Page 95 - IM_FL_Geometry_Print Sample

3. Trace angle on tracing paper along with one of the circular arcs it encloses. In your group, put your tracing papers in order of increasing arc length to radius ratio. Be prepared to share your reasoning.
Student Response
1. Sample response: See diagrams.
2. Let . Then and . Therefore .
3. The order of increasing arc length to radius ratio is smaller acute, larger acute, smaller obtuse, larger obtuse.
Activity Synthesis
One important idea for the discussion is that for a given angle, it doesn’t matter what size the circle is. The ratio of arc length to radius is the same. Another important idea is that angles of di erent sizes have di erent arc length to radius ratios that increase as the angle increases. Select previously identi ed students to share how they quanti ed these ratios by measuring lengths. Also include discussion from students who proved that the arc length to radius ratio for a given angle is the same regardless of the size of the circle. Here are some questions for discussion, along with images to display:
• “If you measure the arc length as 8 centimeters and the radius 6 centimeters, then the ratio of arc length to radius is or about 1.33. What does that number mean in terms of the picture?”
(That number means that the angle always traces out an arc that is 1.33 times the radius.)
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Teacher Guide