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: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles.
problem_activity: Narrative
In this activity, students are given the result of folding a shape along one or more lines of symmetry and asked to reason about the original shape. No lines of symmetry are specified, so students must consider all sides of a folded shape as a possible line of symmetry and visualize the missing half accordingly.
The first question offers opportunities to practice choosing tools strategically (MP5). Some students may wish to trace the half-shapes on patty paper, to make cutouts of them, or to use other tools or techniques to reason about the original shape. Provide access to the materials and tools they might need.
During the activity synthesis, discuss the different ways students approach the second question. Consider preparing cutouts of shapes A–F to facilitate the discussion. (The shapes are provided in the blackline master.)
Action and Expression: Internalize Executive Functions.
Invite students to plan a strategy, including the tools they will use, for the task. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Organization, Attention
Required Materials
Materials to Gather
Paper
Patty paper
Protractors
Rulers or straightedges
Scissors
Materials to Copy
Before and After
Launch
Groups of 2
Give a ruler or a straightedge to each student.
Provide access to protractors, patty paper, scrap paper, and scissors.
Activity
5 minutes: independent work time
2–3 minutes: partner discussion
Monitor for the different strategies students use to identify the original shape of the half-shapes (as noted in the activity narrative).
Student Facing
Mai has a piece of paper. She can get two different shapes by folding the paper along a line of symmetry. What is the shape of the paper before it was folded?
[IMG]
Diego folded a piece of paper once along a line of symmetry and got this right triangle.
[IMG]
Which shapes could the paper have before it was folded? Explain or show how you know.
[IMG]
[IMG]
[IMG]
Student Response
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Activity Synthesis
Invite students to share their responses and strategies.
When discussing the second question, ask students why B, C, and E are not possible shapes of the original piece of paper even though there’s a line that breaks each figure into two right triangles that match the given triangle. (B and E have no line symmetry. C has lines of symmetry but not diagonally from corner to corner. If you fold there, the triangles would not be on top of one another.)


aligns: no