##
* *Reflection: High Expectations
Period and Amplitude - Section 2: Reviewing the graphs of sine and cosine

I have found that expectations need to be clear when teaching students to graph. I do not expect perfect graphs with every point in place as this is unnecessary, but I do have certain expectations. When left to their own devises, students will often draw a sloppy graph with no units labeled. In fact, even with clear expectations, students will sometimes turn in inadequate graphs. This has been a challenge for some of my students this year.

Here are my expectations. There are always key points associated with a function that much be shown. I insist on tick marks with units labeled if a scale other than one is used. I want to graph to extent across their coordinate plane. I also encourage them to use graph paper as this will naturally improve the quality of their graphs as the proportions are already set.

For the parent Graphs of Sine and Cosine, I use the interval of 0 ± π as my point indicators. For sine functions, 0 ± π are the x-intercepts. For cosine, 0 ± π are the local maximum and minimum. By plotting these points, my student’s graphs are more proportional and this really helped when we started changing the period or shifting the graph horizontally.

*High Expectations: Improving Graphs of Sine and Cosine*

# Period and Amplitude

Lesson 9 of 19

## Objective: Students will be able to identify the period and amplitude of a sine or cosine graph or equation.

*59 minutes*

#### Warm-up and Homework Review

*10 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Up- Period and Amplitude, which asks students to compare trigonometric functions to other functions studies this year.

I also use this time to correct and record the previous day's Homework.

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In the previous lesson, students constructed both the sine and cosine graph from a unit circle. The first goal in today's lesson is to draw these graphs in their notes along with all of the major features (except period and amplitude which are the topics of today's lesson). Their graphs need to include the height of each function, the local maximum and minimum values, the y-intercept, and the x-intercepts. For the re-occurring points, this is an important place to discuss how we can write down an expression that represents every one of those points (**Math Practice 8) **I have the students discuss in pairs how you can represent these reoccurring points. The goal is that they come up with ±2π.

#### Resources

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#### Understanding Amplitude

*5 min*

The next section has students look at the amplitude as the height of one of these graphs and then gives the students some examples of finding amplitude. Please note that I am using both degrees and radians as the unit for the x-axis. This practice will reinforce their understanding of the connections between degrees and radians (**Math Practice 7**).

#### Resources

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#### Understanding Period

*15 min*

Period is a bit more involved than amplitude. Instead of giving the students the formula for finding period and having them apply it, we are going to give them an opportunity to build the concept of period as a horizontal stretch or shrink. The first four examples all include both the equation and the graph. The graphs can come up after the student discuss or during as needed. This is a major scaffolding piece in this lesson. The next examples just include the equation. By the end, have the students write a statement on how to find period.

#### Resources

*expand content*

The final portion includes a sine and a cosine function to graph, both with a period and amplitude stretch. I have the students graph these and then model them myself on the board. This is the opportunity to show students what level of accuracy and what specific detail I would like them to include in a graph. I add examples if time allows and it seems like the students need it.

#### Resources

*expand content*

#### Homework

*2 min*

There are four portions to this Homework. The first portion asks the students to identify the period and amplitude from the graph of a sine or cosine function. The section portion asks them to find the same thing from an equation. Next, they are asked to graph some sine and cosine functions. This is the most important part. There is no better way to memorize the important features of a trig graph than draw it given different variations. Half of the problems are in degree and half are in radians. The final question is an extension question asking the students to compare the domain and range of the trig graphs they just completed (**Math Practice 2**).

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#### Exit Ticket

*2 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

This Exit Ticket checks that the students understand the where to find the period and amplitude in a trigonometric function.

#### Resources

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Jennifer,

Thanks for bringing this to my attention! Â I have reloaded it. Â

| one year ago | Reply

Amelia,

I am enjoying your lessons on the graphs of sine and cosine. I am hoping my students will too and that it will increase their understanding. I went to take a look at the homework assignment for this lesson but am unable to see it. If possible, can you upload it again or maybe in word format?

Thanks

| one year ago | Reply##### Similar Lessons

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Angle and Degree Measure
- LESSON 2: Trigonometric Ratios
- LESSON 3: Trigonometric Ratios of General Angles
- LESSON 4: Radians Day 1 of 2
- LESSON 5: Radians Day 2 of 2
- LESSON 6: The Unit Circle Day 1 of 2
- LESSON 7: The Unit Circle Day 2 of 2
- LESSON 8: Graphs of Sine and Cosine
- LESSON 9: Period and Amplitude
- LESSON 10: Period Puzzle
- LESSON 11: Transformations of Sine and Cosine Graphs
- LESSON 12: Graph of Tangent
- LESSON 13: Model Trigonometry with a Ferris Wheel Day 1 of 2
- LESSON 14: Model Trigonometry with a Ferris Wheel Day 2 of 2
- LESSON 15: Modeling Average Temperature with Trigonometry
- LESSON 16: Pythagorean Identity
- LESSON 17: Trigonometric Functions Review Day 1
- LESSON 18: Trigonometric Functions Review Day 2
- LESSON 19: Trigonometric Functions Test