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: Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
problem_activity: Narrative
The purpose of this activity is for students to facilitate the Estimation Exploration they created in the previous activity for another group in the class. Each group should be paired with another group and they will take turns facilitating their Estimation Exploration for the other group. If time allows, students could facilitate their Estimation Exploration with more than 1 group.
MLR8 Discussion Supports.
At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain their Estimation Exploration activity. Invite groups to rehearse what they will say when they share with the whole class.
Advances: Speaking, Conversing, Representing
Required Materials
Materials to Gather
Chart paper
Colored pencils, crayons, or markers
Launch
Groups of 3 or 4
Give each group a piece of chart paper and a marker to record responses.
“Pair up with another group in the class. Each group will present their Estimation Exploration.”
Activity
10–12 minutes: small-group work time
Remind students to switch roles halfway through the time allowed.
Student Facing
Display your image for your classmates.
Ask them, “What is an estimate that’s too high?” “Too low?” “About right?”
Give them a minute of quiet think time.
Give them a minute to discuss together.
Have them share estimates.
Record their ideas.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
“What did you learn as you presented your Estimation Exploration?” (I learned that ranges of estimates might be bigger than you expect. I learned that it’s important to understand the size of the items you are trying to estimate or be familiar with it, otherwise you cannot make good estimates.)


aligns: no