few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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

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


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

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 $P$.


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

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im_task_000870_pos_0

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im_task_001160_pos_0

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standard_description: Construct a tangent line from a point outside a given circle to the circle.
problem_activity: Task
“Exponential decay” describes a situation in which a quantity decreases following a certain pattern. We’re going to investigate a situation that decays exponentially. Read through steps 1-6 below and answer questions a-c before carrying out the investigation.
1. Get your hands on 30 6-sided dice and put them in a container.
2. Find a safe place where you can roll 30 dice at once, like a tray (or the container itself, if it is large enough).
3. Roll the remaining dice (or shake the container thoroughly).
4. Remove any dice that display a "1" and set them aside.
5. Write down how many dice remain.
6. Go back to step 3 and repeat. Stop when you have rolled the dice ten times.
Pre-investigation questions:
When you roll 30 dice, how many 1’s do you predict you are likely to roll? Why?
After you remove the 1’s on the first roll, what fraction of the original number of dice do you predict will remain?
How many times do you predict you might have to roll the dice and remove the 1’s before you get to a roll where you don’t roll any 1’s?
Conduct the Investigation:
Construct a table to record the number of 1’s and the number of dice remaining after each roll, as shown.
|           Roll           | 0  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|--------------------------|----|---|---|---|---|---|---|---|---|---|----|
|      Number of 1’s       | X  |   |   |   |   |   |   |   |   |   |    |
| Number of Dice Remaining | 30 |   |   |   |   |   |   |   |   |   |    |
Now for some mathematical modeling:
Use technology to create a scatter plot with number of dice remaining on the vertical axis and roll number on the horizontal axis. Discuss what you see.
Based on what you know about exponential functions and the behavior of dice, explain in words why an exponential model would be appropriate for this situation.
Based on what you know about exponential functions and 6-sided dice, write a function of the form \(d(x)=ab^x\) to model the relationship between the roll number, x, and the number of dice remaining, d(x). Your job here is to decide what numbers to use in place of a and b. This should not be based on the data you collected -- it is a mathematical model for how dice behave.
Graph your function on the same set of axes as your scatter plot. Compare and contrast the graphed model d(x) to the outcome of the experiment shown in the scatter plot.
Suppose someone repeated this activity, except they started with 100 8-sided dice instead of 30 6-sided dice. Write a new function that would model this situation.
Here is a tool that could be used in this task:


aligns: no