Download SFSD Pre-Calculus Pacing Guide (created 2014

SFSD Pre-Calculus Pacing Guide (created 2014-2015)
Semester 2
Unit 6: Trigonometric Functions
Notes:
Total Suggested Number of Days for the Unit: 25 days
Topic
Unit Circle
Standards
F.TF.1 Understand radian measure of an angle as the length of
the arc on the unit circle subtended by the angle.
F.TF.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.
Converting
between
degrees and
radians
Derive
trigonometri
c ratios on
the unit circle
using special
right triangle
F.TF.1 Understand radian measure of an angle as the length of
the arc on the unit circle subtended by the angle.
F.TF.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.
(+) F.TF.3 Use special right triangles to determine geometrically
I can…
• Define radian, define degree,
explain the need for radian
measure.
● Define and use Initial side,
terminal side, vertex, standard
position, positive angle,
negative angles, coterminal,
central angles.
● Sketch angles in standard
position.
● Find coterminal angles
● Find reference angle
● Convert angles from radians to
degrees
● Convert angles from degrees to
radians
● Construct the unit circle.
● Evaluate trig ratios on the unit
circle
● Explain symmetry (odd and
even) of the six trig functions
using the unit circle.
Resources
Time
Frame
● Section 4.1
● Section 4.2
1-2 days
● Section 4.1
1 days
● Section 4.2
2-3 days
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 2 π-x in terms of their values for x, where x is
any real number.
(+) F.TF.4 Use the unit circle to explain symmetry (odd and even)
and periodicity of trigonometric function.
Determine
trig ratios off
the unit circle
given either a
point or a trig
ratio.
F.TF.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.
G.SRT.7 Explain and use the relationship between the sine and
cosine of complementary angles.
F.TF.8 Prove the Pythagorean identity
sin^2(theta)+cos^2(theta)=1 and use it to find sin(theta),
cos(theta), or tan(theta), given sin(theta), cos(theta), or
tan(theta), and the quadrant of the angle.
(+) F.TF.3 Use special right 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 2 π-x in terms of their values for x, where x is
any real number.
Graph Sine
and Cosine
Functions
F.TF.5 Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency, and midline.
Graphs of
Other
Trigonometric
F.TF.5 Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency, and midline
● Evaluate trig ratios off the unit
circle using triangles.
● Evaluate trig rations using the
Pythagorean identity.
● Use the relationship between
sine and cosine, secant and
cosecant, and tangent and
cotangent to find the
trigonometric ratios of
complementary angles.
● Section 4.3
● Section 4.4
3-4 days
● Graph sine and cosine functions
and find amplitude, frequency,
midline, period, and phase shift.
● Write a sine or cosine function
using given information.
● Use a trigonometric function to
model data.
● Graph tangent, cotangent,
secant, and cosecant functions.
● Find zeros and asymptotes of
● Section 4.5
3-4 days
● Section 4.6
2-3 days
Functions
tangent, cotangent, secant, and
cosecant graphs.
Inverse
Trigonometric
Functions
(+) F.TF. 6 Understand that restricting a trigonometric function
to a domain on which it is always increasing or always
decreasing allows its inverse to be constructed.
Application of
Right Triangles
G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem
to solve right triangles in applied problems.★
● Graph an arcsine, arccosine, and ● Section 4.7
arctangent function by limiting
the domain.
● Evaluate an inverse
trigonometric within the correct
domain.
● Evaluate compositions of
trigonometric functions.
● Use trigonometry to solve
● Section 4.8
problems involving right triangle.
3-4 days
2-3 days
(+) F.TF. 7 Use inverse functions to solve trigonometric
equations that arise in modeling contexts; evaluate the solutions
using technology, and interpret them in terms of the context. *
Semester 2
Unit 7: Analytic Trigonometry
Notes:
Total Suggested Number of Days for the Unit: 25 days
Topic
Trigonometric
Expressions
and Identities
Verifying
Trigonometric
Identities
Standards
I can…
● Factor a trigonometric
expression
● Simplify trigonometric
expressions.
● Rewrite fractional trigonometric
expressions by applying a
conjugate expression
● Verify a trigonometric identity.
Resources
Time
Frame
● Section 5.1
4-5 days
● Section 5.2
4-5 days
Solving
Trigonometric
Equations
(+) F.TF. 7 Use inverse functions to solve trigonometric
● Solve trigonometric equations.
equations that arise in modeling contexts; evaluate the solutions
using technology, and interpret them in terms of the context. *
● Section 5.3
4-5 days
Sum and
Difference
and Multiple
Angle
Identities
(+) F.TF. 9 Prove the addition and subtraction formulas for sine,
cosine, and tangent and use them to solve problems.
● Use sum and difference formulas ● Section 5.4
to evaluate sine, cosine, and
● Section 5.5
tangent expressions.
● Solve trigonometric equations
using double angle formulas.
4-5 days
Semester 2
Unit 8: Vectors
Notes:
Total Suggested Number of Days for the Unit: 18 days
Topic
Standards
I can…
Law of Sines
and Cosines
(+) G.SRT.10 Prove the Laws of Sines and Cosines and use them to
solve problems.
● Derive the law of sines and
cosines.
● Use the law of sines and cosines
to find missing parts of a
triangle.
● Use the law of sines and cosines
in application problems.
(+) G.SRT.11 Understand and apply the Law of Sines and the Law
of Cosines to find unknown measurements in right and non-right
triangles (survey problems, resultant forces, etc.)
Area of
Triangles
(+) G.SRT.9 Derive the formula A = ½ ab sin © for the area of a
● Find the area of triangles.
triangle by drawing an auxiliary line from a vertex perpendicular to
Resources
Time
Frame
● Section 6.1
● Section 6.2
3-4 days
● Section 6.1
● Section 6.2
1-2 days
the opposite side.
Vectors
(+) N.VM.1 Recognize vector quantities as having both magnitude
and direction. Represent vector quantities by directed line
segments, and use appropriate symbols for vectors and their
magnitudes. (e.g., v, |v|,||v||,v)
(+) N.VM.2 Find the components of a vector by subtracting the
coordinates of an initial point from the coordinates of a terminal
point.
(+) N.VM.3 Solve problems involving velocity and other quantities
that can be represented by vectors.
(+) N.VM.4 Add and subtract vectors.
a. Add vectors end-to-end, component-wise, and by the
parallelogram rule. Understand that the magnitude of a sum of
two vectors is typically not the sum of the magnitudes.
b. Given two vectors in magnitude and direction form, determine
the magnitude and direction of their sum.
c. Understand vector subtraction v – w as v + (-w), where –w is the
additive inverse of w, with the same magnitude as w and pointing
in the opposite direction. Represent vector subtraction graphically
by connecting the tips in the appropriate order, and perform
vector subtraction component-wise.
(+) N.VM.5 Multiply a vector by a scalar.
a. Represent scalar multiplication graphically by scaling vectors
and possibly reversing their direction; perform scalar
multiplication component-wise, e.g., as c(vx,
vy)=(cvx,cvy).
b. Compute the magnitude of a scalar multiple cv using
||cv||=|c|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).
(+) N.VM.11 Multiply a vector (regarded as a matrix with one
column) by a matrix of suitable dimensions to produce another
● Find the magnitude and
● Section 6.3
direction of a vector.
● Find the components of a vector
given the initial and terminal
point.
● Add and subtract vectors and
multiply by a scalar.
● Understand graphically how to
add and subtract vectors and
multiply by a scalar.
● Use vectors to solve problems
involving velocity and force.
● Find a transformation of a vector
using matrices.
5-6 days
vector. Work with matrices as transformations of vectors.
Complex
Numbers
(+) N.CN.3 Find the conjugate of a complex number; use
conjugates to find moduli and quotients of complex numbers
(+) N.CN.4 Represent complex numbers on the complex plane in
rectangular and polar form (including real and imaginary
numbers), and explain why the rectangular and polar forms of a
given complex number represent the same number.
(+)N.CN. 5 Represent addition, subtraction, multiplication, and
conjugation of complex numbers geometrically on the complex
plane; use properties of this representation for computation. For
example, (–1+√3i)3=8 because (–1+√3i) has modulus 2 and
argument 120o.
(+)N.CN.6 Calculate the distance between numbers in the complex
plain as the modulus of the difference, and the midpoint of a
segment as the average of the numbers at its endpoints.
● Write complex number in polar
and rectangular form.
● Graph complex number in
rectangular and polar form and
show how they represent the
same number.
● Add, subtract, multiply and
conjugate a complex number.
● Find a power of a complex
number.
● Find the distance and midpoint
between numbers in the
complex plane.
● Section 6.5
● Section 9.5
3-4 days
Semester 2
Unit 9: Polar
Notes:
Total Suggested Number of Days for the Unit: 10 days
Topic
Polar
Coordinates
Standards
I can…
● Plot polar coordinates.
● Find multiple representations of
points.
● Convert from rectangular to
polar coordinates and polar to
rectangular coordinates.
Resources
● Section 9.5
(Consider teaching
with complex
numbers in section
6.5)
Time
Frame
2-3 days
● Convert rectangular to polar
equations and polar to
rectangular equations.
● Plot points to graph a polar
equation.
● Graph polar equations using
technology.
● Describe symmetry of polar
graphs.
● Graph parametric equations by
hand and using technology.
● Write parametric equations to
model a horizontal and vertical
component (projectile motion)
● Eliminate the parameter of
parametric equations.
Polar
Equations
Parametric
Equations
● Section 9.6
2-3 days
● Section 9.4
2-3 days
Semester 2
Unit 11: Limits and Derivatives
Notes:
Total Suggested Number of Days for the Unit: 7 days
Topic
Find Limits
Graphically
Find Limits
Algebraically
Standards
I can…
● Use a table to estimate a limit.
● Use a graph to estimate a limit.
● Explain why a limit exists and
does not exist.
● Find one sided limits.
● Evaluate a limit using
substitution.
● Evaluate limits to infinity.
● Find one sided limits.
Resources
Time
Frame
● Section 11.1
● Section 11.2
2-3 days
● Section 11.1
● Section 11.2
3-4 days