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 addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
problem_activity: Narrative
The purpose of this activity is for students to play Stage 4 in the Shake and Spill center, which was introduced in an earlier unit. This activity supports students in finding the missing part of an equation in a later activity.
Students use 3, 4, or 5 counters. They see some of the counters and determine how many more counters are under the cup. Students write an expression to represent each composition and decomposition.
Representation: Access for Perception.
Synthesis: Use a cup and two-color counters to demonstrate that there is 1 yellow counter hiding under the cup.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Required Materials
Materials to Gather
Cups
Two-color counters
Materials to Copy
Shake and Spill Stage 4 Recording Sheet Kindergarten
Launch
Groups of 2
Give each group of students a cup, 5 two-color counters, and two recording sheets.
“We are going to play Shake and Spill, Cover.”
If needed, review and demonstrate the steps of the center.
“Take turns with your partner spilling and covering the yellow counters. On each turn you can decide to use 3, 4, or 5 counters. Make sure you and your partner agree on how many total counters you are using before you shake, spill, cover, and record.”
Activity
8 minutes: partner work time
Student Facing
Put 3, 4, or 5 counters in the cup.
Shake and spill the counters.
Hide some of the counters under the cup.
Figure out how many of the counters are hidden.
Write an expression.
Student Response
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Activity Synthesis
Display the number 3 and the expression
\(2 + \underline{\hspace{1 cm}}\)
.
“Han and Mai spilled 3 counters and hid the yellow ones. This is what they wrote on the recording sheet. What is missing from the expression? How do you know?” (1 is missing. 2 and 1 are the parts that go together to make 3. If they can see 2 red counters, 1 yellow counter is hiding under the cup.)


aligns: no