SFUSD Mathematics Core Curriculum Development Project · CPM Pre-Calculus with Trigonometry Lesson...

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SFUSD Mathematics Core Curriculum, Algebra 2, Unit A.8: Trigonometric Functions, 2014–2015

SFUSD Mathematics Core Curriculum Development Project

2014–2015

Creating meaningful transformation in mathematics education

Developing learners who are independent, assertive constructors of their own understanding

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Algebra 2

A.8 Trigonometric Functions

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Finding Missing Lengths CCCA2 Notes and Problem 7-15 HW: Special Right Triangles

1 per pair 1 per student 1 per student

4 Lesson Series 1 CPM CCA2 Lesson 7.1.5 (5 pages) HW: CPM PC 4-13 HW: CPM A2C 8-81 CPM Precalculus Lesson 4.1.1 (3 pages) Resource Pages 4.1.1 (3 pages) CPM Precalculus Lesson 4.1.2 – Part 1 (4 pages) HW: Practice Unit Circle (1 page)

1 per pair 1 per student 1 per student 1 per pair 1 per student 1 per pair 1 per student

Rulers Scissors String Circular objects of various diameters

1 Apprentice Task CPM Precalculus Lesson 4.1.2 – Part 2 (2 pages) 1 per pair

6 Lesson Series 2 CPM Precalculus Lesson 4.1.3 (4 pages) Resource Pages 4.1.3 (2 pages) CPM A2C Lesson 8.2.1 (2 pages) HW: CPM A2C Lesson 8.2.1 CPM Precalculus Lesson 4.1.4 (2 pages) HW: CPM Precalculus Lesson 4.1.4 (2 pages) CPM A2C Lesson 8.2.2 (2 pages) HW: A2C Lesson 8.2.2 CPM A2C Lesson 8.2.3 (4 pages) HW: A2C Lesson 8.2.3

1 per pair 1 per student 1 per pair 1 per student 1 per pair 1 per student 1 per pair 1 per student 1 per pair 1 per student

3-inch cans (small soup cans) Dry fettuccine Glue sticks, string, scissors Graphing software

2 Expert Task Ferris Wheel (2 pages) 1 per student

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3 Lesson Series 3 CPM CCA2 Lesson 8.2.4 (4 pages) Modeling Problems

1 per student 1 per student

2 Milestone Task Constructed Response Foxes and Rabbits (3 pages)

Provided by AAO Provided by AAO

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Unit Overview

Big Idea

Model circular and periodic motion using different representations of trigonometric functions extended to all angles.

Unit Objectives

• Students will understand radian measure. • Students will be able to use the unit circle to find the sine and cosine values for all real number inputs, extending the right triangle definitions that

students have seen previously. • Students will recognize sinusoidal functions and be able to graph sine and cosine functions. • Students will be able to identify the period, midline, and amplitude of sine and cosine functions. • Students will be able to translate among the various representations: symbolic, graphical, tabular, and verbal. • Students will be able to prove the Pythagorean identity: 𝑠𝑖𝑛!𝜃 + 𝑐𝑜𝑠!𝜃 = 1. • Students will model situations using trigonometric functions.

Unit Description

In this unit, students will continue their study of trigonometry by first reviewing special right triangles and trigonometric ratios. They will use the special right triangles to extend the definition of angles (in degrees) based on rotation and direction of a ray around the unit circle. Then students will explore what a radian is including conversions to and from degrees. They will use fettuccini to graph the parent sine and cosine functions and explore transformations of the the sine and cosine functions. Finally, they will model real world situations using periodic functions.

CCSS-M Content Standards

Seeing Structure in Expressions Interpret the structure in expressions A.SSE.1 Interpret expressions that represent a quantity in terms of its context★. A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients★. A.SSE.1b 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. ★ A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

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Interpreting Functions Analyze functions using different representations F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ F.IF.7e Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. ★ Trigonometric Functions Extend the domain of trigonometric functions using the unit circle 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. Model periodic phenomena with trigonometric functions F.TF.5 Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.★ Prove and apply trigonometric identities F.TF.8 Prove the Pythagorean identity: 𝑠𝑖𝑛!𝜃 + 𝑐𝑜𝑠!𝜃 = 1 and use it to find given and the quadrant of the angle. ★ A star indicates a modeling standard

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Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

● Define sine, cosine and tangent from geometry

● Use trigonometric ratios to solve right triangle applied problems

● Equation of circle ● Understanding of linear and non-linear

functions

Students will be adding to their repertoire of functions to include a treatment of the sine and cosine functions that includes understanding their key features (period, amplitude, midline). They will use this understanding to model periodic phenomena. Students will also learn about radian measure and how the unit circle enables the extension of the sine and cosine functions to all real numbers. They will prove and apply the Pythagorean identity.

● Define secant, cosecant and cotangent functions

● Graph secant, cosecant and cotangent functions

● Polar coordinates ● Prove other trigonometric identities

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Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit?

1 day 4 days 1 day 6 days 2 days 3 days 2 days Total: 19 days

Lesson Series 1

Lesson Series 2

Lesson Series 3

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Entry Task Finding Missing Length

Apprentice Task What Does the Unit Circle Tell Me?

Expert Task Ferris Wheel

Milestone Task Foxes and Rabbits

CCSS-M Standards

G: SRT

F-TF: 1, 2, 8 A-SSE 1a F-TF 5 F-IF 7e

A-SSE F-IF 7e F-TF: 1, 2, 5, 8

Brief Description of task

Students attack a challenging problem from geometry. The students then see a solution to the problem involving trigonometric ratios and special right triangles.

Students continue to make sense of the unit circle by finding angles and coordinates in different quadrants

After working in pairs on a matching exercise, the students will individually solve the Ferris Wheel problem.

Exam that covers all topics in unit including unit circle, trigonometric ratios and modeling real world situation.

Source Discovering Geometry, 2008 CCCA 2 Connections

CPM Pre-Calculus with Trigonometry

Adapted from FAL Ferris Wheel, Shell Center, 2012

SFUSD Teacher Created

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

F-TF 1, 2, 8 F-IF.7e, F-TF.5 F-IF.7e, F-TF.5, A-SSE.1a, A-SSE.1b, A-SSE.2

Brief Description of Lessons

Students will develop an understanding of the definition of a radian, apply special angles to the unit circle and learn how the definition of sine and cosine extends from an acute angle (of a right triangle) to all angles and develop the Pythagorean Theorem Identity.

Students will graph sine and cosine and transform both functions.

Students have more practice with graphing and modeling, and they understand that sine and cosine are functions that are horizontally shifted.

Sources

CPM Core Connections Algebra 2 CPM Pre-Calculus with Trigonometry

CPM Core Connections Algebra 2 CPM Pre-Calculus with Trigonometry

CPM Core Connections Algebra 2 Foerster – Algebra and Trigonometry

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Entry Task

Finding Missing Lengths

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: Review trigonometric ratios of special right triangles and assess how much students remember from geometry. This will be needed in the next lesson series. CCSS-M Standards Addressed (not from this Unit): • G-SRT: Understand similarity in terms of similarity transformations. Define trigonometric ratios and solve problems involving right triangles.

Launch: This task is an extension of your student’s previous geometry course experience. This is only to discover what your students recall and will serve as a lead into understanding trigonometry. During: If students are struggling, guide students to use previous knowledge about Pythagorean theorem, equilateral triangles, isosceles triangles, and squares. (See CCC2 Problem 7-15) You may need to supplement with a 1-day special right triangle lesson Classwork/Homework – KUTA worksheet, 2-18 even problems. http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/8-Special%20Right%20Triangles.pdf

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Finding Missing Lengths

How will students do this?

Focus Standards for Mathematical Practice: (no more than 2) 1. Make sense of problems and persevere in solving them. Structures for Student Learning: Academic Language Support Vocabulary: right triangles, inscribed, equilateral, area, sine, cosine, tangent Sentence frame: “I think it is an equilateral triangle, because _______________________” Differentiation Strategies

Give angle measurements, give triangle area formula, etc. Participation Structures (group, partners, individual, other)

Groups or pairs

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Lesson Series #1

Lesson Series Overview: Students will develop an understanding of the definition of a radian, apply special angles to the unit circle and learn how the definition of sine and cosine extends from an acute angle (of a right triangle) to all angles and develop the Pythagorean Theorem Identity. Standards: F-TF 1, 2, 8 Time: 3 days

Lesson Overview – Day 1 Resources

Description of lesson: Radians - How else can I measure angles? Students develop understanding of radian. Notes: Materials: rulers, different circular objects, scissors Alternate Activity: Use string to measure the radius, then use the string to mark off radians around the circle.

CPM Algebra 2 Core Connections, Lesson 7.1.5: Problems 7-71 to 7-78, Resource page (Circles) Homework: CPM Algebra 2 Connections, Problem 8-80, 8-81, CPM Pre-Calculus with Trigonometry: problems 4-13


Description of Lesson: UNIT CIRCLE: Do you ever feel like you’re going in circles? Students will learn to create a unit circle of special angles from their understanding of special right triangles Notes: Problems 4-5, 4-6, and 4-7 need to be changed from radians to degrees.

CPM Pre-Calculus with Trigonometry Lesson 4.1.1: Problems 4-2 to 4-8 CPM Pre-Calculus with Trigonometry Resource Page: Lesson 4.1.1 A, B, C

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Lesson Overview – Days 3-4 Resources

Description of Lesson: Students will

• Learn how sine and cosine can be viewed in terms of the unit circle • Develop the Fundamental Pythagorean Identity • Use the unit circle, the Fundamental Pythagorean Identity, and right

triangles, to find the values of trigonometric functions Notes:

• Suggest Warm-up to have problems with students finding different sides of right triangles using sine, cosine and tangent ratios

• For problem 4-15 d,e ask students “if we chose a random point on a circle of radius 1 (centered around the origin), what is the relationship between the coordinates x , y and 1?” Is this true for any point in the circle? If students are stuck, have them consider right triangle and if needed, Pythagorean Theorem.

CPM Pre-Calculus with Trigonometry Lesson 4.1.2: Problems 4-15, 4-16, 4-17 (optional), 4-18 to 4-22 Homework: Practice filling in Unit Circle with radian & degrees angles and their coordinates

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Apprentice Task

What Does the Unit Circle Tell Me?



Math Objectives: • More sense making of unit circle through finding angles and sine and

cosine of angles in different quadrants CCSS-M Standards Addressed: F.TF.1, F.TF.2, F.TF.8 Potential Misconceptions:

• Negative/Positive signs • Students might switch x and y • Distinction between θ vs. sin θ and θ vs. cos θ

CPM Pre-Calculus with Trigonometry Lesson 4.1.2: Problems 4-23, 4-24, 4-25 a, b, c Launch: Students will connect what they know about sine and cosine in right triangles to the unit circle, an extension of what they did yesterday. Allow students to use their unit circle if needed. Students work collaboratively in pairs or groups, but each person does an individual write up on their own paper. These problems could be separated into three strips. Groups or pairs only move on to the next “strip” or problem after each person understands and completes the problem at hand. Require that pairs or groups go through a “check-point” with the teacher before moving on to next problem. During: Monitor and remind students they can us their unit circle. Be mindful of students confused between the angle vs the sine or cosine of that angle.

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What Does the Unit Circle Tell Me?


Focus Standards for Mathematical Practice: 6. Attend to precision. 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support:

Vocabulary: radian, degree, sine, cosine, coordinates, special right triangles, symmetry, quadrants Sentence frames: I know this is a right triangle, because _____________________. I know I can use the Pythagorean Theorem, because _____________________________.

Differentiation Strategies:

• Remind students of Pythagorean Theorem. • Remind students to use Unit Circle.

Participation Structures (group, partners, individual, other): Groups or pairs

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Lesson Series #2

Lesson Series Overview: Students will graph sine and cosine and transform both functions. CCSS-M Standards Addressed: F.IF.7e, F.TF.5 Time: 6 days


Description of Lesson: Fettuccini: How can I build a wave? Graphing sine and cosine function using fettuccini to relate sine and cosine of angles in a unit circle to form a function on the coordinate grid. Notes:

• Read CPM teacher notes for 4.1.3 • Materials: 3-inch cans (standard small soup cans), dry fettuccine, glue sticks,

string, scissors, graphing software • Emphasize the math connection between using the “vertical” length of the pasta

as finding y for sine and the “horizontal” length of the pasta as finding the x for cosine in a unit circle.

• Math Notes on p. 276 defines period and amplitude. Problem 4-38 gives practice on these.

CPM Pre-Calculus with Trigonometry Lesson 4.1.3, Problems 4-32 to 4-36, 4-38 Resource Page: Lesson 4.1.3A, Lesson 4.1.3B Closure/Extension: YouTube video: http://www.youtube.com/watch?v=Ohp6Okk_tww

Lesson Overview – Days 2-3 Resources

Description of Lesson: How can I transform sine and cosine graph? [y = asin(x - h) + k, y = acos(x - h) + k] Notes:

• Materials: Graphing software

Day 2 CPM Algebra 2 Connections, Lesson 8.2.1: Problems 8-113 to 8-115 Homework: CPM Algebra 2 Connections, Lesson 8.2.1: problems through 8-116 to 8-118

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• For 8-115, have students do transformation of cos x as well as sine x • Have students take notice (put in their notes) of the Math Notes 5 point method.

Day 3 CPM Pre-Calculus with Trigonometry Lesson 4.1.4, problems 4-47 to 4-50 Homework: CPM Pre-Calculus with Trigonometry Lesson 4.1.4, problems 4-51, 4-52, 4-53


Description of Lesson: Students learn intuitively the need for another parameter, b: [y = asin b(x - h) + k, y = acos b(x - h) + k] and will be able to graph transformed functions. They will be able to find the period of a function. Notes: Materials: graphing software

• For 8-124, the output is the y coordinate located at each π/6 interval per 2 seconds. (Remember that the x value represents time, not the angle). The first point can be (2, ½) since at 2 seconds, the radar is at π/6 and the y coordinate is ½; since ½ is the distance from the outer end of the radar line to the horizontal axis, the coordinate is the y value. Next points can be (4, √3/2), (6,1), …When students graph these points, they will find that the period is NOT 2π and therefore the general function y = asin(x - h) + k does not work.

CPM Algebra 2 Connections Lesson 8.2.2: Problems 8-123 to 8-126 Homework: CPM Algebra 2 Connections Lesson 8.2.2: Problems 8-127, 8-128


Description of Lesson: Students will understand and be able to find the period of sine and cosine function Notes:

• Omit Problem 8-139 • For Problem 8-138, it is not necessary to use a calculator if there is time to

graph by hand. • This can be done in groups or pairs.

CPM Algebra 2 Connections Lesson 8.2.3: Problems 8-137 to 8-138 Homework: Select other problems from section 8.2.3

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Students will practice with transformations of the sine function with all parameters and some modeling.

CPM Algebra 2 Connections Section 8.2.3: Problems 8-140, 8-141, 8-142, 8-144, 8-145, 8-146

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Expert Task Ferris Wheel



Math Objectives: • Students will model real-world circular situations with a periodic

function. CCSS-M Standards Addressed: A.SSE.1a, F.TF.5, F.IF.7e Notes:

• Problem 3 is modeling with cosine, something they have not done but should be able to figure out. (Don’t worry if they can’t do this, but push as much as possible.)

Launch: Students should work in pairs or groups. Each student does his or her own write up. Remind students that they will need to make sense the problem, in particular the quantities involved. Encourage them to ask lots of questions of their group members to help clarify any misconceptions (e.g., highest height, starting height). Teacher may want to show a video of a Ferris wheel to set up the situation During: For #1 and #2: If students have trouble, have them draw the picture of the Ferris wheel with known quantities, in particular asking them to show the height of axle and the diameter of the wheel. They can also make a table to show points (time, height) that they can deduce from the situation. For #3: The equation is h(t) = 40 - 30 cos (90t) or h(t) = 40 - 30 cos ((π/2)t). The negative may be the hardest to get. A hint could be to think about how the graph is transformed from the parent cosine graph. Closure/Extension: Make sure students connect every part of the function to the situation and can explain in their own words.

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Ferris Wheel


Focus Standards for Mathematical Practice: 4. Model with mathematics. 7. Look for and make use of structure.


Vocabulary: Ferris wheel, diameter, axle, rotates Sentence frames: I think the value a, b, and c will model this situation, because ________________________________.

Participation Structures (group, partners, individual, other):

This should be done as a group; pairs are also an option.

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Lesson Series #3

Lesson Series Overview: Students have more practice with graphing and modeling, and they understand that sine and cosine are functions that are horizontally shifted. CCSS-M Standards Addressed: F.IF.7e, F.TF.5, A.SSE.1a, A.SSE.1b, A.SSE.2 Time: 3 days


Description of Lesson: Students consolidate connections between graphs, equations, and situations Notes: Read the CPM Teacher Notes.

CPM Algebra 2 Connections, Lesson 8.2.4, Problems 8-152 through 8-155 Homework: Problem 8-156


Description of Lesson: Students model with sine and cosine functions.

CPM Algebra 2 Connections, Section 8.2.4, Problems 8-156 to 8-161


Description of Lesson: Students model with sine and cosine functions. Notes: You may want to extend this into two days.

Modeling Problems worksheet You could also Foerster Precalculus, Problems 1, 2, 5, and 6, located in the Resources folder.

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Milestone Task

Foxes and Rabbits



Math Objectives: • Students will find the exact value of trigonometric functions. • Students transform sine and cosine functions. • Students model a real-world situations using periodic functions.

CCSS-M Standards Addressed: A.SSE.1a, F.IF.7e, F.TF.1, F.TF.2, F.TF.5, F.TF.8

Launch: • Go over the rules and expectations for test taking. • Allow students to use unit circles.

During: Allow at least 45 minutes for the assessment.

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Foxes and Rabbits


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 4. Model with mathematics.


Vocabulary: sinusoidal, periodic, amplitude, midline, parameter, radian, reference angle, cyclic function, sine, tangent, trigonometric functions, unit circle, transformations, vertical, horizontal Sentence frames: I think the model is periodic, because __________________________________.

Participation Structures (group, partners, individual, other):

Individual constructed response and performance assessment used for CLA 1.

SFUSD Mathematics Core Curriculum Development Project · CPM Pre-Calculus with Trigonometry Lesson...

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