few_shots: Example:
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.


Your response:
{
  "explanation": "This problem applies trigonometric identities, including the angle addition formula for cosine.",
  "answer": "A"
}

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

HS
options: Reason with shapes and their attributes.

Analyze, compare, create, and compose shapes.

Make geometric constructions.

Understand similarity in terms of similarity transformations.

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

Explain volume formulas and use them to solve problems.

Apply trigonometry to general triangles.

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

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

Understand congruence in terms of rigid motions.

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

Prove theorems involving similarity.

Apply geometric concepts in modeling situations.

Define trigonometric ratios and solve problems involving right triangles.

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

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

Understand and apply the Pythagorean Theorem.

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

Experiment with transformations in the plane.

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

Understand and apply theorems about circles.

Use coordinates to prove simple geometric theorems algebraically.

Prove geometric theorems.

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

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

Find arc lengths and areas of sectors of circles.

Visualize relationships between two-dimensional and three-dimensional objects.
problem_activity: Activity
In the last activity, students started with solids and identified various cross sections. In this activity, students view three-dimensional slabs of a solid between parallel cross sections and try to determine what the original solid was. Being able to visualize the relationship between a solid and its cross sections is important to later work on Cavalieri’s Principle.
Launch
Ask students, “What solid would a stack of all the same coins create?” Display a stack of quarters and note that it creates the shape of a cylinder. Then display, in order, a quarter, a nickel, a penny, and a dime. Ask, “What solid would a stack of coins decreasing in size create?” Make a stack with a few of each type of coin to make a solid that resembles a cone.
Student Facing
Each question shows several parallel cross-sectional slabs of the same three-dimensional solid. Name each solid.
[IMG]
[IMG]
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
3D-printers stack layers of material to make a three-dimensional shape. Computer software slices a digital model of an object into layers, and the printer stacks those layers one on top of another to replicate the digital model in the real world.
[IMG]
Draw 3 different horizontal cross sections from the object in the image.
The layers can be printed in different thicknesses. How would the thickness of the layers affect the final appearance of the object?
Suppose we printed a rectangular prism. How would the thickness of the layers affect the final appearance of the prism?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Ask students to share their predictions for what solids are formed. Then display these images for all to see.
[IMG]
[IMG]
[IMG]
[IMG]
[IMG]
[IMG]
[IMG]
[IMG]
[IMG]
Now focus students’ attention on cross sections that are taken parallel to a solid’s base (for those solids that have bases). Ask students how cross sections can be used to differentiate between prisms and pyramids. (The cross sections of prisms taken parallel to the base are congruent to each other. The cross sections of pyramids taken parallel to the base are similar to each other.)
Speaking: MLR8 Discussion Supports.
To help students respond to the questions for discussion, provide sentence frames such as: “The cross sections of _____ taken parallel to the base are _____ because.…” As students share their responses, press for details by asking students how they know that the cross sections of prisms taken parallel to the base are congruent, while the cross sections of pyramids taken parallel to the base are similar. Ask students to distinguish the meanings of geometric congruence and similarity.
Design Principle(s): Support sense-making; Optimize output (for justification)
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “I noticed _____ so I….”, “How do you know…?”, “That could/couldn’t be true because….”, and “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills


correct_option_index: 26