few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Topic options: 
A. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
B. 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.
C. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
D. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
E. 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.
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. 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. 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.
K. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.


Your response, in json format:
{
  "explanation": "This task involves learning addition, which is a mathematical operation, using objects.",
  "answer": "F"
}

Example 2:
Problem/activity:
Task
 Arrange the fractions in order from least to greatest. Explain your answer with a picture.
 $\frac{1}{5}, \frac{1}{7}, \frac{1}{3} $
 $\frac{2}{5}, \frac{2}{7}, \frac{2}{3} $
 $\frac{5}{6}, \frac{3}{6}, \frac{1}{6} $
 $\frac{5}{12}, \frac{8}{12}, \frac{4}{12} $

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


Your response, in json format:
{
  "explanation": "In this problem, students learn to understand fractions as numbers by comparing their sizes.",
  "answer": "G"
}

Example 3:
Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


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


Your response, in json format:
{
  "explanation": "This set of problems involves performing operations, specifically multiplication, on multi-digit whole numbers in base ten.",
  "answer": "D"
}

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: im_task_001477

K

im_task_000157

3

fl_problem_000674

5
options: 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.

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.

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.

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.

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

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

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

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.

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

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.
problem_activity: Activity
Students design districts in three towns to “gerrymander” the results of elections. In two of the towns, the election results can be skewed to either color. In the third, it isn’t possible to skew the results.
Launch
Arrange students in groups of 2–4.
Explain the history of gerrymandering: Sometimes people in charge of designing districts make them in strange shapes to give the election results they want. One of the first was Elbridge Gerry (governor of Massachusetts in 1812), whose party designed a district that many people thought looked like a salamander. They called a Gerrymander, and the name stuck. It means a very strangely shaped, spread-out district designed to produce a certain result.
Usually districts are required to be connected: a person traveling to all parts of the district should be able to stay inside the district. There should be no “islands” that are separated by parts of other districts.
Engagement: Internalize Self-Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Invite students to choose and respond to four out of six task statements.
Supports accessibility for: Organization; Attention
Student Facing
Smallville’s map is shown, with opinions shown by block in green and gold. Decompose the map to create three connected, equal-area districts in two ways:
Design three districts where
green
will win at least two of the three districts. Record results in Table 1.
[IMG]
Table 1:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}]}]}]}
Design three districts where
gold
will win at least two of the three districts. Record results in Table 2.
[IMG]
Table 2:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}]}]}]}
Squaretown’s map is shown, with opinions by block shown in green and gold. Decompose the map to create five connected, equal-area districts in two ways:
Design five districts where
green
will win at least three of the five districts. Record the results in Table 3.
[IMG]
Table 3:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}, {}, {}]}, {"td": [{"_value": "5"}, {}, {}, {}, {}]}]}]}]}
Design five districts where
gold
will win at least three of the five districts. Record the results in Table 4.
[IMG]
Table 4:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}, {}, {}]}, {"td": [{"_value": "5"}, {}, {}, {}, {}]}]}]}]}
Mountain Valley’s map is shown, with opinions by block shown in green and gold. (This is a town in a narrow valley in the mountains.) Can you decompose the map to create three connected, equal-area districts in the two ways described here?
Design three districts where
green
will win at least 2 of the 3 districts. Record the results in Table 5.
[IMG]
Table 5:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}]}]}]}
Design three districts where
gold
will win at least 2 of the 3 districts. Record the results in Table 6.
[IMG]
Table 6:
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "district"}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["number of blocks", "for gold"]}, {"_attributes": {"scope": "col"}, "br": [{}, {}], "_values": ["percentage", "of blocks", "for green"]}, {"_attributes": {"scope": "col"}, "br": [{}], "_values": ["representative\u2019s", "vote"]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "1"}, {}, {}, {}, {}]}, {"td": [{"_value": "2"}, {}, {}, {}, {}]}, {"td": [{"_value": "3"}, {}, {}, {}, {}]}]}]}]}
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
The white streets on the maps do not disconnect the districts.
However, probably squares that are only connected at their corners should not be considered connected to each other. (But you or the students can make their own rule, since they are in charge of making districts.)
Activity Synthesis
This set of questions is similar to the previous activity, except that the patterns are irregular, so more planning and computation is needed to design districts.
Ask several students with different maps to show and explain their work.
The important result for this problem is that it’s possible to design both fair districts (where the result of the vote is similar to the vote if all individual votes were counted) and unfair districts. According to the rules (equal area and connected), these sets of districts are legal. Encourage students to try to devise some rules for drawing districts that would be more fair.


correct_option_index: 8