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 similarity in terms of similarity transformations.

Prove geometric theorems.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Use coordinates to prove simple geometric theorems algebraically.

Define trigonometric ratios and solve problems involving right triangles.

Visualize relationships between two-dimensional and three-dimensional objects.

Analyze, compare, create, and compose shapes.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Translate between the geometric description and the equation for a conic section.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Make geometric constructions.

Understand and apply the Pythagorean Theorem.

Solve real-world and mathematical problems involving area, surface area, and volume.

Reason with shapes and their attributes.

Explain volume formulas and use them to solve problems.

Experiment with transformations in the plane.

Understand congruence in terms of rigid motions.

Find arc lengths and areas of sectors of circles.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Classify two-dimensional figures into categories based on their properties.

Draw, construct, and describe geometrical figures and describe the relationships between them.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Apply geometric concepts in modeling situations.

Understand and apply theorems about circles.

Apply trigonometry to general triangles.

Prove theorems involving similarity.
problem_activity: Narrative
The purpose of this activity is for students to practice describing geometric attributes of a quadrilateral with increasing precision by playing a game. Students should be encouraged to ask questions like, “Are all the sides the same length?” rather than, “Is it a square?” to keep the focus on attributes of the quadrilateral rather than the name. As students decide which questions to ask they think about important attributes such as side lengths and angles and have an opportunity to use language precisely (MP6, MP7).
Students will use the quadrilaterals from the previous lesson to hide in the “mystery quadrilateral” folder, but will have a copy of all the shapes in their workbook to support them in asking questions to narrow down the shape. Students can also cover shapes in their workbook with counters as they rule out shapes.
MLR8 Discussion Supports.
Synthesis: Think aloud and use gestures to emphasize the attributes that students use to describe the shapes. For example, trace your finger along the angles and sides of the shape as students describe them.
Advances: Listening, Representing
Engagement: Develop Effort and Persistence.
Check in and provide each group with feedback that encourages collaboration and community.
Supports accessibility for: Social-Emotional Functioning
Required Materials
Materials to Gather
Counters
Folders
Materials from a previous lesson
Required Preparation
Each group of 2 needs a set of quadrilateral cards from the previous lesson.
Each group of 2 will need a folder to hide the card during this activity.
Launch
Groups of 2
“Now you’re going to play Mystery Quadrilateral with your partner. Re-read the directions for the game, then think about some words that may be helpful as you play.” (side, angle, right angle, equal, skinny, tall, slanted)
1 minute: quiet think time
Share and record responses.
Give each group a folder containing a set of the quadrilateral cards from the previous lesson.
“How could you use the images of all the quadrilaterals on your paper as you play?” (They can help me think about questions I could ask. I could mark off quadrilaterals as I figure out that they're not the mystery quadrilateral.)
Give students access to counters and let them know they can be used to cover shapes they want to cross out during the game.
Activity
“Play Mystery Quadrilateral with your partner. Be sure to take turns hiding the shape and guessing the shape.”
10–15 minutes: partner work time
Student Facing
Partner A: Choose a shape from the group of quadrilaterals. Place it in the mystery quadrilateral folder without your partner seeing it.
Partner B: Ask up to 5 “yes” or “no” questions to identify the quadrilateral. Then guess which quadrilateral is the mystery quadrilateral.
Partner A: Show your partner the mystery quadrilateral.
Switch roles and play again.
Student Response
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Activity Synthesis
“What shapes were the easiest to figure out and why?” (W was easy because it was so different with one angle pointing in. X was easy because it was the only square resting on a side.)
“What shapes were the most challenging to ask questions about and why?” (FF was challenging because none of the sides were the same length, and it was hard to get more information with “yes” or “no” questions. M and BB were hard to tell apart because they were so similar and it was hard to figure out what questions to ask.)


correct_option_index: 14