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: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_neg_0

HS_2
standard_description: Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
problem_activity: Problem 1
Flares are used to warn cars that there is an obstacle on the road. Flares are often part of cars’ emergency kits and burn for a long time.
Fred’s Flare Company used to make flares that burned for 100 minutes on average. Fred changed the chemical formula in his flares, and he thinks that his new formula will burn for slightly over 120 minutes. Fred wants to make the claim on his packaging that his flares “Now burn for over 2 hours on average!!”
To make this claim, Fred needs to back it up with data. He decides that performing some random sampling is the best way to do this. Since he runs a 24-hour production operation, he decides that taking a sample each hour is the easiest way to do this.
Fred has tried two different ways of sampling, as shown below.
| Sample size: 20 flares                      24 total samples (one per hour of production)            |https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l12_at1.png|  |https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l12_at1.png| image:: https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l12_at1.png | Sample size: 100 flares                      24 total samples (one per hour of production)            |https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l12_at1a.png|  |https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l12_at1a.png| image:: https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l12_at1a.png |
|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
If we further extrapolate that Fred continues to take samples for 30 days in the same manner as above, he charts his results in histograms shown below.
| .. image:: https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l12_at1b.png | .. image:: https://d37su8u1z9ynie.cloudfront.net/uploads/2018/06/05/algiiu8_l12_at1c.png |
|------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------|
The margin of error is the largest expected difference between the estimate of a population parameter and the actual population parameter.
What would you guess is the mean and the margin of error of burn time of Fred’s flares (with the new and improved formula)?
Problem 2
The margin of error can also be calculated in by
$${ME=2\sqrt{{\hat{p}}(1-{\hat{p}})\over n}}$$
, where
$${\hat{p}}$$
is the proportion of a sample of an event occurring,
$$1-{\hat{p}}$$
is the proportion of a sample of an event not occurring,
$$n$$
is the sample size, and
$${\sigma}$$
is the sample standard deviation.
Suppose you draw a random sample of 36 chips from a mystery bag and find 20 red chips. Find:
The sample proportion of red chips
The standard error (standard deviation from the mean)
The margin of error


aligns: no