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: Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
problem_activity: Activity
The goal of this activity is for students to be introduced to the idea that variability is important when analyzing and comparing data sets. Students will compare data sets, and make a decision as to which data set represents a better scenario. This activity prepares students to calculate variability in order to compare data sets in a later lesson. Monitor for students who calculate any
statistics
to inform their decisions. Make note of the students who only calculate a measure of center vs. students who calculate both a measure of center and a measure of variability.
Making statistical technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Students will work individually. Tell them to use statistics to inform their decision.
Student Facing
In order from least to greatest, here are the prices per gallon of gas at two different gas stations over the past 7 days.
Station A:
2.38, 2.68, 2.82, 2.86, 2.99, 3.26, 3.59
Station B:
2.84, 2.85, 2.88, 2.95, 2.98, 3.03, 3.05
Suppose that these gas stations were the closest to your house, but not near each other. Which gas station would you go to for gas? Explain your reasoning.
Student Response
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Activity Synthesis
The purpose of this discussion is to understand that the variability is worth paying attention to. If any students chose to compute the mean, median, IQR, MAD, or range for the two stations, ask them to share their results and how those values help them decide. Interestingly, the mean is the same for the gas stations. But any measure of variability shows that the prices at station B are more stable and predictable than the prices at station A. If no students suggest computing measures of center and variability and comparing them, be prepared to demonstrate.
Here are the summary statistics for each set of data:
For Station A:
Mean: 2.94
Five-number summary: 2.38, 2.68, 2.86, 3.26, 3.59
Range: 1.21
IQR: 0.58
MAD: 0.29
For Station B:
Mean: 2.94
Five-number summary: 2.84, 2.85, 2.95, 3.03, 3.05
Range: 0.21
IQR: 0.18
MAD: 0.07
If time permits, discuss which measures are helpful. Here are sample questions:
“Was the mean helpful in deciding the gas station from which you would purchase gas?”(No, because both means were the same.)
“How did the range (or IQR, MAD, five-number summary) help you decide which gas station you would go to?” (Whichever gas station had the highest range [or IQR, MAD] is the gas station I chose.)
“In the real world, how would you use the gas prices at each gas station to make a decision?” (I would think about which one generally had lower prices and choose that one, or I would see which one had more predictably cheaper prices and choose that one.)


correct_option_index: 0