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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with 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_pos_0

HS_1

im_task_001160_neg_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: Optional activity
In this activity students compute rates to decide which job applicant is working the fastest checking online comments. They compare rates and total number of comments checked, then see that using rates is the more useful information in this situation.
Launch
Keep students in the same groups of 2.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
A company is hiring people to read through all the comments posted on their website to make sure they are appropriate. Four people applied for the job and were given one day to show how quickly they could check comments.
Person 1 worked for 210 minutes and checked a total of 50,000 comments.
Person 2 worked for 200 minutes and checked 1,325 comments every 5 minutes.
Person 3 worked for 120 minutes, at a rate represented by
\(c = 331t\)
,
where
\(c\)
is the number of comments checked and
\(t\)
is the time in minutes.
Person 4 worked for 150 minutes, at a rate represented by
\(t = \left( \frac{3}{800} \right) c\)
.
Order the people from greatest to least in terms of total number of comments checked.
Order the people from greatest to least in terms of how fast they checked the comments.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Write equations for each job applicant that allow you to easily decide who is working the fastest.
Make a table that allows you to easily compare how many comments the four job applicants can check.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ask students which job applicant should get the job and why. If time permits, consider using MLR 1 (Stronger and Clearer Each Time).
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display showing which job applicant should get the job and why.  Allow students time to quietly circulate and analyze the selections and justifications in at least 2 other displays in the room. Give students quiet think time to consider what is the same and what is different and whether or not they agree. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that highlight advantages and disadvantages to each method of determining the top job applicant. This will help students identify situation when comparing rates is more effective than the sum of quantities.
Design Principle(s): Optimize output (for justification); Cultivate conversation


aligns: no