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 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.

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

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.

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.

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.

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.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
problem_activity: Activity
In this activity, students create two-way tables displaying relative frequency. The relative frequency table converts the actual frequency data to percentages which can be useful when comparing groups that include different totals. Finally, students use the relative frequencies to look for a pattern in the data. In the following lesson, students will work with associations in categorical data more explicitly. In this activity, students should use an informal understanding of association to think about whether one is present in the data based on the relative frequency table.
Launch
Keep students in groups of 2–3. After an introduction to relative frequency tables, allow students 3 minutes quiet work time followed by partner discussion and whole-class discussion.
Display the table for all to see.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "watch the news daily"}, {"_attributes": {"scope": "col"}, "_value": "does not watch the news daily"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "younger than 18"}], "td": [{"_value": "30"}, {"_value": "80"}, {"_value": "110"}]}, {"th": [{"_value": "18 or older"}], "td": [{"_value": "10"}, {"_value": "5"}, {"_value": "15"}]}, {"th": [{"_value": "total"}], "td": [{"_value": "40"}, {"_value": "85"}, {"_value": "125"}]}]}]}]}
Ask students, "Based on this data, who is more likely to watch the news daily: someone who is younger than 18 or someone who is 18 or older?"
Tell students that, based on the numbers in the table, there are more younger people who watch the news (30) than older (10). On the other hand, the survey reached out to 110 young people and only 15 older people. Without looking at the whole table, that information may have been missed.
In cases like this, finding a relative frequency including percentages can be more helpful than looking at the actual frequency, which is what they are going to do now.
Use MLR 2 (Collect and Display) to co-create a graphic organizer with students comparing and contrasting the definitions and characteristics of “actual frequency” and “relative frequency” to be referenced and revised throughout this and later activities.
Conversing, Representing, Writing: MLR2 Collect and Display.
As students work in groups, capture the vocabulary and phrases students use when comparing and contrasting the definitions and characteristics of “actual frequency” and “relative frequency.” Listen for students who identify pros and cons for each representation and when one is preferable compared to the other. Record their language on a visual display that can be referenced in future discussions. This routine provides feedback to students that supports sense-making and increases meta-awareness of mathematical language.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
Here is a two-way table that shows data about cell phone usage among children aged 10 to 18.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "has cell phone"}, {"_attributes": {"scope": "col"}, "_value": "does not have cell phone"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "10 to 12 years old"}], "td": [{"_value": "25"}, {"_value": "35"}, {"_value": "60"}]}, {"th": [{"_value": "13 to 15 years old"}], "td": [{"_value": "40"}, {"_value": "10"}, {"_value": "50"}]}, {"th": [{"_value": "16 to 18 years old"}], "td": [{"_value": "50"}, {"_value": "10"}, {"_value": "60"}]}, {"th": [{"_value": "total"}], "td": [{"_value": "115"}, {"_value": "55"}, {"_value": "170"}]}]}]}]}
Complete the table. In each row, the entries for “has cell phone” and “does not have cell phone” should have the total 100%. Round entries to the nearest percentage point.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "has cell phone"}, {"_attributes": {"scope": "col"}, "_value": "does not have cell phone"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "10 to 12 years old"}], "td": [{"_value": "42%"}, {}, {}]}, {"th": [{"_value": "13 to 15 years old"}], "td": [{}, {}, {"_value": "100%"}]}, {"th": [{"_value": "16 to 18 years old"}], "td": [{}, {"_value": "17%"}, {}]}]}]}]}
This is still a two-way table. Instead of showing
frequency
, this table shows
relative frequency
.
Two-way tables that show relative frequencies often don’t include a “total” row at the bottom. Why?
Is there an association between age and cell phone use? How does the two-way table of relative frequencies help to illustrate this?
Student Response
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Student Facing
Are you ready for more?
A pollster attends a rally and surveys many of the participants about whether they associate with political Party A or political Party B and whether they are for or against Proposition 3.14 going up for vote soon. The results are sorted into the table shown.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "for"}, {"_attributes": {"scope": "col"}, "_value": "against"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "party A"}], "td": [{"_value": "832"}, {"_value": "165"}]}, {"th": [{"_value": "party B"}], "td": [{"_value": "80"}, {"_value": "160"}]}]}]}]}
A news station reports these results by saying, “A poll shows that about the same number of people from both parties are voting against Proposition 3.14.”
A second news station shows this graphic.
[IMG]
Are any of the news reports misleading? Explain your reasoning.
Create a headline, graphic, and short description that more accurately represents the data in the table.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may not calculate relative frequencies correctly. Look to see if they are dividing into the total for each row, instead of some other number in the row, or the total for the entire table.
Activity Synthesis
The purpose of this discussion is to help students see the usefulness of two-way tables that display relative frequency.
Some questions for discussion:
“What does the 42% in table mean?” (42% of people 10 to 12 years old have a cell phone.)
“What is the total of the percentages in the first column? What's wrong with that answer?” (205%. It doesn't make sense since it is greater than 100%. It is combining percentages of different wholes, so it doesn't make sense to add them.)
“Did you say there was or was not an association between age and cell-phone use? Explain your reasoning.” (There is an association since the percentages for the 10 to 12 year olds is very different from the other two age groups.)
“What did the relative frequency table show you that was harder to see in the original two-way table?” (It was much easier to see the association with the percentages than the numbers. For example, the number of people without a cell phone for the older 2 age groups was the same, but the percentages were a little different.)


correct_option_index: 8