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: Summarize and describe distributions.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Develop understanding of statistical variability.

Draw informal comparative inferences about two populations.

Understand independence and conditional probability and use them to interpret data.

Investigate patterns of association in bivariate data.

Calculate expected values and use them to solve problems.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Understand and evaluate random processes underlying statistical experiments.

Interpret linear models.

Summarize, represent, and interpret data on a single count or measurement variable.

Investigate chance processes and develop, use, and evaluate probability models.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Use random sampling to draw inferences about a population.

Use probability to evaluate outcomes of decisions.
problem_activity: Activity
Earlier, in the backpack example, students saw a distribution described in terms of where data points are clustered on a dot plot and which values have a large number of occurrences. The shape of that distribution was approximately symmetric. In this activity, they continue to analyze distributions in those terms, and try to identify and interpret the
center
and
spread
of a distribution that is not symmetric. The two distributions used here allow students to contrast a narrow spread and a wide spread and develop a deeper understanding of variability.
As students work, notice how students identify a general location for the center of a data set and the descriptions they use to talk about the spread (e.g., “wide,” “narrow,” or “something in between”). Identify students who connect the size of a spread to how different or alike the data points are; ask them to share later. Additionally, identify students who measure spread as the range of the entire data as well as those who use the distance to the center.
Launch
Arrange students in groups of 2. Give students 5–6 minutes of quiet work time for the first three sets of questions, and another 4–5 minutes to share their responses and then discuss the last two questions with a partner.
Students are asked to find a percentage. If necessary, briefly review how to find a percentage.
Representation: Internalize Comprehension.
Activate or supply background knowledge about calculating percentages. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Representing, Writing: MLR5 Co-Craft Questions.
Invite pairs of students to create one or two mathematical questions that could be answered by the data displayed in the dot plot. Note questions that vary in complexity, making sure to have students share examples that ask about percent, center, or spread, if available. Allow students to ask clarifying questions of their peers regarding how the dot plot could be used to answer the questions, if needed. This will help students to make sense of the data using informal language prior to connecting their existing understanding to formal language.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
Twenty-five sixth-grade students were asked to estimate how many hours a week they spend talking on the phone. This dot plot represents their reported number of hours of phone usage per week.
[IMG]
How many of the students reported not talking on the phone during the week? Explain how you know.
What percentage of the students reported not talking on the phone?
What is the largest number of hours a student spent talking on the phone per week?
What percentage of the group reported talking on the phone for this amount of time?
How many hours would you say that these students typically spend talking on the phone?
How many minutes per day would that be?
How would you describe the
spread
of the data? Would you consider these students’ amounts of time on the phone to be alike or different? Explain your reasoning.
Here is the dot plot from an earlier activity. It shows the number of hours per week the same group of 25 sixth-grade students reported spending on homework.
[IMG]
Overall, are these students more alike in the amount of time they spend talking on the phone or in the amount of time they spend on homework? Explain your reasoning.
Suppose someone claimed that these sixth-grade students spend too much time on the phone. Do you agree? Use your analysis of the dot plot to support your answer.
Student Response
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Anticipated Misconceptions
Students may neglect to change the rate given (from minutes per week to hours per week, or to minutes per day) and may draw incorrect conclusions as a result. Ask them to think about the unit they are using in their responses.
Activity Synthesis
The purpose of the discussion is to help students find good ways to describe a distribution based on center and spread.
Select a few students to answer the questions about the first dot plot. Tell students that the “typical” value for the data is generally considered the center. Ask, “What would you consider the center for the two dot plots shown in this activity?”
Ask students how they thought about the spread of the data. If possible, select students who thought of spread as the range of the entire data and those who thought of it as an interval around the center. Ask students to share their interpretation of the what the spread means in the context of using the phone. Make sure to include previously identified students who connect spread to how alike or different the data points are.
Tell students that distributions are generally described using the center and spread. Select a few students to describe the distributions of the two data sets shown in this activity.


correct_option_index: 0

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