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

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: 
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, 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_neg_0

HS_2
standard_description: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.
problem_activity: Warm-up
The goal of this warm-up is to review the meaning of a function presented graphically. While students do not need to use function notation here, interpreting the graph in terms of the context will prepare them for their work with functions in the rest of the unit.
Launch
Tell students to close their books or devices. Display the graph for all to see. Ask students to observe the graph and be prepared to share one thing they notice and one thing they wonder. Select students to briefly share one thing they noticed or wondered to help ensure all students understand the information conveyed in the graph. Ask students to open their books or devices and answer the questions about the graph. Follow with a whole-class discussion.
Student Facing
Here is a graph of the accumulated rainfall in Las Vegas, Nevada, in the first 60 days of 2017.
[IMG]
Use the graph to support your answers to the following questions.
Is the accumulated amount of rainfall a function of time?
Is time a function of accumulated rainfall?
Student Response
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Anticipated Misconceptions
If students struggle to see from the graph how the accumulated rainfall is a function of time but time is not a function of accumulated rainfall, consider displaying the data in a table. Shown here is the data for the first 20 days of 2017. Help students see that for every value of
\(t\)
, the time in days, there is one value of
\(r\)
, the accumulated rainfall in inches, but this is not true the other way around.
|  \(t\)                  (days)  |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
|---------------------------------|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(r\)                  (inches) |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |   |
Activity Synthesis
Make sure students understand why the accumulated rain is a function of time but not the other way around.
Then, recall the notation for writing, using function notation, accumulated rain fall as a function of time. If
\(r\)
represents the amount of rainfall in inches and
\(t\)
is time in days, then
\(r(t)\)
is the amount of rain that has fallen in the first
\(t\)
days of 2017. For example,
\(r(2) = 0\)
tells us that the accumulated rainfall after in the first two days of the year was 0 inches.
\(r(48)=1\)
means that there was 1 inch of accumulated rain in the first 48 days of the year. Ask students to write and explain the meaning of a few other statements using function notation.


aligns: yes