The following Practice Standards and Literacy Skills will ...€¦ · Maury County Public Schools...

Precalculus TN Ready Performance Standards by Unit

Maury County Public Schools 5/8/18 Office of Instruction Pk-12

Precalculus – Curriculum Overview

The following Practice Standards and Literacy Skills will be used throughout the course:

Standards for Mathematical Practice Literacy Skills for Mathematical Proficiency

1. Make sense of problems and persevere in solving them. 1. Use multiple reading strategies.

2. Reason abstractly and quantitatively. 2. Understand and use correct mathematical vocabulary.

3. Construct viable arguments and critique the reasoning of others. 3. Discuss and articulate mathematical ideas.

4. Model with mathematics. 4. Write mathematical arguments.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

*Unless otherwise noted, all resources are from Glencoe Precalculus.



Precalculus – Curriculum Overview

Green=Major Content Blue=Supporting Content

Quarter 1 Quarter 2 Quarter 3 Quarter 4

Unit 1 14 Days

Unit 2 16 Days

Unit 3 12 Days

Unit 4 16 Days

Unit 5 13 Days

Unit 6 12 Days

Unit 7 7 Days

Unit 8 16 Days

Unit 9 6 Days

Unit 10 14 Days

Functions,

Limits, &

End

Behavior

Power,

Polynomial,

& Rational

Functions

Exponential

&

Logarithmic

Functions

Trigonometric

Functions

Trigonometric

Identities &

Equations

Matrices Conic

Sections

Polar

Graphing &

Vectors

Parametric

Equations

Sequences,

Series, &

Sigma

Notation

Standards (By number)

P.F.BF.A.2

P.F.BF.A.4

P.F.IF.A.1

P.F.BF.A.1

P.F.BF.A.3

P.F.BF.A.5.a

P.F.BF.A.5.b

P.F.BF.A.5.c

P.F.BF.A.5.d

P.F.BF.A.6

F.S.MD.A.3

P.S.MD.A.1

P.S.MD.A.2

P.S.MD.A.3

P.MCPS.6

P.F.IF.A.5

P.F.IF.A.2

P.F.IF.A.4

P.F.IF.A.6

P.F.IF.A.7

P.N.CN.B.6

P.N.CN.B.7

P.N.NE.A.4

P.N.NE.A.5

P.A.REI.A.3

P.A.REI.A.4

P.F.IF.A.2

P.N.NE.A.3

P.N.NE.A.2

P.N.NE.A.1

P.MCPS.2

P.F.TF.A.1

P.G.AT.A.3

P.G.AT.A.4

P.F.TF.A.2

P.F.TF.A.3

P.F.GT.A.1

P.F.GT.A.2

P.F.GT.A.3

P.F.GT.A.4

P.F.GT.A.5

P.F.GT.A.6

P.F.GT.A.7

P.F.GT.A.8

P.G.AT.A.1

P.S.MD.A.3

P.F.TF.A.4

P.MCPS.1

P.G.TI.A.1

P.G.TI.A.2

P.G.AT.A.2

P.G.AT.A.5

P.G.AT.A.6

P.N.VM.C.7

P.N.VM.C.8

P.N.VM.C.9

P.N.VM.C.10

P.N.VM.C.11

P.N.VM.C.12

P.N.VM.C.13

P.A.REI.A.1

P.A.REI.A.2

P.MCPS.5

P.A.C.A.1

P.A.C.A.2

P.A.C.A.3

P.A.C.A.4

P.N.VM.A.1

P.N.VM.A.2

P.N.VM.B.3

P.N.VM.B.4.a

P.N.VM.B.4.b

P.N.VM.B.4.c

P.N.VM.B.5.a

P.N.VM.B.5.b

P.N.VM.B.5.c

P.N.CN.A.4

P.N.CN.A.5

P.N.VM.B.6

P.N.CN.A.1

P.N.CN.A.2

P.G.PC.A.1

P.G.PC.A.2

P.G.PC.A.3

P.N.CN.A.3

P.MCPS.4

P.A.PE.A.1

P.A.PE.A.2

P.MCPS.3

P.A.S.A.1

P.A.S.A.2

P.A.S.A.3.a

P.A.S.A.3.b

P.A.S.A.3.c

P.F.IF.A.8

P.A.S.A.4

P.A.S.A.5



Unit 1

Functions, Limits, & End Behavior

14 Days

Standards “I Can” Statements

Functions

P.F.BF.A.2 Develop an understanding of functions as elements that can be

operated upon to get new functions: addition, subtraction, multiplication,

division, and composition of functions.

I can create functions by adding, subtracting, multiplication, division, and

composition of functions.

P.F.BF.A.4 I can construct the difference quotient for a given function and

simplify the resulting expression.

I can construct the difference quotient for a given function and simplify the

resulting expression.

P.F.IF.A.1 Determine whether a function is even, odd, or neither. I can determine whether a function is even, odd, or neither algebraically and

graphically.

P.F.BF.A.1 Understand how the algebraic properties of an equation

transform the geometric properties of its graph. For example, given a

function, describe the transformation of the graph resulting from the

manipulation of the algebraic properties of the equation (i.e., translations,

stretches, reflections and changes in periodicity and amplitude)

I can describe the transformation of the graph resulting from the manipulation

of the algebraic properties of the equation (i.e., translations, stretches,

reflections, and changes in periodicity and amplitude).

P.F.BF.A.3 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.

I can form a composite function.

I can find the domain of a composite function.

I can recognize the role that domain of a function plays in the combination of

functions by composition of functions.

P.F.BF.A.5 Find inverse functions (including exponential, logarithmic and trigonometric). a. Calculate the inverse of a function, f(x), with respect to each of the functional operations; in other words, the additive inverse, − f(x), the multiplicative inverse, 1/f(x), and the inverse with respect to composition, f-1(x). Understand the algebraic and graphical implications of each type. b. Verify by composition that one function is the inverse of another. c. Read values of an inverse function from a graph or a table, given that the function has an inverse. d. Recognize a function is invertible if and only if it is one-to-one. Produce an invertible function from a non-invertible function by restricting the domain.

I can calculate the inverse of a function with respect to each of the functional

operations.

I can verify by composition that one function is the inverse of another.

I can identify whether a function has an inverse with respect to composition

and when functions are inverses of each other with respect to composition.

I can find an inverse function by restricting the domain of a function that is

not one-to-one.

P.F.BF.A.6 Explain why the graph of a function and its inverse are reflections of one another over the line y = x.

I can explain why the graph of a function and its inverse are reflections of one

another over the line y = x.



P.MCPS.6 Develop the concept of the limit using tables, graphs, and

algebraic properties.

I can explore the properties of a limit by analyzing sequences and series.

I can understand the relationship between a horizontal asymptote and the

limit of a function at infinity.

I can determine the limit of a function at a specified number.

I can find the limit of a function at a number using algebra.

Statistics and Probability

P.S.MD.A.1 ★ Create scatter plots, analyze patterns and describe relationships for bivariate data (linear, polynomial, trigonometric or exponential) to model real-world phenomena and to make predictions.

I can create scatter plots for bivariate data (linear, polynomial, trigonometric

or exponential) to model real-world phenomena.

I can analyze patterns from the scatter plots that I created.

I can describe relationships in the scatter plots.

I can make prediction using the scatter plots.

P.S.MD.A.2 ★Determine a regression equation to model a set of bivariate

data. Justify why this equation best fits the data.

I can explain how to determine the best regression equation model that

approximates a particular data set.

P.S.MD.A.3 ★Use a regression equation modeling bivariate data to make

predictions. Identify possible considerations regarding the accuracy of

predictions when interoplating or extrapolating.

I can find the regression equation that best fits bivariate data.

I can use a regression equation modeling bivariate data to make predictions.

I can identify possible considerations regarding the accuracy of predictions

when interpolating or extrapolating.



Sections/Topics Activities/Resources/Materials

• Pre-assessment

• Section 1 Functions

• Section 2 Analyzing Graphs of Functions

• Section 3 Continuity, End Behavior, and Limits

• Section 4 Extrema and Average Rates of Change

• Quiz

• Section 5 Parent Functions and Transformations

• Section 6 Function Operations and Composition of Functions

• Section 7 Inverse Relations and Functions

• Unit Review

• Unit 1 Test

Extra Resources

• www.shodor.org/interactive

•

Vocabulary

• Set-builder notation

• Interval notation

• Function

• Vertical line test

• Function notation

• Independent variable

• Dependent variable

• Implied domain

• Piecewise-defined function

• Line symmetry

• Point symmetry

• Even function

• Odd function

• Continuous function

• Discontinuous function

• Limit

• Infinite discontinuity

• Jump discontinuity

• Removable discontinuity

• Nonremovable discontinuity

• Continuity test

• Intermediate Value

Theorem

• End behavior

• Increasing

• Decreasing

• Constant

• Critical points

• Extrema

• Relative Extrema

• Absolute Extrema

• Maximum

• Minimum Point of

inflection

• Average rate of change

• Secant line

• Parent function

• Constant function

• Identity function

• Quadratic function

• Cubic function

• Square root function

• Reciprocal function

• Absolute value function

• Step function

• Greatest integer function

• Transformation

• Translation

• Reflection

• Dilation

• Composition of functions

• Inverse relation

• Inverse function

• Horizontal line test

• One-to-one function

http://www.shodor.org/interactive



Unit 2

Power, Polynomial, & Rational Functions

16 Days


Number and Quantity

P.N.CN.B.6 Extend polynomial identities to the complex numbers.

For example, rewrite x2 + 4 as (x + 2i)(x – 2i). I can extend polynomial identities to the complex numbers.

P.N.CN.B.7 Know the Fundamental Theorem of Algebra; show that it is true

for quadratic polynomials.

I know the Fundamental Theorem of Algebra and can show it is true for

quadratic polynomials.

P.N.NE.A.4 Simplify complex radical and rational expressions; discuss and display understanding that rational numbers are dense in the real numbers and the integers are not.

I can simplify complex radical and rational expressions.

P.N.NE.A.5 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

I can add, subtract, multiply, and divide rational expressions.

Algebra

P.A.REI.A.3 Solve nonlinear inequalities (quadratic, trigonometric, conic, exponential, logarithmic, and rational) by graphing (solutions in interval notation if one-variable), by hand and with appropriate technology.

I can solve nonlinear inequalities (quadratic and rational) by graphing

(solutions in interval notation if one-variable), by using a sign chart, and with

appropriate technology.

P.A.REI.A.4 Solve systems of nonlinear inequalities by graphing. I can solve systems of nonlinear inequalities by graphing.

Functions

P.F.IF.A.5

Identify characteristics of graphs based on a set of conditions or on a general

equation such as y = ax2 + c.

I can Identify characteristics of graphs such as direction it opens, vertex based

on a set of conditions or on a general equation

such as y = ax2 + c

P.F.IF.A.2 ★Analyze qualities of exponential, polynomial, logarithmic,

trigonometric, and rational functions and solve real world problems that can

be modeled with these functions (by hand and with appropriate technology).

I can analyze qualities of exponential, polynomial, logarithmic, trigonometric,

and rational functions and solve real world problems that can be modeled

with these functions.

I can identify or analyze the following properties of polynomial, and rational

functions from tables, graphs, and equations.

I can describe the following of a given function: Domain, Range, Continuity,

Increasing/decreasing Behavior, Symmetry, Boundedness, Extrema,

Asymptotes, Intercepts, Holes, End Behavior with Limit Notation, Concavity.

P.F.IF.A.4 Identify the real zeros of a function and explain the relationship

between the real zeros and the x-intercepts of the graph of a function

(exponential, polynomial, logarithmic, trigonometric, and rational).

I can identify the real zeros of the graph of a function (polynomial, rational,

exponential, logarithmic, and trigonometric) in equation or graphical form.

I can explain the relationship between the real zeros and the x-intercept of the

graph of a function (polynomial, rational, exponential, logarithmic, and

trigonometric).



P.F.IF.A.6 Visually locate critical points on the graphs of functions and

determine if each critical point is a minimum, a maximum or point of

inflection. Describe intervals where the function is increasing or decreasing

and where different types of concavity occur.

I can locate critical points on the graphs of polynomial functions and

determine if each critical point is a minimum or a maximum.

I can describe and locate maximums, minimums, increasing and decreasing

intervals, and zeroes given a sketch of the graph.

P.F.IF.A.7 Graph rational functions, identifying zeros, asymptotes (including

slant), and holes (when suitable factorizations are available) and showing

end-behavior.

I can graph rational functions, identifying zeros, asymptotes (including slant),

and holes (when suitable factorizations are available) and showing end-

behavior.


• Pre-assessment

• Section 1 Power and Radical Functions

• Section 2 Polynomial Functions

• Section 3 The Remainder and Factor Theorems

• Quiz

• Section 4 Zeros of Polynomial Functions

• Section 5 Rational Functions

• Section 6 Nonlinear Inequalities

• Unit Review

• Unit 2 Test

Extra Resources


•

Vocabulary

• Power function

• Monomial function

• Radical function

• Extraneous solution

• Polynomial function

• Polynomial function of

degree n

• Leading coefficient

• Degree of a polynomial

• Leading term test

• Quartic function

• Turning points

• Quadratic form

• Repeated zero

• Multiplicity

• Polynomial long division

• Synthetic division

• Depressed polynomial

• Remainder theorem

• Factor theorem

• Rational zero theorem

• Upper bound

• Lower bound

• Upper and lower bound tests

• Descartes’ rule of signs

• Fundamental theorem of

algebra

• Linear factorization theorem

• Conjugate root theorem

• Complex conjugate

• Irreducible over the reals

• Rational function

• Asymptote

• Vertical asymptote

• Horizontal asymptote

• Slant (oblique) asymptote

• Hole

• Polynomial inequality

• Sign chart

• Rational inequality




Unit 3

Exponential & Logarithmic Functions

12 Days


Number and Quantity

P. N.NE.A.3 Classify real numbers and order real numbers that include

transcendental expressions, including roots and fractions of π and e.

I can classify real numbers and order real numbers that include transcendental

expressions, including roots and fractions of π and e.

P.N.NE.A.2 ★Understand the inverse relationship between exponents and

logarithms and use this relationship to solve problems involving logarithms

and exponents.

I can demonstrate understanding of the inverse relationship between

exponents and logarithms.

I can solve problems containing logarithms and exponents.

I can change an equation from logarithmic to exponential form and back.

I can solve exponential equations.

I can solve logarithmic equations.

I can prove basic properties of a logarithm using properties of its inverse and

apply those properties to solve problems.

I can find the inverse of an exponential or a logarithmic function.

P.N.NE.A.1 Use the laws of exponents and logarithms to expand or collect

terms in expressions; simplify expressions or modify them in order to analyze

them or compare them.

I can use the laws of exponents and logarithms to expand or collect terms in

expressions; simplify expressions or modify them in order to analyze them or

compare them.

I can compare exponential and logarithmic expressions.

Functions

P.F.IF.A.2 ★ Analyze qualities of exponential, polynomial, logarithmic, trigonometric, and rational functions and solve real world problems that can be modeled with these functions (by hand and with appropriate technology).

I can identify or analyze the following properties of exponential, logarithmic

and logistic functions: Domain, Range, Continuity, Increasing/Decreasing

Behavior, Symmetry, Boundedness, Extrema, Asymptotes, Intercepts, Holes,

End Behavior with Limit Notation, Concavity.

I can solve real world problems that can be modeled using quadratic,

exponential, or logarithmic functions (by hand and with appropriate

technology).

I can determine what function should be used to model a real-world situation.

I can apply the appropriate function to a real-world situation and then find its

solution.



P.MCPS.2 Apply appropriate techniques to analyze mathematical models and functions constructed from verbal information; interpret the solution obtained in written form using appropriate units of measurement.

I can create and analyze mathematical models that describe situations

including growth and decay and financial applications.


• Pre-assessment

• Section 1 Exponential Functions

• Section 2 Logarithmic Functions

• Section 3 Properties of Logarithms

• Quiz

• Section 4 Exponential and Logarithmic Equations

• Section 5 Modeling with Nonlinear Regression

• Unit Review

• Unit 3 Test

Extra Resources


•

Vocabulary

• Algebraic function

• Transcendental function

• Exponential function

• Natural base

• Compound interest formula

• Continuous compound

interest formula

• Exponential growth function

• Exponential decay function

• Continuous exponential

growth function

• Continuous exponential

decay function

• Logarithmic function with

base b

• Logarithmic form

• Exponential form

• Logarithms

• Properties of logarithms

• Common logarithm

• Natural logarithm

• Change of base formula

• One-to-one property of

exponential functions

• One-to-one property of

logarithmic functions

• Logistic growth function

• Linearizing data




Unit 4

Trigonometric Functions

16 Days


Functions

P.F.TF.A.1 Convert from radians to degrees and from degrees to radians. I can convert from radians to degrees and from degrees to radians.

P.F.TF.A.2 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 + x, and 2π – x in terms of their values for x, where x is any real number.

I can find the reference angle of any angle on the unit circle. I can evaluate

the trig functions of any angle of the unit circle using reference angles.

P.F.TF.A.3 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

I can use the unit circle to explain symmetry of the six trigonometric

functions.

I can describe periodicity of all six trigonometric functions.

P.F.GT.A.1★ Interpret transformations of trigonometric functions. I can match a trigonometric equation with its graph by recognizing the parent

graph and by using transformations.

P.F.GT.A.2★ Determine the difference made by choice of units for angle measurement when graphing a trigonometric function.

I can determine the difference made by choice of units for angle measurement

when graphing a trigonometric function.

P.F.GT.A.3★ Graph the six trigonometric functions and identify characteristics such as period, amplitude, phase shift, and asymptotes.

I can graph the six trigonometric functions (sin, cos, tan, csc, sec, cot) and

identify characteristics such as period, amplitude, phase shift, and

asymptotes.

P.F.GT.A.4★ Find values of inverse trigonometric expressions (including compositions), applying appropriate domain and range restrictions.

I can find values of inverse trigonometric functions, applying appropriate

domain and range restrictions.

P.F.GT.A.5★ Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

I can understand that restricting a trigonometric function to a domain on

which it is always increasing or always decreasing allows its inverse to be

constructed.

P.F.GT.A.6★ Determine the appropriate domain and corresponding range for each of the inverse trigonometric functions.

I can determine identify and list the appropriate domain and corresponding

range for each of the inverse trigonometric functions.

P.F.GT.A.7★ Graph the inverse trigonometric functions and identify their key characteristics.

I can graph the inverse trigonometric functions and identify their key

characteristics.

P.F.TF.A.4 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

I can choose trigonometric functions to model periodic phenomena with

specified amplitude, frequency, and midline.

Geometry

P.G.AT.A.3 Derive and apply the formulas for the area of sector of a circle.

I can derive and apply the formulas for the area of sector of a circle.

P.G.AT.A.4 Calculate the arc length of a circle subtended by a central angle.

I can calculate the arc length of a circle subtended by a central angle.

P.G.AT.A.1 ★Use the definitions of the six trigonometric ratios as ratios of sides in a right triangle to solve problems about lengths of sides and measures of angles.

I can use the definitions of the six trigonometric ratios as ratios of sides in a

right triangle to solve problems about lengths of sides and measures of

angles.



Statistics and Probability

P.S.MD.A.3★ Use a regression equation modeling bivariate data to make predictions. Identify possible considerations regarding the accuracy of predictions when interpolating or extrapolating.

I can use a regression equation modeling bivariate data to make predictions

involving trigonometric functions.

P.MCPS.1 Apply the arc length formula or conversion factors to real world applications.

I can apply linear and angular velocity formulas in real world applications.


• Pre-assessment

• Section 1 Right Triangle Trigonometry

• Section 2 Degrees and Radians, Arc Length, Area of a Sector, Linear

and Angular Speed

• Section 3 Trigonometric Functions on the Unit Circle

• Quiz

• Section 4 Graphing Sine and Cosine Functions

• Section 5 Graphing Tangent, Cosecant, and Secant

• Quiz

• Section 6 Inverse Trigonometric Functions

• Section 7 The Law of Sines and Law of Cosines

• Unit Review

• Unit 4 Test

Extra Resources


•

Vocabulary

• Trigonometric ratios

• Trigonometric functions

• Sine

• Cosine

• Tangent

• Cosecant

• Secant

• Cotangent

• Reciprocal functions

• Inverse trigonometric

function

• Inverse sine

• Inverse cosine

• Inverse tangent

• Angle of elevation

• Angle of depression

• Solving a right triangle

• Vertex

• Initial side

• Terminal side

• Standard position of an

angle

• Radians

• Degrees

• Coterminal angles

• Arc length

• Linear speed

• Angular speed

• Sector

• Area of a sector

• Quadrantal angle

• Reference angle

• Unit circle

• Circular functions

• Periodic functions

• Period

• Sinusoid

• Amplitude

• Frequency

• Phase shift

• Vertical shift

• Midline

• Damped oscillation

• Damped trigonometric

function

• Damping factor

• Damped harmonic motion

• Arcsine function

• Arccosine function

• Arctangent function

• Oblique triangles

• Law of sines

• Law of cosines

• Ambiguous case

• Heron’s formula




Unit 5

Trigonometric Identities & Equations

13 Days


Geometry

P.G.TI.A.1 ★Apply trigonometric identities to verify identities and solve

equations. Identities include: Pythagorean, reciprocal, quotient,

sum/difference, double-angle, and half-angle.

I can recognize and use the following trigonometric identities to verify

identities and solve trigonometric equations: Pythagorean, Reciprocal,

Quotient, Sum/Difference, Double Angle.

P.G.TI.A.2 ★ Prove the addition and subtraction formulas for sine, cosine,

and tangent and use them to solve problems.

I can prove the sum and difference formulas for sine, cosine, and tangent and

apply them in solving problems.

P.G.AT.A.2 ★Derive the formula A = ½ ab sin C for the area of a triangle by

drawing an auxiliary line from a vertex perpendicular to the opposite side.

I can derive the area of triangle formula A = ½ ab sin C by constructing a

drawing to model the situation

P.G.AT.A.5 ★Prove the Laws of Sines and Cosines and use them to solve

problems. I can prove the Law of Sines and Cosines and apply them to solve problems.

P.G.AT.A.6 ★Understand and apply the Law of Sines (including the

ambiguous case) and the Law of Cosines to find unknown measurements in

right and non-right triangles (e.g., surveying problems, resultant forces).

I can apply the Law of Sines (including the ambiguous case) and Cosines in

order to solve right and oblique triangles.

I can solve real work problems (e.g. surveying, navigation.)

I can determine how many solutions are possible for the Ambiguous case of

the Law of Sines.

I can determine when it is appropriate to use A = ½ ab sin C and Heron’s

Law.

I can find areas of triangles using the two area formulas A = ½ ab sin C and

Heron’s Law.




• Pre-assessment

• Section 1 Trigonometric Identities

• Section 2 Verifying Trigonometric Identities

• Quiz

• Section 3 Solving Trigonometric Equations

• Section 4 Sum and Difference Identities

• Section 5 Multiple-Angle and Product-to-Sum Identities

• Unit Review

• Unit 5 Test

Extra Resources


Vocabulary

• Identity

• Trigonometric identity

• Pythagorean identities

• Cofunction identities

• Odd-even identities

• Verify an identity

• Sum and difference

identities

• Reduction identity

• Double-angle identities

• Power-reducing identities

• Half-angle identities

• Product-to-sum identities




Unit 6

Matrices

12 Days


Number and Quantity

P.N.VM.C.7 Use matrices to represent and manipulate data, e.g., to represent

payoffs or incidence relationships in a network.

I can use matrices to represent and manipulate data, e.g., to represent payoffs

or incidence relationships in a network.

P.N.VM.C.8 Multiply matrices by scalars to produce new matrices, e.g., as

when all of the payoffs in a game are doubled.

I can multiply matrices by scalars to produce new matrices, e.g., as when all

of the payoffs in a game are doubled.

P.N.VM.C.9 Add, subtract, and multiply matrices of appropriate dimensions.

I can determine if matrices may be added, subtracted or multiplied by using

their dimensions.

I can add, subtract, and multiply matrices of appropriate dimensions.

P.N.VM.C.10 Understand that, unlike multiplication of numbers, matrix

multiplication for square matrices is not a commutative operation, but still

satisfies the associative and distributive properties.

I can show that matrix multiplication for square matrices is not a

commutative operation, but still satisfies the associative and distributive

properties.

P.N.VM.C.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.

I can show how 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.

I can explain how the determinant of a square matrix is non-zero if and only if

the matrix has a multiplicative inverse.

P.N.VM.C.12 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

I can multiply a vector (regarded as a matrix with one column) by a matrix of

suitable dimensions to produce another vector.

I can work with matrices as transformations of vectors.

P.N.VM.C.13 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

I can work with 2 × 2 matrices as transformations of the plane, and interpret

the absolute value of the determinant in terms of area.

Algebra

P.A.REI.A.1 Represent a system of linear equations as a single matrix

equation in a vector variable.

I can represent a system of equations as a single matrix equation in a vector

variable.

P.A.REI.A.2 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).

I can find the inverse of a matrix if it exists and use it to solve systems of

linear equations. I can use technology when solving system of equations

represented my matrices of dimensions 3 × 3 or greater.

P.MCPS.5 Solve maximum/minimum value problems by converting the

given verbal information into an appropriate mathematical model and

analyzing the graph of that model graphically to answer the questions.

Recognize the approximation necessary when solving graphically.

I can solve challenging optimization problems involving three dimensional

figures, i.e. boxes, cones.

I can describe the solution process by analyzing the graph and constructing

arguments to explain this reasoning.

I can use precise language to write solutions to max/min problems.




• Pre-assessment

• Section 1 Multivariable Linear Systems and Row Operations

• Section 2 Matrix Multiplication, Inverses, and Determinants

• Section 3 Solving Linear Systems Using Inverses and Cramer’s Rule

• Quiz

• Section 4 Partial Fractions

• Section 5 Linear Optimization

• Unit Review

• Unit 6 Test

Extra Resources


•

Vocabulary

• Multivariable linear system

• Row-echelon form

• Gaussian elimination

• Augmented matrix

• Coefficient matrix

• Elementary row operations

• Reduced row-echelon form

• Gauss-Jordan elimination

• Properties of matrix

multiplication

• Identity matrix

• Scalar

• Inverse matrix

• Invertible

• Singular matrix

• Square matrix

• Determinant

• Expansion by minors

• Square system

• Invertible square linear

system

• Cramer’s Rule

• Partial fraction

• Partial fraction

decomposition

• Optimization

• Linear programming

• Objective function

• Constraints

• Feasible region

• Feasible solutions

• Multiple optimal solutions

• Unbounded

• Unbounded region




Unit 7

Conic Sections

7 Days


Algebra

P.A.C.A.1 Display all of the conic sections as portions of a cone. I can display all of the conic sections as portions of a cone.

P.A.C.A.2 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.

I can 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.

P.A.C.A.3 From an equation in standard form, graph the appropriate conic

section: ellipses, hyperbolas, circles, and parabolas. Demonstrate an

understanding of the relationship between their standard algebraic form and

the graphical characteristics.

I can graph ellipses and hyperbolas and demonstrate understanding of the

relationship between their standard algebraic form and the graphical

characteristics.

P.A.C.A.4 Transform equations of conic sections to convert between general

and standard form.

I can transform equations of conic sections to convert between general and

standard form.


• Pre-assessment

• Section 1 Parabolas

• Section 2 Ellipses and Circles

• Quiz

• Section 3 Hyperbolas

• Section 4 Rotations of Conic Sections

• Unit Review

• Unit 7 Test

Extra Resources


•

Vocabulary

• Conic section

• Locus

• Parabola

• Focus directrix

• Axis of symmetry

• Vertex

• Ellipse

• Foci

• Major axis

• Minor axis

• Center

• Vertices

• Co-vertices

• Eccentricity

• Standard form of the

equation of a circle

• Hyperbola

• Transverse axis

• Conjugate axis




Unit 8

Polar Graphing & Vectors

16 Days


Number and Quantity

P.N.VM.A.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).

I can represent vectors graphically with both magnitude and direction.

I can represent vectors by directed line segments and use appropriate symbols

for vectors and their magnitudes.

I can interpret vectors geometrically and their relationship to real life

problems.

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

of an initial point from the coordinates of a terminal point.

I can demonstrate that vectors are determined by the coordinates of their

initial and terminal points, or by their components.

P.N.VM.B.3 Solve problems involving velocity and other quantities that can

be represented by vectors. I can solve velocity problems with vectors.

P.N.VM.B.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, with 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.

I can add and subtract vectors using a variety of methods and multiple

representations.

I can represent vectors and vector arithmetic graphically by creating a

resultant vector.

I can calculate the magnitude and direction angle of a resultant vector.

I can represent vector subtraction graphically.

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

a. Represent scalar multiplication graphically by scaling 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|.

c. 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).

I can multiply a vector by a scalar algebraically and by modeling them

graphically.

I can calculate the magnitude and the direction angle of a scalar multiple of a

vector.

P.N.CN.A.4 Represent addition, subtraction, multiplication, and conjugation

of complex numbers geometrically on the complex plane; use properties of

this representation for computation.

I can represent addition, subtraction, multiplication, and division of complex

numbers geometrically on the complex plane.



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

120°.

P.N.CN.A.5 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.

I can calculate the distance between numbers in the complex plane as the

magnitude or modulus of the difference by finding the absolute value of the

complex number.

I can calculate the midpoint of a segment as the average of the numbers at its

endpoints.

P.N.VM.B.6 Calculate and interpret the dot product of two vectors.

I can interpret the dot product of two vectors

I can use the dot product to find the angle between two vectors.

P.N.CN.A.1 Perform arithmetic operations with complex numbers expressing

answers in the form a + bi.

I can perform arithmetic operations with complex numbers expressing

answers in the form a + bi.

P.N.CN.A.2 Find the conjugate of a complex number; use conjugates to find

moduli and quotients of complex numbers.

I can find the conjugate of a complex number and use them to find moduli

and quotients of complex numbers.

P.N.CN.A.3 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.

I can represent complex numbers on the complex plane in rectangular and

polar form (including real and imaginary numbers).

I can explain why the rectangular and polar forms of a given complex number

represent the same number.

Geometry

P.G.PC.A.1 Graph functions in polar coordinates. I can graph functions in polar coordinates.

P.G.PC.A.2 Convert between rectangular and polar coordinates. I can convert between rectangular and polar coordinates.

P.G.PC.A.3 ★ Represent situations and solve problems involving polar

coordinates.

I can represent complex numbers on the complex plane in rectangular and

polar form.

P.MCPS.4 Apply De Moivre’s Theorem to find powers and roots of complex

numbers.

I can apply De Moivre’s Theorem to find powers and roots of complex

numbers.




• Pre-assessment

• Section 1 Introduction to Vectors

• Section 2 Vectors in the Coordinate Plane

• Section 3 Dot Products and Vector Projections

• Quiz

• Section 4 Vectors in Three-Dimensional Space

• Section 5 Dot Products of Vectors in Space

• Mid-Unit Quiz

• Section 6 Polar Coordinates

• Section 7 Graphs of Polar Equations

• Section 8 Polar and Rectangular Forms of Equations

• Quiz

• Section 9 Polar Forms of Conic Sections

• Section 10 Complex numbers and DeMoivre’s Theorem

• Unit Review

• Unit 8 Test

Extra Resources


•

Vocabulary

• Vector

• Initial point

• Terminal point

• Standard position of a

vector

• Direction of a vector

• Magnitude of a vector

• Quadrant bearing

• True bearing

• Parallel vectors

• Equivalent vectors

• Opposite vectors

• Resultant

• Triangle method

• Parallelogram method

• Zero vector

• Scalar of a vector

• Components

• Rectangular components

• Component form

• Unit vector

• Linear combination

• Dot product

• Orthogonal

• Vector projection

• Work

• Three-dimensional

coordinate system

• z-axis

• Ordered triple

• Cross product

• Torque

• Parallelpiped

• Triple scalar product

• Polar coordinate system

• Pole

• Polar axis

• Polar coordinates

• Polar equation

• Polar graph

• Polar distance formula

• Limaçon

• Cardioid

• Rose

• Lemniscate

• Spiral of Archimedes

• Complex plane

• Real axis

• Imaginary axis

• Argand plane

• Absolute value of a complex

number

• Polar form

• Trigonometric form

• Modulus

• Argument

• DeMoivre’s Theorem

• pth roots of unity




Unit 9

Parametric Equations

6 Days


Algebra

P.A.PE.A.1 ★Graph curves parametrically (by hand and with appropriate

technology). I can graph parametrically by hand and with appropriate technology.

P.A.PE.A.2 ★Eliminate parameters by rewriting parametric equations as a

single equation.

I can eliminate parameters by rewriting parametric equations as a single

equation.

P.MCPS.3 Simulate motion using parametric equations. I can simulate motion using parametric equations.


• Pre-assessment

• Section 1 Graph Parametric Equations

• Section 2 Write Parametric Equations in Rectangular Form

• Unit Review

• Unit 9 Test

Extra Resources


•

Vocabulary

• Parametric equation

• Parameter

• Orientation

• Parametric curve

• Projectile motion

• Polar coordinates

• Rectangular coordinates




Unit 10

Sequences, Series, & Sigma Notation

14 Days


Algebra

P.A.S.A.1 Demonstrate an understanding of sequences by representing them

recursively and explicitly.

I can demonstrate an understanding of sequences by representing them

recursively and explicitly.

P.A.S.A.2 Use sigma notation to represent a series; expand and collect

expressions in both finite and infinite settings. I can use Sigma (Σ) notation to represent a series.

P.A.S.A.3 Derive and use the formulas for the general term and summation of

finite or infinite arithmetic and geometric series, if they exist.

a. Determine whether a given arithmetic or geometric series converges or

diverges.

b. Find the sum of a given geometric series (both infinite and finite).

c. Find the sum of a finite arithmetic series.

I can determine whether a given arithmetic or geometric series converges or

diverges.

I can find the sum of a given geometric series (both infinite and finite).

I can find the sum of a finite arithmetic series.

P.A.S.A.4 Understand that series represent the approximation of a number

when truncated; estimate truncation error in specific examples.

I can understand that the series represent the approximation of a number

when truncated; estimate truncation error in specific examples.

P.A.S.A.5 Know and apply the Binomial Theorem for the expansion of (x +

y)n in powers of x and y for a positive integer n, where x and y are any

numbers, with coefficients determined for example by Pascal’s Triangle.

I can know and apply the Binomial Theorem for the expansion of

(x + y)n in powers of x and y for a positive integer n, where x and y are any

numbers, with coefficients determined, for example, by Pascal’s Triangle.

Functions

P.F.IF.A.8 Recognize that sequences are functions, sometimes defined

recursively, whose domain is a subset of the integers.

For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1,

f(n+1) = f(n) + f(n-1) for n ≥ 1.

I can analyze a variety of types of situations modeled by functions and

recognize that sequences are functions, sometimes defined recursively.




• Pre-assessment

• Section 1 Sequences, Series, and Sigma Notation

• Section 2 Arithmetic Sequences and Series

• Section 3 Geometric Sequences and Series

• Quiz

• Section 4 Mathematical Induction

• Section 5 The Binomial Theorem

• Section 6 Functions as Infinite Series

• Unit Review

• Unit 10 Test

Extra Resources


•

Vocabulary

• Sequence

• Term

• Finite sequence

• Infinite sequence

• Explicit formula

• Recursive formula

• Fibonacci sequence

• Converge

• Diverge

• Series

• Finite series

• Infinite series

• nth partial sum

• Sigma notation

• Arithmetic sequence

• Common difference

• Arithmetic means

• First difference

• Second difference

• Arithmetic series

• Sum of a finite arithmetic

series

• Sum of an infinite

arithmetic series

• Geometric sequence

• Common ratio

• nth term of a geometric

sequence

• Geometric means

• Geometric series

• Sum of a finite geometric

series

• Sum of an infinite geometric

series

• Principle of mathematical

induction

• Anchor step

• Inductive hypothesis

• Inductive step

• Extended principle of

mathematical induction

• Binomial coefficients

• Pascal’s triangle

• Binomial Expansion

• Binomial Theorem

• Power series

• Exponential series

• Euler’s Formula


The following Practice Standards and Literacy Skills will ...€¦ · Maury County Public Schools...

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