Module 7 Resources

The Object of an Angle (Slides 1-5)
Angle Measure as a Fraction of a Circle's Circimference (Degrees) (Slides 5-11)
Radian as a Unit of Measurement (Slides 12-18)
Covariation of Sine and Cosine on a Circle (Slides 19-27)
Evaluation of Sine and Cosine on a Circle (Slide 28)
Inverse Trigonometric Functions (Slides 29-31)
Reference Angles (Slides 32-33)
Periodicity and Amplitues

The Object of an Angle (Slides 1-5): Click for Purpose

The students need to have an image of an object with measurable attributes, in this case an angle. This part of the lesson is intended to motivate the need to define an angle and realize the need to have a common way of measuring an angle.

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Worksheet 1
Notes

Click for Summary/Goals

Create need for angles and angle measure

Click for Implementation Notes

This activity should be given at the beginning of the trig module
Students should be encouraged to discuss how the cannons are set up
This can be done as a small group or whole class exercise
Recommended time: 10-15 minutes

Angle Measure as a Fraction of a Circle's Circimference (Degrees) (Slides 6-11): Click for Purpose

The students should come to understand that a degree subtends 1/360th of the circumference of any circle. This will help prepare students to see angle measure as a measure of a percentage, or fraction, of a circle’s openness.

As the class progresses through these slides, students should continue to describe that one degree refers to an arclength that is 1/360th of the total circle’s circumference. This is intended to promote further student reflection on angle measure in terms of a quantitative relationship between arc-length and circumference while providing an opportunity to gain insights into the students’ developing conceptions of angle measure. This part of the lesson should build an image of angle measurement that includes a circle of any radius.

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Worksheet 1
Notes

Click for Summary/Goals

Consider how to construct a protractor to measure in different units of angle measurement
Reflect on construction of protractor
Continue development of students conceptions of relationship between arclength, circumference, and the measure of an angle’s openness
Continue promotion of students’ reflection on angle measure in terms of a fraction of a circle’s circumference
Further develop the relationship between angular measurement and linear measurement of arclength

Click for Implementation Notes

Page 1 of this activity should be given at the beginning of slide 5 of the trig module
Students should be probed regarding their decision making
To keep the students from focusing on dividing up the area, the teacher may want to instruct students that they may not fold their protractor
Teacher may want to ask about construction method if the size of the protractor is changed
Recommended time: 10-15 minutes
Page 2 of this worksheet should be given at the conclusion of the Protractor Applet Teacher should prompt students to reflect on angle measure as a fraction, or percentage, or circumference
The blank page should be given at the beginning of slide 8
Teacher should focus the students on the use of a circle to construct the angle by determining the linear measurement of arclength that corresponds to both the circle constructed and the angle being produced

Protractor Applet (gsp)
Notes (doc, pdf)

Click for Summary/Goals

Continue investigation of relationship between (a varying) angle measure, the circumference of any circle centered at the vertex of the angle, and the arc-length subtended by the angle
In terms of an increasing radius of the protractor, the students will be promoted to discuss that the angular measurement and proportions remain constant, but the linear measurement of the arc-length cutoff and the linear measurement of the circumference increase.

Click for Implementation Notes

This applet should be used at the beginning of slide 7
This applet should be projected onto a whiteboard
Recommended time: 10-15 minutes

Radian as a Unit of Measurement (Slides 12-18): Click for Purpose

Purpose: Students should come to understand a radian as a unit of measurement. They should see the unit of a radian as the ratio of a linear measurement of the arclength to the length of a radius, thus giving the number of radius lengths or a percentage of one radius. They should realize that there are 2π, or approximately 6.28 radii along the circumference of a circle.

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Worksheet 1 Notes	Click for Summary/Goals Introduce the unit of the radius of a circle as a unit of measurement Determine the number of radii needed to measure the circumference of a circle Click for Implementation Notes Blank page 2 should be given at the conclustion of slide 9 Once this activity is finished, the powerpoint animations on slides 10 and 11 should be used as a summary The Protractor Applet can again be used at the conclusion of slide 13, this time with radians instead of degrees Use similar investigations as were done for degrees
Protractor Applet (gsp) Notes (doc, pdf)	Click for Summary/Goals Continue investigation of relationship between (a varying) angle measure, the circumference of any circle centered at the vertex of the angle, and the arc-length subtended by the angle In terms of an increasing radius of the protractor, the students will be promoted to discuss that the angular measurement and proportions remain constant, but the linear measurement of the arc-length cutoff and the linear measurement of the circumference increase. Click for Implementation Notes The Protractor Applet can again be used at the conclusion of slide 13, this time with radians instead of degrees Use similar investigations as were done for degrees Ask students same questions about changing radius and changing angle measure
Worksheet 2 Notes	Click for Summary/Goals Intended to be a capstone activity Have students participate in activities similar to what has already been done Allow teacher insights into students' understandings Click for Implementation Notes This worksheet should be given at the end of slide 14 The activities are similar to what students have already done The teacher should allow the students to work on the activity with little prompting Can be used as a homework activity Recommended time: 10-15 minutes

Covariation of Sine and Cosine on a Circle (Slides 19-23): Click for Purpose

The intention of this section is to get students to begin to contemplate the relationship between angle measure and location on the unit circle. In particular they should attend to how changes in angle measure result in changes in the vertical and horizontal distance from the origin.

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Worksheet 3
Notes

Click for Summary/Goals

Students should make connection between circular motion of bug and angle measure
Students should see the distance traveled by the bug is the arclength of the circle
Students will create rough graphs for sine and cosine functions
Students will begin to covary an input of angle measure with an output of a distance from the center of a circle (both horizontal, cosine, and vertical, sine)

Click for Implementation Notes

This activity should be given at the beginning of slide 15 of the trig module
Teacher can orient students by asking about how the distance the bug traveled on the fan is related to angle measure
The teacher may also ask about what is created if a ray is put in connection the center of the circle to the bug's starting point and a second ray is used to connect the center with the bug's current location (an angle)
Recommended time: 10-15 minutes
Page 2 of this worksheet should be given at the conclusion of page 1
It may be useful to have a discussion of how angle measure relates to the distance traveled by the bug and what it means in terms of linear distance for the bug to travel so many radians
Students should be allowed to pick any units they want
Students should put their graphs up for display followed by an open discussion on how to create the graphs
Contrast using output measure in feet versus output measured in radians
Push students to discuss the output of the graph for radians in terms of a percentage of the radius
Once the graphs are created use slide 17 to give the formal definitions of the sine and cosine functions
Recommended time: 10-15 minutes

Motion of Fly Applet (gsp)
Notes (doc, pdf)

Click for Summary/Goals

This applet can be used to supplement the motion of the fly around the circle
Can be used to help students identify the covariation of the angle measure and the vertical or horizontal distance from the center of the circle
Intended to promote a connection between numerical, contextual and graphical representations of the covarying quantities
Recognize that the length of a radius can be used as the unit of measurement for both vertical and horizontal positions relative to the center of the circle.

Click for Implementation Notes

Students should work in groups on creating graphs depicting the height of the bug as a function of the angle subtended
Use "Animate Point" to allow applet to run while students work in groups
Probe students to describe the covariation
Once the graphs are created use slide 17 to give the formal definitions of the sine and cosine functions
To focus students on using the radius as a unit of measure, use “Show Coordinates” to give them a coordinate system on the unit circle
Probe students regarding relationship of radius to bug’s location on circle
Recommended time: 10-15 minutes

Evaluation of Sine and Cosine on the Circle (Slides 24-28): Click for Purpose

Purpose: It is important that students recognize the sine and cosine functions that have an input of angle measure, in radians, and an output that is a fraction of a radius. Students will have the opportunity to evaluate sine and cosine for circle's of various sizes in order to further their understanding of what it means to use length of the radius as a unit of measurement.

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Worksheet 4
Notes

Click for Summary/Goals

Create a discussion about evaluating sine and cosine for circle's of various size radii
Understand why it is necessary to multiply the output of trig function by the radius in order to obtain a linear measure

Click for Implementation Notes

This worksheet should be given at the beginning of slide 18 of the trig module
Keep students attentive to the fact that we are always dividing by the radius, even when the radius is 1
Ensure students are differentiating between the point as a measure of distance away from the center versus the distance traveled around the circle
Recommended time: 10-15 minutes

Inverse Trigonometric Functions (Slides 29-31): Click for Purpose

Purpose: Since the focus of the module has been centered around the idea of angle measure and arc length, it is important that students are able to go both ways. In the last section, they learned how to use sine and cosine functions in order to take an angle measure and convert it to arclength or coordinates on their unit circle, now they must learn how to find an angle measure given an arclength or coordinates on their unit circle. They will be introduced to the definition of the inverse trigonometric functions and their domains and ranges.

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Worksheet 4
Notes

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Students should discuss the use of inverse trig functions with multiple radius lengths

Click for Implementation Notes

This activity should be given at the beginning of slide 19 of the trig module
Be sure students remain attentive to the input and output units
Probe students about how the length of the radius is a unit of measurement
Teacher may find it useful to use Motion of Fly GSP applet to lead discussion of domain and range for inverse trig functions
Recommended time: 10-15 minutes

Motion of Fly Applet (gsp)
Notes (doc, pdf)

Click for Summary/Goals

This applet can be used to supplement the motion of the fly around the circle
Help students determine appropriate domain and range for inverse trig functions

Click for Implementation Notes

Students should identify over what intervals of inputs to sine and cosine do they vary over all possible outputs
Ask students how they can determine the arclengths from the standard position for the other coordinates
Recommended time: 10-15 minutes

Reference Angles (Slides 32-33) : Click for Purpose

Purpose: Students need to be aware that there are multiple angle measures that yield the same values on the circle. Often these are described in terms of reference angles. This section and animation on slide 22 are intended to familiarize students with their definition.

Worksheet 4
Notes

Click for Summary/Goals

Create an opportunity for students calculate angles to see how they are related
Teacher should lead a discussion about reference angles

Click for Implementation Notes

Worksheet should be given at the end of slide 20
Teacher should lead a discussion about reference angles
After discussion of handout teacher should go on to animation on slide 22
Recommended time: 10-15 minutes

Slide 22 Animation

Click for Summary/Goals

Lead students through identifying points associated with angles π/6, 5π/6, 7π/6, and 11π/6
Introduce students to definition of reference angles

Click for Implementation Notes

Allow students to see the formal definition of reference angles
Students will have found values on handout
Teacher should lead discussion and have students determine values as s/he goes through the animation
Recommended time: 10-15 minutes

Periodicity and Amplitude (slides 34-46): Click for Purpose

Purpose: Students will come to understand a change in the period of trigonometric functions as a change on the argument to trigonometric functions. This composition view will focus students on the covariation between the input, the argument of the trig function (as a quantity itself) and the output of the trig function. Students will also come to recognize what happens when trig functions are multiplied by a coefficient (amplitude) and how that affects their graphs. Finally, they will create algebraic representations of trig functions given graphical representations helping them become more fluid as they move between representations.

Worksheet 5 Notes	Click for Summary/Goals Introduce students to periodicity Graphing trigonometric functions with different amplitudes and periods Relating circle and arc length to the graphical representation Click for Implementation Notes This activity should be given at the beginning of slide 23 Recommended time: 10-15 minutes
Sine Period Bar Applet (gcf) Notes (doc, pdf)	Click for Summary/Goals This applet can be used to supplement a class discussion on periodicity Help students understand what period is and how it relates to the graph Click for Implementation Notes This applet should be used prior to handout 9 Teacher should lead a whole class discussion Focus students on how the period changes Recommended time: 10-15 minutes
Worksheet 6 Notes	Click for Summary/Goals Students will continue to deepen their understanding of period They will discuss the difference between the input and argument of sine They will determine how the function changes based on changes to the argument of sine Click for Implementation Notes This activity should be given at the conclusion of slide In order to assist students if they are struggling, remind them of what the units of the input and output are Be sure the students are begining to recognize the relationship between the coefficient of the argument and the period This activity will take about 10-15 minutes
Argument Applet (gsp) Notes (doc, pdf)	Click for Summary/Goals This applet can be used to supplement a class discussion on periodicity Help students recognize the relationship between the argument and the coefficient of the input of the trig function They also connect period of a trig function to the unit circle Click for Implementation Notes This applet should be used with handout 9 Teacher should lead a whole class discussion Focus students on how the argument depends on the coefficient of the input of the trig function Focus students on what they should be looking at when thinking about theta and the product of the coefficient and theta Recommended time: 10-15 minutes
Worksheet 7 Notes	Click for Summary/Goals Students will learn to create algebraic representations for graphs of trig functions Click for Implementation Notes This activity should be given prior to slide 26 Allow students to work in groups Push students to identify amplitudes and periods first Push students to describe what the vertical shift represents Recommended time: 10-15 minutes
Worksheet 8 Notes	Click for Summary/Goals Capstone activity Allows students opportunity to apply various concepts from module Click for Implementation Notes This activity should be given at conclusion of slide 28 If time, assign one problem to each group and have them report out when finished Recommended time: 10-15 minutes

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