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: Describe and compare measurable attributes.

Work with time and money.

Geometric measurement: understand concepts of angle and measure angles.

Measure lengths indirectly and by iterating length units.

Convert like measurement units within a given measurement system.

Relate addition and subtraction to length.

Solve problems involving measurement and estimation of intervals of time, liquid volumes, and masses of objects.

Measure and estimate lengths in standard units.

Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit.

Classify objects and count the number of objects in each category.

Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.

Tell and write time.

Represent and interpret data.

Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
problem_activity: Narrative
The purpose of this activity is for students to think about and compare the capacities of containers. Students start by comparing two containers where it is visually obvious which one holds more. They use comparison language such as “The pitcher holds more than the cup.” Then, students consider containers that have capacities that are not easy to compare visually and they brainstorm ways to compare the capacities of the containers. Students may need to see the water poured between the two containers multiple times before they determine and can explain which container has more capacity. As students make predictions and then discuss and justify their comparisons, they share a mathematical claim and the thinking behind it (MP3).
MLR8 Discussion Supports.
Use multimodal examples to clarify what it means for a container to hold liquid. Use verbal descriptions along with gestures or drawings to show the meaning of the word hold in this context.
Advances: Listening, Representing
Required Materials
Materials to Gather
Containers of different sizes
Sticky notes
Required Preparation
Gather a larger pitcher and a small cup to display during the launch.
Gather 2 cups with capacities that are not easy to compare visually, such as a tall stemmed glass and a short, wide cup for the activity.
Launch
Groups of 2
Display a pitcher and a small cup.
“Diego’s class needs a lot of lemonade for a lemonade sale they are going to have at school. Which container do you think they should use to hold the lemonade? Why do you think that?” (The pitcher holds more lemonade because it is bigger. The cup is small.)
30 seconds: quiet think time
1 minute: partner discussion
Share responses.
“We think that the pitcher will hold more lemonade.”
Activity
Display 2 cups and give each student a sticky note.
“Which of these cups do you think would hold more lemonade? Put your sticky note by the cup that you think would hold more lemonade.”
3 minutes: independent work time
“People had different answers about which cup would hold more lemonade. What can we do to figure out which cup can hold more lemonade?”
1 minute: quiet think time
1 minute: partner discussion
Share and record responses.
Demonstrate filling one of the cups with water and then slowly pour that water into the other cup.
“I filled up the red cup and poured the same water into the blue cup, but the blue cup overflowed. Which cup do you think can hold more lemonade?”
30 seconds: quiet think time
1 minute: partner discussion
Share responses.
“The red cup can hold more lemonade than the blue cup.”
Student Response
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Activity Synthesis
“We can use water to help us figure out which cup can hold more lemonade.”


correct_option_index: 0