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
$$-\frac{1}{4}$$


Topic options: 
A. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
D. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
E. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
F. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
G. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
H. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
I. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
J. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
K. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.


Your response:
Thought: This problem teaches students different ways of writing numbers that express the same value. This helps students better understand the complex number system, which includes rational numbers.
Answer: I

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: fl_problem_001097

7
options: Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
problem_activity: Narrative
The purpose of this activity is for students to solve a contextual problem about dividing a fractional amount by a whole number. Students draw a diagram to represent the situation and relate the diagram to a division expression. Because of earlier work in this unit, students may draw one of the familiar square area diagrams showing the product
\(\frac{1}{3} \times \frac{1}{2}\)
. Other students may make a diagram resembling the macaroni and cheese pan and divide it appropriately. The focus in the synthesis is on bringing out how the diagram shows
\(\frac{1}{2} \div 3\)
and how it allows students to answer the question. The relationship between
\(\frac{1}{3} \times \frac{1}{2}\)
and
\(\frac{1}{2} \div 3\)
will be brought out in later lessons. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).
This activity uses
MLR7 Compare and Connect.
Advances: Representing, Conversing.
Representation: Develop Language and Symbols.
Represent the problem in multiple ways to support understanding of the situation. For example, a cut-out or another physical manipulative to represent the pan of macaroni and distribute the “macaroni” to three students or a video that shows the distribution of macaroni and cheese between three people.
Supports accessibility for: Attention, Conceptual Processing
Launch
Groups of 2
Display and read: “Last night, Jada’s aunt baked a pan of macaroni and cheese for dinner. Today, she brought the leftovers to Jada’s home for Jada and her sisters to share.”
“What do you notice? What do you wonder?” (We solved problems about macaroni and cheese before. I wonder how much macaroni and cheese Jada’s aunt brought.)
1–2 minutes: partner discussion
Activity
1–2 minutes: quiet think time
6–8 minutes: partner work time
Monitor for students who:
draw diagrams like the ones in the student responses
explain the situation as
\(\frac {1}{2}\)
divided into 3 equal pieces
recognize each person will get
\(\frac{1}{6}\)
of the whole pan of macaroni and cheese
Student Facing
Jada and her 2 sisters equally share
\(\frac{1}{2}\)
a pan of macaroni and cheese.
Draw a diagram to represent the situation.
Explain how this expression represents the situation:
\(\frac {1}{2} \div 3\)
How much of the whole pan of macaroni and cheese will each person get?
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
MLR7 Compare and Connect
“Create a visual display that shows your thinking about the problems. You may want to include details such as notes, diagrams, drawings, etc., to help others understand your thinking.”
2–5 minutes: independent or group work
5–7 minutes: gallery walk
“How does each representation show
\(\frac {1}{2}\)
a pan of macaroni and cheese?”
“How does each representation show 3 equal pieces?”
30 seconds: quiet think time
1 minute: partner discussion
“How do we know that each person got the same amount of macaroni and cheese?” (I divided the half into 3 equal shares.)
“How much of the whole pan of macaroni and cheese did each person get?” (
\(\frac {1}{6}\)
)
“How do the diagrams show
\(\frac {1}{6}\)
?” (Each
\(\frac {1}{2}\)
of the pan is divided into 3 equal pieces, so each of those pieces is
\(\frac {1}{6}\)
of the whole pan.)


correct_option_index: 6