few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
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 $P$.


Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

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, does the following problem or activity enable students to learn the full intent of the following concept or skill?

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: Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
problem_activity: Optional activity
In this activity, students use the areas of composite shapes from the previous activity to reason about the area of each tangram shape. Students may have recognized previously that the area of one small triangle is
\(\frac12\)
square unit, the area of one medium triangle is 1 square unit, and the area of one large triangle is 2 square units. Here they practice articulating how they know that these observations are true (MP3). The explanations could be written in words, or as clearly-labeled illustrations that support their answers.
As partners discuss, look for two ways of thinking about the area of each assigned triangle: by
composing
copies of the triangle into a square or a larger triangle, or by
decomposing
the triangle or the unit square into smaller pieces and
rearranging
the pieces. Identify at least one student who uses each approach.
Digital
Launch
Arrange students in groups of 2. Assign the first and second questions to one partner and the second and third questions to the other partner. Give each group access to the geometry toolkits and the same set of tangram pieces as used in the earlier activity.
Give students 3–4 minutes of quiet time to find the areas of their assigned triangles and to construct their explanations, followed by a few minutes to share their responses with their partner. Tell students that as one partner explains, the other should listen carefully and either agree or disagree with the explanation. They should then come to an agreement about the answers and explanations.
Classrooms using the digital activities can use an applet to assist in determining the areas of the triangles.
Speaking, Writing: MLR8 Discussion Supports.
Use this routine when students compare areas of triangles and squares to support the use of mathematical language. As students share their responses with their partner, circulate and encourage listeners to push speakers to use the language “compose,” “decompose,” or “rearrange” in their explanations. Look for students who name the square or larger triangles “composite figures” and amplify this language. Encourage students to borrow words and phrases from each other and to use this language in their written responses.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)
Student Facing
Recall that the area of the square you saw earlier is 1 square unit. Complete each statement and explain your reasoning.
The area of the small triangle is ____________ square units. I know this because . . .
The area of the medium triangle is ____________ square units. I know this because . . .
The area of the large triangle is ____________ square units. I know this because . . .
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Assign the first and second questions to one partner and the second and third questions to the other partner. Give each group access to the geometry toolkits and the same set of tangram pieces as used in the earlier activity.
Give students 3–4 minutes of quiet time to find the areas of their assigned triangles and to construct their explanations, followed by a few minutes to share their responses with their partner. Tell students that as one partner explains, the other should listen carefully and either agree or disagree with the explanation. They should then come to an agreement about the answers and explanations.
Classrooms using the digital activities can use an applet to assist in determining the areas of the triangles.
Speaking, Writing: MLR8 Discussion Supports.
Use this routine when students compare areas of triangles and squares to support the use of mathematical language. As students share their responses with their partner, circulate and encourage listeners to push speakers to use the language “compose,” “decompose,” or “rearrange” in their explanations. Look for students who name the square or larger triangles “composite figures” and amplify this language. Encourage students to borrow words and phrases from each other and to use this language in their written responses.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)
Student Facing
Recall that the area of the square you saw earlier is 1 square unit. Complete each statement and explain your reasoning.
The area of the small triangle is _________ square units. I know this because . . .
The area of the medium triangle is _________ square units. I know this because . . .
The area of the large triangle is _________ square units. I know this because . . .
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students initially have trouble determining the areas of the shapes, ask how they reasoned about areas in the previous activity. Have samples of composed and decomposed shapes that form one square unit available for students to reference.
Activity Synthesis
After partners shared and agreed on the correct areas and explanations, discuss with the class:
“Did you and your partner use the same strategy to find the area of each triangle?”
“How were your explanations similar? How were they different?”
Select two previously identified students to share their explanations: one who reasoned in terms of
composing
copies of their assigned triangle into another shape, and one who reasoned in terms of
decomposing
their triangle or the unit square into smaller pieces and
rearranging
them. If these approaches are not brought up by students, be sure to make them explicit at the end of the lesson.


aligns: yes