few_shots: Example:
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:
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

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: fl_problem_001097

7
options: 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.

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.

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.

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.

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

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

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.
problem_activity: Activity
Students dilated a single point in a previous activity. In this activity, they dilate a figure with multiple points and multiple shapes. The goal is for them to practice dilation and begin to confirm some properties of dilation. Students will have additional opportunities to formalize properties of dilation in subsequent lessons (A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. The dilation of a line segment is longer or shorter according to the same ratio given by the scale factor.)
Monitor for students who use the following (incomplete) strategies:
use the two given lines to place
\(D’\)
and
\(B’\)
, and then estimate all of the other shapes and points
measure distances within the figure and try to replicate them, for example calculating the radii of the two circles and trying to place them without scaling
\(A\)
and
\(C\)
along rays through
\(P\)
Monitor for students who use the precise strategy of drawing rays from
\(P\)
through each point in the original figure and placing points in the image halfway between
\(P\)
and the corresponding point.
Launch
It's okay if students do not have time to dilate every point, encourage them to record things they notice and wonder as they draw the dilation.
Engagement: Internalize Self Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. Assign students an appropriate number of points to choose and dilate.
Supports accessibility for: Organization; Attention
Student Facing
Dilate the figure using center
\(P\)
and scale factor
\(\frac12\)
.
[IMG]
What do you notice? What do you wonder?
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?
[IMG]
Dilate segment
\(AB\)
using center
\(P\)
by scale factor
\(\frac12 \)
. Label the result
\(A'B'\)
.
Dilate the segment
\(AB\)
using center
\(Q\)
by scale factor
\(\frac12\)
.
How does the length of
\(A''B''\)
compare to
\(A'B\)
? How would the length of
\(A''B''\)
change if
\(Q\)
was infinitely far away? Explain or show your answer.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The goal of this discussion is to surface some observations. If most students were comfortable drawing a dilation move directly to that. If not, select multiple students who used different strategies (both precise and incomplete) to share their work. Ask students to explain what was
precise
in each student’s method. The key point is that the most precise way to draw a dilation is to draw rays from the center of the dilation through each point of the original figure, and then measure to figure out where the point goes in the image.
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the relative length of segments, or measure of angles do not come up during the conversation, ask students to discuss these ideas.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. As students share their strategies for drawing the dilation of the figure, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped clarify the original explanation such as, “draw rays from the center of dilation through each point of the original figure.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 10