standard_description: Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
problem_activity: Warm-up
This prompt gives students opportunities to see and make use of structure (MP7). The specific structures they might notice is the table and how it relates to a linear relationship between
\(x\)
and
\(y\)
(specifically, that
\(y = 3x + 6\)
).
Monitor for students who:
describe patterns only vertically or only in terms of
\(x\)
or
\(y\)
(e.g., “it” or “
\(y\)
” goes up by 3, “it” starts at 6)
describe
\(y\)
in terms of
\(x\)
describe a relationship between
\(x\)
and
\(y\)
, such as
\(y\)
goes up 3 every time
\(x\)
goes up 1
notice the linear relationship
attempt to identify the missing value in the table
Launch
Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.
Student Facing
What do you notice? What do you wonder?
| \(x\) | \(y\) |
|-------|-------|
|   0   |   6   |
|   1   |   9   |
|   2   |  12   |
|   4   |  18   |
|  10   |  36   |
|  100  |       |
Student Response
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Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the relationship between the table and an equation does not come up during the conversation, ask students to discuss how they can come up with the missing value. It is not necessary to generate an equation like
\(y=3x+6\)
at this time.


aligns: no