HANOVER SCHOOLS

Hanover High School Curriculum - Precalculus The Standards in this document are from the Massachusetts Department of Education: Mathematics Curriculum Framework - Learning Standards for Data Analysis, Statistics, and Probability Note: The parentheses at the end of a learning standard contain the codes for the corresponding standards in the two-year grade spans. Document Key:

M = Mastery; These strands will be formally assessed each grading term using Precalculus Common Assessment Opportunities. D = Developing; The strands are introduced and explored in Precalculus to be continued in Calculus. E = Enrichment; These strands are introduced and explored; however, assessed in the Honors level classes only. R = Review; These topics were addressed in a prior course and will be reviewed in Precalculus.

At Hanover High School, Precalculus students: Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them Select and use appropriate statistical methods to analyze data Develop and evaluate inferences and predictions that are based on data Understand and apply basic concepts of probability Students engage in problem solving, communicating, reasoning, connecting, and representing as they: Standard

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PC.D.1

Design surveys and apply random sampling techniques to avoid bias in the data collection. (12.D.1)

PC.D.1 PC.D.1 PC.D.1 PC.D.1

Understand the integrated processes of designing and conducting quantitative survey research projects Use survey methods: face-to-face interviewing, direct observation, postal and telephone surveys Design surveys and construct data collection instruments Determine sources of error in the survey process and ways of detecting, controlling and minimizing error

PC.D.2

Apply regression results and curve fitting to make predictions from data. (12.D.3)

PC.D.2 PC.D.2 PC.D.2 PC.D.2

Identify appropriate basic functions with which to model real-world problems Produce specific functions to model data, formulas, graphs, and verbal descriptions Use data to find regression equations to model linear, quadratic, power, and other polynomial functions Evaluate exponential expressions and identify and graph exponential and logistic functions Page 1 of 7

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HANOVER SCHOOLS PC.D.2 PC.D.2 PC.D.2 PC.D.2

Use regression to model exponential growth and decay problems Make predictions from data and from equations in any curve fitting situation Use regression to generate equations and graphs of sinusoidal data Make predictions from data and from equations which model sinusoidal behavior

PC.D.3

Apply uniform, normal, and binomial distributions to the solutions of problems. (12.D.4)

PC.D.3 PC.D.3 PC.D.3 PC.D.3 PC.D.3

Distinguish between categorical and quantitative variables Use graphs to display data, including stemplots, histograms, frequency tables, and time plots Use measures of central tendency, the five-number summary, and boxplots to describe data Use standard deviation and normal distribution to describe and analyze data Understand and apply the 68-95-99.7 Rule

PC.D.4

Describe a set of frequency distribution data by spread (variance and standard deviation), skewness, symmetry, number of modes, or other characteristics. Use these concepts in everyday applications. (12.D.5)

PC.D.4 PC.D.4 PC.D.4 PC.D.4

Define mean, median, mode Define standard deviation and variance Calculate standard deviation and variance Describe frequency distribution data by its characteristics

PC.D.5

Compare the results of simulations (e.g., random number tables, random functions, and area models) with predicted probabilities. (12.D.7)

PC.D.5 PC.D.5 PC.D.5 PC.D.5

Identify sample space and calculate probabilities and conditional probabilities in various sample spaces Apply the Binomial Distribution Theorem to determine specific probabilities Define and determine expected value to determine if a game is fair Examine the results of simulations and compare with predicted probabilities

PC.G.1

Demonstrate an understanding of the laws of sines and cosines. Use the laws to solve for the unknown sides or angles in triangles. Determine the area of a triangle given the length of two adjacent sides and the measure of the included angle. (12.G.2)

PC.G.1 PC.G.1 PC.G.1 PC.G.1

Understand the proof of the Law of Sines Use the computational applications of the Law of Sines to solve a variety of problems Apply the Law of Cosines to solve acute and obtuse triangles (SSS, SAS) Determine the area of triangle in terms of the measures of the sides and angles

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PC.G.1 PC.G.1 PC.G.1

Apply the Law of Sines to solve acute and obtuse triangles (AAS, ASA) Apply Heron's Formula to find the area of a triangle Apply Law of Sines and Law of Cosines to determine solutions to real-world problems

PC.G.2

Use the notion of vectors to solve problems. Describe addition of vectors, multiplication of a vector by a scalar, and the dot product of two vectors, both symbolically and geometrically. Use vector methods to obtain geometric results. (12.G.3)

PC.G.2 PC.G.2 PC.G.2 PC.G.2 PC.G.2 PC.G.2

Define standard representation, magnitude, direction angle, resultant, unit vector Perform basic operations on vectors both symbolically (algebraically) and geometrically (pictorially) Use vectors to model a variety of situations Define scalar multiplication, dot product, cross product Determine the angle between vectors Understand the projection of one vector onto another

PC.G.3

Apply properties of angles, parallel lines, arcs, radii, chords, tangents, and secants to solve problems. (12.G.5)

PC.G.3 PC.G.3 PC.G.3 PC.G.3

Find arc lengths, radii, chords, and angles of circles Find the area and perimeter of a sector of a circular Create a geometric connection between secant lines and average rate of change Create a geometric connection between tangent lines and instantaneous rate of change

PC.M.1

Describe the relationship between degree and radian measures, and use radian measure in the solution of problems, in particular problems involving angular velocity and acceleration. (12.M.1)

PC.M.1 PC.M.1 PC.M.1 PC.M.1 PC.M.1 PC.M.1

Evaluate trigonometric expressions using radian measure Convert between radians and degrees Convert to nautical miles Relate angular speed to linear speed Solve problems involving angular speed and linear speed of an object moving on a circular path Solve problems involving angular acceleration of an object moving on a circular path

PC.M.2

Use dimensional analysis for unit conversion and to confirm that expressions and equations make sense. (12.M.2)

PC.M.2 PC.M.2 PC.M.2 PC.M.2 PC.M.2 PC.M.2

Convert time units for exponential and logistic modelling Verify the units an angle measure are correct based upon solution's magnitude Use dimensional analysis for unit conversion for linear measurements Use dimensional analysis for unit conversion for angular measurements Convert units for sinusoidal modelling Determine the physical domain and range of all application problems

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HANOVER SCHOOLS PC.N.1

Plot complex numbers using both rectangular and polar coordinates systems. Represent complex numbers using polar coordinates, i.e., a + bi = r(cosθ + isinθ). Apply DeMoivre's theorem to multiply, take roots, and raise complex numbers to a power.

PC.N.1 PC.N.1 PC.N.1 PC.N.1 PC.N.1 PC.N.1 PC.N.1 PC.N.1 PC.N.1

Write equivalent rectangular and polar forms of points on the coordinate plane Determine multiple polar form representations of a point Identify the pole and the polar axis, and plot points given in polar form Write equivalent forms for rectangular and polar equations Graph polar equations and describe their properties Write equivalent rectangular and polar forms for complex numbers Represent complex numbers in polar form numerically and graphically Determine the product or quotient of two complex numbers in polar form Determine a power or the roots of a complex number using DeMoivre’s theorem

PC.P.1

Use mathematical induction to prove theorems and verify summation formulas.

PC.P.1 PC.P.1 PC.P.1

Define: induction and deduction, the anchor, inductive hypothesis, inductive step Use the principal of mathematical induction to prove mathematical generalizations. Prove summation formulas for sum of integers, squares, and cubes

PC.P.2

Relate the number of roots of a polynomial to its degree. Solve quadratic equations with complex coefficients.

PC.P.2 PC.P.2 PC.P.2 PC.P.2 PC.P.2 PC.P.2 PC.P.2 PC.P.2 PC.P.2 PC.P.2 PC.P.2 PC.P.2

Recognize and graph linear and quadratic functions Use linear and quadratic functions to model situations and solve problems Recognize and graph power functions Use power functions to model situations and solve problems Recognize and graph polynomial functions, predict end behavior, locate zeros and asymptotes Be able to use long division, synthetic division to divide polynomials Apply the Remainder, Factor, and Rational Zeros Theorems, and find upper and lower bounds for zeros Factor polynomials with real coefficients using factors with complex coefficients Describe graphs of rational functions, identify asymptotes, and predict end behavior Solve equations involving fractions using both algebraic and graphical techniques Identify extraneous solutions Solve inequalities involving fractions using both algebraic and graphical techniques

PC.P.3

Demonstrate an understanding of the trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent). Relate the functions to their geometric definitions.

PC.P.3 PC.P.3 PC.P.3 PC.P.3

Define the six trigonometric functions using the lengths of the sides of a right triangle Solve problems involving the trigonometric functions of real numbers Solve problems involving the properties of sine and cosine as periodic functions Relate the concept of inverse functions to trigonometric functions Page 4 of 7

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PC.P.3 PC.P.3

Define Inverse Sine, Cosine, and Tangnet functions Interpret and solve problems involving trigonometric functions

PC.P.4

Explain the identity sin^2θ + sin^2θ = 1. Relate the identity to the Pythagorean theorem.

PC.P.4 PC.P.4 PC.P.4 PC.P.4

Use the fundamental identities to simplify trigonometric expressions Use the fundamental identities to solve trigonometric equations Determine whether an equation is an identity and confirm identities analytically Be able to disprove a non-identity

PC.P.5

Demonstrate an understanding of the formulas for the sine and cosine of the sum or the difference of two angles. Relate the formulas to DeMoivre's theorem and use them to prove other trigonometric identities. Apply to the solution of problems.

PC.P.5 PC.P.5

Apply the identities for the cosine, sine, and tangent of a difference or a sum Apply the double-angle identities, power-reducing identities, and half-angle identities

PC.P.6

Understand, predict, and interpret the effects of the parameters a, ω , b, and c on the graph of y = asin(ω(x - b)) + c; similarly for the cosine and tangent. Use to model periodic processes. (12.P.13)

PC.P.6 PC.P.6 PC.P.6 PC.P.6 PC.P.6 PC.P.6 PC.P.6

Generate graphs of sine, cosine, and tangent functions Explore transformations of the graphs of sine, cosine, and tangent functions Define sinusoid, amplitude, vertical stretch/shrink, period, frequency, and phase shift Model periodic behavior with sinusoids Generate graphs of cosecant, secant, and cotangent functions Explore transformations of the graphs of cosecant, secant, and cotangent functions Graph sums, differences, and other combinations of trigonometric and algebraic functions

PC.P.7

Translate between geometric, algebraic, and parametric representations of curves. Apply to the solution of problems.

PC.P.7 PC.P.7 PC.P.7 PC.P.7 PC.P.7 PC.P.7

Define parameter, parametric curve, and parametric equations Graph curves parametrically Solve application problems using parametric equations Translate a parametic equation into a cartesian equation Translate a cartesian equation into a parametic equation Model motion using parametric equations

PC.P.8

Identify and discuss features of conic sections: axes, foci, asymptotes, and tangents. Convert between different algebraic representations of conic sections.

PC.P.8 PC.P.8 PC.P.8

Define each conic section as a cross section of a cone Define each conic section as a set of points Find the equation, focus, and directrix of a parabola

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PC.P.8 PC.P.8 PC.P.8 PC.P.8

Find the equation, verticies, and foci of an ellipse Find the equation, verticies, foci, and asymptotes of a hyperbola Determine equations for translated and rotated axes for conic sections Understand the general focus-directrix definition of a conic section and write equations in polar form

PC.P.9

Relate the slope of a tangent line at a specific point on a curve to the instantaneous rate of change. Explain the significance of a horizontal tangent line. Apply these concepts to the solution of problems.

PC.P.9 PC.P.9 PC.P.9 PC.P.9 PC.P.9

Define limit, instantaneous velocity, average velocity Calculate instantaneous velocities and derivatives using limits Use secant lines to find average rate of change Use tangent lines to instantaneous rate of change Define average rate of change, derivative at a point, derivative of a function

AII.P.1

Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns such as Pascal’s Triangle. (12.P.1)

AII.P.1 AII.P.1 AII.P.1 AII.P.1 AII.P.1 AII.P.1 AII.P.1 AII.P.1

Define: iterative, recursive, term Generate Pascal's Triangle Relate Pascal's Triangle to combinations Analyze patterns Describe a pattern in words Extend a pattern a certain number of terms Complete patterns Generalize patterns by creating a formula to represent the pattern

AII.P.2

Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the formula for the general term and the sum, recursively and explicitly.(12.P.2)

AII.P.2 AII.P.2 AII.P.2 AII.P.2 AII.P.2 AII.P.2 AII.P.2 AII.P.2 AII.P.2 AII.P.2 AII.P.2 AII.P.2

Define: sequence, series, arithmetic, geometric, finite Contrast: arithmetic vs. geometric, sequence vs. series, recursive vs. explicit Given a sequence determine if it is arithmetic, geometric or neither Create a recursive formula for an arithmetic sequence Create an explicit formula for an arithmetic sequence Create a recursive formula for a geometric sequence Create an explicit formula for a geometric sequence Calculate the nth term of any sequence Create a formula to calculate an arithmetic series Create a formula to calculate a geometric series Define Sigma notation Use Sigma notation to represent a series Page 6 of 7

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HANOVER SCHOOLS AII.P.2

Model real world situations using arithmetic and geometric series and sequences

AII.P.3

Demonstrate an understanding of the binomial theorem and use it in the solution of problems. (12.P.3)

AII.P.3 AII.P.3

Expand binomials using Pascal's Triangle (with varied coefficients, signed numbers, fractions, exponents) Expand binomials using Combinatorics (with varied coefficients, signed numbers, fractions, exponents)

AII.P.4

Demonstrate an understanding of the exponential and logarithmic functions. (12.P.4)

AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4 AII.P.4

Understand exponential growth and decay Compare & contrast exponential growth and decay Graph functions that exhibit exponential growth and decay Solve word problems involving exponential growth and decay Given a graph of an exponential function, determine whether it represents a growth or a decay situation. Define logarithms in all bases Apply the properties of logarithms to simplify expressions Graph logarithmic functions Solve logarithmic equations Solve equations where the variable is the exponent Apply the Change of Base Theorem Show the inverse relationship of exponential and logarithmic functions Introduce the idea of asymptotes

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