few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
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Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Your response:
{
  "explanation": "This task teaches students to understand addition as putting together counters, or adding to already placed counters.",
  "answer": "AB"
}

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Your response:
{
  "explanation": "This problem involves understanding fractions, multiplication, and division.",
  "answer": "A"
}

Example 3:
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.


Your response:
{
  "explanation": "This problem applies trigonometric identities, including the angle addition formula for cosine.",
  "answer": "A"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Summarize, represent, and interpret data on two categorical and quantitative variables.

Interpret linear models.

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

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

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

Investigate patterns of association in bivariate data.

Use probability to evaluate outcomes of decisions.

Summarize and describe distributions.

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

Calculate expected values and use them to solve problems.

Understand and evaluate random processes underlying statistical experiments.

Draw informal comparative inferences about two populations.

Develop understanding of statistical variability.

Use random sampling to draw inferences about a population.

Summarize, represent, and interpret data on a single count or measurement variable.
problem_activity: Activity
The mathematical purpose of this activity is for students to describe how two variables are related, and to determine whether or not there is a
causal relationship
. Students should begin to recognize that some variables may be related, but one does not cause the other to change. At this point, the mathematics of scatter plot analysis cannot decide whether there is a causal relationship. The relationship must be thought through carefully to decide, based on the situation, whether the related variables have a causal relationship. As students analyze the relationships, they are modeling with mathematics (MP4).
Launch
Arrange students in groups of 2. Tell students there are many possible answers for the questions. After quiet work time, ask students to compare their responses to their partner’s and decide if they are both correct, even if they are different. Follow with a whole-class discussion.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. After the first question invite students to suggest language or diagrams to include that will support their understanding of causal relationships. Examples to include can be phrases, such as, “increased rain causes people to wear jackets.” The two variables can be color-coded in two different colors to reveal the relationship. Both the axes and the labels can be highlighted with the color that connects it to the type of variable it addresses.
Supports accessibility for: Conceptual processing; Language
Student Facing
Each of the scatter plots show a strong relationship. Write a sentence or two describing how you think the variables are related.
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During the month of April, Elena keeps track of the number of inches of rain recorded for the day and the percentage of people who come to school with rain jackets.
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A school book club has a list of 100 books for its members to read. They keep track of the number of pages in the books the members read from the list and the amount of time it took to read the book.
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Number of tickets left for holiday parties at a venue and noise level at the party.
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The height and score on a test of vocabulary for several children ages 6 to 13.
Student Response
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Activity Synthesis
The goal is for students to develop an understanding of what it means for a relationship between two variables to be a causal relationship.
Select several students to share their reasoning for the relationship between the variables. For each pair of variables, ask students what might have
caused
the variables to be linked. In some cases (the first two here), one of the variables causes a change in the other. In other cases (the last two here), an additional variable or situation is the cause of the change.
Tell students that, since most people are used to seeing independent variables on the
\(x\)
-axis and dependent variables on the
\(y\)
-axis, the convention is to put the causal variable on the
\(x\)
-axis and the other variable on the
\(y\)
-axis when one of the variables does cause the other to change. The book club scatter plot should probably have the axes switched to meet the convention.
Here are some questions for discussion.
“Why does an increase in precipitation cause an increase in the percentage of people wearing rain jackets?” (When it rains, people usually wear coats to stay dry.)
“An increase in the time it takes to read a book does not cause the number of pages in the book to increase. Does an increase in the number of pages in a book cause the time it takes to read a book to increase? Explain your reasoning.” (Yes, if you switch the axes on the graph, you can see that relationship. It is causal because it takes longer to read more pages.)
“The relationship in the height and test score graphs appears to be strong. Does an increase in height cause an increase in test scores?” (It is not the height that causes the increase in test score, but probably the age or grade level of the students might be causing the increase. Age and grade level are positively related to height, so that likely explains the relationship seen between the two variables.)


correct_option_index: 1