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: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.  For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
problem_activity: Activity
This activity presents students with two graphs without numerical values assigned to any points on the graphs. The only labeled point is where the two graphs intersect. Students will need to use their understanding that tripling each day is a greater growth factor than doubling each day. This is their only means to decide which function represents which situation. Students then interpret the graphs in terms of the situations that they represent.
Launch
Display the graph for all to see. Give students a moment to observe the graph and ask, "What do you notice? What do you wonder?" Invite students to share their observations and questions.
Student Facing
Here are graphs representing two functions, and descriptions of two functions.
[IMG]
Function
\(f\)
: The area of a wall that is covered by Mold A, in square inches, doubling every month.
Function
\(g\)
: The area of a wall that is covered by Mold B, in square inches, tripling every month.
Which graph represents each function? Label the graphs accordingly and explain your reasoning.
When the mold was first spotted and measured, was there more of Mold A or Mold B? Explain how you know.
What does the point
\((p, q)\)
tell us in this situation?
Student Response
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Anticipated Misconceptions
Some students may think that the dashed graph represents the mold that is tripling because it has a greater vertical intercept. Ask these students to think carefully about the second question.
Activity Synthesis
Select students to share their responses and explanations. Focus on how they deciphered the meanings of the graphs. Discuss questions such as:
The solid graph started off lower than the dashed graph. What does that mean in this context? (The mold population represented by the solid graph is initially smaller or covers a smaller area on the wall than the one represented by the dashed graph.)
After some point, the solid graph passes the dashed one. What does that mean in this context? (The mold population that started out covering a greater area on the wall is surpassed by the other population, so it is growing at a slower rate.)
What does the intersection of the two graphs mean? What do
\(p\)
and
\(q\)
represent? (The two mold populations are the same at time
\(p\)
with population covering an area of
\(q\)
square inches.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each explanation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 0