few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

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


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
Concept/skill: 
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.

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 $P$.


Answer: 
Yes. 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.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
problem_activity: Task
A seven-year-old boy has a favorite treat, Super Fruity Fruit Snax.
These "Fruit Snax" come in pouches of 10 snack pieces per pouch, and the pouches are generally sold by the box, with each box containing 4 pouches.
The snack pieces come in 5 different fruit flavors, and usually each pouch contains at least one piece from each of the 5 flavors. The website of the company that manufactures the product says that equal numbers of each of the 5 fruit flavors are produced and that pouches are filled in such a way that each piece added to a pouch is equally likely to be any one of the five flavors.
Of all the 5 fruit flavors, the seven-year-old boy likes mango the best. One day, he was very disappointed when he opened a pouch and there were no (zero) mango flavored pieces in the pouch. His mother (a statistician) assured him that this was no big deal and just happens by chance sometimes.
If the information on the company's website is correct,
What proportion of the population of snack pieces is mango flavored?
On average, how many mango flavored pieces should the boy expect in a pouch of 10 snack pieces?
What is the chance that a pouch of 10 would have no mango flavored pieces? Was the mother's statement reasonable? Explain. (Hint: if none of the 10 independently selected pieces are mango, then all 10 pieces are "not mango.")
The family then finds out that there were in fact no mango flavored pieces in any of the 4 pouches in the box they purchased. Again, if the information on the company's website is correct,
What is the chance that an entire box of 4 pouches would have no mango flavored pieces? (Hint: How is this related to your answer to question (iii) in part (a)?)
Based on your answer and based on the fact that this event of an entire box with "no mangoes" happened to this family, would you be concerned about the company's claims, or would you say that such an event is not surprising given the company's claims? Explain.


aligns: no