context of the data.
Students compute (using technology) and interpret the correlation coefficient of a linear fit.
Students distinguish between correlation and causation.

Probability

Producing Data

Students understand that statistics is a process for making inferences about population parameters based on a random sample from that population.
Students decide if a specified model is consistent with results from a given data-generating process, e.g. using simulation.
Students recognize the purposes of and differences among sample surveys, experiments and observational studies; explain how randomization relates to each.
Students use data from a randomized experiment to compare two treatments; justify significant differences between parameters through the use of simulation models for random assignment.

Random Variables, Binomial, and Geometric

Students define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
Students calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
Students develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
Students develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
Students weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Sample Distributions

Students interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
Students understand that statistics is a process for making inferences about population parameters based on a random sample from that population.
Students use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Inference - 1 or 2 Sample Confidence & Hypothesis Tests

Chi Square Distribution and Tests

Students 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.
Students understand that statistics is a process for making inferences about population parameters based on a random sample from that population.
Students use probabilities to make fair decisions.
Students analyze decisions and strategies using probability concepts.

Inference for Regression

Students represent data on two quantitative variables on a scatter plot, and describe how the variables are related; including fitting a function to the data to solve problems in the context of the data.
Students interpret the slope (rate of change) and the intercept (constant term) of a linear fit in the context of the data.
Students use probabilities to make fair decisions.
Students analyze decisions and strategies using probability concepts.

Probability and Statistics

In Probability and Statistics, student learning focuses on four critical areas: (1) summarizing, representing, and interpreting data on one or two variables; (2) making inferences and justifying conclusions from sample surveys, experiments, and observational studies; (3) using independence and conditional probability to interpret data; and (4) using probability to calculate expected values and evaluate outcomes of decisions.

Units

Statistics Basics

Graph Types: Categorical and Quantitative

Scatterplots and Regression

Students summarize, represent, and interpret data on two categorical and quantitative variables.
Students represent data on two quantitative variables on a scatter plot and describe how the variables are related.
Students interpret the slope (rate of change) and the intercept (constant term) of a linear fit in the context of the data.
Students distinguish between correlation and causation.

Probability

Confidence and Hypothesis Tests

Google Sheets for Statistics

Students use spreadsheets to calculate, graph, organize, and present data in a variety of real-world settings and choose the most appropriate type to represent given data.
Students enter formulas and functions; and use the auto-fill feature in a spreadsheet application
Students use functions (sort, filter, find, etc.), various number formats (percentages, exponentes, etc.) and formatting (repositions rows/columns, naming, etc.) of a spreadsheet application

Pre-Calculus

In Pre-Calculus, student learning focuses on five critical areas: (1) defining trigonometric ratios to generate triangles and solve problems involving right triangles; (2) extending the domain of trigonometric functions using the unit circle; (3) modeling periodic phenomena is trigonometric functions; (4) proving and applying trigonometric functions; and (5) analyzing functions using different representations.

Units

Angles, Right Triangle Trigonometry, Unit Circles and Any Circle

Students understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
Students define trigonometric ratios and solve problems involving right triangles.
Students explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Students use special triangles to determine geometrically the values of sine, cosine, tangent for p/3, p/4 and p/6, and use the unit circle to express the values of sine, cosines, and tangent for x, p + x, and 2p – x in terms of their values for x, where x is any real number.
Students use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Oblique Triangles

Students apply trigonometry to general triangles.
Students derive the formula A = ½ ab sin© for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
Students prove the Laws of Sines and Cosines and use them to solve problems.
Students understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

Graphs of Trigonometric Functions and Inverse Functions

Students explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Students choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.
Students understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
Students use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
Students read values of an inverse function from a graph or a table, given that the function has an inverse.

Using Formulas and Verifying Trigonometric Identities

Students use the structure of an expression to identify ways to rewrite it.
Students prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.
Students rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Students prove the Pythagorean identity sin²(?) + cos²(?) = 1 and use it to calculate trigonometric ratios.

Solving Trigonometric Equations

Families of Functions

Students graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Students write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Students compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)
Students 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.

Limits

Students relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
Students graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Students write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Students find inverse functions.

AP Calculus/ECE Calculus

In AP Calculus/ ECE Calculus, student learning focuses on three critical areas: (1) evaluating limits to find the value of a function as it gets closer to a certain point; (2) building on and applying the concept of derivative to describe function behaviors; and (3) evaluating integrals to determine the area under a function when it is graphed and applying it to the volume of shapes on a three-dimensional coordinate plane.

Units

Limits and Continuity

Students relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.
Students graph functions (linear, quadratic, polynomial, exponential, logarithmic, etc.) expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Students write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Students find inverse functions.

The Derivative

Students build on the concepts of average rate of change of a function using slopes of secant lines, and instantaneous rate of change of functions using slopes of tangent lines.
Students develop the concepts of derivative, general derivative, and its connection to the instantaneous rate of change of a function.
Students determine the general derivative of polynomial, rational and root functions using the Limit Definition of Derivative, and interpret the meaning of higher order derivatives.
Students apply the concept of derivative to Position, Velocity and Acceleration problems.
Students develop the rules for differentiating different types of functions through considering patterns in the structure.
Students determine the derivative of a function expressed in both x and y implicitly.
Students apply the tangent line to estimate functional values using Linear Approximation and estimate the zeros of a function (Newton’s Method).

Applications of Derivatives

Students apply the derivative of an exponential function to derive the general form of an exponential growth and decay model.
Students solve a variety of growth and decay problems in various contexts.
Students develop tools for describing functional behavior.
Students apply the derivative to evaluate when a function's instantaneous rate of change is equal to the average rate of change over an interval and find extremes of a function to solve a variety of problems in multiple contexts.
Students develop tools for evaluating Limits that take on an indeterminate form (0/0, infinity/infinity or 1^infinity) using derivatives.

Antiderivatives and Integration

Students develop a connection between the derivative and the antiderivative of a function and apply the structure and patterns of derivatives of polynomial functions to develop antiderivatives for these functions.
Students develop the integral as the process of determining the antiderivative of a function and compare the Indefinite Integral and the Definite Integral in context of their uses.
Students develop a connection between the integral of a function and the area between the function and an axis.
Students apply estimation techniques to approximate integral values and compare the integral value as a measure of signed or total area
Students develop both parts of the Fundamental Theorem of Calculus and apply them in context.
Students apply u-substitution to indefinite and definite integrals.
Students evaluate the impact of the Chain Rule to derivatives, and develop the u-Substitution method to find the antiderivative of such functions.
Students apply integration techniques to measure the area between curves.

Applications of Integration

Students apply integration and the concept of area between curves to find the volume of various solids not measurable through traditional geometric methods.
Students develop tools for measuring volume of solids of revolution and apply the volume techniques to a variety of problems including solids revolved about an axis parallel to the x or y axis.
Students apply the Fundamental Theorem of Calculus and concepts of Implicit Differentiation to solve separable Differential Equations, with and without initial conditions.
Students create, describe and evaluate slope fields as a map of a differential equation and model functions given an initial condition.
Students apply integration and u-Substitution to evaluate integrals in the form of Inverse Trigonometric Function Derivatives.

Additional Techniques of Integration

Life Skills Math

In Life Skills Math, student learning focuses on four critical areas: (1) exploring and calculating the different ways to earn gross pay; (2) analyzing what can get taken out of a paycheck resulting in net pay; (3) examining the different vocabulary, forms, and tables used to file taxes; and (4) evaluating the different types of student and home loans to determine the best loan to take out in each situation.

Units

Gross Pay

Students create equations and inequalities in one variable and use them to solve problems; including equations arising from linear and quadratic functions, and simple rational and exponential functions.
Students interpret expressions that represent a quantity in terms of its context.
Students investigate different ways to earn gross pay (hourly vs. salary pay, weekly vs. overtime, commission pay, etc.).
Students calculate gross pay in different situations.

Net Pay, Health Insurance & Life Insurance

Students 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.
Students explore different deductions from gross pay resulting in net pay (ie: taxes, medicare, social security, retirement, insurance, etc).
Students calculate net pay based on different types of deductions.

Taxes

Students 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.
Students develop the concept of taxes and the terms that go along with it such as: independent vs. dependent, itemized vs standard deductions, and taxable income.
Students read, comprehend, and fill out portions of tax forms.

Types of Interest and Car Loans

Student Loans and Housing

Students use units as a way to understand problems and to guide the solution of multi-step problems.
Students investigate different types of student and housing loans (i.e. subsidized vs. unsubsidized, consolidation, etc.).
Students explore the loan application process for student loans and housing loans (FAFSA, bank, or other lender).
Students use the concept of interest to calculate the payments for the loan, the portion of the loan that comes from principal or interest, and the total amount paid over the life of the loan.

Departments

English

Mathematics

Science

Social Studies

World Language

Business

Art

Technology Education

Music

Physical Education

Health

Family & Consumer Sciences