options: 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.

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

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.

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.

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.

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.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
problem_activity: Activity
In this activity, students create an arbitrary triangle and use what they know about angle bisectors and constructions to find the incenter. Then they construct segments measuring the distance from the incenter to the sides of the triangle. Finally, they construct the triangle’s inscribed circle, or the circle that is tangent to all 3 sides of the triangle.
Making dynamic geometry software available as well as tracing paper, straightedge, and compass gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
If students have access to dynamic geometry software, suggest that it might be a helpful tool in this activity.
Action and Expression: Internalize Executive Functions.
Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Invite 1–2 students to share how they are constructing the incenter, segments, and circle. Record their thinking on a display for all to see, and keep the work visible as students continue to work.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Mark 3 points and connect them with a straightedge to make a large triangle. The triangle should
not
be equilateral.
Construct the incenter of the triangle.
Construct the segments that show the distance from the incenter to the sides of the triangle.
Construct a circle centered at the incenter using one of the segments you just constructed as a radius.
Would it matter which of the three segments you use? Explain your thinking.
Student Response
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Activity Synthesis
Remind students that the circle they constructed is said to be
inscribed
in the triangle. Students previously discussed inscribed circles in the Constructions and Rigid Transformations unit. Ask students to add this theorem to their reference charts as you add it to the class reference chart:
The 3 angle bisectors of a triangle meet at a single point, called the triangle’s
incenter
. This point is the center of the triangle’s inscribed circle. (
Theorem
)
[IMG]
Display several students’ inscribed circles for different kinds of triangles for all to see. The goal of the discussion is to draw conclusions about inscribed circles. Start by asking, “What do you notice about the triangles and their inscribed circles?” Sample responses:
Some inscribed circles take up a large fraction of the area, while others take up a small fraction.
The circle seems to be tangent to the sides of the triangles.
The original triangles are each divided into 3 smaller triangles. For each small triangle, the base is a side of the original triangle and the height is the radius of the inscribed circle.
The original triangles are each divided into 3 pairs of congruent right triangles with an angle that is half the measure of an angle in the original triangle.
Next, ask students:
“How do we know that an inscribed circle is tangent to all 3 sides of the triangle?” (The radius that represents the distance from the incenter to each side of the triangle is by definition perpendicular to that side. A radius is perpendicular to a line at a point on the circle’s circumference only if the line is tangent to the circle at that point.)
“Is it possible to draw a circle that is larger than the inscribed circle but still fits completely in the triangle?” (No. The circle has already expanded to fit tightly into the triangle. If the radius of the triangle were increased, the circle would push outside the triangle. The inscribed circle is the largest one that can fit inside the circle.)
Conversing: MLR2 Collect and Display.
Listen for and collect the language students use to describe how the circles intersect the sides of the triangles. Capture student language that suggests that the circle is tangent to the sides of the triangle. Write the students’ words and phrases on a visual display. As students review the visual display, ask students to revise and improve how ideas are communicated. For example, a statement such as, “The circle touches the sides of the triangle” can be improved with the statement, “The inscribed circle is tangent to all three sides of the triangle.” This will help students use the mathematical language necessary to precisely describe the features of inscribed circles of triangles.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness


correct_option_index: 6