• Spot check student responses to the question “What is the relationship between a chord and a secant?” on slide 3 for the proper association between line and line segment.
• Observe students’ performance on the Theorem Summary Review Activity, Secants and Tangents Independent Practice worksheet, and the Secant and Tangent Extension Problem for correct correspondence between components of circles.
• Observe students’ performance on the writing application assignment and look for original examples. Explanations should go beyond restating the theorems and include language that reflects understanding of the theorems.

Option 1

Use the Technology Explorations 1, 2, and/or 3 listed in the Related Resources of Lesson 2 to introduce the concepts for Lesson 2. These activities are exploratory in nature and guide students to the discovery of the important theorems and rules discussed in this lesson. After students take time to work through these activities, introduce the lesson presentation in Option 2 to fill in any learning gaps and to verify that students received the appropriate information to work independently.

Option 2

This instruction option skips the introductory activities and begins with direct instruction of concepts. Make sure students have their Concept Builder worksheets (M-G-6-1_Concept Builder.xls). Students will use this chart to record important definitions, formulas, and theorems throughout the unit. This document will help students organize important information so they can more easily locate it when working with the problems. The document will also be a great review and study tool for the assessments during and after the unit.

The instructional portions of this lesson should be taught using the PowerPoint presentation (M-G-6-2_Lesson 2 PowerPoint.pptx) so that you do not have to draw each diagram and write out each definition as it appears in the lesson. All examples and concepts are already in the Lesson 2 PowerPoint. So you can go through the PowerPoint, discuss topics as they come along, give students time to record information, and model the concepts for students. If whiteboard technology is not available, possible alternatives include printing slides onto overhead transparencies or drawing examples on the board.

When students have their Concept Builder worksheets ready, open the Lesson 2 PowerPoint. Introduce the lesson using the first slide: “Today we are going to continue our study of circles and their properties. This lesson is about angles and arc measures made when a circle is intersected by tangents and secants.”

Lesson 2 begins with slide 2, introducing two important vocabulary terms. Display these for students to record on their Concept Builder worksheets.

After the class discusses the definitions on slide 2, slide 3 introduces a critical-thinking question for students to think about and then answer independently. “What is the relationship between a chord and a secant?”

Have students write their thoughts on a piece of paper. After 2 to 4 minutes, have a class discussion about their thoughts. (Possible responses: They are the same thing, but a chord is a segment and a secant is a line. Recalling earlier concepts, a segment is actually a portion of a line, thus a chord is just the inner portion of the secant. Therefore, every secant contains a chord.)

Slides 4–7 introduce the important theorems of the lesson. Make sure to give students time to copy these on the Concept Builder worksheet while you discuss the theorems.

Note: Slide 5 asks students to conjecture about how to apply this theorem to the other side of a circle. This question is checking that students can apply the concept to all circles (since not all circles will have their secants and tangents drawn in the same orientation as the original theorem). Students should come up with = (m arc ACB).

Note: Slide 6 has students draw the three situations mentioned in the theorem. Students can do this on a piece of paper. The goal is to have students translate the verbal description into a visual representation. Thus, when students see slide 7, for example, they can apply the rules and formulas to their drawings. This gives them different perspectives on the rules.

Activity: Theorem Summary Review

Give each student a Circle Angle Relationships Summary worksheet (M-G-6-2_Circle Angle Relationships Summary.xls). The goal of this activity is to have students create their own summary of the theorems discussed in this unit. Students should work with a partner to discuss their learning. Tell students to refer to their Concept Builder worksheet to look up theorems/concepts they might need. During the activity, walk around to observe student discussions and summaries, and guide students in the appropriate direction. After student discussions begin to wind down, bring the class together to have a large group discussion about what students found.

Instruct students to add to their summaries anything that they may have left out earlier. Also inform students that though they may have used different labels and orientations with their circles, that does not mean they are incorrect (M-G-6-2_Circle Angle Relationships Summary KEY.xls).

Continue Lesson 2 with slides 9–11. The slides contain examples of how to use the theorems to solve problems. Print out slides 9–11 for students to follow along with your modeling. Students should follow along while you demonstrate the procedures and associated computations on the board.

Answers to the examples are as follows:

Slide 9: Example 1: 97°

Slide 10: Example 2: 248°

Slide 11: Example 3: 150°

Use slides 12–14 to demonstrate the use of algebra in solving problems similar to the ones in slides 9–11. Provide students with a printout of slides 12–14 so they can follow along with your modeling. Students will need to apply the theorems and their algebra skills to complete the problems accurately. Option: Have students work in groups or pairs to complete the problems and then discuss the answers as a class. If using this option, make sure to walk around to the groups to clear up any inaccuracies.

Slide 12: Example 4: x = 14

Slide 13: Example 5: A = 60°

Slide 14: Example 6: x = 23

Extension:

• Writing application: Have students choose one of the three theorems presented in Lesson 2 and explain it using their own words, as if trying to explain the theorem to a peer who is new to the concept. This can be used as an entrance or exit ticket.
• Problems: Using the Secant and Tangent Extension Problem handout (M-G-6-2_Secant and Tangent Extension Problem.doc and M-G-6-2_Secant and Tangent Extension Problem KEY.doc), have students solve the problems. This problem applies multiple theorems in one diagram and requires students to apply the rules in an unfamiliar context.
• Independent practice: Use the Secants and Tangents Independent Practice handout (M-G-6-2_Secants and Tangents Independent Practice.doc and M-G-6-2_Secants and Tangents Independent Practice KEY.doc) to give students the opportunity to apply the lesson concepts independently. Use it as a homework assignment or in class as independent work.