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

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.


Your response:
{
  "explanation": "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:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Topic options: 
A. 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.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
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. 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. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
H. 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.
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 & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.


Your response:
{
  "explanation": "This problem pertains to functions because it asks students to find the general solution to a trigonometric function.",
  "answer": "B"
}

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

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

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

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

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

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

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.

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.

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.
problem_activity: Activity
This task contains two distinct parts. Depending on what students need and time constraints, you could decide to do either or both.
In the associated Algebra 1 lesson, students will need to evaluate the functions
\(6x^2\)
and
\(3^x\)
for different input values. The purpose of the first part is to prepare them to avoid some common mistakes. Analyzing and correcting incorrect work is an opportunity to critique the reasoning of others (MP3).
The purpose of the second part is to revisit ways to tell whether a function is linear, quadratic, or exponential, and to practice the skills of generating a table and sketching a graph for a given function.
Student Facing
Different students are evaluating two expressions,
\(3\boldcdot 6^x\)
and
\(5^x\)
. Analyze their work, describe any errors made, and then evaluate each expression correctly.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}, {"_attributes": {"scope": "col"}, "_value": "Noah\u2019s work"}, {"_attributes": {"scope": "col"}, "_value": "Mai\u2019s work"}, {"_attributes": {"scope": "col"}, "_value": "corrected work"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "11191"}, "_value": "\\(5^x\\)"}, {"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}], "_values": ["Evaluate", "when", "is 6."]}], "td": [{"p": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "11191"}, "_value": "\\(5^x\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "29657"}, "_value": "\\(5^6\\)"}]}, {"_value": "30"}]}, {"p": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "11191"}, "_value": "\\(5^x\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "29657"}, "_value": "\\(5^6\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(6 \\boldcdot 6 \\boldcdot\u00a06 \\boldcdot\u00a06 \\boldcdot6\\)"}]}, {"_value": "7,776"}]}, {}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "9091"}, "_value": "\\(3 \\boldcdot 6^x\\)"}, {"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}], "_values": ["Evaluate", "when", "is 2."]}], "td": [{"p": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "failed-conversion"}, "_value": "\\(3 \\boldcdot\u00a06^x\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "29795"}, "_value": "\\(3\\boldcdot 6^2\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "29846"}, "_value": "\\(3 \\boldcdot 12\\)"}]}, {"_value": "36"}]}, {"p": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "9091"}, "_value": "\\(3 \\boldcdot 6^x\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "29893"}, "_value": "\\(3 \\boldcdot 6^2\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24976"}, "_value": "\\(18^2\\)"}]}, {"_value": "324"}]}, {}]}]}]}]}
Here are three functions. For each function:
Complete the table of values.
Sketch a graph.
Decide whether each function is linear, quadratic, or exponential, and be prepared to explain how you know.
\(f(x)=3 \boldcdot 2^x\)
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}], "td": [{"_value": "-1"}, {"_value": "0"}, {"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "5"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "287"}, "_value": "\\(f(x)\\)"}]}], "td": [{}, {}, {}, {}, {}, {}]}]}]}]}
[IMG]
\(g(x)=3 \boldcdot x^2\)
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}], "td": [{"_value": "-1"}, {"_value": "0"}, {"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "5"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "970"}, "_value": "\\(g(x)\\)"}]}], "td": [{}, {}, {}, {}, {}, {}]}]}]}]}
[IMG]
\(h(x)=3 \boldcdot 2x\)
{"table": [{"_attributes": {"border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}], "td": [{"_value": "-1"}, {"_value": "0"}, {"_value": "1"}, {"_value": "2"}, {"_value": "3"}, {"_value": "5"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "1834"}, "_value": "\\(h(x)\\)"}]}], "td": [{}, {}, {}, {}, {}, {}]}]}]}]}
[IMG]
Student Response
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Activity Synthesis
For the first part, invite students to describe the errors they see, and share how to correctly evaluate each expression.
For the second part, display the tables and graphs for all to see. Ask students to share some ways they can tell a function is linear, exponential, or quadratic using the different representations. For example:
The graph of the linear function is a straight line.
The table shows the exponential function has a growth factor of 2. Each time
\(x\)
increases by 1, the
\(f(x)\)
values increase by a factor of 2.
The table shows the quadratic function does not have a constant rate of change or a constant growth factor.


correct_option_index: 2