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.


Your response:
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.


Your response:
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.


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

Reason with shapes and their attributes.

Understand and apply the Pythagorean Theorem.

Understand and apply theorems about circles.

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

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

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

Understand similarity in terms of similarity transformations.

Experiment with transformations in the plane.

Explain volume formulas and use them to solve problems.

Use coordinates to prove simple geometric theorems algebraically.

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

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

Apply geometric concepts in modeling situations.

Apply trigonometry to general triangles.

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

Make geometric constructions.

Analyze, compare, create, and compose shapes.

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

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

Prove theorems involving similarity.

Define trigonometric ratios and solve problems involving right triangles.

Find arc lengths and areas of sectors of circles.

Understand congruence in terms of rigid motions.

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

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

Prove geometric theorems.
problem_activity: Activity
The purpose of this activity is for students to practice solving problems that involve volumes of pyramids and cones. Monitor for various problem-solving strategies—in particular, for the problem in which students find the radius or height of a cone given a particular volume. Some students may substitute values into a formula and solve the resulting equation, and others may follow a process of deductive reasoning.
Launch
Provide students access to scientific calculators. Arrange students in groups of 2. Ask students to discuss their thinking with their partner, and if they disagree, work to reach an agreement.
Representation: Internalize Comprehension.
Activate or supply background knowledge about the formulas for the area of a triangle, square, and circle. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Calculate the volume of each solid. Round your answers to the nearest tenth if necessary.
A
[IMG]
B
[IMG]
C
[IMG]
D: height 12 cm; area of base 32 cm
2
[IMG]
A particular cone has radius
\(r\)
and height
\(h\)
.
Write an expression for the volume of this cone in terms of
\(r\)
and
\(h\)
.
What is the height of a cone whose volume is
\(16\pi\)
cubic units and whose radius is 3 units?
What is the radius of a cone whose volume is
\(16\pi\)
cubic units and whose height is 3 units?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
The Pyramid of Giza is 455 feet tall. The base is square with a 756-foot side length. How many Olympic-size swimming pool volumes of water can fit inside the Pyramid of Giza? Explain or show your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
For the pyramid with a triangular base, students may forget that the triangle area formula involves dividing by 2, resulting in a base area of 12 square units. Prompt them to draw and label the base prior to calculating its area.
Activity Synthesis
Ask students to share their strategies for volume calculations. For example, for the cone, did they calculate the area of the base first, and substitute it into the formula
\(V=\frac13 Bh\)
? Or, perhaps they wrote out
\(V=\frac13 \pi r^2 h\)
and performed the calculations all at once.
Select previously identified students to share their thought process on the “working backward” cone problem. If possible, find one student who solved an equation algebraically and another who used backward reasoning rather than formally solving an equation.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to enhance the whole-class discussion about strategies for volume calculation. After students calculate the volume of each solid and determine the radius or height of each cone, invite them to create a visual display of their work for one of the solids. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their strategies. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast strategies for volume calculation.
Design Principle(s): Cultivate conversation


correct_option_index: 9