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: Work with radicals and integer exponents.

Multiply and divide within 100.

Use the four operations with whole numbers to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Understand the connections between proportional relationships, lines, and linear equations.

Work with equal groups of objects to gain foundations for multiplication.

Use properties of operations to generate equivalent expressions.

Use polynomial identities to solve problems.

Generate and analyze patterns.

Represent and solve equations and inequalities graphically.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with addition and subtraction equations.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Represent and solve problems involving addition and subtraction.

Solve equations and inequalities in one variable.

Write and interpret numerical expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand properties of multiplication and the relationship between multiplication and division.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Analyze patterns and relationships.

Create equations that describe numbers or relationships

Understand and apply properties of operations and the relationship between addition and subtraction.

Reason about and solve one-variable equations and inequalities.

Represent and solve problems involving multiplication and division.

Write expressions in equivalent forms to solve problems.

Add and subtract within 20.

Gain familiarity with factors and multiples.

Interpret the structure of expressions.

Solve systems of equations.

Perform arithmetic operations on polynomials.

Rewrite rational expressions.

Understand the relationship between zeros and factors of polynomials.

Understand solving equations as a process of reasoning and explain the reasoning.
problem_activity: Narrative
The purpose of this activity is for students to visualize and make sense of the context of problems they will solve in the next activity. They will also consider representations that can be used to model the quantities and actions in the situation (MP4) and try creating them. Students will not have time to make a complete representation, so the focus is on making sense of the situation and persevering in finding effective representations (MP1).
Required Materials
Materials to Gather
Coins
Index cards
Paper
Two-color counters
Launch
Groups of 3 or 4
Display the image of the lockers.
“In your small group, discuss what you notice and wonder about the picture.”
1 minute: quiet think time
1 minute: group discussion
Share and record responses. If students are not familiar with lockers, explain that lockers are used to hold students’ coats, books, and backpacks in some schools.
Activity
Read the problem aloud to students.
“Take some time on your own to make sense of the problem. You may want to try to visualize it or reread the problem.”
2 minutes: quiet think time
Give students access to two-color counters, index cards, coins, and other materials that may be helpful to them.
“Based on what you know, create a representation of the problem.”
5 minutes: independent work time
Monitor for and select 3 different representations that:
represent lockers in different ways
show instances of contact with lockers (or of opening and closing)
show correspondence between each student and the lockers touched
Student Facing
The picture shows lockers in a school hallway.
The 20 students in Tyler’s fourth-grade class are playing a game in a hallway that is lined with 20 lockers in a row.
[IMG]
[IMG]
The first student starts with the first locker and goes down the hallway and opens all the lockers.
The second student starts with the second locker and goes down the hallway and shuts every other locker.
The third student stops at every third locker and opens the locker if it is closed or closes the locker if it is open.
This process continues until all 20 students in the class have touched the lockers.
Create a representation to show what you understand about this problem. Consider:
How does your representation show lockers?
How does your representation keep track of students who touch lockers?
How does your representation show which lockers are open or closed?
Student Response
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Activity Synthesis
While discussing each representation, ask students:
“Where are the lockers in this representation?” (Numbers or pictures used to indicate lockers)
“How can we tell if a locker is open or closed in this representation?” (Symbols or colors used to indicate open or closed)
“How does this representation show which student is touching the lockers?” (Students and their actions listed in a table or a T-chart)
“In the next activity, you will answer questions about the game. You may use your representation or one shared by a classmate to help you think through the problems.”


correct_option_index: 27