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

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.

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.

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.

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.

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

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.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
problem_activity: Warm-up
The mathematical purpose of this activity is for students to match bivariate data with its context. Students should think about whether they might expect a strong correlation or not as well as whether the relationship has a positive or negative correlation. Monitor for students who discuss linear relationships or variability in the data.
Launch
Arrange students in groups of two to four. Provide access to the scatter plots and contexts for the warm-up.
Student Facing
Match the variables to the scatter plot you think they best fit. Be prepared to explain your reasoning.
|    | \(x\)                  variable | \(y\)                  variable |
|----|---------------------------------|---------------------------------|
| 1. |                                 |                                 |
| 2. |                                 |                                 |
| 3. |                                 |                                 |
| 4. |                                 |                                 |
A
[IMG]
B
[IMG]
C
[IMG]
D
[IMG]
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
Students may struggle with matching the pairs of variables with a scatter plot. Encourage students to think about how related the variables are and how the
\(y\)
variable might change as the
\(x\)
variable increases.
Activity Synthesis
The goal of this discussion is for students to discuss how the characteristics of the scatter plots allowed them to determine the context. For each context, select groups to share their match and reasoning. Select groups who used linear model and variability in their small group discussions.
Here are some questions for discussion.
“How did the concept of linear relationships help you to make a match?” (For the height in inches and the height in feet, I knew that the data would be very linear. I chose scatter plot A because it was the only one that appeared linear. I did wonder why scatter plot A was not perfectly linear. Maybe it had to do with rounding or measurement error.)
“How did you use the concept of linearity to help you to make a match?” (First, I expected the temperature and cereal context relationship to be totally random rather than linear at all, so that fit with scatter plot C. Second, I knew that problem 3 matched with scatter plot A, because I knew it should be almost perfectly linear. For a given height in feet, there is only one height in inches.)
“Which matches were the most difficult? What helped you figure them out?” (It was hardest to find the matches for scatter plots B and D because they were pretty scattered. I noticed that the data in scatter plot D displayed a negative relationship, so I looked for a context that would have a negative relationship. It makes sense that customer satisfaction decreases as wait time increases.)


correct_option_index: 5