Download Module 2 Guidance

Algebra II Module 2
Trigonometric Functions
Topic A
The Story of Trigonometry and Its Contexts
13 Days
Topic B
Understanding Trigonometric Functions
8 Days
Overview
Module 2 builds on students’ previous work with units (N-Q.A.1) and with functions (F-IF.A.1, F-IF.A.2, F-IF.B.4, F-IF.C.7e, F-BF.A.1,
F-BF.B.3) from Algebra I and with trigonometric ratios and circles (G-SRT.C.6, G-SRT.C.7, G-SRT.C.8) from high school Geometry.
Included in Topic A is preparation for extension standard F-TF.A.3. Extension standard F-TF.C.9 is also discussed in Topic B as
preparation for the Precalculus and Advanced Topics course.
Topic A starts by asking students to graph the height of a passenger car on a Ferris wheel as a function of how much rotation it has
undergone and uses that study to help define the sine, cosine, and tangent functions as functions from all (or most) real numbers to
the real numbers. A precise definition of sine and cosine (as well as tangent and the co-functions) is developed using
transformational geometry. This precision leads to a discussion of a mathematically natural unit of measurement for angle
𝜋 𝜋 𝜋 𝜋
measures, a radian, and students begin to build fluency with values of sine, cosine, and tangent at 6 , 4 , 3 , 2 , 𝜋, etc. The topic
concludes with students graphing the sine and cosine functions and noticing various aspects of the graph, which they write down as
simple trigonometric identities.
In Topic B, students make sense of periodic phenomena as they model them with trigonometric functions. They identify the
periodicity, midline, and amplitude from graphs of data and use them to construct sinusoidal functions that model situations from
both the biological and physical sciences. They extend the concept of polynomial identities to trigonometric identities and prove the
Pythagorean identity; this identity is then used to solve problems.
1
Lesson
Big Idea
Emphasize
Suggested
Problems
Exit
Ticket
# of Days
TOPIC A
1
Students apply geometric concepts to
graph a periodic relationship.
4
Students define sine and cosine as
functions for degrees of rotation of
the ray formed by the positive 𝑥-axis
up to one full turn
Include drawing a diagram of an
angle in standard position.
Note: The second objective of this
lesson falls under a “+” (additional)
standard.
+Students use special triangles to
geometrically determine the values
of sine and cosine for 30, 45, 60, and
90 degrees.
5
Students evaluate the sine and cosine
functions at multiples of 30 and 45.
Exploratory Challenges 1-2
Exercises 1-5
Problem Set 1-2
eMath Instruction
Unit 11 lesson 1
Yes
Opening Exercise
Examples 1-2
Exercises 1-5
Problem Set 1 ,3-8
eMath Instruction
Unit 11 Lesson 3
Yes
2
3
eMath Instruction
Unit 11 lesson 4
The values of sine and cosine for
angles greater than 360 are
determined based on the fact that
these functions are periodic.
2
Examples 1-2
Exercises 1, 2, 6, 7
Problem Set 1, 3, 4, 5
Yes
1
Lesson
6
Big Idea
Emphasize
Students define the tangent function.
Note: The second objective of this
lesson falls under a “+” (additional)
standard.
+Students use special triangles to
determine geometrically the values
of the tangent function for 30°, 45°,
and 60°.
7
Secant and the Co-functions
Suggested
Problems
Opening Exercise
Example 1
Exercises 1-8
Problem Set #1
eMath Instruction
Unit 11 lesson 10
Exit
Ticket
Yes
# of Days
2
Do not place large emphasis on this
lesson.
eMath Instruction
Unit 11 lesson 11
No
1
Exploratory Challenges 1-2
eMath Instruction
Unit 11 Lesson 6
Examples 1-4
Exercise 6
Problem Set 2,3,6-9
eMath Instruction
Unit 11 Lesson 2
Yes
2
Yes
2
Note: We are currently unable to
verify if this item will be tested on
the Regents exam. Reciprocal trig
functions do not explicitly appear in
any of the standards.
8
Students graph the sine and cosine
functions.
Key features of the graphs including
intercepts, period and amplitude.
9
Students convert between degrees
and radians.
Include finding the length s of an arc
intercepted by a central angle of
measure 𝜃 in radians using 𝑠 = 𝑟𝜃.
3
Lesson
Big Idea
Emphasize
Suggested
Problems
Exit
Ticket
# of Days
TOPIC B
11
Students transform the graphs of the
sine and cosine function.
The effect of changing the
coefficients in the equation on the
graphs of the sine and cosine
functions.
Exercises a-f
Problem Set 1, 2
eMathInstruction
Unit 11 Lesson 7
Yes
3
eMathInstruction
Unit 11 Lesson 8
12
Students use trig functions to model
cyclical behavior.
Look at released sample items for
modeling problems.
Exercises 1-4
Problem Set 1,3
eMathInstruction
Unit 11 Lesson 9
Yes
2
14
Students graph the tangent function.
Use graphing calculator to graph
tangent function.
Exercises 1-5
Yes
1
15
Students use the Pythagorean
identity to find sin(𝜃), cos(𝜃), or
tan(𝜃), given sin(𝜃), cos(𝜃), or tan(𝜃)
and the quadrant of the terminal ray
of the rotation.
Sample Item: Angle 𝜃 is in Quadrant
II, and sin 𝜃 = 4/5. What is the
value of cos 𝜃?
Exercises 1a-b
Problem Set 4, 5
eMath Instruction
Unit 11 Lesson 5
No
2
4