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.


Thought: 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.


Thought: 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.


Thought: 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: 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.

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

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

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

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

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

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.

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.

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.
problem_activity: Activity
In a previous lesson, students had a primer on histograms—how they are drawn, how they differ from dot plots, and what information they can tell us. In this activity, students practice drawing a histogram for a given data set and using it to answer statistical questions. To help students understand the lengths involved in the data set, students are asked to draw various lengths used to group the worms in the first histogram.
As students organize the data set and draw their histogram, notice any challenges or questions students come across. If a question is raised by multiple groups, consider discussing it with the whole class. Also pay attention to how students use the histogram to identify a “typical” length—some might describe it in terms of the size of the bins (e.g., “a typical length is between 20 and 40 mm”); others might choose a value within a bin or a boundary between bins. Invite them to share their reasonings later.
Launch
Arrange students in groups of 2. Provide access to centimeter rulers.
Consider giving students a brief overview of the context for the problems in the activity. Tell students that there are nearly 6,000 species of earthworms in the world.  Some earthworms help the environment, while others (generally not native to the region in which they are found) may harm the environment.  Earthworms that are native to a particular region of the world are often raised, by farmers, in terrariums (a container or bin similar to an aquarium but it contains soil and leaves). The terrarium-raised earthworms provide bait for people who fish, provide food for various wildlife, and decompose food waste into soil.  Food waste and water are added to the terrariums as food for raising and growing worms.  Soil produced by the worms as they eat the food waste is often used as fertilizer.
Explain that the lengths of the worms in the bins provide information about the ages of the worms, which can be useful for the farmer. In this activity, students will organize the lengths of the earthworms in several terrariums or bins.
Give students 8–10 minutes of quiet work time, and then 3–4 minutes to discuss their work and complete the activity with a partner.
Action and Expression: Internalize Executive Functions.
Activate or supply background knowledge by demonstrating how to use a ruler to draw a line segment for a given length.
Supports accessibility for: Memory; Conceptual processing
Student Facing
An earthworm farmer set up several containers of a certain species of earthworms so that he could learn about their lengths. The lengths of the earthworms provide information about their ages. The farmer measured the lengths of 25 earthworms in one of the containers. Each length was measured in millimeters.
[IMG]
Using a ruler, draw a line segment for each length:
20 millimeters
40 millimeters
60 millimeters
80 millimeters
100 millimeters
Here are the lengths, in millimeters, of the 25 earthworms.
6
11
18
19
20
23
23
25
25
26
27
27
28
29
32
33
41
42
48
52
54
59
60
77
93
Complete the table for the lengths of the 25 earthworms.
|                   length                    | frequency |
|---------------------------------------------|-----------|
|  0 millimeters to less than 20 millimeters  |           |
| 20 millimeters to less than 40 millimeters  |           |
| 40 millimeters to less than 60 millimeters  |           |
| 60 millimeters to less than 80 millimeters  |           |
| 80 millimeters to less than 100 millimeters |           |
Use the grid and the information in the table to draw a histogram for the worm length data. Be sure to label the axes of your histogram.
[IMG]
Based on the histogram, what is a typical length for these 25 earthworms? Explain how you know.
Write 1–2 sentences to describe the spread of the data. Do most of the worms have a length that is close to your estimate of a typical length, or are they very different in length?
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?
Here is another histogram for the earthworm measurement data. In this histogram, the measurements are in different groupings.
[IMG]
Based on this histogram, what is your estimate of a typical length for the 25 earthworms?
Compare this histogram with the one you drew. How are the distributions of data summarized in the two histograms the same? How are they different?
Compare your estimates of a typical earthworm length for the two histograms. Did you reach different conclusions about a typical earthworm length from the two histograms?
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
When determining frequencies of data values, students might lose track of their counting. Suggest that they use tally marks to keep track of the number of occurrences for each bin.
When drawing the histogram, students might mistakenly use bar graphs as a reference and leave spaces between the bars. Ask them to look at the bars in other histograms they have seen so far and to think about what the gaps might mean considering that the bars are built on a number line.
Activity Synthesis
Ask one or two students to display their completed histograms for all to see and briefly describe the overall distribution. Then, select a few other previously identified students to share their responses and explanations for the last two questions.
Focus the discussion on how identifying center and spread using a histogram is different than doing so using a dot plot. Discuss:
“In a histogram, are we able to see clusters of values in the distribution?”
“Can we see the largest and smallest values? Can we tell the overall spread?”
“How do we identify the center of a distribution?”
From the various estimates that students give for a typical earthworm length (and from earlier exercises), students should begin to see that identifying a typical value of a distribution is not a straightforward or precise process so far. Explain that in upcoming lessons they will explore how to describe a typical value and characterize a distribution more systematically.
Representing, Conversing: MLR7 Compare and Connect.
As the selected students show and describe their histograms, invite pairs to discuss: “What is the same and what is different?” about their own histograms and reasoning. This will help students better understand there are multiple valid ways for describing center and spread.
Design Principle(s): Maximize meta-awareness; Support sense-making


correct_option_index: 8