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.

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.

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.

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.

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.

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.

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 this activity is to have students identify corresponding parts in congruent triangles.
Identify students who:
make general statements such as “The lines look parallel.”
make arguments based on properties of rotations
make arguments based on alternate interior angles being congruent
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, mark congruent angles with the same color.
Supports accessibility
for:
Visual-spatial processing
Student Facing
Draw a triangle.
Find the midpoint of the longest side of your triangle.
Rotate your triangle
\(180^{\circ}\)
using the midpoint of the longest side as the center of the rotation.
Label the
corresponding
parts and mark what must be congruent.
Make a conjecture and justify it.
What type of quadrilateral have you formed?
What is the definition of that quadrilateral type?
Why must the quadrilateral you have fit the definition?
Student Response
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Anticipated Misconceptions
If students are struggling to perform the rotation, encourage them to use whatever tools they are most comfortable with, including sketching.
Activity Synthesis
The purpose of this discussion is to share examples of convincing arguments.
Select previously identified students to share in this order:
A student who made a general statement such as “The lines look parallel.”
A student who used the fact that rotation by
\(180^{\circ}\)
using a point not on the line as the center takes lines to parallel lines
A student who used corresponding parts to establish that the lines were parallel because alternate interior angles are congruent
If no student used corresponding parts to establish alternate interior angles as congruent, ask students to look for structure by looking at the congruence marks and seeing whether they match any diagrams on the reference chart.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. As students share their justification for why the opposite sides must be parallel, 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 to clarify the original statement. 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: 1