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: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
problem_activity: Task
A function machine takes an input, and based on some rule produces an output.
[IMG]
The tables below show some input-output pairs for different functions. For each table, describe a function rule in words that would produce the given outputs from the corresponding inputs. Then fill in the rest of the table values as inputs and outputs which are consistent with that rule.
Input values can be any English word. Output values are letters from the English alphabet.
$\ \ \ \ \ $
{"table": [{"_attributes": {"class": ["regular"]}, "tbody": [{"tr": [{"th": [{"_value": "input"}], "td": [{"_value": "cat"}, {"_value": "house"}, {"_value": "you"}, {"_value": "stem"}, {"_value": "$\\ \\ \\ \\ \\ $"}, {"_value": "$\\ \\ \\ \\ \\ $"}, {"_value": "$\\ \\ \\ \\ \\ $"}]}, {"th": [{"_value": "output"}], "td": [{"_value": "t"}, {"_value": "e"}, {"_value": "u"}, {}, {"_value": "z"}, {}, {}]}]}]}]}
Input values can be any rational number. Output values can be any rational number.
{"table": [{"_attributes": {"class": ["regular"]}, "tbody": [{"tr": [{"th": [{"_value": "input"}], "td": [{"_value": "$2$"}, {"_value": "$5$"}, {"_value": "$-1.53$"}, {"_value": "$0$"}, {"_value": "$-4 $"}, {"_value": "$\\ \\ \\ \\ \\ $"}, {"_value": "$\\ \\ \\ \\ \\ $"}]}, {"th": [{"_value": "output"}], "td": [{"_value": "$7$"}, {"_value": "$10$"}, {"_value": "$3.47$"}, {"_value": "$5$"}, {}, {"_value": "$8$"}, {}]}]}]}]}
Input values can be any whole number. Output values can be any whole number.
{"table": [{"_attributes": {"class": ["regular"]}, "tbody": [{"tr": [{"th": [{"_value": "input"}], "td": [{"_value": "$1$"}, {"_value": "$2$"}, {"_value": "$3$"}, {"_value": "$4$"}, {"_value": "$5$"}, {"_value": "$6$"}, {"_value": "$7$"}]}, {"th": [{"_value": "output"}], "td": [{"_value": "$2$"}, {"_value": "$1$"}, {"_value": "$4$"}, {"_value": "$\\ \\ \\ \\ \\ $"}, {}, {}, {}]}]}]}]}
Input values can be any whole number between 1 and 365. Output values can be any month of the year.
{"table": [{"_attributes": {"class": ["regular"]}, "tbody": [{"tr": [{"th": [{"_value": "input"}], "td": [{"_value": "25"}, {"_value": "365"}, {"_value": "35"}, {"_value": "95"}, {"_value": "330"}, {"_value": "66"}, {}]}, {"th": [{"_value": "output"}], "td": [{"_value": "January"}, {"_value": "December"}, {"_value": "February"}, {"_value": "April"}, {"_value": "November"}, {"_value": "$\\ \\ \\ \\ \\ \\ \\ \\ $"}, {"_value": "October"}]}]}]}]}
For at least one of the tables, describe a second rule which fits the given pairs but ultimately produces different pairs than the first rule for the rest of the table.


aligns: no