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:
{
  "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: Represent and solve problems involving multiplication and division.

Write expressions in equivalent forms to solve problems.

Work with addition and subtraction equations.

Reason about and solve one-variable equations and inequalities.

Use the four operations with whole numbers to solve problems.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Create equations that describe numbers or relationships

Represent and solve problems involving addition and subtraction.

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

Use polynomial identities to solve problems.

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

Solve systems of equations.

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

Write and interpret numerical expressions.

Add and subtract within 20.

Perform arithmetic operations on polynomials.

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

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

Rewrite rational expressions.

Solve equations and inequalities in one variable.

Understand the relationship between zeros and factors of polynomials.

Work with radicals and integer exponents.

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

Gain familiarity with factors and multiples.

Use properties of operations to generate equivalent expressions.

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

Generate and analyze patterns.

Represent and analyze quantitative relationships between dependent and independent variables.

Represent and solve equations and inequalities graphically.

Multiply and divide within 100.

Interpret the structure of expressions.

Analyze patterns and relationships.

Understand the connections between proportional relationships, lines, and linear equations.
problem_activity: Warm-up
The purpose of an Estimation warm-up is to practice the skill of estimating a reasonable answer based on experience and known information, and also help students develop a deeper understanding of the meaning of standard units of measure. It gives students a low-stakes opportunity to share a mathematical claim and the thinking behind it (MP3). Asking yourself, “Does this make sense?” is a component of making sense of problems (MP1), and making an estimate or a range of reasonable answers with incomplete information is part of modeling with mathematics (MP4).
Launch
Display the image for all to see. Ask students to silently think of a number they are sure is too low, a number they are sure is too high, and a number that is about right, and write these down. Then, write a short explanation for the reasoning behind their estimate.
Student Facing
[IMG]
The book in the center is 276 pages long. What is the mean number of pages of the 3 books?
Record an estimate that is:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "too low"}, {"_attributes": {"scope": "col"}, "_value": "about right"}, {"_attributes": {"scope": "col"}, "_value": "too high"}]}]}], "tbody": [{"tr": [{"td": [{}, {}, {}]}]}]}]}
Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select a few students to share their estimate and their reasoning. If a student is reluctant to commit to an estimate, ask for a range of values. Display these for all to see in an ordered list or on a number line. Add the least and greatest estimate to the display by asking, “Is anyone’s estimate less than
\(\underline{\hspace{.5in}}\)
? Is anyone’s estimate greater than
\(\underline{\hspace{.5in}}\)
?” If time allows, ask students, “Based on this discussion, does anyone want to revise their estimate?”
Then, reveal the actual value and add it to the display. The books are 830, 276, and 32 pages long, so the average number of pages in the books is about 379 pages. Ask students how accurate their estimates were, as a class.
“Is the actual value inside the range of values?”
“Is 379 near the middle of the estimates?”
“How variable are the estimates?”
“What are the sources of errors?”
Consider developing a method to record a snapshot of the estimates and actual value so students can track their progress as estimators over time.


correct_option_index: 4