Trigonometry. With approval from the mathematics department … WG Precalculus.pdf ·...

Pre-Calculus

Course Description Precalculus (WG) Grade Level: 9, 10, 11, 12 Prerequisite: Successful completion of Honors Algebra II or Algebra III and Trigonometry. With approval from the mathematics department chair, students may enroll with the successful completion of Algebra II during the school year and Trigonometry during Rockwood Summer Academy.

1 Credit This accelerated course is designed for students interested in pursuing a math related curriculum in college. Topics include functions, series, sequences, matrices, complex numbers, conic sections, polar and parametric equations, vectors, applications of trigonometry, and an introduction to Calculus. Since this course is designed to prepare students for Calculus, the focus will be on problem solving using mathematical models to represent real world situations. Technology will be incorporated throughout the curriculum. The grade for this course is weighted. NOTE: A grade of “B” or better in Honors Algebra II or Algebra III and Trigonometry is recommended for success in this course.

Pre-Calculus

CORE CONCEPTUAL OBJECTIVES

I. The student will solve, graph, and analyze various functions. II. The student will interpret and utilize conic sections, parametric equations, and

polar equations using their properties to model mathematical and real-world problem situations.

III. The student will apply the concepts of trigonometry to the Cartesian coordinate

system.

IV. The student will apply the principles of two-dimensional vectors.

V. The student will apply the concepts of sequences and series.

VI. The student will apply the concepts of limits and continuity.

Pre-Calculus CCO I

I. CORE CONCEPTUAL OBJECTIVE:

The student will solve, graph, and analyze various functions. (Addresses NCTM, Number and Operation, Algebra and Geometry) A. CONTENT AND SKILLS:

By the end of Pre-Calculus, all students should be able to:

Essential Skills

Missouri Show-Me Standards

NCTM Standards

1) identify and utilize the

properties of families of functions. a) linear b) absolute value c) quadratic d) cubic e) polynomial f) greatest integer g) piecewise defined h) rational i) sine j) cosine k) secant l) cosecant m) tangent n) cotangent o) exponential p) logarithmic

E 1.4, 2.7 MA1, MA2,

MA4

Algebra, Geometry

2) utilize graphs to determine the characteristics of functions. a) domain/range b) max/min c) increasing/decreasing d) intercepts e) vertical/horizontal/oblique

asymptotes f) holes g) even/odd h) end behavior i) other characteristics

according to function

E

1.4, 2.7 MA1, MA4

Algebra

Pre-Calculus 3) evaluate and apply functions.

a) composite b) inverse

E 1.4, 2.7, 3.2, 3.7

MA1, MA4

Algebra

4) utilize the theorems associated with polynomial functions. a) Remainder Theorem b) Factor Theorem c) Rational Root Theorem d) Fundamental Theorem of

Algebra e) Intermediate Value

Theorem

E 1.4, 1.10, 2.7 MA1, MA4

Algebra

5) utilize data to model functions.

E 1.4, 1.10, 2.7 MA1, MA4

Algebra

6) utilize theorems and properties associated with exponential and logarithmic functions.

a) exponential notation b) logarithmic notation c) properties of logs d) evaluating logs e) characteristics of

exponential graphs f) characteristics of

logarithmic graphs g) solving exponential and

logarithmic equations

E

1.4, 1.10, 2.7, MA1, MA4

Algebra

7) find characteristics of graphs using the graphing calculator functions

a) maximum b) minimum c) zeros d) value e) intersection

E MA6 Algebra

Pre-Calculus

Pre-Calculus CCO I

B. FACILITATING ACTIVITIES RECALL 1) See attached family of functions identification worksheet. C/S (1) (T) SKILL/CONTENT 2) Find the roots of 5 4 3 2( ) 7 20 12P x x x x x x= − − − − − and sketch a graph. C/S (2, 4) (T)

3) Give a graphical analysis of 23 27( )

3xf xx−

=+

. C/S (2) (T)

4) The Beaver, Pennsylvania, Borough Municipal Authority has the following rates per 1000 gallons of water used. C/S (5)

Usage X

Cost per 1000 gallons C(x)

First 100,000 gallons 1.557 Next 900,000 gallons 1.040

Over 1,000,000 gallons .689

a.) Write a function that models the charges.

b.) Graph the function

c.) Predict the cost for 120,000 gallons.

d.) Predict the cost of 1,100,000 gallons. 5) See attached Postage Stamp Problem. C/S (5) STRATEGIC THINKING 6) Given h(x) = f(g(x)) and h(x) =3 7x − . Determine reasonable functions for f(x) and g(x). Defend your answer. C/S (3) EXTENDED THINKING 7) Compile a set of data and formulate a model to make predictions for future outcomes. C/S (5) (R)

Pre-Calculus Family of Functions

Function Basic Linear (Identity) Basic Quadratic

Formula ( )f x x= 2( )f x x=

Sketch

Domain

Range

x-intercept(s) (zeros)

y-intercept

Continuous/Discontinuous

Symmetry

Even/Odd/Neither

Bounded above/Bounded below

Increasing/Decreasing/Constant

Extrema

Horizontal Asymptotes

Vertical Asymptotes

Holes

Oblique Asymptotes

End Behavior

One-to-One

Concavity

Pre-Calculus

Function Basic Cubic (Cubing) Absolute Value

Formula 3( )f x x= ( )f x x=

Sketch

Domain

Range


y-intercept


Symmetry

Even/Odd/Neither



Extrema


Vertical Asymptotes

Holes

Oblique Asymptotes

End Behavior

One-to-One

Concavity

Pre-Calculus

Function Reciprocal Square Root

Formula 1( )f xx

= ( )f x x=

Sketch

Domain

Range


y-intercept


Symmetry

Even/Odd/Neither



Extrema


Vertical Asymptotes

Holes

Oblique Asymptotes

End Behavior

One-to-One

Concavity

Pre-Calculus

Function (Natural) Exponential (Natural) Logarithmic

Formula ( ) xf x e= ( ) lnf x x=

Sketch

Domain

Range


y-intercept


Symmetry

Even/Odd/Neither



Extrema


Vertical Asymptotes

Holes

Oblique Asymptotes

End Behavior

One-to-One

Concavity

Pre-Calculus

Function Sine Cosine

Formula ( ) sinf x x= ( ) cosf x x=

Sketch

Domain

Range


y-intercept


Symmetry

Even/Odd/Neither



Extrema


Vertical Asymptotes

Holes

Oblique Asymptotes

End Behavior

One-to-One

Concavity

Pre-Calculus

Function Tangent Cotangent

Formula ( ) tanf x x= ( ) cotf x x=

Sketch

Domain

Range


y-intercept


Symmetry

Even/Odd/Neither



Extrema


Vertical Asymptotes

Holes

Oblique Asymptotes

End Behavior

One-to-One

Concavity

Pre-Calculus

Function Secant Cosecant

Formula ( ) secf x x= ( ) cscf x x=

Sketch

Domain

Range


y-intercept


Symmetry

Even/Odd/Neither



Extrema


Vertical Asymptotes

Holes

Oblique Asymptotes

End Behavior

One-to-One

Concavity

Pre-Calculus

Function [ ][ ]xxf =)( 2

1( )f xx

=

Formula

Sketch

Domain

Range


y-intercept


Symmetry

Even/Odd/Neither



Extrema


Vertical Asymptotes

Holes

Oblique Asymptotes

End Behavior

One-to-One

Concavity

Pre-Calculus CCO I

Postage Stamp Problem 5) Believe it or not, in 1885 the cost of a first-class stamp was 2 cents. In 1995 the cost

rose to 32 cents. This is a 16 -fold increase in 90 years. Let's take a look at the data from 1965 until the present.

a) Graph the data. b) Find an appropriate mathematical model to fit the data. c) Use your model to extrapolate the cost of a postage stamp in 1885. How does it

compare with the known value? d) Use your model to extrapolate a cost for the year 2018. Does this value seem

realistic? e) What is the real world meaning of the slope in the linear model? f) What is the real world meaning of the y-intercept in the model?

Year Price (cents)

1965 5 1966 5 1967 5 1968 6 1969 6 1970 6 1971 6 1972 8 1973 8 1974 8

Year Price (cents)

1975 10 1976 10 1977 13 1978 13 1979 15 1980 15 1981 15 1982 18 1983 20 1984 20

Year Price (cents)

1985 22 1986 22 1987 22 1988 25 1989 25 1990 25 1991 29 1992 29 1993 29 1994 29 1995 32

Pre-Calculus CCO I

C. APPLICATION LEVEL ASSESSMENT Ickum High School The senior class of Ickum High School is planning to raffle off a one week vacation at the beach as a way of making money for class activities. A member of the Parents’ Club is willing to donate a condominium for a week. The seniors want to know at what price they should sell the raffle tickets. The higher the price, the fewer tickets they will sell, but the more money they will make on each ticket. A survey is sent out to 1100 parents asking what they would be willing to pay for a ticket. The results are as follows:

Price of a Ticket Number Willing to Purchase

$2.50 508

$5.00 420

$7.50 389

$10.00 293

$12.50 245

$15.00 152

$17.50 90

$20.00 63

a) Graph the profit vs. the price on your graphing calculator. Make a sketch of the graph below. Remember, the condo is donated, so all revenue is profit.

Pre-Calculus CCO I

C. Application Level Assessment b) Pick three points from the chart and determine the quadratic function which passes through these three points. Show all work below. Leave your answer in vertex form. Show the work that leads to your answer. c) Use your calculator to determine the best fit quadratic function and record the results below. d) Determine the price the seniors should charge for each ticket. Support your answer

using the information from parts a and b.

Pre-Calculus CCO I

Scoring Guide

Criteria 4 3 2 1 Hand calculated quadratic function

Reasonable model exhibiting transformations with all supporting work/explanation

Reasonable model exhibiting transformations, some support

Either dilation or translation is correct

No attempt or both dilation and translation incorrect

Calculator regression curve

Correct calculator regression in ax 2 +bx+c form

Only provide values for a, b, c

or substitute values incorrectly

Major flaws in calculator regression

No calculator regression

Analysis of optimization

Reasonable answer with supporting explanation

Reasonable answer with some support

Reasonable answer with no support

No attempt or

Unreasonable answer

Pre-Calculus

CCO II

II. CORE CONCEPTUAL OBJECTIVE: The student will interpret and utilize conic sections, parametric and polar equations using their properties to model mathematical and real-world problem situations.

(Addresses NCTM, Number and Operation, Algebra; ACT) A. CONTENT AND SKILLS:


Essential Skills


NCTM Standards

1) classify conic relations given

general form.

E 1.4, 1.6, 2.7, 3.2, 3.3

MA1, MA4, MA5

Algebra

2) define the parts of a parabola and utilize them to graph. a) vertex b) focus c) directrix d) axis of symmetry e) equation: y – k = a(x – h)2

x – h = a(y-k)2

E 1.4, 1.6, 2.7, 3.2, 3.3

MA1, MA4, MA5

Algebra

3) define the parts of a circle and utilize them to graph. a) center b) radius c) equation: (x – h)2 + (y – k)2 = r2

E 1.4, 1.6, 2.7, 3.2, 3.3

MA1, MA4, MA5

Algebra

4) define the parts of an ellipse and utilize them to graph. a) center b) major/minor axis c) foci d) vertices e) endpoints f) equations

E 1.4, 1.6, 2.7, 3.2, 3.3

MA1, MA4, MA5

Algebra

Pre-Calculus

CCO II

5) define the parts of a hyperbola and utilize them to graph. a) center b) transverse/conjugate axis c) asymptotes d) foci e) vertices f) equations

1.4, 1.6, 2.7, 3.2, 3.3 MA1, MA4, MA5

Algebra

6) write the equation in standard form given the graph of a conic section.

E 1.4, 1.6, 2.7, 3.2, 3.3 MA1, MA4, MA5

Algebra

7) express points in the plane in both rectangular and polar forms.

E 1.6, 3.1, 3.2 MA2, MA4

Algebra Geometry

8) find equivalent representations for points and curves, including the conics, in both rectangular and polar forms.

E 1.6, 3.2, 3.4 MA2, MA4

Algebra Geometry

9) use parametric equations to represent situations involving motion in the plane.

E 1.6, 3.2, 3.4 MA2, MA4

Algebra Geometry

10) convert between a pair of parametric equations and an equation in x and y to interpret the situation represented.

E 1.6, 3.2, 3.4 MA2, MA4

Algebra Geometry

11) analyze planar curves, including those given in parametric form.

E 1.6, 3.2, 3.4 MA2, MA4

Algebra Geometry

Pre-Calculus

CCO II B. FACILITATING ACTIVITIES RECALL 1) Identify each equation as representing one of the following: a line, a circle, an

ellipse, a hyperbola or a parabola. C/S (1)

a. 8x2 – 27y2 = 236 b. 2x – y2 = 26 c. 8x + 3y = 1 d. (x-8)2 + (y+4)2 = 25

e. ( ) ( )2 23 71

9 49x y+ −

+ =

2) Identify each conic and list the critical information. C/S (2) a. 6(x - 1)2 + 6(y + 2)2 = 24

b. y = (x - 3)2 + 4

c. ( ) ( )2 23 21

16 4x y+ −

+ =

d. 2 2

116 25y x

− =

e. x2 + y2 + 4x + 12y + 39 = 0

3) Plot points in the polar coordinate system, recognizing multiple representations for each point. C/S (7, 8)

a. (r, )θ = ( 2, 3π ) c. (r, )θ = (-3, )

611π

b. (r, )θ = (3, 6π− ) d. (r, )θ = (-2, )

35π−

4) Convert rectangular to polar and polar to rectangular. C/S (7, 8)

Coordinate Conversion Convert each point to polar. Convert each point to rectangular.

a. (-1, 1) c. (2, )π

b. (0, 2) d. ( )6

,3 π

Pre-Calculus CCO II

Equation Conversion

Convert the rectangular equation to polar form. a. 4922 =+ yx b. y = 4 c. x = 3 d. 3x - 6y + 2 = 0 Convert the polar equation to rectangular form.

a. r = 4sinθ b. r = 2cosθ

c. θ = 6π

d. r = 10

SKILL/CONCEPT 5) Make a poster board presentation comparing the different conic sections. C/S (1, 2, 3, 4, 5) (W) 6) Solve the following system graphically and algebraically compare your results.

Assess which method is most efficient and explain why. C/S (3)

=+=−

3416

22

22

xyxy

7) Utilize a graphing calculator or a computer program to graph each equation. Compare and analyze the graphs. C/S (2) (T)

a. (x-1)2 + (y+2)2 = 9

b. ( ) ( )2 23 11

49 25y x− +

+ =

c. x = (y-4)2 – 1

d. ( ) ( )2 21 21

16 81y x− +

− =

Pre-Calculus

8) Graph polar equations. C/S (7, 8)

a. Sketch r = 4 sin θ , by using a table of values across the interval 0 πθ 2≤≤ . b. Sketch r = 3 + 2cosθ , by using symmetry. c. Sketch r = 1-2cosθ , by finding zeroes, and maximum r-values.

9) Sketch the curve given by the parametric equations x = 42 −t , and y = 2t , for

-2 3≤≤ t . C/S (10)

10) Convert from conic form to parametric form. Find a set of parametric equations to represent the graph of y = 1 - x 2 , using the following parameters. C/S (10)

a. t = x and b. t = 1 - x

STRATEGIC THINKING 11) Tape equations of conic sections to the backs of students and have the students tell each other characteristics of their conic section. After several minutes, students draw the graph of the equation on their foreheads. C/S (1, 2)

12) Analyze the graph of r = 2 cos3θ , using symmetry, the maximum r-value, and the

zeroes of the equation. C/S (11) 13) Use a graphing utility in parametric mode to graph the curves represented by the

parametric equations. Then state for which curve is y a function of x? C/S (11) (Use -4 )4≤≤ t a. x = t 2 b. x = t c. x = t 2 y = t 3 y = t 3 y = t

14) Sketch the curve represented by x = 3 cos θ and y = 4 sin θ , πθ 20 ≤≤ , by

eliminating the parameter. C/S (11)

Pre-Calculus

CCO II EXTENDED THINKING 15) Create interesting or entertaining graphic pictures. On the grid below, graph each equation. Review with students how they would graph an equation such as y = 22,42 ≤≤−− xwherex .

a. y = x2 - 4, where –2 ≤ x ≤ 2 b. (x + 5)2 + (y - 7)2 = 9 c. x = y2 + 13, where 12 ≤ x ≤ 13

d. (x - 5)2 + (y - 7)2 = 9

e. x = y2 – 13, where –13 ≤ x ≤ -12

f. ( )22 91

16 4yx +

+ =

g. 2 2

1144 225x y

+ =

After students have completed this “Conic Man”, allow them to create their own graphic pictures that utilize equations for conics as well as for straight lines. By creating a grid with nails or tacks and using different colored thread or string, students can even make a display of their art. C/S (1-5) (W)

Pre-Calculus

CCO II

C. APPLICATION LEVEL ASSESSMENT

Every year some high school Social Studies students go to Washington D. C. to participate in the Close Up program. One of their experiences is a tour of the United States Capital. Ms. Math, the Honors Algebra II teacher, says she will excuse all homework if her Close Up students will complete this problem and fax it to her after they have determined it can actually be done.

The U.S. Capital contains an elliptical room 96 feet in length and 46 feet in width. If you stand at the appropriate points, you can hear the words another person is whispering. Complete this problem for Ms. Math.

1. Graph the figure. Assume that the center is the origin and the major axis is horizontal.

2. Write an equation to describe the shape of the room. 3. Write a short explanation of how you determined your equation and

your graph.

Pre-Calculus CCO II

Scoring Guide

Criteria 4 3 2 1

Graph

Accurate and appropriate

Minor errors Major errors None or inappropriate

Equation

Accurate Minor errors Major errors None or inappropriate

Explanation

Complete, accurate and clearly explained

Accurate and complete, but not clearly explained

Inaccurate and/or lacks clarity

Incorrect or incomplete or not written

Pre-Calculus CCO III

III. CORE CONCEPTUAL OBJECTIVE:

The student will apply the concepts of trigonometry to the Cartesian coordinate system. (Addresses NCTM, Number and Operation, Algebra, Geometry, and

Measurement) A. CONTENT AND SKILLS:


Essential Skills


NCTM Standards

1) evaluate degree and radian

measure.

E 1.4, 2.7

MA1, MA2, MA4

Algebra, Geometry,

Measurement 2) determine arc length and area

of a sector. E 1.4, 2.7

MA1, MA2, MA4

Algebra, Geometry,

Measurement 3) evaluate angular and linear

velocity. E 1.4, 2.7

MA1, MA2, MA4

Algebra, Geometry,

Measurement 4) calculate the area of a triangle.

a) geometric b) trigonometric c) Heron’s formula

E 1.4, 2.7

MA1, MA2, MA4

Algebra, Geometry,

Measurement

5) graph functions and their transformations.

E 1.4, 2.7

MA1, MA2, MA4

Algebra, Geometry,

Measurement 6) utilize trigonometric identities.

a) sum/difference b) double angle c) half angle d) sum/product

E 1.4, 2.7

MA1, MA2, MA4

Algebra, Geometry,

Measurement

7) verify identities.

E 1.4, 2.7

MA1, MA2, MA4

Geometry

8) calculate inverse trigonometric functions.

E 1.4, 2.7

MA1, MA2, MA4

Algebra, Geometry,

Measurement 9) solve trigonometric equations.

E 1.4, 2.7

MA1, MA2, MA4

Algebra, Geometry,

Measurement

CCO III

Pre-Calculus

B. FACILITATING ACTIVITIES RECALL 1) San Francisco and Seattle are on the same meridian; that is, Seattle is due north of San Francisco. If the latitude of San Francisco is 37° 47’ and that of Seattle is 47° 37’, find the distance between the two cities if the radius of the Earth is about 3960 miles. C/S (1) (T) (R) 2) While traveling across flat land, you notice a mountain directly in front of you. The angle of elevation to the peak is 3.5°. After you drive 13 miles closer to the mountain, the angle of elevation is 9°. Approximate the height of the mountain. C/S (2, 3, 4) (T) SKILL/CONCEPT 3) The rate of change of the function f(x)= -sinx + csc x with respect to change in the variable x is given by the expression csc x – csc x tan x. Show that the expression for the rate of change can also be given by –cos x cot2 x. C/S (6, 7) 4) The diagram below shows the dimensions for a sail on a wooden model ship. Find the area of the sail to the nearest square inch. If the scale of the length of the real ship to that of the model ship is 1 foot: ½ inch, find the area of the real sail in square feet. C/S (1, 2) (T) (R)

12.5 in

18 in

16.5 in

26 in

75°

Pre-Calculus

CCO III STRATEGIC THINKING 5) The branch of physics that involves the behavior of liquid at rest and in motion is called hydraulics. A reservoir is a part of a scientific apparatus in which liquid is held. An important formula in analyzing reservoirs involves the capillary pressure, P. If t is the tension between two fluids, r is the radius of the capillary tube, and θ is the angle

between the interface and the capillary wall, then 2 costPr

θ= . A certain reservoir with

an interfacial tension between fluids of 75 d/cm has a capillary tube of radius 0.02 cm. C/S (8, 9) (T) (R)

a. Draw a graph of the capillary pressure in the apparatus for 0θ = ° to 90°. b. What is the amplitude of this curve? c. Write a few sentences to explain the meaning of the graph.

6) Describe the relationship between periodic and one-to-one functions. Can a function be both periodic and one-to-one? Explain why. C/S (5) EXTENDED THINKING 7) A weight is attached to a spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is described by the model y = 1.5 sin 8t - 0.5 cos 8t, where y is the distance from equilibrium measured in feet and t is the time in seconds. C/S (9) (T) a. Write a model in the form y = ).sin(22 CBtba ++ b. Use a graphing utility to graph the model. c. Find the amplitude of the weight’s oscillations. d. Find the frequency of the weight’s oscillations.

Pre-Calculus

CCO III


The Ferris Wheel, a classic amusement park ride, was invented by George Ferris. Mr. Ferris was an American engineer who debuted his wheel at the 1893 World’s Fair in Chicago.

Suppose that you are 4 feet off the ground in the bottom car of a Ferris Wheel and ready to ride. If the radius of the wheel is 25 feet and it makes 2 revolutions per minute:

a) Sketch a graph that shows your height (in feet) above the ground at time t (in seconds) during the first 45 seconds of your ride.

b) Give a possible equation for your curve. Show all work leading to the equation.

c) Suppose that the radius of the wheel is decreased and that the wheel still makes two revolutions per minutes. What would be the effect on the period and amplitude of the graph and on the speed of the passengers?

Pre-Calculus CCO III

Scoring Guide

Criteria 43 3 2 1

Graph

Graph is completely correct and includes labeled axes

Graph is mostly correct with only minor errors

Graph is periodic in nature but contains two or more errors

Graph is completely incorrect (not periodic or not there)

Equation

Completely correct

Mostly correct with only one minor error

Used a sin/cos function but has two mistakes

Equation is completely incorrect (not even periodic) or not there

Explanation of part c

Effect on period, amplitude, and speed of passengers is correct

Correctly identifies two of the three quantities correctly

Correctly identifies only one of the three quantities correctly

None of the information is correct

Pre-Calculus

Pre-Calculus

CCO IV

V. CORE CONCEPTUAL OBJECTIVES The student will apply the principles of two-dimensional vectors. (Addresses NCTM, Number and Operation, and Algebra) A. CONTENT AND SKILLS

By the end of Pre-Calculus,

all students should be able to:

Essential

Skills


NCTM

Standards

1) create directed graphs.

E MA1, MA6 Algebra

2) utilize operations to calculate vectors.

a) addition b) subtraction c) scalar multiplication d) magnitude

E 1.4, 1.10, 2.7

MA1, MA6

Number and Operation,

Algebra

3) calculate the dot product of two vectors.

E 1.4, 1.10, 2.7

MA1, MA6


Algebra

4) determine if vectors are parallel, orthogonal, or neither.

E MA1, MA6 Algebra

Pre-Calculus CCO IV

B. FACILITATING ACTIVITIES RECALL 1) Let v = -2, 5 and w = 4,3 , find each of the following vectors. C/S (2) a. v + w b. 2v c. w – v d. v + 2w 2) Find the component form and length of vector V that has initial point (-4, 7) and terminal point (-1, 5). C/S (2) 3) Find the magnitude of vector V from question (2). C/S (2) SKILL/CONCEPT 4) Are vectors PQ and RS equivalent? (same magnitude and direction) P (0, 0), Q (3, 2), R (1, 2), and S ( 4, 4). C/S (1, 2) 5) Find each dot product. Let U = 3,1− , V = 4,2 − , and W = 2,1 − . C/S (3) a. VU • b. WVU )( • c. VU 2• 6) Find a unit vector in the direction of V = 5,2− , and verify that the result has a length of 1. C/S (2) STRATEGIC THINKING 7) Find the angle between and determine if the vectors are orthogonal, parallel, or neither. C/S (4)

a. U = 3,4 and V = 5,3

b. U = 4,8 and V = 1,2 −−

c. U = 3,2 − and V = ,6 4

Pre-Calculus

CCO IV 8) Revenue. The vector V = 2600,1245 gives the number of units of two

products produced by a company. The vector U = 50.8,20.12 gives the price (in dollars) of each unit, respectively. Find the dot product U•V, and explain what information it gives. C/S (3) (W)

EXTENDED THINKING 9) Write a program for your graphing utility that graphs two vectors and their difference given the vectors in component form. C/S (2) (T)

Pre-Calculus

CCO IV C. APPLICATION LEVEL ASSESSMENT

A pilot wants to land his plane on the runway as shown in the figure below. The true wind is coming from the front right of the airplane. This wind can be broken down into a crosswind component and a headwind component. If the crosswind component has a magnitude greater than 15 knots, the pilot will not be able to land the plane safely on this runway and will need to fly to a place where the runway is more directly aligned with the wind direction. Determine if the pilot can land on this runway safely if the wind speed is 19 knots. (Although this problem could be solved with trigonometry, use vectors to solve it).

Pre-Calculus

CCO IV

Scoring Guide

Criteria 4 3 2 1 Vector Diagram

Complete Diagram

Diagram with one minor error

Diagram with two minor errors

Incomplete or diagram with three or more errors

Formulas

Used Vector formulas

Correct formulas with calculation error

Used trigonometry with triangles

No attempt

Pre-Calculus

CCO V

VI. CORE CONCEPTUAL OBJECTIVE: The student will apply the concepts of sequences and series. (Addresses NCTM, Number and Operations, Algebra)

A. CONTENT AND SKILLS:


Essential Skills


NCTM Standards

1) generate the terms of a

sequence. a) arithmetic b) geometric c) explicitly defined d) recursively defined

E 1.4, 2.7

MA1, MA4

Algebra

2) utilize summation notation.

E 1.4, 2.7

MA1, MA4

Algebra

3) calculate the sums of series.

E 1.4, 2.7

MA1, MA4


Algebra

4) apply the Binomial Theorem in problem solving situations.

E 1.4, 2.7, 3.2

MA1, MA4

Algebra

5) evaluate the limits of sequences and series.

E 1.4, 2.7

MA1, MA4

Algebra

Pre-Calculus

CCO V

B. FACILITATING ACTIVITIES RECALL 1) Give both the explicit and the recursive formula for the sequence 4, 9, 14, 19,… Then find the seventy-seventh term. C/S (1) (T)

2) In a geometric sequence, the third term is 163

and the fifth term is 6427

. Find

an explicit formula for the sequence. Then find the seventh term of the sequence. C/S (1) (T) SKILL/CONCEPT 3) Each row of an auditorium has four more seats than the preceding row. Find the seating capacity of the auditorium if the front row seats 30 people and there are 20 rows. C/S (2, 3) (T) 4) On the day you were born, your grandmother put $50,000 in a trust fund for you with the following two stipulations: 1) you could give yourself a $1500 birthday gift each year and 2) the rest of the money would remain untouched until the investment reached $1,000,000. How old will you be when the trust fund reaches $1,000,000 if interest is compounded yearly at 8%. C/S (4) (T) STRATEGIC THINKING 5) Create a PowerPoint presentation, flyer, or web page demonstrating at least three patterns found in Pascal’s Triangle or at least three instances of the Fibonacci sequence in nature. Include in your final product a section of references cited. C/S (5) (T) (R) (W)

EXTENDED THINKING 6) One 8 ounce cup of coffee doesn't have an adverse effect on you but more than one cup causes your hands to tremble from all of the caffeine. If your kidneys purify 1/4 of your blood every 4 hours, how long will it take your hands to stop trembling after drinking 3 cups of coffee? C/S (3, 4) (T) (R)

Pre-Calculus CCO V


See following two pages

Drug Modeling Problem

Romainian Gymnast

Andrea Raducan stripped of All-Around Gold Medal

2000 Olympics, Sydney after coach gave her two

over-the-counter pills containing the drug pseudoephedrine for her

cold/fever

US Shot Putter Randy Barnes

banned for life from US Track and Field events for use of methyltestosterone (steroid)

Pre-Calculus

Skiers Johann Muehlegg (Spain), Olga Danilova (Russia), and Larissa Lazutina (Russia)

Lazutina stripped of Gold Medal in 30 km cross country race, 2002 Salt Lake City Games Muehlegg stripped of Gold Medal in 50 km classical race, 2002 Salt Lake City Games

Danilova thrown out of Oylmpic competition, 2002 Salt Lake City Games all tested positive for use of darbepoetin

Pre-Calculus CCO V

In today’s increasingly competitive world, some athletes feel the need to take performance enhancing drugs despite the strong consequences involved. Aside from the vast range of physical ill-effects, there is also the emotional price one must pay when drug use is discovered. Take for example, the female athlete who was stripped of her world record in the 1972 Olympics when she used an asthma medication. In the Seoul Olympics of 1988, sprinter Ben Johnson was stripped of his gold medal and world record in the 100 meter dash after failing a urine test for steroid use. In the Sydney Olympics of 2000, gymnast Andreea Raducan was stripped of her All-Around Gold Medal after failing a drug test because she took some over-the-counter cold medicine.

More and more organizations, both athletic and business, are implementing drug screening. Through this activity, you will develop a model for the amount of a given drug remaining in a system after a certain number of hours.

A) A high school athlete has a very bad cough/cold and her doctor has prescribed a codeine cough syrup. Suppose the initial dose is 2 teaspoons of cough syrup. The function of the kidneys is to remove impurities from the blood. Assume that during a 4 hour period the kidneys purify ¼ of the blood. Make a table showing the amount of cough syrup left in the bloodstream after 10 four hour periods. Include “time 0” in your table. B) Given that a teaspoon is equivalent to 5 mL, how long will it take for the level

to reach 0.1 mL of cough syrup in the system. C) You are an athlete and you are required to take a drug test before a big

competition. If the level of drug in your system exceeds 0.05 mL, you fail the test and will be disqualified. How long should you wait to ensure that you will pass the test?

D.) Will there ever be a time when the bloodstream is completely free of cough syrup? Support your answer.

Pre-Calculus CCO V

Scoring Guide

Criteria 4 3 2 1

Part A Table

Correctly makes table including time 0 and 10 more points along with correct labels for each column

Entries correct but missing labels OR labels correct but missing one entry

Not enough points in table

No table present

Part B

Correct answer with correct label

Correct answer but without label

No attempt OR incorrect answer

Part C

Correct answer with correct label

Correct answer but without label

No attempt OR incorrect answer

Part D Reasonable answer with supporting explanation

Reasonable answer with no supporting explanation

No attempt or unreasonable answer

Pre-Calculus CCO VI

VII. CORE CONCEPTUAL OBJECTIVE:

The student will apply the concepts of limits and continuity. (Addresses NCTM, Number and Operation, Algebra; ACT) A. CONTENT AND SKILLS:


Essential Skills


NCTM Standards

1) evaluate the limit of a function.

E 1.4, 2.7

MA1, MA4


Algebra

2) evaluate the continuity of a function.

E 1.4, 2.7

MA1, MA4


Algebra

3) evaluate the slope of a line that is tangent to a curve.

E 1.4, 2.7

MA1, MA2, MA4


Algebra

Pre-Calculus CCO VI

B. FACILITATING ACTIVITIES RECALL 1) Have students examine lines that are continuous and those that have breaks by drawing several curves on the board or overhead. Examine also functions with a single point of discontinuity. C/S (2) (R) 2) Graph several functions on a graphing utility and then sketch the functions by hand to determine if you can complete the sketch without lifting the pencil from the paper. C/S (2) (T) (R) SKILL/CONCEPT 3) Utilize a graphing calculator to graphically verify the limit of a graph after completing a table of values for the function. C/S (1) (T)

4) Examine a group of functions and discuss the characteristics of the function. Determine if the limit of the function could best be solved numerically, graphically, or analytically. C/S (1, 2, 3) STRATEGIC THINKING 5) Create with the class or have students work in groups to create a strategy for finding limits. C/S (1, 2, 3)

6) Assign pairs of students two problems to complete. Problems should include those that are approximating the slope of the graph. Each student does a different problem and then orally explains to the partner their process and justifies their work. The partner verifies the other student’s answer. C/S (3)

Pre-Calculus EXTENDED THINKING 7) Sketch the graph of a function f(x) that satisfies all of the following conditions. Also discuss the discontinuities found in the following function and point out which condition(s) of continuity is/are violated. C/S (1)

∞→=

xxf 5)(lim

23)(lim

→=

xxf

2)2( −=f

−∞→−=

xxf 4)(lim

1)(lim

−→xexistnotdoesxf

a) Summarize the different ways the graph could be drawn at x = -1. b) Compare )(lim.)(lim

11xfvsxf

xx −→−→

Pre-Calculus

CCO VI

C. APPLICATION LEVEL ASSESSMENT 1) Approximate the surface area of the lake with successive approximations using 1, 2, 4, and 8 rectangles. Assume the side of each square represents 10 yards. 2) Use your findings from part A to estimate the exact area. Support your estimate.

Pre-Calculus CCO VI

Scoring Guide

Criteria 4 3 2 1

Approximation of the area

All approximations are completely correct

Minor errors in the approximations

Major errors in the approximations

Incorrect approximations or did not attempt the work

Estimation of the area

Accurate estimations

Inaccurate support or did not attempt the work

Support of the estimate

Reasonable explanation which completely supports the estimate

Fairly complete support of the estimate with some information missing

Support presented but lacking foundation and accuracy

Incorrect support or did not attempt the work

Pre-Calculus

Differentiation Topics

I. Matrices a. ) dimensions b. ) addition c. ) subtraction d. ) scalar multiplication e. ) matrix multiplication f. ) equality of matrices g. ) operations using a graphing calculator II. Rotation of Conics

Trigonometry. With approval from the mathematics department … WG Precalculus.pdf ·...

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