Lesson 10

The purpose of this warm-up is to elicit the idea that the more sides an inscribed polygon has the closer it comes to approximating a circle, which will be useful when students calculate perimeter and eventually \(\pi\) in later activities. While students may notice and wonder many things about these images, approximating a circle is the important discussion point.

This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

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 the things they notice and wonder with their partner, followed by a whole-class discussion.

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 respectfully disagree, ask for clarification, or point out contradicting information.

If the word inscribed does not come up during the conversation, ask students if they remember the word and record the definition near a relevant item on the list of things students noticed. (We say a polygon is inscribed in a circle if it fits inside the circle and every vertex of the polygon is on the circle. We say a circle is inscribed in a polygon if it fits inside the polygon and every side of the polygon is tangent to the circle.) Leave the responses visible throughout the lesson, as students will refer to them during the next activity and the lesson synthesis.

Demonstrate for students how the inscribed polygons change as the number of sides increases using the applet.

During this activity students are asked to compute the perimeter of inscribed regular polygons given only the radius. They will need to figure out that drawing in the altitude will form a right triangle. Drawing this auxiliary segment shows students recognize the important structure of right triangles to calculate missing information (MP7). In this activity students are building towards an equation to approximate the value of \(\pi\) which they will complete in the next lesson.

Provide the definitions of a regular polygon (a polygon where all of the sides are congruent and all the angles are congruent), pentagon (5-sided polygon), and decagon (10-sided polygon) as needed.

Invite students to share their strategies. Focus on students who repeated the same strategy for multiple questions.

Use the applet to demonstrate what happens with the perimeter of the inscribed polygon as the number of sides increases. 

Ask students what they noticed. Connect the generalizations they make with the noticing and wondering they did in the warm-up. If not mentioned by students, ask students what the circumference of the circle is. (\(2\pi\)) Build anticipation for calculating the value of \(\pi\) in the next lesson.

If students are not using workbooks tell them to be extra careful to put this work in a safe place because they will use it in the next lesson.

This activity requires more interpretation than a basic right triangle problem. First, students need to draw the diagram and figure out how to label it. Second, students need to recognize that the units do not match and figure out how to convert to a common unit. At this point the problem looks like a basic right triangle (with large numbers) and students will be ready to use their usual techniques of trigonometry and the Pythagorean Theorem.

Monitor for groups that convert to feet and groups that convert to miles.

If angle of descent isn’t familiar vocabulary, after 1 minute of quiet work time ask students which side will be longer, the horizontal or the vertical? (Horizontal because it is in miles.) Draw a diagram. Invite students to discuss how to label the diagram with important information including the angle of descent.

Prompt students to attend to precision with units.

Invite the previously selected groups to share their process of converting to one of the common units. Acknowledge that both methods are valid. Ask which unit is clearer for answering the question of how far the plane traveled (miles).