Westerville City Schools COURSE OF STUDY Precalculus and Honors Precalculus MA304 … · 2020. 8....

Westerville City Schools

COURSE OF STUDY Precalculus and Honors Precalculus

MA304 and MA314

Course Description

Recommended Grade Level:

10 - 12

Course Length:

1 year

Credits:

1.0 mathematics credit

Course Weighting:

1.0 for MA304 and 1.125 for MA314

Recommended Supplies:

Graphing calculator (TI-84 family preferred)

Course Rationale

The State of Ohio requires that all students graduate from high school with proficiency

in Algebra 2. All students who enter this course have already met that standard;

therefore this course will offer college level material at a high school pace. The content

is considered non-remedial for any student entering a state-supported college in Ohio.

This course is a response to three distinct but similar sources. The three are the Ohio

Standards for Mathematics, The State of Ohio transfer module for Precalculus, and the

ACT College and Career Readiness Standards.

Course Information

This course is intended to prepare students to succeed in Calculus, either in Advanced

Placement at the high school or at a postsecondary institution. This course extends

algebraic concepts necessary for higher mathematics. Considerable time is devoted to

analytic geometry, trigonometry, sequences, series, limits, vectors, logarithmic and

2

exponential functions, and the use of appropriate technology. This course is theory-

oriented. Students enrolled in this course often have plans to continue their education at

the collegiate level.

Considerations for Cultural Relevancy/Inclusivity/Diversity

Mathematics classrooms are being transformed by the Ohio Mathematics Standards for

Mathematical Practices. The teacher is tasked with creating a classroom that embraces

these practices, a classroom that is inviting to all students. One of the inherent ideas in

these practices is that all students have valuable contributions and diverse

representations to bring to the mathematics classroom investigations. One of the

mathematical practices is to have students construct viable arguments and critique the

reasoning of others. Heterogeneous collaboration will help all students gain an

understanding and appreciation of all their classmates have to offer. Further, as the

students collaborate throughout the school year, students will gain an understanding

and appreciation of all their classmates, no matter what their background.

Scope and Sequence

Topics of Study Estimated Time

1 Transformations,Trigonometry I, Area under a

Curve, Exponential and Logarithmic

Functions

8-10 weeks

2 Trigonometry II, Limits I, Periodic Functions 8-10 weeks

MIDTERM ASSESSMENT

3 College Algebra, Limits II, Rates of Change 8-10 weeks

4 Vectors and Parametric Functions, Polar

Functions, Matrices, Conic Sections

8-10 weeks

FINAL ASSESSMENT

Primary Materials Recommendation

Text:

Other Resources:

3

● Ohio Resource Center course four standards:

http://www.ohiorc.org/standards/ohio/grade/mathematics/grade12.aspx

● Common Core State Standards:

http://www.corestandards.org/Math/

● Common Core State Standards Mathematical Practices:

http://www.corestandards.org/Math/Practice/

Topics of Study

Note: In the Instructional Strategies section the Standards for Mathematical Practice are

listed by name and then more fully defined and explained in Table 1 at the end of this

document.

Topic of Study #1: Functions

Content Standards

F.IF.1 Understand that a function from one set (called the

domain) to another set (called the range) assigns to each

element of the domain exactly one element of the range. If f is a

function and x is an element of its domain, then f(x) denotes the

output of f corresponding to the input x. The graph of f is the

graph of the equation y = f(x).

F.IF.2 Use function notation, evaluate functions for inputs in

their domains, and interpret statements that use function

notation in terms of a context.

F.LF.4 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.

F.IF.5 Relate the domain of a function to its graph and, where

applicable, to the quantitative relationship it describes. For

example, 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.

F.IF.7 Graph functions expressed symbolically and show key

http://www.ohiorc.org/standards/ohio/grade/mathematics/grade12.aspx

http://www.corestandards.org/Math/

http://www.corestandards.org/Math/Practice/

4

features of the graph, by hand in simple cases and using

technology for more complicated cases.

a. Graph linear and quadratic functions and show

intercepts, maxima, and minima.

b. Graph square root, cube root, and piecewise-defined

functions, including step functions and absolute value

functions.

c. Graph polynomial functions, identifying zeros when

suitable factorizations are available, and showing end

behavior.

d. Graph rational functions, identifying zeros and

asymptotes when suitable factorizations are available,

and showing end behavior.

e. Graph exponential and logarithmic functions, showing

intercepts and end behavior, and trigonometric functions,

showing period, midline, and amplitude.

F.BF.3 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.

Essential Questions

1. What are the essential properties of a function? 2. What are the characteristics of a complete graph of a

function? 3. What are the different ways in which functions can be

represented?

Enduring Understandings

1. Functions are a tool that can be used to assist in our understanding our world.

2. Through tables, graphs, and equations, functions show the structure of a problem situation.

3. Using a function to represent a situation brings the power of mathematics to help solve the problem.

Key Concepts/ Vocabulary

function definition, domain and range, end behavior,

asymptotes,increasing and decreasing, continuity, points of

inflection, concavity, even and odd, graphs of parent functions

and their transformations, inverse functions, operations with

functions, transformations of non-parent functions, polynomial

functions, periodic functions, rational functions, parametric

functions, piecewise functions

5

Content Elaborations

Graph and transform quadratic, piecewise, exponential,

logarithmic, rational, and trigonometric functions.

Write equations of functions from their graphs.

Determine whether a function is even, odd, or neither.

Define the trigonometric functions of sine and cosine in terms of

the unit circle.

Define other trigonometric functions in terms of sine and cosine.

Apply exponential functions to model real-world situations.

Define a log function and explore its graph and properties.

Learning Targets I can: ● use function operations to combine expressions. ● use transformations to graph piecewise, exponential,

logarithmic, rational, and trigonometric functions. ● find the inverse of a function algebraically and

graphically. ● identify whether a given graph or equation is a function. ● determine whether a function is even, odd, or neither

algebraically or graphically. ● identify key aspects of a graph, including domain, range,

intercepts, asymptotes, where it is increasing and decreasing, its concavity, and whether it is continuous.

Instructional Strategies and

Materials

Use the mathematical practices with a focus on: 1) Make sense of problems and persevere in solving them. 2) Reason abstractly and quantitatively. 3) Construct viable arguments and critique the reasoning of

others. 4) Model with mathematics. 5) Use appropriate tools strategically. 6) Attend to precisions. 7) Look for and make use of structure. 8) Look for an express regularity and repeated reasoning.

Considerations

for Interventions

and Acceleration

Please see attached documents addressing the challenges and response to exception students of all levels.

Assessments ● Perform operations on functions.

● Graph and/or transform any given parent graph.

Topic #2: Trigonometry

6

Content Standards F.TF.3 Use special triangles to determine geometrically the

values of sine, cosine, tangent for π/3, π/4 and π/6, and use

the unit circle to express the values of sine, cosine, and

tangent for π–x, π

F.TF.4 Use the unit circle to explain symmetry (odd and even)

and periodicity of trigonometric functions.

F.TF.6 Understand that restricting a trigonometric function to

a domain on which it is always increasing or always

decreasing allows its inverse to be constructed

F.TF.7 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.

F.TF.9 Prove the addition and subtraction formulas for sine,

cosine, and tangent and use them to solve problems.

F.BF.1c Write a function that describes a relationship between

two quantities.

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.

Essential Questions

1. How can we use the unit circle to evaluate and graph trig functions?

2. How can we use the parent graphs of trig functions to graph their inverse functions?

3. How do we find all solutions when solving trig equations?


Trigonometry is used to model many biological and physical relationships. Using the relationships between the trigonometric functions assists in making sense of the regularities in our world. Trigonometry is a key tool for scientific investigations.


angles and Coordinates in the Unit Circle, Sine and Cosine in

the Unit Circle, The Three Fundamental Pythagorean

Identities, Using Right Triangles to Find Trigonometric Ratios,

Graphs and Transformations of Sine and Cosine, The Five-

Point Method for Graphing Sine and Cosine, The Reciprocal

Trig Functions, Other Trigonometric Functions in the Unit

Circle, Simplifying Complex Fractions, Angular Frequency and

Period, Verifying Identities, Modeling with Periodic Functions

Laws of Sine and Cosine

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● Solve trigonometric Equations.

● Solve the SSA case of a triangle.

● Model and solve more complex periodic applications.

● Simplify expressions involving more than one angle.


Trigonometry, using the unit circle, all six functions, identities,

inverse functions, continuing graphical analysis, solving

equations

Find the values of coordinates of key points on the unit circle

and see how they can be used to find the values of sine and

cosine for any angle.

Generate parent graphs for sine and cosine and use them to

sketch various transformations.

Define other trigonometric functions in terms of sine and

cosine.

Model situations using periodic functions

Solving Trig Equations; Inverse Sine and Inverse Cosine;

Ambiguous Case for the Law of Sines; Tangent and Inverse

Tangent; Graphing Trig Functions of the Form 𝑦 = 𝑎𝑠𝑖𝑛(𝑏(𝑥 −

ℎ)) + 𝑘; Angle Sum and Difference Formulas; Modeling with

Trig Functions; Double and Half Angle Formulas; Solving

Complex Trig Equations

Learning Targets I can:

● use the unit circle to find exact values of trigonometric

functions.

● graph various sinusoidal functions.

● define the reciprocal trigonometric functions: secant,

cosecant, and cotangent.

● use identities to simplify and verify expressions.

● begin modeling using periodic functions. ● solve trigonometric equations.

● solve triangles that have more than one solution.

● model periodic functions that have both a shift and a

period other than 2�.

● use the angle sum and difference formulas and double-angle formulas.

Instructional Strategies

and Materials

Use the mathematical practices with a focus on: 1) Make sense of problems

3) Construct viable arguments

4) Model with mathematics

8

Considerations for

Interventions and

Acceleration


Assessments ● Develop fluency with angles and values in the unit

circle

● Generate the parent graphs for sine and cosine and

use them in a variety of transformations

● Define and use trig functions

● Use identities to simplify and verify expressions

● Solve trig equations over a variety of domains

● Define the inverse functions for sine, cosine, and

tangent

● Solve triangles that have more than one solution

● Model real-life problems by using periodic functions

● Develop the sum and difference, double-angle, and

half-angle formulas

Topic #3: Sequences and Series

Content Standards F.BF.1c Write a function that describes a relationship

between two quantities.

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.

F.BF.2 Write arithmetic and geometric sequences both

recursively and with an explicit formula, use them to model

situations, and translate between the two forms.

Essential Questions 1. How are arithmetic and geometric patterns alike and different?


Real world growth can be modeled by Arithmetic and Geometric sequences. Functions can help to understand and make predictions about real world situations. They will also begin to see the difference between a model and the complexity of real phenomena.


summation (Sigma) notation, Pascal’s Triangle, finite and infinite series, arithmetic sequence, geometric sequence

9


Develop a formula for the sums of arithmetic and geometric

series.

Expand binomials with Pascal’s Triangle.

Use Sigma Notation to express the area under a curve.


● determine whether a sequence is arithmetic or geometric and find particular terms of a sequence.

● write and evaluate sigma notation for a series ● find the sum of arithmetic and geometric sequences

intuitively, using a formula, and using limits. ● write the next n terms given the first few terms. ● write a rule for the nth term of an arithmetic or

geometric sequence. ● use Pascal’s triangle to calculate binomial coefficients

and expand binomial expressions.


Materials

Use the mathematical practices with a focus on: 1) Make sense of problems and persevere in solving

them. 2) Reason abstractly and quantitatively. 3) Construct viable arguments and critique the reasoning

of others. 4) Model with mathematics. 5) Use appropriate tools strategically. 6) Attend to precisions. 7) Look for and make use of structure. 8) Look for an express regularity and repeated reasoning.

Considerations for

Interventions and

Acceleration


Assessments ● Write and evaluate Summation Notation

● Find the Area Under a Curve

● Solve problems using arithmetic and geometric

sequences and their rules

● Use algebraic techniques to transform equations and

expressions into more useful forms

● Find the sum of finite arithmetic and geometric series.

Topic #4: Vectors, Polar, and Parametric Equations

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Content Standards N.CN.3 Find the conjugate of a complex number; use

conjugates to find moduli and quotients of complex numbers.

N.CN.4 Represent complex numbers on the complex plane

in rectangular and polar form (including real and imaginary

numbers), and explain why the rectangular and polar forms

of a given complex number represent the same number.

N.CN.5 Represent addition, subtraction, multiplication, and

conjugation of complex numbers geometrically on the complex

plane; use properties of this representation for computation. For

example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus 2 and

argument 120°.

N.CN.6 Calculate the distance between numbers in the

complex plane as the modulus of the difference, and the

midpoint of a segment as the average of the numbers at its

endpoints

N.VM.1 Recognize vector quantities as having both

magnitude and direction. Represent vector quantities by

directed line segments, and use appropriate symbols for

vectors and their magnitudes (e.g., v, |v|, ||v||, v).

N.VM.2 Find the components of a vector by subtracting the

coordinates of an initial point from the coordinates of a

terminal point.

N.VM.3 Solve problems involving velocity and other

quantities that can be represented by vectors.

N.VM.4 Add and subtract vectors.

a. Add vectors end-to-end, component-wise, and by the

parallelogram rule. Understand that the magnitude of

a sum of two vectors is typically not the sum of the

magnitudes.

b. Given two vectors in magnitude and direction form,

determine the magnitude and direction of their sum.

c. Understand vector subtraction v – w as v + (–w),

where –w is the additive inverse of w, wi

d. th the same magnitude as w and pointing in the

opposite direction. Represent vector subtraction

graphically by connecting the tips in the appropriate

order, and perform vector subtraction component-

wise.

N.VM.5 Multiply a vector by a scalar.

a. Represent scalar multiplication graphically by scaling

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vectors and possibly reversing their direction; perform

scalar multiplication component-wise, e.g., as c(vx,

vy) = (cvx, cvy).

b. Compute the magnitude of a scalar multiple cv using ||cv|| =

|c|v. Compute the direction of cv knowing that when |c|v ≠

0, the direction of cv is either along v (for c > 0) or against

v (for c < 0)

Essential Questions 1. Can you model real world phenomena using vectors and parametric equations?

2. Can you convert between polar and rectangular coordinates and vice versa?


Vectors offer the student another tool to use in their understanding of how mathematics can be used to understand the structure of our world. In particular, vectors are used in physics and all navigation. The map applications on their phones and in their cars are dependant technology involving vectors.


polar coordinates, conversions between polar and

rectangular forms, polar graphs: common forms, rotations,

complex numbers: simplifying, graphing, polar forms of

complex numbers, multiplying and dividing complex

numbers,powers and roots of complex numbers, DeMoivre’s

Theorem,

● Devine vectors in standard and component form

● Use vectors to solve common physics problems

● Find the angle between vectors using the dot product,

Define the motion of a point using two dependent

variable

● Use parametric equations to solve real world

problems.


Vectors, vector operations, polar coordinates, parametric

equations

Vector Addition; Magnitude and Standard Angle of a Vector;

Component Form of a Vector; Unit Vectors; Applications of

Vectors; Dot Product; Parametric Equations; Vector

Equations; Inverses and Parametric Equations;

Applications of Parametric Equations

Learning Targets

I can:

● use geometry to define and perform operations using

vectors.

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● write vectors in component form.

● use vectors to solve common physics and calculus

problems.

● define and use the motion of an object using

parametric equations.

● use parametric equations to solve common physics

and calculus problems.

● plot points and graph equations using polar

coordinates

● convert between polar and rectangular equations

● investigate various families of polar functions


Materials



6) Tend to precision

Considerations for

Interventions and

Acceleration


Assessments ● Introduce and use vectors in both component and

standard forms.

● Use vectors to solve problems common to calculus

and physics.

● Define and graph parametric equations and common

polar equations.

● Use parametric equations to solve problems involving

motion.

● Use a coordinate system based on an angle and a

distance.

● Make conversions between polar and rectangular

systems.

Topic #5: Complex Numbers

Content Standards N.CN.3 Find the conjugate of a complex number; use

conjugates to find moduli and quotients of complex

numbers.

N.CN.4 Represent complex numbers on the complex

plane in rectangular and polar form (including real and

13

imaginary numbers), and explain why the rectangular

and polar forms of a given complex number represent the

same number.

N.CN.5 Represent addition, subtraction, multiplication, and

conjugation of complex numbers geometrically on the complex

plane; use properties of this representation for computation.

For example, (–1 + √3 i)3 = 8 because (–1 + √3 i) has modulus

2 and argument 120°.

N.CN.6 Calculate the distance between numbers in the

complex plane as the modulus of the difference, and the

midpoint of a segment as the average of the numbers at

its endpoints.

Essential Questions 1. What are Polar Coordinates and how are they related to complex numbers?


Complex numbers arise from the structure of Algebra

and were invented to respond to a purely mathematical

problem, but about three centuries later these numbers

were essential to model the properties of electricity.


polar coordinates, cartesian coordinates, polar and rectangular form of an equation, complex numbers, DeMoivre’s Theorem

Content Elaborations Complex numbers, complex plane (tie back to polar

coordinates), DeMoivre’s theorem

Work with complex numbers

Graph complex numbers

Use complex numbers to find roots of functions

Find powers and roots of complex numbers

Polar forms of complex numbers

Multiplying and dividing complex numbers

Powers and roots of complex numbers

Learning Targets I can: ● plot points and graph equations using polar

coordinates. ● convert between polar and rectangular equations. ● investigate various families of polar functions. ● Iwork with complex numbers in both standard and

polar form. ● use DeMoivre’s Theorem to find powers and roots

of complex numbers. ● graph complex numbers. ● use complex numbers to find roots of functions.

Instructional Strategies Use the mathematical practices with a focus on:

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and Materials 1) Make sense of problems and persevere in solving them.

2) Reason abstractly and quantitatively. 3) Construct viable arguments and critique the

reasoning of others. 4) Model with mathematics. 5) Use appropriate tools strategically. 6) Attend to precisions. 7) Look for and make use of structure. 8) Look for an express regularity and repeated

reasoning.

Considerations for

Interventions and

Acceleration


Assessments ● Polar coordinates and complex numbers.

Topic #6: Exponential and Logarithmic Functions

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Content Standards

A.REI.8 Represent a system of linear equations as a

single matrix equation in a vector variable.

A.REI.9 Find the inverse of a matrix if it exists and use it

to solve systems of linear equations (using technology for

matrices of dimension 3 × 3 or greater).

F.BF.1 Write a function that describes a relationship

between two quantities.

c. 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.

F.IF.7 Graph functions expressed symbolically and show

key features of the graph, by hand in simple cases and

using technology for more complicated cases.

a. Graph linear and quadratic functions and show

intercepts, maxima, and minima.

b. Graph square root, cube root, and piecewise-

defined functions, including step functions and

absolute value functions.

c. Graph polynomial functions, identifying zeros

when suitable factorizations are available, and

showing end behavior.

d. Graph rational functions, identifying zeros and

asymptotes when suitable factorizations are

available, and showing end behavior.

e. Graph exponential and logarithmic functions,

showing intercepts and end behavior, and

trigonometric functions, showing period, midline,

and amplitude.

Essential Questions 1. What real world problems are solved using logarithmic and exponential equations?

2. How are exponential and logarithmic equations related? And which came first?


Exponential and Logarithmic functions arise in the modeling of many real world phenomena.


transformations of kf(x) and f(kx), applications of

exponential functions, equivalent transformations, inverse

functions, logarithm, log graphs, laws of logarithms,

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solving exponential and logarithmic equations, LN vs

LOG, The number e, Pert and applications of e

Content Elaborations Exponential and logarithmic functions, properties of

logarithms, e and natural log, and solving equations

Teach students how to shift and stretch graphs, both

horizontally and vertically.

Show students that for some functions, two different

transformations can give the same result.

Have students apply previous knowledge about

exponential functions to model real-world situations.

Define the log function and explore its graph and

properties.


● shift and stretch graphs, both horizontally and vertically.

● explore equivalent transformations.

● apply exponential functions to model real-world

situations.

● define a log function and explore its graph and

properties.

● define the number e and apply it in applications of

exponential functions.

Instructional Strategies and Materials



6) Tend to precision

Considerations for

Interventions and

Acceleration


Assessments ● Transformations of exponential and logarithmic

functions.

● Show that two different transformations yield the

same result.

● Use exponential functions to model real-world

situations.

● Explore the properties of log functions to simplify

and solve log equations.

Topic #7: Matrices

17

Content Standards

N.VM.6 Use matrices to represent and manipulate data,

e.g., to represent payoffs or incidence relationships in a

network.

N.VM.7 Multiply matrices by scalars to produce new

matrices, e.g., as when all of the payoffs in a game are

doubled.Multiply matrices by scalars to produce new

matrices, e.g., as when all of the payoffs in a game are

doubled.

N.VM.8 Add, subtract, and multiply matrices of appropriate

dimensions.

N.VM.9 Understand that, unlike multiplication of numbers,

matrix multiplication for square matrices is not a

commutative operation, but still satisfies the associative and

distributive properties.

N.VM.10 Understand that the zero and identity matrices play

a role in matrix addition and multiplication similar to the role

of 0 and 1 in the real numbers. The determinant of a square

matrix is nonzero if and only if the matrix has a multiplicative

inverse.

N.VM.11 Understand that the zero and identity matrices play

a role in matrix addition and multiplication similar to the role

of 0 and 1 in the real numbers. The determinant of a square

matrix is nonzero if and only if the matrix has a multiplicative

inverse.

N.VM.12 Work with 2 × 2 matrices as transformations of the

plane, and interpret the absolute value of the determinant in

terms of area.

Essential Questions 1. How can matrices assist in solving systems of equations?

2. Can I perform basic operations using matrices? 3. How can I use technology to assist in the solving of

matrices?


Matrices are the utility infielder of mathematics. Matrices

can be used to conceptualize many different mathematical

operations from systems of equations to geometric

transformations. Matrices’ versatility is key to seeing the

overall structure of mathematics and the concepts that cut

across many mathematical concepts.

18


matrices, matrix operations, identity matrix, inverse matrix

linear transformation, rotation matrix, vectors, composition of

matrices, eigenvalues

Content Elaborations Matrices, solving systems of equations

Matrices

Matrix Operations

Identity Matrix

Inverse Matrix

Linear Transformation

Rotation Matrix

Vectors and Matrices

Composition of Matrices

Eigenvalues


● solve systems of equations using matrices on a calculator.

● examine theorems relating scalars and matrices.

● define linear transformations and relate them to

matrices.

● compose two linear transformations


Materials

Use the mathematical practices with a focus on: 4) Model with mathematics

5) Use appropriate tools strategically

7) Look for and make use of structure

Considerations for

Interventions and

Acceleration

Please see attached documents addressing the challenges and response to exception students of all levels

Assessments ● Perform basic operations with matrices

● Solve systems using matrices

Topic #8: Conic Sections

Content Standards G.GPE.1 Derive the equation of a circle of given center

and radius using the Pythagorean Theorem; complete

the square to find the center and radius of a circle given

by an equation.

G.GPE.2 Derive the equation of a parabola given a

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focus and directrix.

G.GPE.3 Derive the equations of ellipses and

hyperbolas given the foci, using the fact that the sum or

difference of distances from the foci is constant.

Essential Questions 1. In what context is the model of a circle close but

not accurate?

2. How can circles be revised to fit these contexts?

3. How are these new shapes related to circles and

to each other?

Enduring

Understanding

A double cone can be cut to produce a number of useful

shapes.

Conic sections are all around us.

Key

Concepts/Vocabulary

conic sections, circles, ellipses, hyperbolas, parabolas,

2nd degree equations, eccentricity,foci


Conics (graphs that might not be functions), and

solving equations

Define and investigate properties of conic sections.

Solve problems involving conic sections.

Define and use eccentricity.


● use formal definitions and properties of conic

sections to determine their equations.

● learn the structure and characteristics of a

college math text.

● work with second-degree equations and

eccentricity to determine the shape of a conic.

Instructional Strategies

and Materials

Use the mathematical practices with a focus on:


5) Use appropriate tools strategically

Considerations for

Interventions and

Acceleration

Please see attached documents addressing the

challenges and response to exception students of all

levels.

Assessments

● Identify conic sections

● Graph conic sections

20

● Solve problems involving conic sections

Appendix A

CRITERIA TO DISTINGUISH REGULAR FROM HONORS

Precalculus (MA304) Honors Precalculus (MA314)

Scope and Sequence

· Pacing is such that the average

student can keep up on a daily basis

For example…

· Students learn what limits are and

Scope and Sequence

· More rigorous and in-depth coverage

at faster pace

For example…

· Students apply the concepts of limits

and derivatives to graph functions by

finding critical and inflection points

21

compute simple limits and derivatives

Assignments

· Students will do a variety of problems

that not only help them to understand

the process but also to analyze the

process. Typically these will include

only the core problems

Assignments

· Students will do a variety of problems

that not only help them to understand

the process but also to analyze the

process. This course will go beyond

the core problems.

Assessments

· Students show the ability to problem

solve with the graphing calculator

and verify algebraically (without the

calculator)

· · Assessment problems are more like

what the student experiences in

homework problems

Assessments

· Students show the ability to problem

solve with the graphing calculator

and verify algebraically (without the

calculator)

· Assessment problems require

deeper thought and may not have

been seen before

Support for Student Learning

More time taken to review concepts

from previous units and courses.

Support for Student Learning

· Students are expected to apply

concepts to different types of

questions

Appendix B

The Standards for Mathematical Practice

The Standards for Mathematical Practice describe varieties of expertise that

mathematics educators at all levels should seek to develop in their students. These

practices rest on important “processes and proficiencies” with longstanding importance

in mathematics education. The first of these are the NCTM process standards of

problem solving, reasoning and proof, communication, representation, and connections.

The second are the strands of mathematical proficiency specified in the National

Research Council’s report Adding It Up: adaptive reasoning, strategic competence,

conceptual understanding (comprehension of mathematical concepts, operations and

relations), procedural fluency (skill in carrying out procedures flexibly, accurately,

efficiently and appropriately), and productive disposition (habitual inclination to see

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mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and

one’s own efficacy).

1 .Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a

problem and looking for entry points to its solution. They analyze givens, constraints,

relationships, and goals. They make conjectures about the form and meaning of the

solution and plan a solution pathway rather than simply jumping into a solution attempt.

They consider analogous problems, and try special cases and simpler forms of the

original problem in order to gain insight into its solution. They monitor and evaluate their

progress and change course if necessary. Older students might, depending on the

context of the problem, transform algebraic expressions or change the viewing window

on their graphing calculator to get the information they need. Mathematically proficient

students can explain correspondences between equations, verbal descriptions, tables,

and graphs or draw diagrams of important features and relationships, graph data, and

search for regularity or trends. Younger students might rely on using concrete objects or

pictures to help conceptualize and solve a problem. Mathematically proficient students

check their answers to problems using a different method, and they continually ask

themselves, “Does this make sense?” They can understand the approaches of others to

solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in

problem situations. They bring two complementary abilities to bear on problems

involving quantitative relationships: the ability to decontextualize—to abstract a given

situation and represent it symbolically and manipulate the representing symbols as if

they have a life of their own, without necessarily attending to their referents—and the

ability to contextualize, to pause as needed during the manipulation process in order to

probe into the referents for the symbols involved. Quantitative reasoning entails habits

of creating a coherent representation of the problem at hand; considering the units

involved; attending to the meaning of quantities, not just how to compute them; and

knowing and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions,

and previously established results in constructing arguments. They make conjectures

and build a logical progression of statements to explore the truth of their conjectures.

They are able to analyze situations by breaking them into cases, and can recognize and

use counterexamples. They can justify their conclusions, communicate them to others,

and respond to the arguments of others. They reason inductively about data, making

plausible arguments that take into account the context from which the data arose.

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Mathematically proficient students are also able to compare the effectiveness of two

plausible arguments, distinguish correct logic or reasoning from that which is flawed,

and—if there is a flaw in an argument—explain what it is. Elementary students can

construct arguments using concrete referents

such as objects, drawings, diagrams, and actions. Such arguments can make sense

and be correct, even though they are not generalized or made formal until later grades.

Later, students learn to determine domains to which an argument applies. Students at

all grades can listen or read the arguments of others, decide whether they make sense,

and ask useful questions to clarify or improve the arguments.

4. Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve

problems arising in everyday life, society, and the workplace. In early grades, this might

be as simple as writing an addition equation to describe a situation. In middle grades, a

student might apply proportional reasoning to plan a school event or analyze a problem

in the community. By high school, a student might use geometry to solve a design

problem or use a function to describe how one quantity of interest depends on another.

Mathematically proficient students who can apply what they know are comfortable

making assumptions and approximations to simplify a complicated situation, realizing

that these may need revision later. They are able to identify important quantities in a

practical situation and map their relationships using such tools as diagrams, two-way

tables, graphs, flowcharts and formulas. They can analyze those relationships

mathematically to draw conclusions. They routinely interpret their mathematical results

in the context of the situation and reflect on whether the results make sense, possibly

improving the model if it has not served its purpose.

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a

mathematical problem. These tools might include pencil and paper, concrete models, a

ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical

package, or dynamic geometry software. Proficient students are sufficiently familiar with

tools appropriate for their grade or course to make sound decisions about when each of

these tools might be helpful, recognizing both the insight to be gained and their

limitations. For example, mathematically proficient high school students analyze graphs

of functions and solutions generated using a graphing calculator. They detect possible

errors by strategically using estimation and other mathematical knowledge. When

making mathematical models, they know that technology can enable them to visualize

the results of varying assumptions, explore consequences, and compare predictions

with data. Mathematically proficient students at various grade levels are able to identify

relevant external mathematical resources, such as digital content located on a website,

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and use them to pose or solve problems. They are able to use technological tools to

explore and deepen their understanding of concepts.

6. Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to

use clear definitions in discussion with others and in their own reasoning. They state the

meaning of the symbols they choose, including using the equal sign consistently and

appropriately. They are careful about specifying units of measure, and labeling axes to

clarify the correspondence with quantities in a problem. They calculate accurately and

efficiently, express numerical answers with a degree of precision appropriate for the

problem context. In the elementary grades, students give carefully formulated

explanations to each other. By the time they reach high school they have learned to

examine claims and make explicit use of definitions.

7. Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young

students, for example, might notice that three and seven more is the same amount as

seven and three more, or they may sort a collection of shapes according to how many

sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5

+ 7 × 3, in preparation for learning about the distributive property. In the expression x2 +

9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the

significance of an existing line in a geometric figure and can use the strategy of drawing

an auxiliary line for solving problems. They also can step back for an overview and shift

perspective. They can see complicated things, such as some algebraic expressions, as

single objects or as being composed of several objects. For example, they can see 5 –

3(x – y)2 as 5 minus a positive number times a square and use that to realize that its

value cannot be more than 5 for any real numbers x and y.

8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for

general methods and for shortcuts. Upper elementary students might notice when

dividing 25 by 11 that they are repeating the same calculations over and over again,

and conclude they have a repeating decimal. By paying attention to the calculation of

slope as they repeatedly check whether points are on the line through (1, 2) with slope

3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the

regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and

(x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a

geometric series. As they work to solve a problem, mathematically proficient students

maintain oversight of the process, while attending to the details. They continually

evaluate the reasonableness of their intermediate results.

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Appendix C

Connecting the Standards for Mathematical Practice to the Standards for

Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student

practitioners of the discipline of mathematics increasingly ought to engage with the

subject matter as they grow in mathematical maturity and expertise throughout the

elementary, middle and high school years. Designers of curricula, assessments, and

professional development should all attend to the need to connect the mathematical

practices to mathematical content in mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and

understanding. Expectations that begin with the word “understand” are often especially

good opportunities to connect the practices to the content. Students who lack

understanding of a topic may rely on procedures too heavily. Without a flexible base

from which to work, they may be less likely to consider analogous problems, represent

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problems coherently, justify conclusions, apply the mathematics to practical situations,

use technology mindfully to work with the mathematics, explain the mathematics

accurately to other students, step back for an overview, or deviate from a known

procedure to find a shortcut. In short, a lack of understanding effectively prevents a

student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are

potential “points of intersection” between the Standards for Mathematical Content and

the Standards for Mathematical Practice. These points of intersection are intended to be

weighted toward central and generative concepts in the school mathematics curriculum

that most merit the time, resources, innovative energies, and focus necessary to

qualitatively improve the curriculum, instruction, assessment, professional development,

and student achievement in mathematics.

Acknowledgements

It is through the hard work and dedication of the middle school/high school Mathematics

team that the Westerville City Schools’ Precalculus Course of Study is presented to the

Board of Education. Sincere appreciation is extended to the following individuals for

their assistance and expertise.

Central HS North HS South Hs District

Deanna Caviccia

Jill Crawford

Rebecca Riccobono

Amie Ward

Rodger Elander

Lawrence Pack

Tim Bates

George Crooks

Michael Huler

Westerville City Schools COURSE OF STUDY Precalculus and Honors Precalculus MA304 … · 2020. 8....

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