few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Concept/skill: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
- An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
- An angle that turns through n one-degree angles is said to have an angle measure of n degrees.
problem_activity: Narrative
In a previous lesson, students described an angle to a peer so that they might draw the angle accurately without seeing it. Students learned what an angle is and have reasoned about how to describe the size of an angle.
In this activity, students use the features of an analog clock (minute hand, hour hand, and position of numbers) to explain how to draw a given angle. In doing so, they use language related to turning one or both hands on the clock. In future activities, students will relate this idea to turning a ray around a fixed point.
The hands on an actual clock are designed so that moving the minute moves the hour hand. The examples here disregard this constraint. For the purposes of this activity, it is assumed that both hands can be moved freely, or that one hand can be held in place while the other moved freely.
Required Materials
Materials to Gather
Rulers or straightedges
Launch
Groups of 2
Give students access to rulers or straightedges.
Activity
“In an earlier lesson, we played a game where you described an angle to your partner to help them draw it without seeing it.”
“Read Andre’s explanation for how to draw an angle. Try to draw his angle, and then think of another way to describe how to draw the same angle.”
3 minutes: independent work time on the first two problems
“Share your drawing and description with your partner.”
“Discuss how you would explain how to draw the other angles.”
3–5 minutes: partner discussion and work time
Monitor for students who clearly describe turning a hand around the clock and who connect the hands to the rays of the angles.
Student Facing
Andre used the hands of a clock to explain how to draw an angle to his partner.
“Imagine both hands are pointing at the 12. Turn the minute hand so it's pointing at the 3.”
[IMG]
Draw Andre’s angle.
What is another way to describe how to draw the same angle using the clock?
Use Andre’s way to explain how to draw these angles:
[IMG]
[IMG]
[IMG]
Student Response
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Activity Synthesis
“Let’s listen to some different directions for how to draw these angles. As you listen, try to sketch the angle and guess which one they are describing how to draw.”
Invite 2–3 students to share the directions they wrote for drawing one of the angles. Choose students to share in a different order than the angles are presented in the task.
For each, consider asking:
“Which angle did they describe (a, b, or c) and how do you know?”
“What words help you picture how to draw the angle?” (turn right, turn left, point at 4)
“We can describe the size of an angle by explaining how much one ray has turned from the other.”


aligns: yes