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: Model periodic phenomena with trigonometric functions.

Analyze functions using different representations.

Build a function that models a relationship between two quantities.

Define, evaluate, and compare functions.

Use functions to model relationships between quantities.

Build new functions from existing functions.

Construct and compare linear and exponential models and solve problems.

Prove and apply trigonometric identities.

Interpret functions that arise in applications in terms of the context.

Extend the domain of trigonometric functions using the unit circle.

Interpret expressions for functions in terms of the situation they model.

Understand the concept of a function and use function notation.
problem_activity: Activity
In the previous lesson, students learned the definition of amplitude and midline by examining sine graphs produced to model the vertical position of a point on a rotating windmill. This activity keeps the windmill context and asks students to interpret both cosine and sine equations. They will need to focus on three different aspects of the equations:
The amplitude of the sine graph (length of the windmill blade)
The midline of the sine graph (height of the center of the windmill blades)
The direction the windmill spins (clockwise or counterclockwise).
Launch
Arrange students in groups of 2. Display an image of the blades of a windmill for all to see, such as this one from a previous lesson:
[IMG]
Tell students that these blades extend 4 meters from the center and that the center of the windmill is 20 meters off the ground. The windmill rotates counterclockwise.
Ask students to write an equation to represent the height in meters of the point
\(W\)
, relative to the ground, as a function of the angle
\(\theta\)
of rotation (
\(h = 4\sin(\theta) + 20\)
). Select 2–3 students to share how they wrote their equation. Tell them that in this activity they are going to work the opposite direction. That is, they are going to start with an equation giving the vertical position of a point
\(W\)
at the tip of a windmill blade and use it to describe the windmill.
Conversing: MLR8 Discussion Supports.
Use this routine to help students describe the three different windmills based on their equations. In their groups of 2, invite Partner A to begin with this sentence frame: “The windmill _____ because . . .”  and “The positive of negative sign tells me _____ because . . . .” Invite the listener, Partner B, to press for additional details and ask clarifying questions, such as “What is the amplitude of the windmill?” and “What is the midline of the windmill?” Students should switch roles for each equation. This will help students interpret sine equations in the context of a windmill by referring to specific features of the equations.
Design Principle(s): Support sense-making; Cultivate conversation
Representation: Access for Perception.
Provide students with a physical copy of the image displayed and information provided during the Launch. Check for understanding by inviting students to rephrase the information in their own words. Invite students to label the image based on the information provided. Keep the display of the image and information visible throughout the activity.
Supports accessibility for: Language; Memory
Student Facing
Here are three equations for three different windmills. Each equation describes the height
\(h\)
, in feet above the ground, of a point at the tip of a blade of a different windmill. The point is at the far right when the angle
\(\theta\)
takes the value 0. Describe each windmill and how it is spinning.
\(h = 2.5 \sin(\theta) + 10\)
\(h = \frac{4}{5} \sin(\theta) + 3\)
\(h = \text-1.5 \sin(\theta) + 5\)
Student Response
Teachers with a valid work email address can
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Anticipated Misconceptions
If students struggle with the negative amplitude in the final question, encourage them to substitute in a few angle values
\(\theta\)
and plot them.
Activity Synthesis
Display the 3 equations for all to see. Begin the discussion by asking, “What is the midline and amplitude of each function and what do these values mean in this context?” (The midline is the height of windmill blades’ center and the amplitude is the length of the blades.) Record student responses next to the equations.
Ask students how they interpreted the sign in front of the sine on the last equation (it means that the windmill is spinning in the clockwise direction as the angle
\(\theta\)
increases). To help students visualize the impact of the negative sign in front of the expression, note that while
\(\sin(\theta)\)
increases from 0 to 1 as
\(\theta\)
increases from 0 to
\(\frac{\pi}{2}\)
,
\(\text-\sin(\theta)\)
decreases from 0 to -1 as
\(\theta\)
increases from 0 to
\(\frac{\pi}{2}\)
. In terms of the windmill, this would mean that instead of going toward the highest spot on its circle of rotation, the point
\(W\)
begins by going to the lowest spot on its circle of rotation. This means that the windmill is rotating in a clockwise direction instead of a counterclockwise direction.


correct_option_index: 0