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: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Answer: 
No. This task does not involve an equation in two variables. There is only 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_neg_0

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im_task_001160_pos_0

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standard_description: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
problem_activity: Activity
This task combines concepts of decomposition, cylinder and cone volume formulas, and solids of rotation.
If students choose to use 3D graphing technology, reviewing the Axis of Rotation lesson may be helpful. Making this technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Student Facing
Suppose this two-dimensional figure is rotated 360 degrees using the vertical axis shown. Each small square on the grid represents 1 square inch.
[IMG]
Draw the solid that would be traced out. Label the dimensions of the solid.
Find the volume of the solid. Round your answer to the nearest tenth if needed.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Anticipated Misconceptions
Students may struggle to visualize and draw the solid of rotation. Suggest that they divide the two-dimensional figure into two pieces horizontally and think about what each would look like rotated using the vertical axis. If necessary, they can draw the two resulting solids (cone and cylinder) separately.
Activity Synthesis
Ask students to compare and contrast the cylinder and the cone. They each have the same radius measurement, but their heights are different.
Invite students to summarize the kinds of solids that can be traced out through rotating a two-dimensional figure: spheres, cones, and cylinders can be created through spinning half-circles, triangles, and rectangles, but pryamids and prisms can’t be created through rotation. In general, a solid created through rotation will have circular cross sections.
Speaking: MLR8 Discussion Supports.
As students share the solids that can be formed by rotating a two-dimensional figure, press for details by asking students to elaborate or give an example. Also, ask students how they know that prisms and pyramids cannot be created through rotation. This will help students justify their reasoning for the solids that can and cannot be formed by rotating a two-dimensional figure.
Design Principle(s): Support sense-making; Optimize output (for justification)
Representation: Internalize Comprehension.
Provide examples of actual three-dimensional models of cylinders, cones, spheres, and prisms for students to view or manipulate. Ask students which solids can be formed by rotating the two-dimensional figure around the vertical axis.
Supports accessibility for: Visual-spatial processing; Conceptual processing


aligns: yes