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

HS_1

im_task_001160_pos_0

HS_2
standard_description: Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
problem_activity: Warm-up
The purpose of this warm-up is for students to recall how to calculate an average rate of change from two points. Students will use this skill throughout the lesson and return to the context and data table in the last activity of the lesson.
Launch
Ask students to close their books or devices. Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time, and then select 3–5 students to share something they notice or wonder about the table  and record and display their responses for all to see. If not pointed out, ask students, "Describe how the data is changing as the value of
\(t\)
is increasing." (The values of
\(p\)
are decreasing at a slower rate.) Ask students to reopen their books or devices and complete the activity. Follow with a whole-class discussion.
Student Facing
Let
\(p\)
be the function that gives the cost
\(p(t)\)
, in dollars, of producing 1 watt of solar energy
\(t\)
years after 1977. Here is a table showing the values of
\(p\)
from 1977 to 1987.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "166"}, "_value": "\\(t\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "13322"}, "_value": "\\(p(t)\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {"_value": "80"}]}, {"td": [{"_value": "1"}, {"_value": "60"}]}, {"td": [{"_value": "2"}, {"_value": "45"}]}, {"td": [{"_value": "3"}, {"_value": "33.75"}]}, {"td": [{"_value": "4"}, {"_value": "25.31"}]}, {"td": [{"_value": "5"}, {"_value": "18.98"}]}, {"td": [{"_value": "6"}, {"_value": "14.24"}]}, {"td": [{"_value": "7"}, {"_value": "10.68"}]}, {"td": [{"_value": "8"}, {"_value": "8.01"}]}, {"td": [{"_value": "9"}, {"_value": "6.01"}]}, {"td": [{"_value": "10"}, {"_value": "4.51"}]}]}]}]}
Which expression best represents the average rate of change in solar cost between 1977 and 1987?
\(p(10)-p(0)\)
\(p(10)\)
\(\dfrac{p(10) - p(0)}{10-0}\)
\(\dfrac{p(10)}{p(0)}\)
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students might confuse finding an average of the values in the table and finding an average rate of change. Help them see that average usually involves one unit, such as average number of cookies. Average rate of change involves comparing how one quantity changes when another quantity changes by 1, such as cost ($) per year.
Activity Synthesis
The goal of this discussion is for students to recall the meaning of average rate of change for a function and how it is calculated. Select students to describe what the value of each expression represents in this context. For example,
\(p(10)-p(0)\)
is the difference in cost for a solar cell between 1977 and 1987 while
\(\dfrac{p(10)}{p(0)}\)
could be used to identify the percent change from 1977 to 1987. An important takeaway for students is that the actual expression for the average rate of change,
\(\dfrac{p(10)-p(0)}{10-0} \approx \text-7.55\)
, tells us that the price decreased by -
$
7.55 each year on average since the expression looks at the total difference in price between the 1977 and 1987 divided by the total number of years that passed.


aligns: no