few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
Concept/skill: 
Solve quadratic equations in one variable.

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


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

Example 3:
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.

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 $P$.


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

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: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
problem_activity: Provide a written explanation for each question below.
a.   Is it true that
$${\left(1000^{1\over3}\right)^3=(1000^3)^{1\over3}}$$
? Explain or show how you know.
b.   Is it true that
$${\left(4^{1\over2}\right)^3=(4^3)^{1\over2}}$$
? Explain or show how you know.
c.   Suppose that
$$m$$
and
$$n$$
are positive integers and
$$b$$
is a real number so that the principal
$$n^{th}$$
root of
$$b$$
exists. In general, does
$$\left(b^{1\over n}\right)^m=(b^m)^{1\over n}$$
? Explain or show how you know.


aligns: no