few_shots: Example 1:
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.


Thought: This problem pertains to functions because it asks students to find the general solution to a trigonometric function.
Answer: B

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


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. 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. 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. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
F. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
G. 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.
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. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
J. 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.
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 task asks students to reason with equations, specifically a quadratic equation. Since the problem asks for as many different methods as possible, it's possible that students would discover solution methods that include graphing both sides of the equation with a graphing calculator and finding the x coordinates of where the two functions intersect.
Answer: B, G

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


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


Thought: This task involves geometric concepts including angles, circles, and rays.
Answer: H

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: fl_problem_001894

HS_0

im_task_000870

HS_1

im_task_001160

HS_2
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.

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

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

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.

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.

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.

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.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
problem_activity: Activity
This activity gives students an opportunity to put into practice some things they already know about finding percent rates. Additionally, the idea of a fraction of a percent appears for the first time. Encourage students to use any representation they would like to calculate the percentage of appetizers, entrees and desserts. Monitor for students who used various representations and ask them to share during the discussion. The main focus should be on the fractional percentages they encounter in this problem for the first time.
Digital
Launch
Tell students they will be finding some more percentages. Encourage them to use any representation they understand, for example, a double number line or a table. Students in groups of 2. Give students 1–2 minutes of quiet work time, followed by partner then whole-class discussion.
If using the digital activity, students will use an applet to find and check percentages.
Representation: Internalize Comprehension
. Activate or supply background knowledge about diagrams that can be used to represent percent rates such as tables and double number line diagrams.
Supports accessibility for: Memory; Conceptual processing
Writing and Listening: MLR5 Co-Craft Questions.
Before students begin work, display the waiter's situation without revealing the questions. Ask students to write down possible mathematical questions that might be asked about the situation. Invite pairs to compare their questions, and then ask for a few to be shared in a whole-class discussion. Reveal the actual questions about the waiter's situation that students will answer. This will help students make sense of the problem before attempting to solve it.
Design Principle(s): Optimize output for explanation
Student Facing
During one waiter’s shift, he delivered appetizers, entrées, and desserts. What percentage of the dishes were desserts? appetizers? entrées? What do your percentages add up to?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Tell students they will be finding some more percentages. Encourage them to use any representation they understand, for example, a double number line or a table. Students in groups of 2. Give students 1–2 minutes of quiet work time, followed by partner then whole-class discussion.
If using the digital activity, students will use an applet to find and check percentages.
Representation: Internalize Comprehension
. Activate or supply background knowledge about diagrams that can be used to represent percent rates such as tables and double number line diagrams.
Supports accessibility for: Memory; Conceptual processing
Writing and Listening: MLR5 Co-Craft Questions.
Before students begin work, display the waiter's situation without revealing the questions. Ask students to write down possible mathematical questions that might be asked about the situation. Invite pairs to compare their questions, and then ask for a few to be shared in a whole-class discussion. Reveal the actual questions about the waiter's situation that students will answer. This will help students make sense of the problem before attempting to solve it.
Design Principle(s): Optimize output for explanation
Student Facing
During one waiter’s shift, he delivered 13 appetizers, 17 entrées, and 10 desserts.
What percentage of the dishes he delivered were:
desserts?
appetizers?
entrées?
What do your percentages add up to?
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
If students round to the nearest percentage, they will get that 33% of the dishes were appetizers and 43% of the dishes were entrées. Along with the 25% desserts, their percentages will sum to 101%. Point out that all of the dishes taken as percentages should sum to 100% and encourage them to be critical of their method and try to figure out where the extra 1% came from.
Activity Synthesis
Select students to share the percentages they calculated for each type of dish the waiter delivered. Depending on the outcome of the warm-up, it may be appropriate once again to display different representations of percentages as rates per 100.
Double number line:
[IMG]
Table:
| number of dishes | percentage |
|------------------|------------|
|        10        |     25     |
|        1         |    2.5     |
|        13        |    32.5    |
|        17        |    42.5    |
Equation: Students may have previously learned to represent relationships like this using an equation in a form
\(y=kx\)
. For example, to find what percent 13 is of 40, they might write
\(13=k \boldcdot 40\)
, and find that
\(k\)
is 0.325 by evaluating
\(13 \div 40\)
. 0.325 is the rate per 1, so 32.5 is the rate per 100.
Students may have never seen a percentage that was not a whole number. Spend a few minutes making sense of this. Ask students:
What do you notice that is different about these percentages from the ones you have looked at before? (Some of these percentages are not whole numbers.)
What do the percentages add up to? (Exactly 100.)
What does 32.5% of 40 mean? (It’s halfway between 32% and 33% of 40.)


correct_option_index: 10