Algebra II : Unit #9 - Trigonometric Functions : Lesson #8

Graphs of Sine and Cosine

Objective: Students will be able to graph the parent function for sine and cosine.
Standards: HSF-IF.C.7e HSF-TF.A.2 MP1 MP2 MP3 MP5 MP7
Subject(s): Math
60 minutes
1 Section 1: Warm Up and Homework Review - 10 minutes

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-Graphs of Sine and Cosine, which asks students to describe the trigonometric information obtained from the point (p,q) on a unit circle.

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

The Unit Circle is Very Important For Graphing Sine and Cosine Functions
Intervention and Extension

The goal of this warm up was to remind students about the relationship between a point on the unit circle and the trigonometric ratios.  This is the power of the unit circle and is fundamental to the success of today’s lesson where we will take these amounts to graph sine and cosine.  I received several warm ups like Warm Up 2 where they described the three trig ratios in terms of p and q which is exactly where I wanted them to go.    Others gave me responses that were correct but unimportant to the lesson (Warm Up 3) or purely incorrect (Warm Up 1).  I really think that the key to this lesson on trigonometric functions is a solid understanding of the unit circle.   Some students were more comfortable after today’s activity but I think next year I will spend more time on the unit circle to ensure that more students are really confident.  

2 Section 2: Building a Sine or Cosine Graph - 25 minutes

This lesson give students the opportunity to physically build the graphs of sine and cosine using the unit circle.  Several tools are needed  for this investigation.  Each pair of students needs:

  • ruler
  • two legal papers taped together length wise, or the regular papers taped together length wise ( I have TAs create these before hand)
  • marker
  • 10ish sticks of spaghetti
  • 2 foot length of yarn
  • glue (I used Elmer's)
  • completed unit circle (Unit Circle)

The lesson begins with a review of the fact that the x value of any coordinate on a unit circle is also the cosine of the angle extending through that point (Math Practice 7).  The sine is the same as the y value.  This is key to the remainder of the lesson so I spend enough time here to make sure that the majority of the students are confident. 

Each pair of students will be paired with another group.  Out of each group of four, one pair creates a graph of sine and the other creates cosine.   Since my students are already in pairs in desks next to each other, I pick two pairs that are in the same column to ease communication.  My students won't be moving, we  use the room as is.  If your students are not already grouped, pick the pairs and then pairs of pairs beforehand.  If there are an odd number of pairs, one group has three pairs where two pairs do the same graph. I assign each pair either sine or cosine before getting into the investigation.

All directions to the investigation are on the PowerPoint.  I like to model steps either by doing a sample myself or by highlight a student's work.  The first step of the investigation is to wrap the yarn around their unit function and mark each significant angle on the yarn with their marker (Yarn on Unit Circle). 

Next they create the x- and y- axes on their paper.  The x-axis should be along the center of the long length of the paper.  The students label  it θ.  The y-axis should be placed 1/3 away from the left end of the paper and should be labeled either sinθ or cosθ.  

The yarn is then laid across the positive portion of the x-axis with the end representing 0 radians at the origin (Yarn on Coordinate Plane).    Each of the important angles from the unit circle is marked and labeled on the x-axis in both radians and degrees.  Radians are a mathematically more significant unit, however, my students are just transitioning to radians so it will be helpful to them to see both.

The next step is the key to entire lesson.  These students have never seen the graph of sine or cosine.  They use pasta noodles to measure either the sine or cosine of each of the major angles on the unit circle (Pasta on Unit Circle).  My goal is that they solidify and deepen their understanding of the fact that the x and y values are also the cosine and sine respectively on the unit circle.  This is also a nice introduction to the shape and major features of these graphs in a very physical way (Math Practice 5).  The noodles are glued above or below each appropriate angle on the coordinate plane (Pasta on Coordinate Plane).  Once they have the first several lengths glued, I stop them for a second and ask them to think about how the negative sines or cosines should be represented on the graph.  They talk with their partner and we discuss it as a class (Math Practice 3).   They then continue their graphs.  I circulate to ensure that the graphs are being properly constructed.  (Pasta Sine Curve and Pasta Sine Curve 2)

Once they have finished gluing their pastas for the entire unit plane, I have them draw a curve using their marker.  (Final Pasta Sine Curve)  We do a think-pair-share on: Why is the function curve wider than the unit circle?   Next, I ask them:  What if we extended the graph to include negative angles? (Math Practice 2)  Use your yarn and the pasta to graph the major negative angles.  This can be cut if time is becoming an issue.  Remind them watch the sign of their trig ratio and circulate to ensure that students are properly creating their graph.

I used the Activity Created by Michael D. Sturdivant to build this lesson.

Tips For A Smooth Lesson
Developing a Conceptual Understanding

There are several hints that I have to help this lesson run smoothly.  First, ensure that your materials are well organized as time is definitely at a premium in this lesson.  Keeping the class moving is also important because of time.  I was very careful in picking which groups did the Sine graph and which did Cosine.  Cosine is more challenging as you have to take the length of the x-value to find the height (or the y-value).  I assigned it to pairs that I felt could make that conceptual leap easier.  Even so, I had to go around and give some support to those groups.   When measuring the angles around the unit circle with the yarn, it is important to be clear that students start at 0o.  I had students start in other places which can mess up the graph.   The negative values also tripped some students up.  I tried to catch them all before they started gluing but a few groups had to pull them up.  

3 Section 3: Comparing Graphs - 10 minutes

At this point the students get into their group of four and compare their graphs.  First, they demonstrate how they found their graph.  This should only take 2 or so minutes.  The rest of the time is spent making a list of the similarities and differences between the two graphs (Math Practice 1).  Some things that could be mentioned include the intercepts, the shape, any equivalent coordinates.  Once the students have made their list, we compile a class list.

Finally, I ask the students whether this is the entire graph of sine or cosine.  If no one volunteers the fact that angles can go forever in either direction, I remind them of this fact or demonstrate by wrapping a piece of yarn around a unit circle multiple times.  

Short On Time

We ran very short on time at the end of this lesson.  I really wanted them to have an opportunity to do some comparison of the sine and cosine graph so I had list it on actual graph itself.  Here are several samples; 1, 2, and 3.  While brief, there were some important connections made like the similarity between the curves but the difference in their y-intercepts.  I think a good start was made here that we will solidify in the next lesson where we will discuss these similarities and differences as a class.  

4 Exit Ticket - 3 minutes

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

Today's Exit Ticket asks students to list one similarity and one difference between the graphs of sine and cosine.

5 Homework - 0 minutes

This Homework is meant to solidify the student's understanding of the shape and basic features of both the sine and cosine graphs.  They are asked to find the domain and range of the sine graph.  They also apply two basic transformations, one vertical translation and one horizontal translation, to the sine graph as well as determine any changes that may have occurred to the domain and range.  The final extension question asks how a person could use the graph to find angles that have the same sine? (Math Practice 1 and 7

Quality Work
High Expectations

In looking at the assignments that the students turned in (Student Assignment), I realized that their graphs for the first problem were unacceptable.  So rather than accepting this work, I decided to have the students redo this graphs in this problem as a class during the beginning section of the next lesson.  I think doing this really firmed in the ideas from the graphing activity.  As you can see from my sample (Student Assignment 2), I had them reproduce the graph with all of the important points.  We talked about the repetitive nature of the y-coordinates (Math Practice 8).  This took a bit of time but I really think it was worth it in the students’ conceptual growth.