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: Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

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

Use functions to model relationships between quantities.

Construct and compare linear and exponential models and solve problems.

Build a function that models a relationship between two quantities.

Define, evaluate, and compare functions.

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

Understand the concept of a function and use function notation.

Prove and apply trigonometric identities.

Build new functions from existing functions.

Analyze functions using different representations.
problem_activity: Activity
This activity presents students with two graphs without numerical values assigned to any points on the graphs. The only labeled point is where the two graphs intersect. Students will need to use their understanding that tripling each day is a greater growth factor than doubling each day. This is their only means to decide which function represents which situation. Students then interpret the graphs in terms of the situations that they represent.
Launch
Display the graph for all to see. Give students a moment to observe the graph and ask, "What do you notice? What do you wonder?" Invite students to share their observations and questions.
Student Facing
Here are graphs representing two functions, and descriptions of two functions.
[IMG]
Function
\(f\)
: The area of a wall that is covered by Mold A, in square inches, doubling every month.
Function
\(g\)
: The area of a wall that is covered by Mold B, in square inches, tripling every month.
Which graph represents each function? Label the graphs accordingly and explain your reasoning.
When the mold was first spotted and measured, was there more of Mold A or Mold B? Explain how you know.
What does the point
\((p, q)\)
tell us in this situation?
Student Response
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Anticipated Misconceptions
Some students may think that the dashed graph represents the mold that is tripling because it has a greater vertical intercept. Ask these students to think carefully about the second question.
Activity Synthesis
Select students to share their responses and explanations. Focus on how they deciphered the meanings of the graphs. Discuss questions such as:
The solid graph started off lower than the dashed graph. What does that mean in this context? (The mold population represented by the solid graph is initially smaller or covers a smaller area on the wall than the one represented by the dashed graph.)
After some point, the solid graph passes the dashed one. What does that mean in this context? (The mold population that started out covering a greater area on the wall is surpassed by the other population, so it is growing at a slower rate.)
What does the intersection of the two graphs mean? What do
\(p\)
and
\(q\)
represent? (The two mold populations are the same at time
\(p\)
with population covering an area of
\(q\)
square inches.)
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each explanation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 7