few_shots: Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Topic options: 
A. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
D. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
E. 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.
F. 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.
G. 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.
H. 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.
I. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
J. 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.
K. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.


Thought: This problem asks students to reason about ratios of butter amounts to sugar amounts.
Answer: D

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


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.


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

Example 3:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Topic options: 
A. 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.
B. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
C. 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.
D. 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.
E. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
F. 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.
G. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
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. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
K. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.


Thought: This set of problems asks students to reason with equations, specifically linear equations, such as the one shown in the table in Problem 1.
Answer: F

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: 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.

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.

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

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

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

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.

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

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

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

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.
problem_activity: Narrative
The purpose of this activity is for students to solve multi-step distance problems using centimeters, meters, and kilometers. This gives students an opportunity to think about which units are most helpful for communicating a distance (MP6). When the distance is short, like the length of a single footstep, centimeters or meters both work well. For a longer distance, like the distance a person walks in a day, it is reasonable to use meters or kilometers, but the number of centimeters is very large and more difficult to visualize.
To add movement or make this activity interactive, consider providing groups of 2 or 4 with a centimeter ruler or meter stick to measure their or a classmate’s step before working on the task. While students could use the measurements of their own steps to complete the table, the arithmetic may be more complex and, as a result, it may be harder to observe patterns.
Engagement: Provide Access by Recruiting Interest.
Optimize meaning and value. Invite students to share the places they could walk in the classroom, at home, in the community, and so on, that would be equivalent to the distances presented in the chart.
Supports accessibility for: Conceptual Processing; Visual-Spatial Processing
Required Materials
Materials to Gather
Metersticks
Launch
Groups of 2 or 4
“If someone wants to measure the length of one of their footsteps, which unit do you think they should use? Millimeters, centimeters, meters, or kilometers?”
Highlight that millimeters are too small and kilometers are too large. Both centimeters and meters make sense and the first part of this activity will use those two units of measure.
“There are 100 centimeters in a meter. About how many steps are in 1 meter?” (2 or 3)
“There are 1,000 meters in a kilometer. About how many steps are in a kilometer?” (2,000 or 3,000)
Give students access to a meter stick.
Activity
5 minutes: independent work time
5 minutes: small-group work time
Monitor for students who use the following strategies when determining how many kilometers Lin walks during the day:
multiply 50 centimeters by 8,500 steps to determine the distance in centimeters that Lin walked and divide 425,000 by 100,000
multiply 0.5 meters by 8,500 steps to determine the distance in meters that Lin walked and then divide 4,250 by 1,000
Student Facing
Lin has a watch that counts the number of steps she takes during the day and displays those steps in centimeters, meters, or kilometers.
Here is a list of activities Lin did on Monday. Next to each activity, write whether it would make sense to display the distance in cm, m, or km.
walked to her friend’s desk
walked to the front of the classroom
walked from her classroom to the bus
ran twice around the playground
The table shows the amount of steps Lin’s watch displayed for each activity. If each of her steps is 50 centimeters, how many centimeters and meters did Lin walk for each activity?
|               activity               | number of steps | distance (cm) | distance (m) |
|--------------------------------------|-----------------|---------------|--------------|
|     walked to her friend’s desk      |        5        |               |              |
| walked to the front of the classroom |       12        |               |              |
| walked from her classroom to the bus |       250       |               |              |
|   ran twice around the playground    |      1,000      |               |              |
At the end of the day, Lin’s watch displayed 8,500 steps. Would it make sense for her watch to record the distance in centimeters, meters, or kilometers? Why?
How many kilometers did Lin walk that day?
[IMG]
Student Response
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Activity Synthesis
“How did you determine how many kilometers Lin walked during the day?”
Ask previously selected students to share their solutions.
Display student work or write these equations for all to see:
\(8,\!500 \times 50= 425,\!000\)
\(8,\!500 \times 0.5 = 4,\!250\)
“How does each of these equations represent the situation?” (
\(8,\!500 \times 50= 425,\!000\)
represents 8,500 steps that are each 50 centimeters long so that would be a total of 425,000 centimeters.
\(8,\!500 \times 0.5 = 4,\!250\)
represents 8,500 steps that are each 0.5 meter long so that would be a total of 4,250 meters.)
“What is the same about these equations?” (They have the same number of steps. They are both multiplication equations.)
“What is different about these equations?” (One multiplies the number of steps by 50 centimeters and one multiplies the number of steps by 0.5 meter. The products are different. 4,250 is 100 times smaller than 425,000 because it represents meters instead of centimeters.)
Display:
\(425,\!000 \div 100,\!000= 4.25\)
\(4,\!250 \div 1,\!000 = 4.25\)
“How does each of these equations represent the situation?” (They both represent the number of kilometers that Lin walked. 425,000 is the distance she walked in centimeters so if you divide it by 100,000, you get the number of kilometers she walked. 4,250 is the distance in meters that she walked so we only have to divide by 1,000 to figure out how many kilometers she walked.)
Display: 4.25 km, 4,250 m, 425,000 cm
“Which of these do you think best communicates how far Lin walked this day?” (4.25 kilometers because I can picture how long a kilometer is.)


correct_option_index: 8

9