Unit 7, Lesson 5
Lines, Angles, and Curves
Integrated Math 2
5.1
Warm-up
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to elicit ideas about different kinds of lines and angles in circles, which will be useful when students explore inscribed angles, chords, and tangent lines in later activities. While students may notice and wonder many things about these images, the idea there may be many kinds of lines and angles in circles is the important discussion point.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). Students might first propose less formal or imprecise language and then restate their observation with more precise language in order to communicate more clearly.
Launch
Arrange students in groups of 2. Display the images for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
NoneWhat do you notice? What do you wonder?
Activity Synthesis
The purpose of this discussion is for students to share their thinking about the components of circles and to consider the ideas of others.
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 observe what is on display, and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the fact that there may be many kinds of lines and angles in circles does not come up during the conversation, ask students to discuss this idea.
5.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
The goal of this activity is for students to strengthen their understanding of the vocabulary terms from the previous activity. Students prove a property of congruent chords. In the process, they draw central angles and think about the relationships between radii, central angles, and arcs. The ability to understand and visualize central angles and arcs will be helpful when students study the relationship between central angles and inscribed angles in an upcoming activity.
Launch
Tell students that we define the measure of an arc as the measure of the central angle formed by drawing radii from the endpoints of the arc. Display this image for all to see:
Ask students, “What is the measure of the highlighted arc from point to point ?” (60 degrees.) “What is the measure of the highlighted arc from point to point ?” (120 degrees.) “What is the measure of the arc that goes from point to point through point ?” (180 degrees.) “What is the measure of the other arc that goes from point to point ?” (It also measures 180 degrees.)
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code the radii in the same color to clarify that they are all the same length.
Supports accessibility for: Visual-Spatial Processing
Supports accessibility for: Visual-Spatial Processing
Lesson Synthesis
The goal is to practice interpreting diagrams that include chords, arcs, and measures of central angles. Display this image for all to see:
Tell students that segment is a diameter. Then ask:
- “How many chords are drawn in this picture? Name them all.” (There are 3 chords. Segments and are all chords.)
- “Is angle a central angle? Why or why not?” (No. The vertex of angle is point , which is not the center of the circle.)
- “What arc and angle measures can you find?” (The chords and are congruent. This means that angles and are congruent because these central angles are created by drawing radii from the endpoints of the chords. So angle measures 50 degrees, as do the arcs between and and between and . The measure of the highlighted arc from to is 80 degrees because central angle measures 80 degrees.)
Student Lesson Summary
Diameters and radii are two types of line segments that appear in circles. Here are some additional geometric objects associated with circles.
A chord is a line segment with both endpoints on a circle. A central angle in a circle is an angle whose vertex is at the center of the circle. An arc is a portion of a circle's circumference that is between two points on the circle. The measure of an arc is defined as the measure of the central angle formed by the radii drawn to the endpoints of the arc. For example, in the image, the highlighted arc between points and measures 45 degrees because the central angle measures 45 degrees.
chord
central angle $PQR$
arc ; ;