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: Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
problem_activity: Narrative
The purpose of this activity is for students to have an opportunity to apply their place value understanding to estimate quantities of objects and accurately count familiar objects.
Required Materials
Materials to Gather
Bags
Collections of objects
Required Preparation
Each group of 2 needs a bag of a collection of small objects.
Launch
Groups of 2
Display for all to see approximately 15–25 beans or other small objects.
“How many objects do you think are in this pile?”
1 minute: partner discussion
Share responses.
“How could we find out exactly?” (Count them.)
Activity
“How many objects are in 2 handfuls? Let's do an experiment.”
Give each group a bag of objects.
“Take turns and grab a handful. Estimate how many objects you both grabbed altogether. Then find out how many you have exactly.”
“You will do this experiment three times.”
5 minutes: partner work time
Monitor for students who:
count by ones
group the objects into groups of 10 and then count the tens and ones
Student Facing
Experiment 1: How many objects are in 2 handfuls?
Record an estimate that is:
|                    too low                     |                  about right                   |                    too high                    |
|------------------------------------------------|------------------------------------------------|------------------------------------------------|
| \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) |
Now find the exact number. ____________
Experiment 2: How many objects are in 2 handfuls?
Record an estimate that is:
|                    too low                     |                  about right                   |                    too high                    |
|------------------------------------------------|------------------------------------------------|------------------------------------------------|
| \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) |
Now find the exact number. ____________
Experiment 3: How many objects are in 2 handfuls?
Record an estimate that is:
|                    too low                     |                  about right                   |                    too high                    |
|------------------------------------------------|------------------------------------------------|------------------------------------------------|
| \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) |
Now find the exact number. ____________
Student Response
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Activity Synthesis
Invite previously identified students to share.
“Did your estimates get better with practice?”
“How did organizing your objects help you to count?”
“Imagine I have 72 objects. How many would you have if you have 10 fewer than I do? What if you had 10 more?”
“If you had 72 objects, how could they be arranged in piles of 10 and ones? Are there any other ways?" (7 tens and 2 ones, 6 tens and 12 ones)


aligns: yes