Lawrence High School Pre-calculus Curriculum Map...LAWRENCE HIGH SCHOOL PRE-CALCULUS CURRICULUM MAP 2015-2016 Updated 8/10/15 + denotes a Honors level concept The following are a list

LAWRENCE HIGH SCHOOL PRE-CALCULUS CURRICULUM MAP 2015-2016

Updated 8/10/15 + denotes a Honors level concept

Pre

-Cal

culu

s

Quarter 1

8/24-10/30

Chapter 1 - Analyzing Trigonometric Functions

1A - The Cosine and Sine Functions

1B - Other Trigonometric Functions

1C - Sinusoidal Fnctions and their Graphs

Chapter 2 - Complex Numbers and Trigonometry

2A* - Graphing Complex numbers

Quarter 2

11/2-1/22

2B - Trigonometric Identites

Chapter 3 - Analysis of Functions

3A - Polynomial Functions

3B - Rational Functions

3C - Exponential and Logarithmic Functions

Quarter 3

1/25-4/8

Chapter 4 - Combinatorics

4B* - Permutations and Combinations

Chapter 5 - Functions and Tables

5B - Fitting Functions to Tables - Newton's Difference Formula

Chapter 6 - Analytic Geometry

6A - Coordinate Geometry

6B - Conic Sections

Quarter 4

4/11-6/14

Chapter 8 - Ideas of Calculus

8A - Finding Areas of Shapes

8B* - Finding Areas Under Curves

8C* - A Function Emerges



The following are a list of essential standards for this course and a brief map of where they will be addressed.

Standard Quarter 1 Quarter 2 Quarter 3 Quarter 4

F.TF.1 X

F.TF.2 X

F.TF.3 X

F.TF.4 X

F.TF.5 X

F.TF.6 X

F.TF.7 X

F.TF.8 X X

F.TF.9 X

F.IF.4 X X

F.IF.5 X

F.IF.6 X X

F.IF.7 X X

F.IF.8 X X

F.BF.1 X X X X

F.BF.2 X

F.BF.3 X X

F.BF.4 X X

F.BF.5 X

A.SSE.1 X X

A.SSE.2 X

A.SSE.3 X X X

A.SSE.4 X

A.REI.1 X

A.REI.4 X X

G.SQRT.7 X X

+N.CN.1 X



Standard Quarter 1 Quarter 2 Quarter 3 Quarter 4

+N.CN.2 X

+N.CN.3 X

N.CN.5 X X

A.APR.2 X

A.APR.3 X

+A.APR.4 X

A.APR.6 X

A.APR.7 X

A.REI.2 X X

A.REI.4 X X

A.REI.10 X

A.REI.11 X X

A.CED.2 X X

A.CED.4 X

F.LE.1 X

F.LE.4 X

F.LE.5 X

N.VM.6 X

N.RN.2 X

G.GPE.1 X

G.GPE.2 X

G.GPE.3 X

G.GPE.4 X

G.GMD.4 X

N.Q.2 X

Note: Quarter Four covers introductory topics in Calculus which are not covered in the common core.



Chapter 1: Analyzing Trigonometric Functions

Learning Goals: Students will be introduced to radian measure. Students will review the graphs of the cosine and sine functions, and develop an understanding of a periodic function. They also begin to solve equations that involve cosine and sine. Students review the graphs of tangent, secant, cosecant, and cotangent. They look at how they fit on the unit circle and work with some basic identities. Students will also learn how domain, range, inverse, and one-to-one apply to these functions. Students will be able to apply the rules of functions to all sinusoidal functions and will be able to transform them on the coordinate plane.

Essential Questions

Where are the turning points of the cosine and sine functions? What is a radian? How can you use a graph to estimate solutions? How are the six trigonometric functions defined? Given the maximum and minimum values of cosine and sine function, how do you find the amplitude and vertical displacement? How can you make a sinusoidal function that has a specific period? How can you use sinusoidal functions to model periodic phenomena?

Objectives SWBAT:

Understand the relationship between degree and radian measure as the arc on the unit circle subtended by a central angle

Relate the motion of an object around a circle to the graphs of the cosine and sine functions

Solve equations that involve sine and cosine

Understand several relationships between the tangent function and the unit circle

Sketch and describe the graph of the tangent function

Recognize three other trigonometric functions: secant, cosecant, and cotangent

Make sense of sinusoidal functions in the context of previous experience

Understand the geometry of sinusoidal functions

Model with sinusoidal functions

Standards F.TF.1; F.TF.2; F.TF.3; F.TF.4; F.TF.5; F.TF.6; F.TF.7; F.TF.8; F.IF.4; F.IF.5; F.IF.6; F.IF.7; F.BF.1; F.BF.3; F.BF.4; A.SSE.1; A.SSE.3;

A.REI.1; A.REI.4; G.SQRT.7

Tier II Vocabulary increasing; decreasing; maximum; minimum; Tier III Vocabulary amplitude; arc; asymptote; central angle; inverse function; period; periodic; phase shift; Pythagorean Identity; radian;

secant line; sinusoidal function; turning point; vertical displacement Assessments CIA: 10/26-10/30/15 Data Meeting:

investigation reflections; mid chapter test; end of unit test Summative assessments: Formative assessments: Common Prompts:



11/9/15 Rubrics: Grading:

21st Century Learning Expectations

Academic: Effective communication, evaluate information, solve problems, collaborate, support claims, use technology Social: Act with persistence when facing challenging tasks, responsible and respectful behavior, goal setting Civic: Utilize networking skills and engage inclusively with others

RETELL Strategies 7-step Vocab; posted word walls; Think Aloud; Partner Reading; Write Around Texts/Resources Precalculus: common core CME Project Notes:



Chapter 2: Complex Numbers and Trigonometry

Learning Goals: +Students will explore polar form as a way to represent complex numbers. +This chapter will highlight the connection between complex numbers and transformational geometry. Students will explore ways to build and prove trigonometric formulas and identities.

Essential Questions

+How can you write a complex number using trigonometry? +What are the magnitude and argument of a complex number, and how do you find them? +How do you use geometry to add and multiply complex numbers? How can you see if an equation might be an identity? How can you use complex numbers to find formulas for cos 2𝑥 and sin2𝑥? How can you use identities to prove other identities?

Objectives SWBAT:

+Represent complex numbers using both rectangular coordinates and polar coordinates

+Determine the magnitude and argument of any complex number

+Decide when its best to use either rectangular or polar coordinates to represent complex numbers

Test trigonometric equations to predict

Show the basic addition rules for cosine and sine using the Multiplication Law for complex numbers

Use Pythagorean identities and algebra to prove that a trigonometric equation is an identity

Standards +N.CN.1; +N.CN.2; +N.CN.3; +A.APR.4; +A.REI.4; F.TF.8; F.TF.9; N.CN.5; A.SSE.1; A.SSE.2; A.CED.4; G.SRT.7 Tier II Vocabulary magnitude; norm; rectangular coordinates; identically equal; identity; Tier III Vocabulary absolute value of a complex number; argument; conjugate; modulus; polar coordinates; polar form of complex

numbers; rectangular form of complex numbers Assessments CIA: 10/26-10/30/15 Data Meeting: 11/9/15

investigation reflections; mid chapter test; end of unit test Summative assessments: Formative assessments: Common Prompts: Rubrics: Grading:






Chapter 3: Analysis of Functions

Learning Goals: Students will explore the graphs of many different polynomial functions and develop a general description of the properties of these functions. Students will explore the graphs of rational functions. Students will analyze exponential and logarithmic functions.

Essential Questions

How can you graph a polynomial function given its factored form? How can you determine a polynomial’s behavior at very large or very small inputs? How can you use long division to find equations of secant or tangent lines to the graph of a polynomial function? What happens to a rational function as x gets larger and larger? Why do some rational functions look so different from each other? How can you find tangent lines to rational functions? What happens when interest is compounded more and more frequently? What are some reasons to introduce the number e? How can you relate any exponential or logarithmic function to 𝑓(𝑥) = 𝑒𝑥 and 𝑔(𝑥) = ln 𝑥?

Objectives SWBAT:

State the Change of Sign Theorem and the Intermediate Value Theorem for Polynomials, and use them to analyze the graphs of

polynomial functions

Find the equation of a line secant to a polynomial function and the average rate of change of a function between two points

Write the Taylor expansion for a polynomial function about a point

Find the equation of the tangent to a polynomial curve at a point

Sketch the graph of a rational function, including asymptotes and holes

Evaluate limits of rational expressions

Find the equation of the tangent to the graph of a rational function at a point

State and use the limit and factorial definitions of 𝒆 and 𝒆𝒙

Use the inverse relationship between 𝒆𝒙 and 𝐥𝐧 𝒙 to slove equations

Find an equation for the line tangent to the graph of 𝒚 = 𝒆𝒙or 𝒚 = 𝐥𝐧𝒙 at a point

Standards A.SSE.3; A.SSE.4; A.APR.2; A.APR.3; A.APR.6; A.APR.7; A.CED.2; A.REI.2; A.REI.11; F.IF.4; F.IF.6; F.IF.7; F.IF.8; F.BF.1; F.BF.3; F.BF.4; F.BF.5; F.LE.1; F.LE.4; F.LE.5; N.VM.6; N.RN.2

Tier II Vocabulary average rate of change; continuous; hole; instantaneous speed; power function; rational function; removable discontinuity;

Tier III Vocabulary continuously compounded interest; determinant; 𝒆; infinite discontinuity; linear fractional transformation; natural logarithm; reciprocal function; secant line; structure preserving map; tangent line; Taylor expansion

Assessments Midterms: 1/19-1/22/16 Data Meeting:

investigation reflections; mid chapter test; end of unit test Summative assessments: Formative assessments: Common Prompts:



2/1/16 Rubrics: Grading:






Chapter 4 Combinatorics

Learning Goals: +Students will be formally introduced to permutations and combinations

Essential Questions

+In how many ways can you pick a certain number of objects, in order, from a defined set of distinct objects? +How many ways can you arrange a defined set of objects if the objects can be repeated? +How many ways can you arrange a defined set of objects if the objects cannot be repeated?

Objectives SWBAT:

+Develop and use formulas for finding the number of permutations of n objects, taken k at a time.

+Find a formula for the number of combinations of n objects, taken k at a time.

Find the number of anagrams for a word.

Standards A.CED.2; Prepares for S.CP.9

Tier II Vocabulary combination; permutation;

Tier III Vocabulary anagram; nCk; nPk

Assessments CIA: 4/4-4/8/16 Data Meeting: 4/25/16

Investigation quiz or test Summative assessments: Formative assessments: Common Prompts: Rubrics: Grading:






Chapter 5: Functions and Tables

Learning Goals: Students will explore difference tables and learn about the connections between how difference tables are built and how Pascal’s Triangle is constructed.

Essential Questions

What does a constant difference tell you about a function? What are the Mahler polynomials? How can you use differences to find a polynomial function that fits a table?

Objectives SWBAT:

Find a polynomial function that fits a difference table.

Explain how the up-and-over rule of difference tables relates to Pascal’s Triangle

Quickly find rules for summations, like the sum of the first n squares.

Standards F.BF.1; F.BF.2; F.IF.8

Tier II Vocabulary difference table; hockey-stick property; up-and-over property

Tier III Vocabulary Mahler Polynomials








Chapter 6: Analytic Geometry

Learning Goals: Students will make connections between algebra and geometry by using algebraic techniques to prove geometric results. Students explore conic sections from many different perspectives.

Essential Questions

What is the set of points equidistant from the x-axis and a specific point? How can you use coordinates to prove geometric statements? How can you find the center and radius of a circle with an equation written in normal form? What does the intersection of a 3-D object and a plane look like? What is a locus?

Objectives SWBAT:

Sketch the graphs of equations in two variables.

Use distance and slope relationships to prove geometric results

Evaluate and use the signed power of a point with respect to a circle

Visualize each of the conic sections as the intersection of a plane with an infinite double cone

Give a locus definition for each of the conic sections

Identify the equations for the graphs of the conic sections, and sketch their graphs.

Standards A.CED.2; A.REI.2; A.REI.4; A.REI.10; A.REI.11; A.SSE.3; G.GPE.1; G.GPE.2; G.GPE.3; G.GPE.4; G.GMD.4; N.Q.2

Tier II Vocabulary axis; conic sections; double cone; ellipse; focus; generator; major axis; midline; minor axis; parabola; parallelogram; point-tester; rhombus; trapezoid;

Tier III Vocabulary apex; coordinatize; Dandelin sphere; directrix; eccentricity; hyperbola; locus; perpendicular bisector; power of a point; vertex


investigation reflections; mid chapter test; end of unit test Summative assessments: Formative assessments: Common Prompts: Rubrics: Grading:






Chapter 8: Ideas in Calculus

Learning Goals: Students look at the area of familiar shapes and think about areas of irregular shapes.

Essential Questions

How can you find the area of an irregularly-shaped figure? How can you estimate the area under a curve?

Objectives SWBAT:

Estimate the areas of irregularly-shaped objects.

Estimate the area under the graph of 𝒚 = 𝒙𝒎 between 𝒙 = 𝟎 and 𝒙 = 𝟏.

Calculate the area under the graph of 𝒚 = 𝒙𝒎 between 𝒙 = 𝟎 and 𝒙 = 𝟏.

Find the area under the graph 𝒚 = 𝒆𝒙 between 𝒙 = 𝟎 and 𝒙 = 𝟏

Standards Extends A.CED.3

Tier II Vocabulary area; sum

Tier III Vocabulary lower sum; upper sum Assessments Finals: 6/7-6/10/16*




RETELL Strategies 7-step Vocab; posted word walls; Think Aloud; Partner Reading; Write Around Texts/Resources Precalculus: common core CME Project Notes: *Dates may be adjusted according to inclement weather cancellations

Documents

Lawrence High School Pre-calculus Curriculum Map...LAWRENCE HIGH SCHOOL PRE-CALCULUS CURRICULUM MAP 2015-2016 Updated 8/10/15 + denotes a Honors level concept The following are a list