Unit 7, Lesson 6
The Pythagorean Identity (Part 2)
Algebra 2
6.1
Warm-up
Math Talk: Which Quadrant?
Materials
None
Activity Narrative
This Math Talk focuses on determining the sign of the cosine, sine, and tangent of an angle
Up until now, students’ understanding of tangent has been limited to either right triangles or as the value of sine divided by the value of cosine. While the latter of these works for angles beyond those found in right triangles, during the whole-class discussion, students learn another way to think about tangent, using the unit circle, expanding their understanding of tangent beyond the angles possible within a right triangle.
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation, before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
If needed, remind students at the start of the Warm-up that one definition of tangent of an angle
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking,
- “Who can restate
’s reasoning in a different way?” - “Did anyone have the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to
’s strategy?” - “Do you agree or disagree? Why?”
In particular, highlight any strategies in which students used a unit circle diagram or sketched their own unit circle showing an angle
End the discussion by displaying this image:
Say, “We learned earlier that we can think of
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
Advances: Speaking, Representing
- “First, I
because .” - “I noticed
, so I .”
Advances: Speaking, Representing
Lesson Synthesis
The goal of this discussion is to call student attention to the four points on the unit circle where we cannot draw in a right triangle. For this lesson, we will address what this means for the value of tangent at these four angles, which students will study in more depth in a future lesson when they work with the graph of the tangent function.
Display a unit circle, such as the one here, with 4 points marked where the circle intersects the axes.
Here are some questions for discussion:
- “What is the value of tangent at
and ?” ( does not exist since a vertical line does not have a slope.) - “What is the value of tangent at
and ?” ( - “Are there any other points on the unit circle where the value of tangent is either 0 or does not exist?” (None, because all other points on the unit circle have non-zero
- and -coordinates.)
Student Lesson Summary
Suppose we know that
Since we can think of
The sketch helps us see that the
Now that we know the value of cosine, we can calculate the value of tangent with some division:
We can use one piece of information and the structure of the unit circle to figure out a whole bunch more, similar to how we used the value of one side length, the hypotenuse, and the structure of right triangles in the past to figure out the other side length of the right triangle.
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