2014-2015 curriculum blueprint grade: 9-12 course: pre-calculus unit 1… · 2015-01-16 ·...

Updated 1-16-15

Time Allowed:

15 Days

2014-2015 Curriculum Blueprint Grade: 9-12 Course: Pre-Calculus

Unit 1: Conic Sections

Instructional Focus Benchmarks

The below benchmark(s) is linked to the CPALMS site that contains the Specifications to include the Content limits, Attributes/Stimulus, and additional information. EduSoft Mini-Assessment(s): Date Range: Given during the instruction per the outline in this section Key Vocabulary:

Conic Section Circle Ellipse Hyperbola Parabola Focus Vertex Branches Directrix Latus Rectum Degenerate Conic Sections Eccentricity

Learning Goal:

Students will be able to model, graph, analyze, and derive equations of conic sections and solve problems involving real-world applications.

Objectives:

Derive equations for conic sections given key features.

Recognize, graph, analyze, and transform equations of conic sections, with and without graphing calculators.

Evaluate, analyze, model, and solve real-world applications of conic sections.

Benchmarks/Standards Supporting Florida Standards MACC.912.G-GPE.1.1 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. MACC.912.G-GPE.1.2 Derive the equation of a parabola given a focus and directrix. MACC.912.G-GPE.1.3 Derive the equations of

ellipses and hyperbolas given the foci and

directrices.

Essential Content & Understanding:

A. Analyze and graph equations of conic

sections, with and without graphing

calculators

B. Write the equations of conic sections in

standard form and general form

Completing the square

C. Identify type of conic section from its

equation

D. Derive equations of conic sections given key

features

Circle, given center and radius

Parabola, given a focus and directrix

Ellipse, given foci and directrices

Hyperbola, given foci and directrices

E. Identify the geometric properties of a conic

section using its equation and graph

Circle: center, radius, intercepts

Ellipse: center, foci. vertices, major axis,

minor axis

Hyperbola: center, foci, vertices,

asymptotes, transverse axis, conjugate

axis

Parabola: Focus, vertex, directrix, axis of

symmetry, latus rectum

Essential Questions:

How are the shapes, known as conic sections,

created?

How are equations of conic sections analyzed

and graphed?

What are the similarities and differences in

the equations of conic sections and the key

features of their graphs?

How can the general form of a conic section

equation be converted to standard form?

What real-world issues could be analyzed

and solved using equations and graphs of

conic sections?

Resources/Links: High School Flip Book on CCSSM - A user-friendly resource for understanding the specifications of the Common Core Standards with examples, misconceptions, and connections Supplemental Resources: http://illuminations.nctm.org/ Standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers. http://www.cpalms.org/Courses Resources aligned to the new Florida Standards Writing Links: Writing Template Tasks template tasks designed from the Mathematical Practice Standards.

http://www.cpalms.org/Public/PreviewStandard/Preview/5627

http://www.cpalms.org/Public/Rummage/Search?hierarchyTag=Searches.RelatedStandards&anchorEntityTag=Searches.Courses&anchorEntityId=10352




http://katm.org/wp/wp-content/uploads/flipbooks/High-School-CCSS-Flip-Book-USD-259-2012.pdf

http://illuminations.nctm.org/

http://www.cpalms.org/Public/PreviewCourse/Preview/10352


http://www.utdanacenter.org/wp-content/uploads/math_template_and_teaching_tasks.pdf

Updated 1-16-15

Time Allowed:

15 Days


Unit 1: Conic Sections

F. Solve real-world problems involving conic

sections

Glencoe: Teaching Today Article An article outlining tips for teachers to use when planning writing tasks in math. Higher Order Questioning: Mathematical Practice Standards Rubric Questions, writing, and tasks aligned to the Mathematical Practice Standards Teaching & Learning HOQ Reference Sheets Tool to assist teachers in developing higher order questioning and student tasks Webb’s DOK Guide Tools to promote classroom discourse aligned to higher levels of cognitive demand. Teaching Channel Video 2 min video with focus on Improving Participation with Talk Moves (Personalized Learning Opportunity). Remediation & Enrichment Resources: Khanacademy.org free resources for teachers and

students with interactive challenges, assessments,

and videos from any computer with access to the

web

http://www.glencoe.com/sec/teachingtoday/subject/int_writing_math.phtml

http://intranet.staff.lakeschools.local/cms/lib/FL01000800/Centricity/Domain/11/math/Rubric%20for%20Measuring%20Rigor%20in%20the%20Math%20Classroom.pdf

http://intranet.staff.lakeschools.local/cms/lib/FL01000800/Centricity/Domain/11/math/WebsDepthofKnowledgeFlipChart.pdf

https://www.teachingchannel.org/videos/student-participation-strategy?fd=1

http://www.khanacademy.org/math/precalculus

Updated 1-16-15

Time Allowed:

20 Days


Unit 2: Unit Circle and Trigonometric Ratios



Sine Cosine Tangent Cotangent Secant Cosecant Reference Angle Unit Circle Quadrantal Angle Radian Arc Length Coterminal Angles

Learning Goal:

Students will be able to sketch angles, define and use radian measures, explain characteristics of trigonometric functions, and evaluate trigonometric

ratios and their inverses using special right triangles and the unit circle.

Objectives:

Illustrate angles and trigonometric ratios in the rectangular coordinate system.

Evaluate trigonometric ratios and their inverses using right triangles and the unit circle, with and without graphing calculators.

Explain characteristics of trigonometric functions using the unit circle.

Benchmarks/Standards Supporting Florida Standards MACC.912.F-TF.1.1 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle. MACC.912.F-TF.1.2 Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. MACC.912.F-TF.1.3 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+x, and 2π–x in terms of their values for x, where x is any real number. MACC.912.F-TF.1.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.


A. Sketch angles in standard position using

protractors

Define terminal and initial sides

Identify the direction for angles

B. Define the radian measure of an angle on the

unit circle

Convert between degree and radian

measures

Explain the relationship between radian

measure of an angle along the unit circle,

terminal coordinate of that angle, and the

associated real number

C. Explain how the unit circle allows the

extension of trigonometric functions to all

real numbers

D. Identify and use reference angles, reference

triangles, and their quadrants to evaluate

trigonometric ratios

E. Determine sine, cosine, and tangent of angles

using the unit circle

F. Find and use exact values of trigonometric

functions for special angles using degree and


What is trigonometry?

How are angles illustrated and measured?

What is meant by one radian?

What is the difference between positive and

negative measures of angles?

How can trigonometric ratios be evaluated?

How can diagrams be drawn to determine

values of trigonometric ratios?

What real-world issues might be analyzed

using trigonometric ratios?


http://www.cpalms.org/Standards/PublicPreviewBenchmark5589.aspx









Updated 1-16-15

Time Allowed:

20 Days


Unit 2: Unit Circle and Trigonometric Ratios

radian measures

G. Evaluate inverse trigonometric ratios with

and without graphing calculators

H. Explain symmetry (odd and even) and the

periodicity of trigonometric functions using

the unit circle




web






Updated 1-16-15

Time Allowed:

15 Days


Unit 3: Triangles and Trigonometric Ratios



Cofunctions Angle of Depression Angle of Elevation Oblique Representations Ambiguous Case

Learning Goal:

Students will be able to prove the Pythagorean Identity, the Law of Sines, and the Law of Cosines and be able to use them in solving right and oblique

triangles through the context of real-world applications.

Objectives:

Prove the Pythagorean Identity.

Formulate strategies and use them to solve right and oblique triangles.

Prove the Laws of Sines and Cosines and derive the sine area of a triangle formula.

Solve real-world applications of right and oblique triangles.

Benchmarks/Standards Supporting Florida Standards MACC.912.F-TF.3.8 Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

MACC.912.G-SRT.3.8 Use trigonometric ratios and

the Pythagorean Theorem to solve right triangles in

applied problems.

MACC.912.G-SRT.4.9 Derive the formula A = 1/2 ab

sin(C) for the area of a triangle by drawing an

auxiliary line from a vertex perpendicular to the

opposite side.

MACC.912.G-SRT.4.10 Prove the Laws of Sines and Cosines and use them to solve problems.

MACC.912.G-SRT.4.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).


A. Prove the Trigonometric Pythagorean

Identity

B. Use trigonometric ratios to solve triangles

Right triangles

o Special right triangles

o Two sides given

o One side and one acute angle given

Oblique triangles

o Using Law of Sines

o Using Law of Cosines

C. Derive the area of a triangle formula which

uses the sine ratio

D. Prove the Law of Sines and Cosines

E. Solve real-world problems involving right

and oblique triangles, including but not

limited to angle measures, side lengths, and

area of triangles and similar triangles


What are the similarities and differences

between the sine and cosine ratios?

What are trigonometric identities and how

are they proven?

How can the Pythagorean Identity be

manipulated to include other versions?

What methods can be used to solve right and

oblique triangles?

What real-world problems can be analyzed

and solved using right and oblique triangles?


















Updated 1-16-15

Time Allowed:

15 Days


Unit 3: Triangles and Trigonometric Ratios




web






Updated 1-16-15

Time Allowed:

20 Days


Unit 4: Trigonometric Functions and Their Graphs



Sinusoidal Graphs Period Frequency Amplitude Phase Shift Vertical Shift Midline Scale Oscillate Inverse Trigonometric Functions

Learning Goal:

Students will be able to model, graph, and analyze real-world and mathematical periodic phenomena using trigonometric functions and their inverses,

with and without graphing calculators.

Objectives:

Sketch all six parent trigonometric functions, including transformations, with and without graphing calculators, and analyze their key features.

Model periodic phenomena with trigonometric functions and solve their real-world applications.

Understand why domains of trigonometric functions must be restricted in order to construct inverses and sketch their graphs.

Benchmarks/Standards Supporting Florida Standards MACC.912.F-TF.2.5 Choose trigonometric

functions to model periodic phenomena with

specified amplitude, frequency, and midline.

MACC.912.F-TF.2.6 Understand that restricting a

trigonometric function to a domain on which it is

always increasing or always decreasing allows its

inverse to be constructed.


A. Graph trigonometric functions on graph paper

and identify and interpret key features, with

and without graphing calculators

Domain and range

Intercepts

Period

Frequency

Amplitude

Midline

Phase shift

Vertical shift

Vertical asymptotes

Setting the appropriate window on the

graphing calculator

B. Solve real-world problems involving

applications of trigonometric functions

C. Explain why real-world or mathematical

phenomena exhibit periodicity and model that

behavior with appropriate trigonometric

functions

D. Sketch graphs for inverse trigonometric

functions and understand how the domains of

the functions must be restricted


What are the key characteristics of the sine

curve?

What is the relationship between the sine

function and the cosine function?

How are the graphs of sine and cosine

functions used to obtain graphs of cosecant

and secant functions?

Why can trigonometric functions be used to

model periodic behavior?

Why must the domains of trigonometric

functions be restricted to allow

trigonometric functions to have inverses?

What real-world issues can be analyzed and

solved using trigonometric functions?

Resources/Links: High School Flip Book on CCSSM - A user-friendly resource for understanding the specifications of the Common Core Standards with examples, misconceptions, and connections Supplemental Resources: http://illuminations.nctm.org/ Standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers. http://www.cpalms.org/Courses Resources aligned to the new Florida Standards







Updated 1-16-15

Time Allowed:

20 Days


Unit 4: Trigonometric Functions and Their Graphs

Writing Links: Writing Template Tasks template tasks designed from the Mathematical Practice Standards. Glencoe: Teaching Today Article An article outlining tips for teachers to use when planning writing tasks in math. Higher Order Questioning: Mathematical Practice Standards Rubric Questions, writing, and tasks aligned to the Mathematical Practice Standards Teaching & Learning HOQ Reference Sheets Tool to assist teachers in developing higher order questioning and student tasks Webb’s DOK Guide Tools to promote classroom discourse aligned to higher levels of cognitive demand. Teaching Channel Video 2 min video with focus on Improving Participation with Talk Moves (Personalized Learning Opportunity). Remediation & Enrichment Resources: Khanacademy.org free resources for teachers and



web







Updated 1-16-15

Time Allowed:

15 Days


Unit 5: Trigonometric Identities and Equations



Identity Verifying an Identity Equivalent Expressions Substitution

Learning Goal:

Students will be able to verify trigonometric identities, prove addition and subtraction formulas, use formulas to evaluate trigonometric functions, and

solve trigonometric equations through real-world applications.

Objectives:

Create logically valid algebraic steps for verifying trigonometric identities and proving trigonometric formulas.

Evaluate sine, cosine, and tangent functions using special angle formulas.

Solve trigonometric equations through real-world applications, with and without graphing calculators.

Benchmarks/Standards Supporting Florida Standards MACC.912.F-TF.3.9 Prove the addition and

subtraction formulas for sine, cosine, and tangent

and use them to solve problems.

MACC.912.F-TF.2.7 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.


A. Use the basic trigonometric identities and

algebraic methods to verify other identities

Rewrite as equivalent expressions

Prove the sum and difference, double-

angle, and half-angle formulas for sine,

cosine, and tangent

B. Evaluate sine, cosine, and tangent functions

using trigonometric formulas given known

angles or one trigonometric relationship

and its quadrant

Sum and difference formulas

Double-angle and half-angle formulas

C. Solve trigonometric equations through real-

world applications, with and without

graphing calculators

Using algebraic methods

Using substitution of identities and

formulas

Using inverse trigonometric functions


What strategies are used in verifying an

identity?

What are similarities and differences

between verifying an identity and writing a

formal proof?

How can the double angle formulas be

derived?

What are solutions to trigonometric

equations?

What methods used in verifying an identity

can be used to solve a trigonometric

equation?

Resources/Links: High School Flip Book on CCSSM - A user-friendly resource for understanding the specifications of the Common Core Standards with examples, misconceptions, and connections Supplemental Resources: http://illuminations.nctm.org/ Standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers. http://www.cpalms.org/Courses Resources aligned to the new Florida Standards Writing Links: Writing Template Tasks template tasks designed








Updated 1-16-15

Time Allowed:

15 Days


Unit 5: Trigonometric Identities and Equations

from the Mathematical Practice Standards. Glencoe: Teaching Today Article An article outlining tips for teachers to use when planning writing tasks in math. Higher Order Questioning: Mathematical Practice Standards Rubric Questions, writing, and tasks aligned to the Mathematical Practice Standards Teaching & Learning HOQ Reference Sheets Tool to assist teachers in developing higher order questioning and student tasks Webb’s DOK Guide Tools to promote classroom discourse aligned to higher levels of cognitive demand. Teaching Channel Video 2 min video with focus on Improving Participation with Talk Moves (Personalized Learning Opportunity). Remediation & Enrichment Resources: Khanacademy.org free resources for teachers and



web






Updated 1-16-15

Time Allowed:

15 Days


Unit 6: Polar Coordinates and Complex Numbers



Modulus Argument Polar Form

Learning Goal:

Students will be able to represent points and equations in both rectangular and polar forms with and without graphing calculators, represent complex

numbers in both rectangular and polar forms, and operate with complex numbers in polar form.

Objectives:

Convert points and equations from rectangular to polar form and vice versa.

Graph polar equations, with and without graphing calculators.

Represent complex numbers in rectangular and polar forms, and explain why they represent the same number.

Represent operations of complex numbers geometrically on the complex plane and use them to operate with complex numbers.

Apply DeMoivre’s Theorem to operations with complex numbers.

Benchmarks/Standards Supporting Florida Standards MACC.912.N-CN.1.3 Find the conjugate of a complex

number; use conjugates to find moduli and quotients

of complex numbers.

MACC.912.N-CN.2.4 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.

MACC.912.N-CN.2.5 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°.


A. Define polar coordinates and relate them to

Cartesian coordinates

B. Represent equations given in rectangular

coordinates in terms of polar coordinates

and vice versa

C. Graph equations in the polar coordinate

plane, with and without graphing

calculators

D. Convert complex numbers written in

rectangular form to trigonometric form and

vice versa

E. Represent operations of complex numbers

geometrically on the complex plane and use

those representations for computation

F. Apply DeMoivre’s Theorem to operations

with complex numbers


How are the coordinates of points written in

rectangular form converted to polar form?

How are equations written in rectangular

form converted to polar form?

How is graphing in the polar coordinates

different from graphing in the rectangular

system?

How are complex numbers converted to

polar form?

How do you operate on complex numbers

written in polar form?

Resources/Links: High School Flip Book on CCSSM - A user-friendly resource for understanding the specifications of the Common Core Standards with examples, misconceptions, and connections Supplemental Resources: http://illuminations.nctm.org/ Standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers. http://www.cpalms.org/Courses Resources aligned to the new Florida Standards Writing Links:








Updated 1-16-15

Time Allowed:

15 Days


Unit 6: Polar Coordinates and Complex Numbers

Writing Template Tasks template tasks designed from the Mathematical Practice Standards. Glencoe: Teaching Today Article An article outlining tips for teachers to use when planning writing tasks in math. Higher Order Questioning: Mathematical Practice Standards Rubric Questions, writing, and tasks aligned to the Mathematical Practice Standards Teaching & Learning HOQ Reference Sheets Tool to assist teachers in developing higher order questioning and student tasks Webb’s DOK Guide Tools to promote classroom discourse aligned to higher levels of cognitive demand. Teaching Channel Video 2 min video with focus on Improving Participation with Talk Moves (Personalized Learning Opportunity). Remediation & Enrichment Resources: Khanacademy.org free resources for teachers and



web







Updated 1-16-15

Time Allowed:

5 Days


Unit 7: Vectors



Vector Magnitude Position Vector Zero Vector Unit Vector Scalar Resultant Vector Velocity Vector

Learning Goal:

Students will be able to represent vectors and their operations using sketches, symbols, rectangular components, and in terms of its magnitude and

direction. Students will be able to determine magnitude of vectors, their sums and scalar multiples, and solve real-world applications of vectors.

Objectives:

Represent vectors using sketches, symbols, rectangular components, and in terms of its magnitude and direction.

Add and subtract vectors using their components, their geometric representations, and the parallelogram rule.

Multiply vectors by scalars.

Calculate magnitudes for vectors, sums of vectors, and scalar multiples of vectors.

Solve real-world applications of velocity and other quantities that can be modeled by vectors.

Benchmarks/Standards Supporting Florida Standards MACC.912.N-VM.1.1 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).

MACC.912.N-VM.1.2 Find the components of a vector

by subtracting the coordinates of an initial point

from the coordinates of a terminal point.

MACC.912.N-VM.1.3 Solve problems involving

velocity and other quantities that can be represented

by vectors.

MACC.912.N-VM.2.4 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


A. Sketch vectors using directed line segments

and appropriate symbols for them and their

magnitudes

B. Find the components of vectors

C. Perform vector operations

Add vectors

o Represent sums graphically by placing

vectors end-to-end

o Find sums using components

o Find sums using the parallelogram rule

o Determine the magnitude and

direction of the sum of vectors

Subtract vectors

o Represent differences graphically by

connecting the tips in the appropriate

order

o Find differences using components

Multiply a vector by a scalar

o Represent scalar multiplication

graphically

o Perform scalar multiplication using

components

o Compute the magnitude of a scalar


How are vectors represented and graphed?

How can vectors be added?

Why is the magnitude of a sum of two vectors

typically not the sum of the two magnitudes?

How can a vector written in rectangular

components be written in terms of its

magnitude and direction?

What real-world issues can be analyzed and

solved using vectors?












Updated 1-16-15

Time Allowed:

5 Days


Unit 7: Vectors

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.

MACC.912.N-VM.2.5 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 = .

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).

multiple

D. Solve real-world problems involving

velocity and other quantities that can be

modeled with vectors




web







Updated 1-16-15

Time Allowed:

10 Days


Unit 8: Exponential and Logarithmic Functions



Logarithmic Function Exponential Function Product Property Quotient Property Power Property Exponential Growth Exponential Decay Common Bases Natural Log

Learning Goal:

The student will demonstrate applying concepts to solve problems of exponential growth and decay, apply concepts to solve exponential and logarithmic

equations, analyze and transform graphs of exponential and logarithmic graphs, applying properties of exponential and logarithmic expression for

expanding and condensing.

Benchmarks/Standards Supporting Common Core Standards

MACC.912.A-SSE.1.1: (DOK 2) Interpret expressions that represent a quantity in terms of its context.

A. Interpret parts of an expression, such as terms, factors, and coefficients.

B. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

MAFS.912.F-BF.2.a: (DOK 1) Use the change of base formula.

MACC.912.F-BF.2.3: (DOK 2) Identify the effect on

the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and

f(x+k)for specific values of k (both positive and

negative); find the value of k given the graphs.

Experiment with cases and illustrate an explanation

of the effects on the graph using technology.

MAFS.912.F-IF.2.5: Relate the domain of a function to

its graph and, where applicable, to the quantitative

relationship it describes.

MACC.912.F-IF.3.7: (DOK 2) Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

MAFS.912.F-IF.3.8: Write a function defined by an

expression in different but equivalent forms to reveal


A. Write, Evaluate, and Graph exponential

functions

Define Exponential Function

Examine and Graph Growth

Examine and Graph Decay

Identify Domain and Range

Transform

B. Write, Evaluate, and Graph Logarithmic

Functions

Define Logarithmic Function

Identify Domain and Range

Identify Asymptotes

Identify Logs as Inverse of Exponential

C. Use properties to condense and expand logs

Product Property

Quotient Property

Power Property

D. Solve Exponential Equations

Real World

Inverse Properties (Different Bases)

Common Bases

E. Solve Logarithmic Equations

Properties

Real World

Inverses

Use change of base formula


What is the relationship between exponents

and logarithmic?

How can the relationship between logs and

exponents help solve problems of growth and

decay?

Resources/Links: High School Flip Book on CCSSM - A user-friendly resource for understanding the specifications of the Common Core Standards with examples, misconceptions, and connections Supplemental Resources: http://illuminations.nctm.org/ Standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers. http://www.cpalms.org/Courses Resources aligned to the new Florida Standards Writing Links: Writing Template Tasks template tasks designed from the Mathematical Practice Standards. Glencoe: Teaching Today Article An article outlining tips for teachers to use when planning writing tasks in math. Higher Order Questioning: Mathematical Practice Standards Rubric Questions, writing, and tasks aligned to the Mathematical Practice Standards Teaching & Learning HOQ Reference Sheets Tool














Updated 1-16-15

Time Allowed:

10 Days


Unit 8: Exponential and Logarithmic Functions

and explain different properties of the functions.

A. Use properties of exponents to interpret

expressions for exponential functions.

MACC.912.F-LE.1.4: (DOK 2) For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. Recognize the laws and properties of logarithms,

including change of base. Recognize and describe the key features of

logarithmic functions. Recognize and know the definition of logarithm

base b. Evaluate a logarithm using technology. For exponential models, express as a logarithm,

the solution to =d, where a, c, and d are numbers and the base b is 2, 10, or e.

MAFS.912.F-LE.2.5: Interpret the parameters in an

exponential function in terms of context.

Recognize linear or exponential function including: vertical and horizontal shifts, vertical and horizontal dilations.

Recognize rates of change and intercepts as parameters in linear or exponential functions.

Interpret the parameters in a linear or exponential function in terms of a context.

MAFS.912.N-Q.1.2: Define appropriate quantities for the purpose of descriptive modeling. Define descriptive modeling. Determine appropriate quantities for the

purpose of descriptive modeling.

to assist teachers in developing higher order questioning and student tasks Webb’s DOK Guide Tools to promote classroom discourse aligned to higher levels of cognitive demand. Teaching Channel Video 2 min video with focus on Improving Participation with Talk Moves (Personalized Learning Opportunity). Remediation & Enrichment Resources: Khanacademy.org free resources for teachers and



web







Updated 1-16-15

Time Allowed:

15 Days


Unit 9: Polynomial Identities and Rational Expressions



Fundamental Theorem of Algebra Closure Pascal’s Triangle Binomial Theorem

Learning Goal: Students will be able to verify and use polynomial identities, expand a binomial raised to a power, understand the relationship between the sets of rational expressions and rational numbers, and use that understanding to operate and simplify rational expressions.

Objectives:

Create logically valid algebraic steps to verify polynomial identities and use those identities to describe numerical relationships.

Understand the Fundamental Theorem of Algebra, and show that it is true for quadratic polynomials.

Expand binomials raised to a power.

Understand the relationship between rational expressions and rational numbers, and use that understanding to operate and simplify rational

expressions.

Benchmarks/Standards Supporting Florida Standards MACC.912.A-APR.3.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x + y) = (x y) + (2xy) can be used to generate Pythagorean triples. MACC.912.N-CN.3.9 Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. MACC.912.A-APR.3.5 Know and apply the Binomial Theorem for the expansion of (x 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. MACC.912.A-APR.4.6 Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a


A. Use basic identities and algebraic methods to

verify polynomial identities and use

polynomial identities to describe numerical

relationships

B. Understand the Fundamental Theorem of

Algebra and show that it is true for quadratic

polynomials

C. Expand a binomial raised to a power

Pascal’s Triangle

Binomial Theorem

D. Operate and simplify rational expressions

Understand the relationship between the

sets of rational expressions and rational

numbers, including closure

Add rational expressions

Subtract rational expressions

Multiply rational expressions


What is the Fundamental Theorem of Algebra, and how can it be verified for quadratic polynomials?

What patterns are found in the expansion of a binomial raised to a power?

How can Pascal’s Triangle and the Binomial Theorem be used to expand a binomial raised to a power?

How can Pascal’s Triangle and the Binomial Theorem be used to determine a particular term in a binomial expansion without finding the entire product?

How is operating on the set of rational expressions like operating on rational numbers?

What is the meaning of a set being closed under an operation?

What restrictions must be placed when dividing rational expressions?

How can a polynomial be divided by a monomial?

How can the Division Algorithm be used to verify a quotient and a remainder in long division?












Updated 1-16-15

Time Allowed:

15 Days


Unit 9: Polynomial Identities and Rational Expressions

computer algebra system. MACC.912.A-APR.4.7 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.

Divide rational expressions

o Inspection

o Long Division

o Computer Program

Resources/Links: High School Flip Book on CCSSM - A user-friendly resource for understanding the specifications of the Common Core Standards with examples, misconceptions, and connections Supplemental Resources: http://illuminations.nctm.org/ Standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers. http://www.cpalms.org/Courses Resources aligned to the new Florida Standards Writing Links: Writing Template Tasks template tasks designed from the Mathematical Practice Standards. Higher Order Questioning: Mathematical Practice Standards Rubric Questions, writing, and tasks aligned to the Mathematical Practice Standards Teaching & Learning HOQ Reference Sheets Tool to assist teachers in developing higher order questioning and student tasks Webb’s DOK Guide Tools to promote classroom discourse aligned to higher levels of cognitive demand. Teaching Channel Video 2 min video with focus on Improving Participation with Talk Moves (Personalized Learning Opportunity). Remediation & Enrichment Resources: Khanacademy.org free resources for teachers and



web













Updated 1-16-15

Time Allowed:

10 Days


Unit 10: Building Functions



Explicit Expression Recursive Formula Composition Inverse Functions Horizontal Line Test

Learning Goal:

Students will be able to build new functions by arithmetically combining functions, composing functions, and producing inverses of functions and using

them to model, analyze, and solve real-world applications of functions.

Objectives:

Combine functions using arithmetic operations and compositions.

Produce inverse functions, and verify those using equations, compositions, tables, and graphs.

Solve real-world applications of arithmetically combined functions, compositions of functions, and inverses of functions.

Benchmarks/Standards Supporting Florida Standards MACC.912.F-BF.1.1 Write a function that describes a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. 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. MACC.912.F-BF.2.4 Find inverse functions. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x or f(x) = (x+1)/(x1) for x 1. Verify by composition that one function is the inverse of another. Read values of an inverse function from a graph or a table, given that the function has an inverse. Produce an invertible function from a non-invertible function by restricting the domain.


A. Build functions

Combine functions using arithmetic

operations

Compose functions

Build functions that model real-world

situations and use them to solve

associated problems

B. Find inverse functions, including producing

an invertible function from a non-invertible

function by restricting the domain

Using algebraic method

Using mental math and the

understanding of inverses

Using tables

Using graphs

C. Verify inverse functions by composing the

functions in both orders


How can functions be combined? How are domains affected when functions

are combined arithmetically or composed? How can a function’s inverse be determined

and graphed?

What is the Horizontal Line Test, and what does it show?

How can inverse functions be verified? What real-world situations could be analyzed

and solved by combining functions? What real-world situations could be analyzed

and solved by composing functions? What real-world situations could be analyzed

and solved by producing inverse functions?

Resources/Links: High School Flip Book on CCSSM - A user-friendly resource for understanding the specifications of the Common Core Standards with examples, misconceptions, and connections Supplemental Resources: http://illuminations.nctm.org/ Standards-based resources for teaching and learning mathematics, including interactive tools for students and instructional support for teachers. http://www.cpalms.org/Courses Resources aligned to the new Florida Standards Writing Links:










Updated 1-16-15

Time Allowed:

10 Days


Unit 10: Building Functions

Writing Template Tasks template tasks designed from the Mathematical Practice Standards. Glencoe: Teaching Today Article An article outlining tips for teachers to use when planning writing tasks in math. Higher Order Questioning: Mathematical Practice Standards Rubric Questions, writing, and tasks aligned to the Mathematical Practice Standards Teaching & Learning HOQ Reference Sheets Tool to assist teachers in developing higher order questioning and student tasks Webb’s DOK Guide Tools to promote classroom discourse aligned to higher levels of cognitive demand. Teaching Channel Video 2 min video with focus on Improving Participation with Talk Moves (Personalized Learning Opportunity). Remediation & Enrichment Resources: Khanacademy.org free resources for teachers and



web







Updated 1-16-15

Time Allowed:

15 Days


Unit 11: Limits



Limit Continuity Discontinuity One-Sided Limit

Learning Goal:

Students will be able to find limits analytically, graphically, and numerically, as well as understand continuity as defined in terms of limits.

Objectives:

Understand the concept of a limit, estimate limits from graphs and tables, and apply the Intermediate Value and the Extreme Value Theorems.

Find limits using algebraic techniques.

Find the one-sided limit of a function.

Determine if a function is continuous at a point, and name the discontinuities of a function.

Benchmarks/Standards Supporting Florida Standards MA.912.C.1.2 Find limits by substitution. MA.912.C.1.3 Find limits of sums, differences, products, and quotients. MA.912.C.1.4 Find limits of rational functions that are undefined at a point. MA.912.C.1.5 Find one-sided limits. MA.912.C.1.1 Understand the concept of limit and estimate limits from graphs and tables of values.

MA.912.C.1.10 Decide if a function is continuous at a point.

MA.912.C.1.11 Find the types of discontinuities of a function.

MA.912.C.1.12 Understand and use the Intermediate Value Theorem on a function over a closed interval. MA.912.C.1.13 Understand and apply the Extreme Value Theorem: If f(x) is continuous over a closed interval, then f has a maximum and a minimum on the interval. MA.912.C.1.9 Understand continuity in terms of

limits.


For examples and teaching materials for this

Unit, please use all of Chapter 1 from the Larson

Calculus book.

A. Use algebraic techniques to find limits

Substitution

Sums, differences, products, and

quotients

Finding the limit of an average rate of

change

B. Understand limits intuitively

Graphical approach Numerical approach Analytical approach

o Intermediate Value Theorem

o Extreme Value Theorem

C. Determine the continuity and discontinuity

of functions

Finding one-sided limits

Determining continuity of functions

o Key functions

o Rational functions

o Piecewise functions

Naming the types of discontinuity


How can tables and graphs be used to find

limits?

How do the limits of functions help to describe the graphical representations of these functions?

What processes are necessary to find the limits of functions, and what does it mean when limits fail to exist?

What key functions are continuous at every number in the domain?

What key functions are not continuous at every point, and how can those domain values of discontinuity be found?

How are one-sided limits found? What are the types of discontinuity of a

function and how are they determined?

Resources/Links: TI-nSpire Calculator – TI’s website of activities that can be completed using the TI-nSpire calculator to teach Pre-Calculus topics. TI-84 Calculator – a website from TI with activities to teach Calculus and Pre-Calculus topics with the calculator. High School Flip Book on CCSSM - A user-friendly resource for understanding the specifications of the Common Core Standards with examples, misconceptions, and connections Supplemental Resources: http://illuminations.nctm.org/ Standards-based resources for teaching and learning mathematics,
















http://education.ti.com/en/timathnspired/us/precalculus

http://education.ti.com/en/timath/us/precalculus



Updated 1-16-15

Time Allowed:

15 Days


Unit 11: Limits

including interactive tools for students and instructional support for teachers. http://www.cpalms.org/Courses Resources aligned to the new Florida Standards Writing Links: Writing Template Tasks template tasks designed from the Mathematical Practice Standards. Glencoe: Teaching Today Article An article outlining tips for teachers to use when planning writing tasks in math. Higher Order Questioning: Mathematical Practice Standards Rubric Questions, writing, and tasks aligned to the Mathematical Practice Standards Teaching & Learning HOQ Reference Sheets Tool to assist teachers in developing higher order questioning and student tasks Webb’s DOK Guide Tools to promote classroom discourse aligned to higher levels of cognitive demand. Teaching Channel Video 2 min video with focus on Improving Participation with Talk Moves (Personalized Learning Opportunity). Remediation & Enrichment Resources: Khanacademy.org free resources for teachers and



web









Updated 1-16-15

Supporting Florida Standards

MATHEMATICS PRACTICE STANDARDS: MACC.K12.MP.1.1: 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. MACC.K12.MP.2.1: 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. MACC.K12.MP.3.1: 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 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. 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. MACC.K12.MP.4.1: 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. MACC.K12.MP.5.1: 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






Updated 1-16-15


various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. MACC.K12.MP.6.1: 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. MACC.K12.MP.7.1: 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. MACC.K12.MP.8.1: 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. LAFS.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements, or performing technical tasks, attending to special cases or exceptions defined in the text.

LAFS.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in a specific scientific or technical context relevant to grades 9–10 texts and topics.

LAFS.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation)

into words.

LAFS.910.SL.1.1:

Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9–10 topics, texts, and issues, building on others’

ideas and expressing their own clearly and persuasively.

a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or

issue to stimulate a thoughtful, well-reasoned exchange of ideas.

b. Work with peers to set rules for collegial discussions and decision-making (e.g., informal consensus, taking votes on key issues, presentation of alternate views), clear goals and deadlines,

and individual roles as needed.

c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and








Updated 1-16-15


clarify, verify, or challenge ideas and conclusions.

d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make

new connections in light of the evidence and reasoning presented.

LAFS.910.SL.1.2:

Integrate multiple sources of information presented in diverse media or formats (e.g., visually, quantitatively, orally) evaluating the credibility and accuracy of each source.

LAFS.910.SL.1.3:

Evaluate a speaker’s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence.

LAFS.910.SL.2.4:

Present information, findings, and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning and the organization, development, substance, and style are

appropriate to purpose, audience, and task.

LAFS.910.WHST.1.1:

Write arguments focused on discipline-specific content.

a. Introduce precise claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that establishes clear relationships among the claim(s), counterclaims,

reasons, and evidence.

b. Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-

appropriate form and in a manner that anticipates the audience’s knowledge level and concerns.

c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and

between claim(s) and counterclaims.

d. Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing.

e. Provide a concluding statement or section that follows from or supports the argument presented.

LAFS.910.WHST.2.4:

Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience.

LAFS.910.WHST.3.9:

Draw evidence from informational texts to support analysis, reflection, and research.







2014-2015 curriculum blueprint grade: 9-12 course: pre-calculus unit 1… · 2015-01-16 ·...

Documents