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 ratio concepts and use ratio reasoning to solve problems.

Analyze proportional relationships and use them to solve real-world and mathematical problems.
problem_activity: Activity
In this activity, students are given a percent increase and use it to calculate the new value, rather than being given the original and new values to calculate the percent increase. Students can solve the problems using the double number lines but the discussion that follows will be the connection to what the words specifically mean (MP6).
As students work, monitor for students who complete the double number line diagram and any student that uses other methods to correctly reason about the problem.
Launch
Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by partner then whole-class discussion.
Student Facing
​​
A cereal box says that now it contains 20% more. Originally, it came with 18.5 ounces of cereal. How much cereal does the box come with now?
[IMG]
​​
The price of a shirt is \$18.50, but you have a coupon that lowers the price by 20%. What is the price of the shirt after using the coupon?
[IMG]
Student Response
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Anticipated Misconceptions
Have students use the double number line diagram if they need help figuring out 20% more.
Activity Synthesis
Select students to share their reasoning for each problem. Start with a student that used the double number line diagram to solve the problem. Then, have students share other methods they used to solve the problem, such as multiplying by 1.2 and 0.8.
Ask students:
Did the number of ounces of cereal in the cereal box increase or decrease?
What percentage of the original amount of cereal is the new amount of cereal?
Did the price of the shirt increase or decrease?
What percentage of the original price of the shirt is the new price of the shirt?
Tell students that the change on the cereal box is an example of a
percent increase
. The discount on the shirt is an example of a
percent decrease
. You might want to mention that in both cases, 100% always corresponds to the original amount before the change; however, future activities will address this concept in depth.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of percent increase and percent decrease.
Supports accessibility for: Conceptual processing; Language
Writing: MLR3 Clarify, Critique, Correct.
Before students share their reasoning for each problem, present an incomplete strategy for finding the discount price for the shirt. For example, “The discount is 20%, so the price of the shirt is $3.70.” Ask students to critique the reasoning, and work with a partner to write an improved explanation. Listen for and amplify the language students use to make sense of what is displayed (e.g., “I think they ___ because ____.”), as well as the mathematical language that students use that strengthen their explanations. This will support student understanding of mathematical language related percent increase and percent decrease.
Design Principle(s): Maximize meta-awareness


correct_option_index: 1