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

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, 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_neg_0

HS_2
standard_description: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
problem_activity: Activity
First, students examine several examples of functions and non-functions from the perspective of “Can we figure out . . . ?” Then, they decide which of the functions can be represented by a given expression.
Launch
Allow quiet think time to consider the situations in the first question and answer yes or no. Ensure students understand which ones are functions, and which are not. Then, give students time to complete the table.
Student Facing
For each question, answer yes or no.
It is 50 miles to Tucson. Can we figure out how many kilometers it is to Tucson?
It is 200 kilometers to Saskatoon. Can we figure out how many miles it is to Saskatoon?
A number is -3. Can we figure out its absolute value?
The absolute value of a number is 8. Can we figure out the number?
A circle has a diameter of 8 cm. Can we figure out its circumference?
A circle has a circumference of
\(10\pi\)
cm. Can we figure out its diameter?
A square has a side length of 6 units. Can we figure out its perimeter?
A rectangle has a perimeter of 30 meters. Can we figure out its width?
Which of the relationships are functions?
For each function definition in the table, match it with the situation, write a statement explaining which variable depends on which, and write an example using function notation. An example is done for you.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "function definition"}, {"_attributes": {"scope": "col"}, "_value": "situation"}, {"_attributes": {"scope": "col"}, "_value": "statement"}, {"_attributes": {"scope": "col"}, "_value": "example"}]}]}], "tbody": [{"tr": [{"td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "23857"}, "_value": "\\(m(x) = 0.62x \\)"}]}, {"_value": "You know kilometers and want to find miles."}, {"_value": "Distance in miles depends on distance in kilometers, or, distance in miles is a function of distance in kilometers."}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "23898"}, "_value": "\\(m(100)=62\\)"}]}]}, {"td": [{"p": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "23943"}, "_value": "\\(f(x) = x \\boldcdot \\pi\\)"}]}, {}]}, {}, {}, {}]}, {"td": [{"p": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "23991"}, "_value": "\\(g(x) = 1.6x\\)"}]}, {}]}, {}, {}, {}]}, {"td": [{"p": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "24038"}, "_value": "\\(h(x) = 4x\\)"}]}, {}]}, {}, {}, {}]}, {"td": [{"p": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "24087"}, "_value": "\\(k(x) = |x|\\)"}]}, {}]}, {}, {}, {}]}]}]}]}
Student Response
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Activity Synthesis
Invite students to share their responses in the table, and how they know. Emphasize that there are many correct examples one could choose—the thing to understand is how function notation communicates an input and output to the function (or a value of the independent variable and corresponding value of the dependent variable.)


aligns: yes