Teacher Instructions Time: 4 double block 80 minute classes Prerequisite: Students should have spent the previous class working with periodic functions including basic Ferris Wheel problems considering a person’s position over time. Purpose: This is a multi-day discovery activity that creates a trig foldable. The purpose is to use the unit circle to discover the following and include this information in their foldable:

Radians Coordinates of special angles on the unit circle Use scatter plot to discover the general shape of Sine, Cosine, and Tangent and their corresponding reciprocal graphs.

Materials:

• Manila folder (one per student) • Brad (one per student) • Student packet printed one sided because they will have to cut out the pieces to glue • Card stock (I use a manila folder – 1 per 4 students) • Glue • Scissors • Protractor

Overview: This activity is best designed to let the students discover on their own the basic building blocks of trigonometry. Although there are times that it seems to move slowly, the reward in how well the students connect the unit circle to the sine and cosine function is well worth the journey. The activity is designed to be done in groups with each student creating their own foldable. Each step below is given with basic teacher instructions and student instructions. The student instructions are the information on the student handout. The teacher instructions are how you should guide the students through the discovery. At the end of each step the class should be brought together to discuss the concept developed. Homework should be given to solidify the concept. A student example is included after the instructions.

Activities Activity 1 Purpose: Activity 1 actually has no “math” purpose. It is the setup for the “Ferris wheel” needed for all the following activities. Student Instructions:

1. Go to the page labeled Unit Circle. Paste the unit circle on a piece of card stock. Cut out the circle but be careful as you cut out the circle to make sure that the dots around the circumference of the circle are still visible.

2. Go to the page labeled Cartesian Plane with Unit Circle and cut out the Cartesian plane. Do not cut out the circle, cut out the graph paper square containing the circle.

3. Glue the graph paper on the top left position of the file folder (this is important to conserve space).

4. Poke a hole through the center of the cut out unit circle and the circle on the graph paper. 5. Push the brad clip through the center of both. This should create a disk that spins or you

“Ferris wheel” Teacher instructions: Let the students know that they are beginning a project that will create a foldable reference for the basic information they need for trigonometry. Although I tell the students this is their foldable and needs to contain what they feel is relevant, it is important that they put the Cartesian plane in the top left corner of the manila folder. There is a lot of information that will be added and space needs to be conserved. Notes:

Activity 2 Purpose: Activity 2 is designed to discover the relationship between degree measurements and radians. Students use the unit circle to generate the distance traveled around the “Ferris wheel” after they have traveled a specific angle rotation. All angles of rotation are special angle measurements. Students will need to figure out that they need to measure the degrees traveled with their protractor and use these values to find the fraction of degree/360 and use the unit circle to find the radian measurements for all the special angles. Student Instructions: This circle has a radius of unit 1. Use this information to label the outside of the circle as follows:

1. Mark the outside of your Cartesian plane circle with the same marks that are on the “Ferris wheel” circle

2. You task is to enter the “Ferris wheel” at the 3:00 position and travel around the circle in a counter-clockwise direction. You will stop the “Ferris wheel” at each marked position

3. Once you stop at a location, calculate the distance traveled along the circle from where you entered the Ferris wheel to your current location.

4. Label the circle with the degree measurement and length around the circle you have traveled at each location. Make sure you put your values on the outside of the circle at the place you stopped.

5. Leave your answers with π in the solution and in the numerator. 6. All fractions MUST be reduced.

****You must enter your Ferris Wheel from the right side at 3:00****

Teacher instructions: Make sure the students “enter” the “Ferris wheel” on the right hand side or 3:00. This should be the positive x-axis but should not be identified as the x-axis unless necessary for clarification. When the students figure out that they need to measure the angle traveled with their protractor, make sure they are not being too exact with their measurements and should round to numbers divisible by 5. They should be measuring all the special angles (30-45-60-90). Have the students physically rotate the “Ferris wheel” around and stop at each location. They should calculate and mark at the location on the Cartesian plane the distance traveled around the unit circle. Some students may calculate the height from the ground, instead of the distance around the circle. If you notice students doing this, try to use guided questions to relate the distance around the curve to circumference. Some students will find a pattern for the values, this is great but encourage them to verify their predictions if they start to just label the circle. Students should be labeling the values they find on the outside of the circle not on the inside of the circle as it rotates.

Make sure students are calculating the distance from the x-axis not from the previous stopping location. Modification: Instead of making the students measure the angles with a protractor, you can let them know the different degrees of rotation traveled. This saves time and you don’t have to worry about students measuring angles incorrectly. After Activity At this point it is time to discuss what radians are. I usually ask the students why we use degrees. It is “arbitrary” and related to the number of days in the year. Since radians are based on the unit circle and pi, it is universal. I then have them generate a formula for calculating radian to degrees, then degrees to radians. We put student formulas on the board. Students then write the conversion formulas in their foldables to the right of the unit circle. They should use the formulas to calculate other angle measurements that are not the special values. Part of my conversation with the students at this time is that although this is new concept for them, they need to get away from thinking as the number in degrees but instead understand where the radians are located. I relate it to learning Spanish. When you first learn Hola you have to think about it being “hi” but after you use it for a while you don’t translate it anymore. This is their goal with radians. The final part of the discussion is radians that are more than one rotation and negative radians. I give them several possible values and ask them to find where they think they belong. Then we discuss as a class. Notes:

Activity 3 – Radians Defined Purpose: Activity 3 is to show the students that radians are actual numbers. Too often students think π/3 is the same number as 60°. The second purpose for the activities is we discuss how many radii fit around the circle. This again relates the name radians to its origin. Student Instructions:

1. Mark the location of 1 radian on the outside of the circle. 2. Wait to discuss location with the class. 3. Now mark 2 – 6 radians on your wheel 4. How many radians fit around the circle?

Teacher instructions: Ask the students to mark where they think 1 radian is located on their circle. Let them discuss this in their groups. Listen carefully because this is a time where you can pick up misconceptions concerning radians. Some will go to 1 degree (which is somewhat difficult). Usually there are heated discussions in the groups. At the end of the discussion I let someone from one of the groups put 1 radian on a circle on the board. At this point, I let the class finish the discussion in case someone else is still struggling with the concept. After 1 radian is marked, I then ask the students to predict how many radians will fit around the whole circle. You would think everyone would say 2π but some still need to make this connection. At this point, let them mark the rest of the radians around the circle. The final question that should be poised at this time is what would happen if you had a different circle? How far around the circle is 1 radius? How does this relate to the angle measurement in radians? These questions should be used to generate the arc length and sector area formula for a circle. Once groups come up with their formulas I let them put them on the board and we decide as a class which is the best. These formulas should be placed to the right of the unit circle along with the radian/degree formulas. Modification: This can be removed from the student handout and instead the teacher can just present the questions to the class. Notes:

Activity 4 – Coordinates on the unit circle Purpose: Activity 4 is used to generate the coordinates of each point on the unit circle. Students will use their knowledge of right angle trig to calculate these values. These are all based on special right triangles and students do not need a calculator to find the values. Coordinates should contain radicals. Student Instructions:

1. Use your knowledge of right angle trig and label the coordinates at each mark for your special angles.

2. Be careful that you consider which quadrant you point is located when determining the sign of your coordinate values.

Teacher instructions: Have students calculate the coordinates on the Cartesian plane for each mark they have made where the “Ferris wheel” has stopped. The students should not use a calculator for this if they are preAP students and preferable for no students. Some students will rationalize the denominator and some will not. I do not require either form. What I do require is that my students recognize that both forms are equivalent and can identify them as equivalent. It is easier to use the values if denominator is rationalized and many students move towards that representation. After Activity: Once they have completed the activity ask them if they can generalize how to find any value on the unit circle. Some students will need to work through this but they should come up and write that y-coordinate = sine Θ and x-coordinate = cosine Θ. This formula should be added to the foldable. Have them use their calculator to generate estimated coordinates of non-special angles using their calculators. Finally the students should generate a general equation for the coordinates given a circle centered at the origin and any radius. Notes:

Activity 5 – Sine, Cosine, Tangent graph Purpose: Activity 5 is used to generate the sine and cosine curve, as well as the tangent graph. It is done by the students generating a scatter plot of values and connecting the dots to form a smooth curve. There are other ways to do this activity and if you already have a great way to teach the shape of the curves (for example with spaghetti) that activity could be substituted here. Student Instructions: Generate scatter plots.

1. Find the value for each function given each angle measurement and fill in the Function Value Table.

2. Use the information to create three scatter plots on the coordinate graph paper. 3. The first one will have the input value radians and output value of sinθ. 4. The second will have the input value radians and output value of cosθ. 5. The third will have the input value radias and output value of tanθ. 6. Connect the dots on the scatter plot to form smooth curves.

∗∗∗∗Create these scatter plots twice, one on each graph, for each function**** Teacher instructions: Make sure the students understand that the input value for all of these functions is the angle position on the unit circle and the output is the x or y coordinate or their ratio. Don’t let the students enter functions and graph them in their calculator. I do let them enter the exact values to find the estimate that matches it. For example, find the estimate of !

! . Watch for the students that just graph the max, min, and zeros of the functions. They need to at least do a ½ of a period by plotting each point. Some students in your group may already know the basic shape of the function from other classes. Encourage them to verify the shape by plotting the points. After Activity: When you have completed the scatter plots, allow them to graph the functions in their calculators. Make sure you show them the trig view on their calculator. Also, show them how they can change their steps and scaling by using π. Once students have created the graphs, use the transformation rules to sketch different versions of each graph.

Notes:

Activity 6 – Organizing Graphs in foldable Purpose: Activity 6 is just used to get the graph and table into the foldable. It can be done in conjunction with activity 5. The graphs do need to be glued so the student can lift them up and look under them and be as close to the fold of the folder as possible. The students will be gluing the inverse function under it and the reciprocal graphs next to them, Student Instructions:

1. Glue the table below your “Ferris wheel” 2. On the other side glue down the three different scatter plots on the coordinate plane.

Glue the graphs as close to the fold line as possible. Another set of graphs will be going to the right of these graphs.

3. Glue such that you are able to lift the graph so that you can see under it. Another graph will go here at a later date.

4. Label each graph with its name, domain, range, period, midline, and amplitude Teacher instructions: Graphs should be pasted on the right side of the manila folder as close to the seam as possible

Notes:

Activity 7 Purpose: Activity 7 generates the scatter plot for the reciprocal graphs Student Instructions:

1. Generate the reciprocal values in the table for secant, cosecant, and cotangent. 2. Use the generated a scatter plot graph for secΘ, cscΘ, and cotΘ 3. Connect the dots on the scatter plot to form smooth curves (be careful about division by

zero) 4. Glue the graph next to their reciprocal graph on the foldable 5. Label each graph with its name, domain, range, period, and asymptotes

Teacher instructions: Before the students begin this activity the reciprocal functions must be defined. Let them know which functions match up and have them write the reciprocal relationship in their folders. It is beneficial write both csc = 1/sin as well as sin = 1/csc. Students often forget the second relationship.

Students then generate cosecant, secant, and cotangent in the same manner they did cos, sin, and tan. Pay special attention to cot. They need to go back to the definition of the reciprocal and that cot = cos/sin. If not, they will get stuck where tan equals zero and is undefined. Some students will forget that you can’t divide by zero. After Activity: After the student finish, again build all the transformations into the graph. It is beneficial to show the students that if you graph the reciprocal functions (for example sine) as a dotted graph, you can see the max/min of the function and where the asymptotes will be located. I find with period changes, it is easier for them to see the asymptotes if they do that step first. Students should glue the reciprocal function next to it related function. Notes:

Additional Items: Inverse Functions: Inverse functions are not listed as an activity. I have an activity called Inverse Function Lab that is used to generate the shape of the functions and the limit on their range. Once the inverse functions are discussed they are then glued under its inverse function. Final thoughts: As the students continue to learn trig identities and other aspects of trig, they always ask if they can add it to the foldable. I strongly encourage my students to put their identities in the foldable and anything else they feel is relevant. It is theirs to use. My students hold on to this foldable and always use it in calculus, not because I require them to, but because they find it a valuable tool. I have had students that took calculus in college let me know that they have used it as a reference.

.

UNIT CIRCLE

Unit Circle on a Cartesian Plane

f(x)=(1-x^2)^(1/2)

f(x)=-(1-x^2)^(1/2)

-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

y

Degrees Radians sin θ cos θ tan θ csc θ sec θ cot θ O°

30°

45°

60°

90°

120°

135°

150°

180°

210°

225°

240°

270°

300°

315°

330°

360°

Coordinate Plan graph paper