few_shots: Example:
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, in json format:
{
  "explanation": "This problem involves understanding fractions, multiplication, and division.",
  "answer": "A"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001097

7
options: Interpret the structure of expressions.

Reason about and solve one-variable equations and inequalities.

Write expressions in equivalent forms to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Use properties of operations to generate equivalent expressions.

Work with radicals and integer exponents.

Add and subtract within 20.

Understand the connections between proportional relationships, lines, and linear equations.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Understand properties of multiplication and the relationship between multiplication and division.

Multiply and divide within 100.

Understand solving equations as a process of reasoning and explain the reasoning.

Represent and solve problems involving addition and subtraction.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Write and interpret numerical expressions.

Analyze patterns and relationships.

Understand and apply properties of operations and the relationship between addition and subtraction.

Perform arithmetic operations on polynomials.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Rewrite rational expressions.

Represent and solve equations and inequalities graphically.

Represent and solve problems involving multiplication and division.

Gain familiarity with factors and multiples.

Solve equations and inequalities in one variable.

Create equations that describe numbers or relationships

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Work with equal groups of objects to gain foundations for multiplication.

Work with addition and subtraction equations.

Solve systems of equations.

Understand the relationship between zeros and factors of polynomials.

Generate and analyze patterns.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Use the four operations with whole numbers to solve problems.

Use polynomial identities to solve problems.
problem_activity: Problem 1
Expand each expression below as a product of two linear binomials and then multiply to write the product in standard form.
a.
$${(x-5)^2}$$
b.
$${(x+4)^2}$$
c.
$${(x-3)^2}$$
d.
$${(2x+1)^2}$$
e.
$${(3x-2)^2}$$
Describe any patterns you notice between the square of the linear binomial and the resulting quadratic trinomial.
Problem 2
Solve the quadratic equation and then sketch a graph of the parabola.
$${y=(x-4)^2}$$
Problem 3
What is the value of
$$c$$
in the equation below such that the quadratic equation is a perfect square trinomial?
$$y=x^2+12x+c$$


correct_option_index: 0

2