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.

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

Apply trigonometry to general triangles.

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

Understand and apply theorems about circles.

Reason with shapes and their attributes.

Use coordinates to prove simple geometric theorems algebraically.

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

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.

Prove theorems involving similarity.

Find arc lengths and areas of sectors of circles.

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

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

Apply geometric concepts in modeling situations.

Make geometric constructions.

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

Analyze, compare, create, and compose shapes.

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

Experiment with transformations in the plane.

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

Understand congruence in terms of rigid motions.

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

Explain volume formulas and use them to solve problems.

Prove geometric theorems.

Understand and apply the Pythagorean Theorem.
problem_activity: Narrative
The purpose of this activity is for students to identify the attributes that make a quadrilateral a rectangle, a rhombus, or a square. They do so by studying examples and non-examples, looking for features that each set has in common and drawing conclusions accordingly (MP7). The goal is not to craft the most precise definition for each, but to develop an understanding of the defining geometric attributes.
MLR8 Discussion Supports.
Display sentence frames to support small-group discussion: “I notice _____ so I think . . .”, “_____ and _____ are the same/alike because . . .”, “_____ and _____ are different because . . . .”
Advances: Conversing, Representing
Engagement: Provide Access by Recruiting Interest.
Optimize meaning and value. Invite students to share, with a partner, the attributes needed in order to be an example of this type of quadrilateral.
Supports accessibility for: Social-Emotional Functioning, Attention
Required Preparation
Create a chart with labels showing a rectangle, rhombus, and square for the lesson synthesis.
Launch
Groups of 2–3
Display the examples and non-examples of right triangles.
“The triangles on the left are right triangles. The ones on the right are not. Take a minute to think about their differences.”
“Then, choose all the shapes that are right triangles from the set labeled A–H. Write down what you think makes a shape a right triangle.”
1–2 minutes: quiet think time
1–2 minutes: partner discussion
Share responses.
“All triangles on the left and triangle E in the set are right angles. A triangle with a right angle is a right triangle.”
Activity
“Work with your group to analyze what makes a shape a rectangle, a rhombus, or a square. Each group will look at one type of quadrilateral.”
Assign each group one quadrilateral to analyze.
3–5 minutes: small-group work time
“Now, share your work with another group that worked on the same quadrilateral. Be sure to ask any questions you have about each other’s work.”
3–5 minutes: small-group discussion
Student Facing
These are right triangles.
[IMG]
These are not right triangles.
[IMG]
Which of the following are right triangles? Circle them.
[IMG]
What makes a shape a right triangle?
These are rectangles.
[IMG]
These are not rectangles.
[IMG]
Which of the following are rectangles? Circle them.
[IMG]
What makes a shape a rectangle?
These are rhombuses.
[IMG]
These are not rhombuses.
[IMG]
Which of the following are rhombuses? Circle them.
[IMG]
What makes a shape a rhombus?
These are squares.
[IMG]
These are not squares.
[IMG]
Which of the following are squares? Circle them.
[IMG]
What makes a shape a square?
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
Invite 1–2 groups who analyzed rectangles to share their responses. Record their responses.
Ask the class:
“Do you agree with the shapes that are marked as rectangles (or rhombuses or squares)?”
“Do you agree with the descriptions?”
“Is there any part of the descriptions you’d say differently?”
“Is there anything you'd add to the description?”
When students have finished describing rectangles, highlight that: “A shape is a rectangle only if it is a quadrilateral with only right angles, and the sides that are across from each other are the same length.”
Repeat the discussion with rhombuses and squares.


correct_option_index: 5