Moon Shadows, Cones, and Pyramids
Lesson 4 of 6
Objective: SWBAT calculate the dimensions of cross-sections of cones and pyramids. Students will understand how the cross-section of a cone or pyramid varies with distance from the apex.
Opener and Goal Setting
The purpose of today's warm-up problem is to help students see how a cone can be generated by dilating a plane figure from a point of dilation that is not in the plane. Viewed this way, every parallel section of the cone is a dilated image of the base.
Students are asked to describe the solid that results from transforming a circle in each of three ways. Two—translating the circle perpendicular to the plane in which the figure lies and rotating the circle around its diameter—were covered in an earlier unit. The third—dilate the circle using a point not in the plane, is a new idea.
This lesson opener follows our Team Warm-up routine.
Following the warm-up problem, I use the slide show to display the agenda and learning targets for the lesson. I tell the class that today we will be learning about the properties of cones and pyramids. We will use those properties to understand the geometries of a solar eclipse and of a laser beam.
The purpose of this activity is to get students to think about the properties of cones as they use them to analyze a real-world phenomenon (MP4). In particular, I want students to see that a cone is formed when a plane figure (like the great circle of the Moon) is dilated using a center of dilation not in the plane. (In the case of the Moon’s shadow, the center of dilation and apex of the cone is a point in space determined by the relative positions of the Sun and Moon.) The parallel sections of a cone are the dilated images of the original plane figure. (For example, the region on Earth from which a total eclipse can be viewed is approximately a circle—actually, it is a curved surface—and it varies in size as the distance between the Earth and Moon varies.)
Using the slide show, I display three questions about the geometry of solar eclipses. I tell the class that they will be drawing diagrams to answer those questions following the video clip. After students have a minute to read the questions, I show the NASA Spitzer Space Telescope video using the link in the slide show (about 3 minutes). As the video is playing, I distribute the materials for the activity: a copy of the Shadow Cones and Light Cones Activity handout (the 3 questions), the Geometry of Eclipses blog, and the Eclipses in 2013 Article.
I obtained the video from the Internet at NASA Spitzer Space Telescope.
Following the video, I display the rules for the activity using the slide show. I give the class 15 minutes to work and start a timer. The video does a good job explaining the geometry of a solar eclipse, but students will need to extract information from the blog and article to answer the questions.
This is an unstructured cooperative learning activity. To answer all three questions within the time limit, teams will have to get organized and divide up the work. If students are working together to interpret the written material and understand the geometry, they are getting what they need from this activity (MP1). As an incentive, I offer a chance to earn a reward (bonus points, candy, etc.) for presenting an accurate explanation to the class.
At the end of the 10 minute time limit, I call the class together. I choose teams at random to present answers to the questions (if all teams are working well) or based on the quality of the
answers they have prepared (if it is one of those days). Teams use the classroom document
camera to display their drawings to the class. Explanations can be written or oral. The class decides whether the answer is understandable, while I am the final judge of whether the answer is accurate. Ideally, at least one team will come up with an accurate answer to each question, but I am prepared to provide explanations using drawings in the slide show.
Before class: Make one copy of the article and the blog for each cooperative learning team (plus a couple extra, in case a student wants to take one). I make one copy of the questions per 2 teams and cut them into half-sheets. I test the video clip to be sure that the PC I use for classroom presentations will play it.
Bouncing Lasers Off the Moon
Because I expect to give students a lot of help in using geometric models to explain why the discs of the sun and moon appear to be the same size, this activity gives them an opportunity to solve a different problem using similar geometry (MP4).
The activity is conducted in pairs using a variation of the Team Jigsaw format. One student sketches and labels a diagram to illustrate the geometry of the situation. A second student is responsible for setting up the problem and computing an answer. Both students check each
I consulted the following website for information on lunar ranging:
Before Class: Make one copy of the Bouncing Lasers Off the Moon Activity handout for every two students and cut into half-sheets.
During this activity, we summarize what we have learned using the guided notes for the unit. Those notes--Properties of Solids Notes, Relating Volumes Notes--are uploaded as a resource under the heading of the unit. During this lesson, we will complete the notes on properties of cones and pyramids only.
I distribute copies of the guided notes for the unit (one member of each team retrieves copies from the Resource Center), while I display them on the front board. I use a Mimeo Teach system with an LCD projector. I tell the class that we are going to summarize a few facts about cones and pyramids, most of which they have already encountered. Note-taking using guided notes is a classroom routine, so my students already know what to do. More information on how I use guided notes in my classroom can be found in my Strategies folder.
Before class: Reproduce one copy of the notes (both documents, stapled, 3-hole punched) for each student.
The homework provided with this lesson consists of three problems. The first problem asks students to find the radii and areas of different sections of a cone. The second problem asks students to perform this skill in a real-world context. The third problem asks students to use the geometry of a light cone to explain the relationship between light intensity and distance from the source.
I display the Lesson Close prompt using the slide show for this lesson. Students write their answers in their learning journals. I remind students to turn in their work from the Bouncing Lasers Off the
Moon activity as an exit slip. After class, I look over the students’ work to see who is getting this and what problems others are having.