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: Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.
problem_activity: Narrative
The purpose of this activity is for students to analyze a collection of connecting cubes that is arranged in towers of 10. Students analyze another student’s thinking about a representation of 48 cubes in towers of 10. When students explain that they disagree with Noah because a ten must include 10 ones, they show their understanding of a ten and the foundations of the base-ten system (MP3, MP7).
MLR8 Discussion Supports.
Synthesis: At the appropriate time, give students 2–3 minutes to make sure that everyone in their group can explain why they agree or disagree with Noah. Invite groups to rehearse what they will say when they share with the whole class.
Advances: Speaking, Conversing, Representing
Required Materials
Materials to Gather
Connecting cubes in towers of 10 and singles
Launch
Groups of 2
Give students access to connecting cubes in towers of 10 and singles.
“Noah counted a collection of connecting cubes. He says there are 50 cubes. Do you agree or disagree? Explain how you know. You will have a chance to think about it on your own and talk to your partner about Noah’s thinking before you write your response.”
Activity
1 minute: quiet think time
“Share your thinking with your partner.”
2 minutes: partner discussion
“Explain why you agree or disagree with Noah. Write the word ‘agree’ or ‘disagree’ in the first blank. Then write why you agree or disagree.”
3 minutes: independent work time
Student Facing
Noah organized his collection of connecting cubes.
[IMG]
He counts and says there are 50 cubes.
Do you agree or disagree?
Explain how you know:
I ____________________ with Noah because
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Advancing Student Thinking
If students attempt to check Noah's count by counting all the cubes by one, consider asking:
“How are you checking to see if there are 50 cubes?”
“How could you use what you know about groups of ten to check Noah’s thinking?”
Activity Synthesis
Invite students to share whether they agree or disagree, and their reasons.
If needed, ask “How do you know there are not 50 cubes?” (50 is 5 tens. There are only 4 tens here. The last tower isn’t 10. There are 4 tens and 8 ones.)


aligns: no