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

Draw informal comparative inferences about two populations.

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

Use random sampling to draw inferences about a population.

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

Use probability to evaluate outcomes of decisions.

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

Calculate expected values and use them to solve problems.

Summarize and describe distributions.

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

Understand and evaluate random processes underlying statistical experiments.

Develop understanding of statistical variability.

Investigate patterns of association in bivariate data.

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

Interpret linear models.
problem_activity: Optional activity
The activity provides further practice in finding probabilities of events.
In this activity, students see an experiment that has two steps where the result of the first step influences the possibilities for the second step. Often this process is referred to as doing something “without replacement.” At this stage, students should approach these experiments in a very similar way to all of the other probability questions they have encountered, but they must be very careful about the number of outcomes in the sample space (MP6).
Launch
Keep students in groups of 2. Give students 5–7 minutes of quiet work time followed by partner and whole-class discussion.
Identify students who are not noticing that it is impossible to draw the same color twice based on the instructions. Refocus these students by asking them to imagine drawing a red card on the first pick and thinking about what’s possible to get for the second card.
Representation: Access for Perception.
Provide access to concrete manipulatives. Provide five different colored cards for students to view or manipulate. These hands-on models will help students identify characteristics or features, and support finding outcomes for calculating probabilities.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing: MLR5 Co-Craft Questions.
Display the initial task statement that begins, “Imagine there are five cards...", before revealing the questions that follow. Ask pairs to write down possible mathematical questions that can be answered about the situation. As pairs share their questions with the class, listen for and highlight questions that ask how the “without putting the card back” part of the scenario would change the outcome of the sample space. This will help students consider the differences between this problem and previous problems and how that impacts the sample space.
Design Principle(s): Optimize output; Maximize meta-awareness
Student Facing
Imagine there are 5 cards. They are colored red, yellow, green, white, and black. You mix up the cards and select one of them without looking. Then, without putting that card back, you mix up the remaining cards and select another one.
Write the sample space and tell how many possible outcomes there are.
What structure did you use to write all of the outcomes (list, table, tree, something else)? Explain why you chose that structure.
What is the probability that:
You get a white card and a red card (in either order)?
You get a black card (either time)?
You do not get a black card (either time)?
You get a blue card?
You get 2 cards of the same color?
You get 2 cards of different colors?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
In a game using five cards numbered 1, 2, 3, 4, and 5, you take two cards and add the values together. If the sum is 8, you win. Would you rather pick a card and put it back before picking the second card, or keep the card in your hand while you pick the second card? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may misread the problem and think that they replace the card before picking the next one. Ask these students to read the problem more carefully and ask the student, “What is possible to get when you draw the second card while you already have a red card in your hand?”
Activity Synthesis
The purpose of the discussion is for students to compare the same context with replacement and without replacement.
Consider asking these questions for discussion:
“What would change about your calculations if the experiment required replacing the first card before picking a second card?” (There would be 25 outcomes in the sample space. The probability of getting the same color twice would be
\(\frac{5}{25}\)
. The probability of getting different colors would be
\(\frac{20}{25}\)
. The probability of getting red and white would be
\(\frac{2}{25}\)
. The probability of getting a black card would be
\(\frac{9}{25}\)
and not getting a black card would be
\(\frac{16}{25}\)
. It would still be impossible to get a blue card, so its probability would be 0.)
“What do you notice about the sum of the probability of getting a black card and the probability of not getting a black card?” (They have a sum of 1.)
“Explain why these outcomes might have probabilities with this relationship.” (Since you either get a black card or not, together their probabilities should be 1 or 100%.)


correct_option_index: 2