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: Understand and evaluate random processes underlying statistical experiments.

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

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

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

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

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

Use random sampling to draw inferences about a population.

Summarize and describe distributions.

Draw informal comparative inferences about two populations.

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

Interpret linear models.

Develop understanding of statistical variability.

Use probability to evaluate outcomes of decisions.

Calculate expected values and use them to solve problems.

Investigate patterns of association in bivariate data.
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


correct_option_index: 9