Illustrative Mathematics Algebra 2 Unit 6 Resources
Illustrative Mathematics Algebra 2 Unit 6 Resources
Table of Contents
- Introduction
- Purpose
- Standards
- Unit Circle
- Measuring Circles Using the Radius - 2.6.3
- Pythagorean Identity - 2.6.5
- Periodic Functions
- Introduction to Trigonometric Functions - 2.6.9
- Beyond 2 π - 2.6.10
- Trigonometric Transformations
- Amplitude and Midline - 2.6.13
- Modeling Circular Motion - 2.6.18
Purpose
The purpose of this Geogebra Book is to accompany Unit 6 of the Algebra 2 Illustrative Mathematics curriculum. The activities are not intended to entirely replace Unit 6, but instead to supplement/replace certain activities within the unit. It is intended to be a resource for teachers who can assign activities at specific times within lessons or the unit (see "Sequence" column below).[br][br][br][table][tr][td]Activity[/td][td]Learning Targets[/td][td]Sequence[/td][/tr][tr][td]Measuring Circles Using the Radius[br]2.6.3[/td][td][list][*]I understand that a radian angle measurement is the ratio of the arc length to the radius of the circle.[/*][/list][/td][td]In place of Activity #3 in Lesson #3[br](2.6.3.3)[/td][/tr][tr][td]Pythagorean Identity[br]2.6.5[/td][td][list][*]I understand that cosine and sine are defined as the x- and y-coordinates of a point on the unit circe at [math]\theta[/math] radians.[/*][/list][/td][td]In place of Lesson #5[br](2.6.5)[/td][/tr][tr][td]Introduction to Trigonometric Functions[br]2.6.9[/td][td][list][*]I can use the coordinates of points on the unit circle to graph the cosine and sine functions.[/*][/list][/td][td]In place of Lesson #9[br](2.6.9)[/td][/tr][tr][td]Beyond [math]2\pi[/math][br]2.6.10[/td][td][list][*]I understand how to find the values of cosine and sine for inputs greater than [math]2\pi[/math] radians.[/*][/list][/td][td]In place of Lesson #10[br](2.6.10)[/td][/tr][tr][td]Amplitude and Midline[br]2.6.13[/td][td][list][*]I can write a trigonometric function to represent situations with different amplitudes and midlines.[/*][/list][/td][td]In place of Activities #2&3 in Lesson 13[br](2.6.13.2-1.6.13.3)[/td][/tr][tr][td]Modeling Circular Motion[br]2.6.18[/td][td][list][*]I can represent a circular motion situation using a graph and an equation[/*][/list][/td][td]In place of Activities #1&2 in Lesson 18[br](2.6.18.1-2.6.18.2)[/td][/tr][/table]
Measuring Circles Using the Radius - 2.6.3
Use the slider below and observe what happens. What do you notice? What do you wonder?
About how many radii does it take to go halfway around the circle?
About how many radii does it take to go all the way around the circle?
Why doesn’t the number of radii that fit around the circumference of a circle depend on the length of the radius of the circle? [br]
What is the exact number of radii that fit around the circumference of the circle? Explain how you know. For a hint, type "Give me a hint!" and click "check my answer"
A bicycle wheel has a 1 foot radius. The wheel rolls to the left (counterclockwise).
What is the circumference of the wheel when the radius is 1 foot?[br]
Introduction to Trigonometric Functions - 2.6.9
Use the slider below to plot points on the graphs below. You may need to reduce fractions.
Use the Point tool to plot the values of y=cos(θ), where θ is the measure of an angle in radians.
Use the point tool to plot the values of y=sin(θ), where θ is the measure of an angle in radians.
Check your graphs by clicking on "Algebra" on each graph and typing in the corresponding function next to the plus sign: y=cos(x) on the first graph and y=sin(x) on the second graph.
What do you notice about the two graphs?
Could these graphs represent functions? Explain your reasoning.
[size=150]Looking at the graphs of [math]y=\cos\left(\theta\right)[/math] and [math]y=\sin\left(\theta\right)[/math], at what values of [math]\theta[/math] do [math]\cos\left(\theta\right)=\sin\left(\theta\right)[/math]? [br][/size]
Where on the unit circle do these points correspond to?
Amplitude and Midline - 2.6.13
Use the windmill animation to answer the questions below:
Let's say the windmill has a radius of 1 meter and its center is 8 meters off the ground. Write a function describing the relationship between the height [math]h[/math] of Point [math]k[/math] and the angle of rotation [math]\theta[/math]. Explain your reasoning. For a hint, write "Give me a hint!" and click "Check my answer"[br]
Describe how your function and its graph would change if the windmill blade has length 3 meters.[br]
Describe how your function and its graph would change if the windmill blade has length 0.5 meter.[br]
Describe how your function and its graph would change if the windmill were 11 meters off the ground.
Test your predictions by graphing the functions below:
Here is the graph of a different function describing the relationship between the height y, in feet, of the tip of a blade and the angle of rotation θ made by the blade.
Describe the windmill.