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


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

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


Standard options: 
A. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
B. Solve quadratic equations in one variable.


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

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 $P$.


Standard options: 
A. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
B. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.


Thought: This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.
Answer: A

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
problem_activity: Optional activity
Earlier in this lesson, students found the area of regions around circles and the area of fractions of circles in separate problems. In this activity, students combine these two strategies to find the area of a complex real-world object. Students engage in MP2 as they decide how to decompose the running track into measurable pieces and how to use the given information about the dimensions of the track to calculate areas.
Launch
Arrange students in groups of 2. Give students 3–4 minutes of partner work time followed by small-group and whole-class discussions.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to organize the information provided in the problem and to structure their problem-solving strategy. The graphic organizer should include the prompts: “What do I need to find out?”, “What do I need to do?”, “How I solved the problem.”, and “How I know my answer is correct.”
Supports accessibility for: Language; Organization
Student Facing
The field inside a running track is made up of a rectangle 84.39 m long and 73 m wide, together with a half-circle at each end. The running lanes are 9.76 m wide all the way around.
[IMG]
What is the area of the running track that goes around the field? Explain or show your reasoning.
Student Response
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Anticipated Misconceptions
Some students may think they can calculate the area of the running track by multiplying half of the perimeter times the radius, as if the shape were just a circle. Prompt them to see that they need to break the overall shape into rectangular and circular pieces.
Activity Synthesis
Pair the groups of 2 to create groups of 4. Have students compare answers and explain their reasoning until they reach an agreement.
As a whole group, discuss how finding the area of the track in this activity was similar or different to solving the two area problems in the previous activity. Make sure to highlight these points:
Like the first problem in the previous activity, finding the area of the track can be done by finding the area of the larger shape (the track and the field inside) and then taking away the area of the field inside.
Like the second problem in the previous activity, there are semicircles whose area can be found by composing them to make a full circle or by taking half the area of the corresponding full circle.
Representing, Speaking, Listening: MLR7 Compare and Connect.
Ask students to prepare a visual display that shows how they calculated the area of the running track that goes around the field, and look for students with different strategies for calculating the area of the curved parts of the track. As students investigate each other’s work, ask students to share what worked well in a particular approach. During this discussion, listen for and amplify any comments that clarify that the curved parts of the running track are half circles with a smaller circle missing from the inside. Then encourage students to make connections between the quantities representing areas and the shapes in the diagram. Ask questions such as, “What does the quantity 6,719.561 represent in the diagram?” and “What does the quantity 4,183.265 represent in the diagram?” During this discussion, listen for and amplify language students use to reason that these quantities represent the areas of larger and smaller circles, respectively. This will foster students’ meta-awareness and support constructive conversations as they compare strategies for finding the area of a complex real-world object.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 0