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: 
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).

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


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

HS_1

im_task_001160_pos_0

HS_2
standard_description: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
problem_activity: Activity
Students design districts in three towns to “gerrymander” the results of elections. In two of the towns, the election results can be skewed to either color. In the third, it isn’t possible to skew the results.
Launch
Arrange students in groups of 2–4.
Explain the history of gerrymandering: Sometimes people in charge of designing districts make them in strange shapes to give the election results they want. One of the first was Elbridge Gerry (governor of Massachusetts in 1812), whose party designed a district that many people thought looked like a salamander. They called a Gerrymander, and the name stuck. It means a very strangely shaped, spread-out district designed to produce a certain result.
Usually districts are required to be connected: a person traveling to all parts of the district should be able to stay inside the district. There should be no “islands” that are separated by parts of other districts.
Engagement: Internalize Self-Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to four out of six task statements.
Supports accessibility for: Organization; Attention
Student Facing
Smallville’s map is shown, with opinions shown by block in green and gold. Decompose the map to create three connected, equal-area districts in two ways:
Design three districts where
green
will win at least two of the three districts. Record results in Table 1.
[IMG]
Table 1:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
Design three districts where
gold
will win at least two of the three districts. Record results in Table 2.
[IMG]
Table 2:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
Squaretown’s map is shown, with opinions by block shown in green and gold. Decompose the map to create five connected, equal-area districts in two ways:
Design five districts where
green
will win at least three of the five districts. Record the results in Table 3.
[IMG]
Table 3:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
|    4     |                                                      |                                                     |                                                                                    |                                                 |
|    5     |                                                      |                                                     |                                                                                    |                                                 |
Design five districts where
gold
will win at least three of the five districts. Record the results in Table 4.
[IMG]
Table 4:
| district | number of blocks                       for green | number of blocks                       for gold | percentage                       of blocks                       for green | representative’s                       vote |
|----------|--------------------------------------------------|-------------------------------------------------|----------------------------------------------------------------------------|---------------------------------------------|
|    1     |                                                  |                                                 |                                                                            |                                             |
|    2     |                                                  |                                                 |                                                                            |                                             |
|    3     |                                                  |                                                 |                                                                            |                                             |
|    4     |                                                  |                                                 |                                                                            |                                             |
|    5     |                                                  |                                                 |                                                                            |                                             |
Mountain Valley’s map is shown, with opinions by block shown in green and gold. (This is a town in a narrow valley in the mountains.) Can you decompose the map to create three connected, equal-area districts in the two ways described here?
Design three districts where
green
will win at least 2 of the 3 districts. Record the results in Table 5.
[IMG]
Table 5:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
Design three districts where
gold
will win at least 2 of the 3 districts. Record the results in Table 6.
[IMG]
Table 6:
| district | number of blocks                           for green | number of blocks                           for gold | percentage                           of blocks                           for green | representative’s                           vote |
|----------|------------------------------------------------------|-----------------------------------------------------|------------------------------------------------------------------------------------|-------------------------------------------------|
|    1     |                                                      |                                                     |                                                                                    |                                                 |
|    2     |                                                      |                                                     |                                                                                    |                                                 |
|    3     |                                                      |                                                     |                                                                                    |                                                 |
Student Response
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for free access to Student Response.
Anticipated Misconceptions
The white streets on the maps do not disconnect the districts.
However, probably squares that are only connected at their corners should not be considered connected to each other. (But you or the students can make their own rule, since they are in charge of making districts.)
Activity Synthesis
This set of questions is similar to the previous activity, except that the patterns are irregular, so more planning and computation is needed to design districts.
Ask several students with different maps to show and explain their work.
The important result for this problem is that it’s possible to design both fair districts (where the result of the vote is similar to the vote if all individual votes were counted) and unfair districts. According to the rules (equal area and connected), these sets of districts are legal. Encourage students to try to devise some rules for drawing districts that would be more fair.


aligns: yes