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: 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.

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.

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.

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.

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.
problem_activity: Activity
The purpose of the task is for students to compare signed numbers in a real-world context and then use inequality signs accurately with negative numbers (MP2). The context should help students understand “less than” or “greater than” language. Students evaluate and critique another's reasoning (MP3).
Launch
Allow students 5–6 minutes quiet work time followed by whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge. Some students may benefit from access to partially completed or blank number lines. Consider preparing blank number lines with just the tick marks to get students started.
Supports accessibility for: Visual-spatial processing; Organization
Student Facing
Here are the low temperatures, in degrees Celsius, for a week in Anchorage, Alaska.
|     day     | Mon | Tues | Weds | Thurs | Fri | Sat | Sun |
|-------------|-----|------|------|-------|-----|-----|-----|
| temperature |     |      |      |       |     |     |     |
Plot the temperatures on a number line. Which day of the week had the lowest low temperature?
The lowest temperature ever recorded in the United States was -62 degrees Celsius, in Prospect Creek Camp, Alaska. The average temperature on Mars is about -55 degrees Celsius.
Which is warmer, the coldest temperature recorded in the USA, or the average temperature on Mars? Explain how you know.
Write an inequality to show your answer.
On a winter day the low temperature in Anchorage, Alaska, was -21 degrees Celsius and the low temperature in Minneapolis, Minnesota, was -14 degrees Celsius.
Jada said, “I know that 14 is less than 21, so -14 is also less than -21. This means that it was colder in Minneapolis than in Anchorage.”
Do you agree? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Another temperature scale frequently used in science is the
Kelvin scale
. In this scale, 0 is the lowest possible temperature of anything in the universe, and it is -273.15 degrees in the Celsius scale. Each
\(1\text{ K}\)
is the same as
\(1^\circ\text{C}\)
, so
\(10\text{ K}\)
is the same as
\(\text-263.15^\circ\text{C}\)
.
Water boils at
\(100^\circ \text{C}\)
. What is this temperature in
\(\text{K}\)
?
Ammonia boils at
\(\text-35.5^\circ \text{C}\)
. What is the boiling point of ammonia in
\(\text{K}\)
?
Explain why only positive numbers (and 0) are needed to record temperature in
\(\text{K}\)
.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Some students may have difficulty comparing numbers on the negative side of the number line. Have students plot the numbers on a number line in order to sequence them from least to greatest. Make it evident that temperatures become warmer (i.e., greater) as they move from left to right on the number line.
Activity Synthesis
Ask students to explain how they could tell which of the two negative numbers was greater in problem 2. After gathering 1 or 2 responses, display the following image for all to see:
[IMG]
Explain to students that as we go to the right on the number line, we can think of the temperature as getting hotter, and as we go to the left, we can think of the temperature as getting colder. Numbers don’t always describe temperature, though. So we use the word “greater” to describe a number that is farther to the right, and “less” to describe numbers that are farther to the left. For example, we would write
\(6 > \text-50\)
and say “6 is greater than -50 because it is farther to the right on the number line.” Equivalently, we could write
\(\text-50 < 6\)
and say “-50 is less than 6 because -50 is farther to the left on the number line.”
Ask students to make up their own list of negative numbers and plot on a number line. Then ask them to write several inequality statements with their numbers. As time allows, have students share one inequality statement with the class to see if all agree that it is true. If time allows, have students swap their list with a partner to plot and check their partner’s number lines and inequality statements.
Speaking: MLR7 Compare and Connect.
Use this routine to support students’ understanding of both the language and the symbolic notation of inequalities. Ask students to consider what is the same and what is different between the following sentences: “6 degrees is warmer than -50 degrees,” “6 is greater than -50,” and “
\(6 > \text-50\)
.” This helps students make the connection between the language of the context, the written inequality statement and the symbolic inequality statement.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness


correct_option_index: 5