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.

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.

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.

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

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.

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.

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.

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
The purpose of this activity is for students to compare the possible ways of constructing equilateral triangles. Look for student work that illustrates a variety of methods and sizes of triangles.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Encourage students to construct the equilateral triangles using different methods rather than repeating the same method twice.
Student Facing
Use straightedge and compass moves to construct at least 2 equilateral triangles of different sizes.
Student Response
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Student Facing
Are you ready for more?
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Examine the figure carefully. What different shapes is it composed of? Be specific.
Figure out how to construct the figure with a compass and straightedge.
Then, cut it out, and see if you can fold it up into a container like this.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students get stuck, ask them to refer to the warm-up to see whether they can spot equilateral triangles. Then encourage them to do the compass and straightedge moves required to make those equilateral triangles.
Activity Synthesis
Display several student responses for all to see. Ask students to explain their methods for producing equilateral triangles and justify how they know each triangle is equilateral. If not mentioned by students, conjecture that, in some of these constructions, the entire construction remains the same even when rotated
\(\frac13\)
of a full turn (120 degrees) around the center. This means that each side can be rotated onto the other sides, and each angle can be rotated onto the other angles. This plants ideas about symmetry that will be explored in later lessons.
Speaking, Representing: MLR8 Discussion Supports.
Use this routine to help students prepare for the whole-class discussion. Arrange students in groups of 2. Invite students to take turns explaining their method for constructing equilateral triangles of different sizes. This will provide all students with additional opportunities to clarify their thinking and to produce language.
Design Principle(s): Support sense-making; Cultivate conversation
Action and Expression: Develop Expression and Communication.
Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy and ideas. For example, “First, I _____ because _____. Then, I....” and “I know _____ because….”
Supports accessibility for: Language; Organization


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