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:
Thought: 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: Write and interpret numerical expressions.

Add and subtract within 20.

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

Use properties of operations to generate equivalent expressions.

Create equations that describe numbers or relationships

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

Represent and solve problems involving addition and subtraction.

Rewrite rational expressions.

Use polynomial identities to solve problems.

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

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

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

Work with addition and subtraction equations.

Interpret the structure of expressions.

Represent and solve equations and inequalities graphically.

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

Multiply and divide within 100.

Solve systems of equations.

Generate and analyze patterns.

Gain familiarity with factors and multiples.

Understand the relationship between zeros and factors of polynomials.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use the four operations with whole numbers to solve problems.

Work with radicals and integer exponents.

Solve equations and inequalities in one variable.

Reason about and solve one-variable equations and inequalities.

Represent and analyze quantitative relationships between dependent and independent variables.

Perform arithmetic operations on polynomials.

Represent and solve problems involving multiplication and division.

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

Analyze patterns and relationships.

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

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

Write expressions in equivalent forms to solve problems.
problem_activity: Warm-up
Students reflect on equivalent expressions that represent the area of a shaded rectangle which is part of a larger rectangle of unknown width.
Launch
Allow students 2 minutes of quiet work time, followed by a whole-class discussion.
Student Facing
A rectangle with dimensions 6 cm and
\(w\)
cm is partitioned into two smaller rectangles.
Explain why each of these expressions represents the area, in cm
2
, of the shaded region.
\(6w-24\)
\(6(w-4)\)
[IMG]
Student Response
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Activity Synthesis
Ask students to share their reasoning.
\(6w-24\)
should be straighforward: the area of the entire rectangle is
\(6w\)
, and the area of the unshaded portion is
\(6 \boldcdot 4\)
or 24, so the area of the shaded portion is
\(6w-24\)
.
Students may have more trouble with
\(6(w-4)\)
. The key is to understand that the longer side of the shaded portion can be represented by
\(w-4\)
.
[IMG]


correct_option_index: 21