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: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

Now, does the following problem or activity 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_neg_0

HS_2
standard_description: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: Activity
In this activity, students apply the expression they previously generated to find the areas of various triangles. Each diagram is labeled with two or three measurements. Before calculating, students think about which lengths can be used to find the area of each triangle.
As students work, notice students who choose different bases for Triangles B and D. Invite them to contribute to the discussion about finding the areas of right triangles later.
Launch
Explain to students that they will now practice using their expression to find the area of triangles without a grid. For each triangle, ask students to be prepared to explain which measurement they choose for the base and which one for the corresponding height and why.
Keep students in groups of 2–4. Give students 5 minutes of quiet think time, followed by 1–2 minutes for discussing their responses in their group.
Representing: MLR2 Collect and Display.
While pairs are working, circulate and listen to student talk about identifying the bases and corresponding heights for each of triangles. Write down common or important phrases you hear students say about each triangle, specifically focusing on how students make sense of the base and height of each triangle. Record the words students use to refer to each triangle and display them for all to see during the whole-class discussion.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
For each triangle, circle a base measurement that you can use to find the area of the triangle. Then, find the area of any
three
triangles. Show your reasoning.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
The extra measurement in Triangles C, D, and E may confuse some students. If they are unsure how to decide the measurement to use, ask what they learned must be true about a base and a corresponding height in a triangle. Urge them to review the work from the warm-up activity.
Activity Synthesis
The aim of this whole-class discussion is to deepen students’ awareness of the base and height of triangles. Discuss questions such as:
For Triangle A, can we say that the 6-cm segment is the base and the 5-cm segment is the height? Why or why not? (No, the base of a triangle is one of its sides.)
What about for Triangle C:  Can the 3-cm segment serve as the base? Why or why not? (No, that segment is not a side of the triangle.)
Can the 3.5-cm side in Triangle C serve as the base? Why or why not? (Yes, it is a side of the triangle, but because we don't have the height that corresponds to it, it is not helpful for finding the area here.)
More than two measurements are given for Triangles C, D, and E. Which ones are helpful for finding area? (We need a base and a corresponding height, which means the length of one side of the triangle and the length of a perpendicular segment between that side and the opposite vertex.)
When it comes to finding area, how are right triangles—like B and D—unique? (Either of the two sides that form the right angle could be the base or the height. In non-right triangles—like A, C, and E—the height segment is not a side of the triangle; a different line segment has to be drawn.)


aligns: yes