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: Analyze, compare, create, and compose shapes.

Find arc lengths and areas of sectors of circles.

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

Make geometric constructions.

Understand congruence in terms of rigid motions.

Prove theorems involving similarity.

Apply trigonometry to general triangles.

Explain volume formulas and use them to solve problems.

Understand similarity in terms of similarity transformations.

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

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

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

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.

Define trigonometric ratios and solve problems involving right triangles.

Experiment with transformations in the plane.

Use coordinates to prove simple geometric theorems algebraically.

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

Reason with shapes and their attributes.

Prove geometric theorems.

Apply geometric concepts in modeling situations.

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

Understand and apply the Pythagorean Theorem.

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

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

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

Understand and apply theorems about circles.
problem_activity: Warm-up
In this warm-up, students compare and contrast two ways of decomposing and rearranging a parallelogram on a grid such that its area can be found. It serves a few purposes: to reinforce the work done in the previous lesson; to allow students to practice communicating their observations; and to shed light on the features of a parallelogram that are useful for finding area—a base and a corresponding height.
The flow of key ideas—to be uncovered during discussion and gradually throughout the lesson—is as follows:
There are multiple ways to decompose a parallelogram (with one cut) and rearrange it into a rectangle whose area we can determine.
The cut can be made in different places, but to compose a rectangle, the cut has to be at a
right angle
to two opposite sides of the parallelogram.
The length of one side of this newly composed rectangle is the same as the length of one side of the parallelogram. We use the term
base
to refer to this side.
The length of the other side of the rectangle is the length of the cut we made to the parallelogram. We call this segment a
height
that corresponds to the chosen base.
We use these two lengths to determine the area of the rectangle, and thus also the area of the parallelogram.
As students work and discuss, identify those who make comparisons in terms of the first two points so they could share later. Be sure to leave enough time to discuss the first four points as a class.
Digital
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet think time and access to geometry toolkits. Ask them to share their responses with a partner afterwards.
If using the digital activity, have students explore the applet for 3 minutes individually, and then discuss with a partner. Students will be able to see the cuts by dragging the point on the segment under the parallelogram.
Student Facing
Elena and Tyler were finding the area of this parallelogram:
[IMG]
Move the slider to see how Tyler did it:
Move the slider to see how Elena did it:
How are the two strategies for finding the area of a parallelogram the same? How they are different?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet think time and access to geometry toolkits. Ask them to share their responses with a partner afterwards.
If using the digital activity, have students explore the applet for 3 minutes individually, and then discuss with a partner. Students will be able to see the cuts by dragging the point on the segment under the parallelogram.
Student Facing
Elena and Tyler were finding the area of this parallelogram:
[IMG]
Here is how Elena did it:
[IMG]
Here is how Tyler did it:
[IMG]
How are the two strategies for finding the area of a parallelogram the same? How they are different?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Ask a few students to share what was the same and what was different about the methods they observed.
Highlight the following points on how Elena and Tyler's approaches are the same, though do not expect students to use the language. Instead, rely on pointing and gesturing to make clear what is meant. If any of these are not mentioned by the students, share them.
The rectangles are identical; they have the same side lengths. (Label the side lengths of the rectangles.)
The cuts were made in different places, but the length of the cuts was the same. (Label the lengths along the vertical cuts.)
The horizontal sides of the parallelogram have the same length as the horizontal sides of the rectangle. (Point out how both segments have the same length.)
The length of each cut is also the distance between the two horizontal sides of the parallelogram. It is also the vertical side length of the rectangle. (Point out how that distance stays the same across the horizontal length of the parallelogram.)
[IMG]
Begin to connect the observations to the terms
base
and
height
. For example, explain:
“The two measurements that we see here have special names. The length of one side of the parallelogram—which is also the length of one side of the rectangle—is called a
base
. The length of the vertical cut segment—which is also the length of the vertical side of the rectangle—is called a
height
that corresponds to that base.”
“Here, the side of the parallelogram that is 7 units long is also called a base. In other words, the word
base
is used for both the segment and the measurement.”
Tell students that we will explore bases and heights of a parallelogram in this lesson.


correct_option_index: 17