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


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

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


Mathematical concept/skill options: 
A. Apply and extend previous understandings of arithmetic to algebraic expressions.
B. Write expressions in equivalent forms to solve problems.
C. Understand solving equations as a process of reasoning and explain the reasoning.
D. Use the four operations with whole numbers to solve problems.
E. Generate and analyze patterns.
F. Understand and apply properties of operations and the relationship between addition and subtraction.
G. Use properties of operations to generate equivalent expressions.
H. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
I. Represent and analyze quantitative relationships between dependent and independent variables.
J. Understand properties of multiplication and the relationship between multiplication and division.
K. Work with equal groups of objects to gain foundations for multiplication.
L. Represent and solve problems involving multiplication and division.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Work with radicals and integer exponents.
O. Interpret the structure of expressions.
P. Solve equations and inequalities in one variable.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Multiply and divide within 100.
S. Represent and solve problems involving addition and subtraction.
T. Use polynomial identities to solve problems.
U. Represent and solve equations and inequalities graphically.
V. Understand the connections between proportional relationships, lines, and linear equations.
W. Reason about and solve one-variable equations and inequalities.
X. Rewrite rational expressions.
Y. Understand the relationship between zeros and factors of polynomials.
Z. Perform arithmetic operations on polynomials.
AA. Create equations that describe numbers or relationships
AB. Analyze and solve linear equations and pairs of simultaneous linear equations.
AC. Gain familiarity with factors and multiples.
AD. Write and interpret numerical expressions.
AE. Add and subtract within 20.
AF. Work with addition and subtraction equations.
AG. Solve systems of equations.
AH. Analyze patterns and relationships.


Thought: This task asks students to solve an equation in one variable. Since the problem asks for many different solution methods, students are also encouraged to solve the problem graphically.
Answer: P, U

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 $P$.


Mathematical concept/skill options: 
A. Apply trigonometry to general triangles.
B. Understand similarity in terms of similarity transformations.
C. Analyze, compare, create, and compose shapes.
D. Prove geometric theorems.
E. Make geometric constructions.
F. Graph points on the coordinate plane to solve real-world and mathematical problems.
G. Understand congruence and similarity using physical models, transparencies, or geometry software.
H. Understand congruence in terms of rigid motions.
I. Draw, construct, and describe geometrical figures and describe the relationships between them.
J. Translate between the geometric description and the equation for a conic section.
K. Reason with shapes and their attributes.
L. Prove theorems involving similarity.
M. Explain volume formulas and use them to solve problems.
N. Find arc lengths and areas of sectors of circles.
O. Define trigonometric ratios and solve problems involving right triangles.
P. Use coordinates to prove simple geometric theorems algebraically.
Q. Visualize relationships between two-dimensional and three-dimensional objects.
R. Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.
S. Understand and apply theorems about circles.
T. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
U. Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.
V. Classify two-dimensional figures into categories based on their properties.
W. Apply geometric concepts in modeling situations.
X. Solve real-world and mathematical problems involving area, surface area, and volume.
Y. Experiment with transformations in the plane.
Z. Understand and apply the Pythagorean Theorem.
AA. Draw and identify lines and angles, and classify shapes by properties of their lines and angles.


Thought: This task asks students to make geometric constructions, such as bisecting an angle.
Answer: E

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
options: Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Draw informal comparative inferences about two populations.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Develop understanding of statistical variability.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Understand independence and conditional probability and use them to interpret data.

Understand and evaluate random processes underlying statistical experiments.

Summarize, represent, and interpret data on a single count or measurement variable.

Use probability to evaluate outcomes of decisions.

Investigate chance processes and develop, use, and evaluate probability models.

Use random sampling to draw inferences about a population.

Calculate expected values and use them to solve problems.

Summarize and describe distributions.

Investigate patterns of association in bivariate data.

Interpret linear models.
problem_activity: Task
A famous arcade in a seaside resort town consists of many different games of skill and chance.  In order to play a popular “spinning wheel” game at Fred's Fun Factory Arcade, a player is required to pay a small, fixed amount of 25 cents each time he/she wants to make the wheel spin. When the wheel stops, the player is awarded tickets based on where the wheel stops -- and these tickets are then redeemable for prizes at a redemption center within the arcade.
Note: this particular game has no skill component; each spin of the wheel is a random event, and the result from each spin of the wheel is independent of the results of previous spins.
The wheel awards tickets with the following probabilities:
|  1 ticket   | 35% |
|-------------|-----|
|  2 tickets  | 20% |
|  3 tickets  | 20% |
|  5 tickets  | 10% |
| 10 tickets  | 10% |
| 25 tickets  | 4%  |
| 100 tickets | 1%  |
(Note: A picture of a wheel fitting these parameters is included.)
If a player were to play this game many, many times, what is the expected (average) number of tickets that the player would win from each spin?
The arcade often provides quarters to its customers in $5.00 rolls. Every day over the summer, a young boy obtains one of these quarter rolls and uses all of the quarters for the spinning wheel game. In the long run, what is the average number of tickets that this boy can expect to win each day using this strategy?
One of the redemption center prizes that the young boy is playing for is a trendy item that costs 300 tickets.  It is also available at a store down the street for $4.99. Without factoring in any enjoyment gained from playing the game or from visiting the arcade, from a strictly monetary point of view, would you advise the boy to try and obtain this item based on arcade ticket winnings or to just go and buy the item at the store?  Explain.
The histogram below summarizes the results of 90 summer days of the boy playing the game using his "$5 roll per day" strategy. The first bar in the histogram represents those days where 40 to 59 tickets were won.
[IMG]
For approximately what percentage of the 90 days did the boy earn fewer than 100 tickets in a day?
For approximately what percentage of the 90 days did the boy earn 200 or more tickets in a day?
For approximately what percentage of the 90 days did the boy earn 300 or more tickets in a day?
To maintain the spinning wheel game machine, the arcade manager adopts a strategy of emptying the money box (that's where the quarters go after they are inserted in the machine) each time she refills the machine with a new roll of tickets. The ticket refill rolls contain 5000 tickets, and the machine is designed to hold $1300 in quarters in its money box. Assume that the machine was fully loaded with 5000 tickets and had an empty money box when it was first used. Using the manager's maintenance strategy, is there any chance that the money box could become completely full with quarters or overflow with quarters? Explain.
Arcade Wheel
| Color  | Tickets |
|--------|---------|
| Orange |    1    |
| Green  |    2    |
|  Blue  |    3    |
| Yellow |    5    |
| Purple |   10    |
| Black  |   25    |
|  Red   |   100   |
Graphics Note: The wheel diagram was developed in Microsoft Excel using its "Pie Chart" graph building feature.  The intent is that each wedge represents 1% of the pie (3.6 degrees).  There are 35 orange wedges (each representing a win of 1 ticket) to correspond to the 35% probability of obtaining 1 ticket in a spin, 20 green wedges (each representing a win of 2 tickets) to correspond to the 20% probability of obtaining 2 tickets in a spin, and so on.
[IMG]


correct_option_index: 8

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