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

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

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.

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.

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

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.

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.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
problem_activity: Narrative
The purpose of this activity is for students to observe and use the structure of diagrams to find areas of shaded regions with non-unit fraction side lengths. Students build on what they learned in the previous activity, solidifying their understanding of why the numerator of a product of two fractions is the product of the numerators and the denominator of a product of fractions is the product of the denominators.
Action and Expression: Internalize Executive Functions.
Invite students to verbalize their strategy for writing multiplication expressions to represent the area of the shaded rectangle in each figure before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language
Launch
Groups of 2
Activity
“Start working on completing the table independently. After a couple minutes, you'll work with your partner to complete the table and answer the rest of the questions.”
1–2 minutes: independent work time
5-8 minutes: partner work time
Monitor for students who:
notice the area of the shaded regions is always twentieths
write the expression
\(\frac{6}{5}\times\frac{4}{5}\)
to represent the shaded region of the last diagram in the table
explain that the expression
\(\frac{6\times4}{5\times4}\)
represents the shaded part of the last diagram in the table because
\(6\times4\)
represents the number of pieces that are shaded and
\(4 \times 5\)
represents the number of those pieces in the unit square
Student Facing
Complete the table.
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | multiplication                       expression | shaded area                       (square units) |
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------|--------------------------------------------------|
| A                |https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad|  |https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad| image:: https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad |                                                 |                                                  |
| B                |https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92|  |https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92| image:: https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92 |                                                 |                                                  |
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | multiplication                       expression | shaded area                       (square units) |
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------|--------------------------------------------------|
| C                |https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e|  |https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e| image:: https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e |                                                 |                                                  |
| D                |https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d|  |https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d| image:: https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d |                                                 |                                                  |
What patterns do you notice in the table?
Explain or show how the expression
\(\frac{6\times4}{5\times4}\)
represents the last diagram in the table.
Student Response
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Advancing Student Thinking
If students do not explain that the product of the numerators represents the number of pieces in the shaded region and the product of the denominators represents the number of pieces in the whole, consider asking: “What is the same and what is different about diagrams A and B?”
Activity Synthesis
Ask previously selected students to share their reasoning.
“How do the expressions in the table represent the number of pieces shaded in and the size of the pieces shaded in?” (If we multiply the numerators, we get the number of pieces that are shaded in. If we multiply the denominators, we get the size of the pieces.)
Refer to diagrams and draw on each diagram to show how the multiplication of the numerators and denominators represents the number of shaded pieces and the size of the shaded pieces (that is, the number of those pieces in the whole).


correct_option_index: 7