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: Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
problem_activity: Task
Many researchers have studied chimpanzees to learn about their problem solving skills. In 1978, researchers Premack and Woodruff published an article in Science magazine, reporting findings from a study on an adult chimpanzee named Sarah, who had been raised in captivity and had received extensive training using photos and symbols. In one experiment, Sarah was shown videotapes of eight different situations in which a human being was faced with a problem. After each videotape showing, Sarah was presented with two photographs, one of which depicted a possible solution to the problem. The researchers ensured that the order in which the photographs were presented was randomized (for example, the correct answer was not always presented first, etc.) and that the photographs had similar visuals (for example. similar colors, etc.) Of the eight problems, Sarah picked the photograph with the correct solution seven times. Could Sarah have been merely guessing and just lucky with her responses, or is there evidence that Sarah does better than just guessing?
Give two possible explanations for why Sarah might have answered seven out of eight correctly.
If Sarah were just guessing, and was just likely to pick one photograph compared to the other, how many would you expect her to get right out of eight problems?
Give an example of how you could use basic classroom tools (coins, dice, calculators, cards, etc.) to simulate one trial of Sarah “just guessing” to pick a photograph for one problem.
A student, James, decides to use simulation to investigate whether the study data provide evidence that Sarah was doing better than just randomly guessing, and so James tosses a coin eight times, and obtains six heads. Explain why James should repeat the process of tossing the coin eight times and recording the number of heads, many times.
James repeats the process of eight coin-tosses 100 times, each time recording the number of heads on the eight coin-tosses. The following is a dotplot of his results.
[IMG]
Based on the above dotplot, what was the most common result for “number of heads” in eight coin-tosses? Why does this make sense?
Based on this dotplot, would you say that a score of 7 out of 8 would be unusual if Sarah has just been guessing? Why or why not?
Which of the following is a possible explanation for Sarah’s performance?
Sarah had been just guessing and got lucky with her responses.
Sarah does better than just guessing.
Both (i) and (ii) are possible explanations.
Based on the simulation results, which of the following appears to be a plausible (likely) explanation for Sarah’s performance?
Sarah had been just guessing and got lucky with her responses.
Sarah does better than just guessing.
Both (i) and (ii) are possible explanations.
Based on the results of this study, would it be reasonable to say that all chimpanzees do better than just guessing when faced with the kind of problems posed to Sarah? Explain why or why not.


correct_option_index: 0