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
The purpose of this activity is for students to identify the attributes that make a quadrilateral a rectangle, a rhombus, or a square. They do so by studying examples and non-examples, looking for features that each set has in common and drawing conclusions accordingly (MP7). The goal is not to craft the most precise definition for each, but to develop an understanding of the defining geometric attributes.
MLR8 Discussion Supports.
Display sentence frames to support small-group discussion: “I notice _____ so I think . . .”, “_____ and _____ are the same/alike because . . .”, “_____ and _____ are different because . . . .”
Advances: Conversing, Representing
Engagement: Provide Access by Recruiting Interest.
Optimize meaning and value. Invite students to share, with a partner, the attributes needed in order to be an example of this type of quadrilateral.
Supports accessibility for: Social-Emotional Functioning, Attention
Required Preparation
Create a chart with labels showing a rectangle, rhombus, and square for the lesson synthesis.
Launch
Groups of 2–3
Display the examples and non-examples of right triangles.
“The triangles on the left are right triangles. The ones on the right are not. Take a minute to think about their differences.”
“Then, choose all the shapes that are right triangles from the set labeled A–H. Write down what you think makes a shape a right triangle.”
1–2 minutes: quiet think time
1–2 minutes: partner discussion
Share responses.
“All triangles on the left and triangle E in the set are right angles. A triangle with a right angle is a right triangle.”
Activity
“Work with your group to analyze what makes a shape a rectangle, a rhombus, or a square. Each group will look at one type of quadrilateral.”
Assign each group one quadrilateral to analyze.
3–5 minutes: small-group work time
“Now, share your work with another group that worked on the same quadrilateral. Be sure to ask any questions you have about each other’s work.”
3–5 minutes: small-group discussion
Student Facing
These are right triangles.
[IMG]
These are not right triangles.
[IMG]
Which of the following are right triangles? Circle them.
[IMG]
What makes a shape a right triangle?
These are rectangles.
[IMG]
These are not rectangles.
[IMG]
Which of the following are rectangles? Circle them.
[IMG]
What makes a shape a rectangle?
These are rhombuses.
[IMG]
These are not rhombuses.
[IMG]
Which of the following are rhombuses? Circle them.
[IMG]
What makes a shape a rhombus?
These are squares.
[IMG]
These are not squares.
[IMG]
Which of the following are squares? Circle them.
[IMG]
What makes a shape a square?
Student Response
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Activity Synthesis
Invite 1–2 groups who analyzed rectangles to share their responses. Record their responses.
Ask the class:
“Do you agree with the shapes that are marked as rectangles (or rhombuses or squares)?”
“Do you agree with the descriptions?”
“Is there any part of the descriptions you’d say differently?”
“Is there anything you'd add to the description?”
When students have finished describing rectangles, highlight that: “A shape is a rectangle only if it is a quadrilateral with only right angles, and the sides that are across from each other are the same length.”
Repeat the discussion with rhombuses and squares.


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