Satellite Overflight 1

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Objective

SWBAT describe the relationship between the perimeter or area of an image under dilation and the perimeter or area of the pre-image using an equation or a verbal rule. SWU how dilating a shape affects the perimeter and area of the figure.

Big Idea

Students learn how dilating a shape affects perimeter and area as they design a sign large enough to show up on Google Earth.

Lesson Opener and Goal Setting

10 minutes

Lesson Opener

Today's lesson opener has two purposes, which I explain in the video narrative for this section.

The lesson opener follows our Team Warm-up routine.

 

Motivating the lesson.  

I ask students if they have ever looked at satellite imagery of their home or of the school using Google Earth.  I ask if any student knows how that satellite imagery gets updated.  I show the website for the NASA Earth Imaging Program (using the links found in the slide show for the lesson) and briefly describe the ASTER program.  I show the calendar for the ASTER program for the current month.  (Look! We can see where the satellite will be collecting imagery on any particular day of the year.)  I introduce the problem students are about to try to solve.  Suppose you learned that t a NASA satellite will be taking photos of your community next month, and the imagery will be used to update Google Earth.  What if a group of students conceived the idea of spelling “GO HUSKIES” on the football field on the day the satellite is scheduled to fly over?


Review learning targets and the agenda for the lesson (2 minutes).   I display the agenda and learning goals for the lesson as I distribute the problem.  I ask students to read them over and invite questions.  

Challenge Problem and Strategy Discussion

35 minutes

Have students work individually on the problem (10 minutes).  Tell students that today they are going to be practicing the Standards for Mathematical Practice as they work through a problem such as they might encounter in the real world  (MP1).  Display the Standards for Mathematical Practice, highlighting the standards to be highlighted in this lesson.  Remind students that real-world problems are not always as straightforward as they seem.  Display the list of problem-solving strategies.  Tell students that if they are not sure where to start, they should try one of these strategies.

Most of the problem-solving strategies listed in the presentation come from Georg Polya’s book, How to Solve It.  

Ask students to keep working hard for 10 minutes.  Students who complete the problem in less time should try to verify their solution by another method.   As students work, circulate through the classroom observing what they are doing.  Offer encouragement and help students get started, but do not be distracted from your primary purpose, which is to plan the class discussion which is to follow.  You may choose to use the accompanying planner (MTPGeometry_Activity6_0_4_StrategyTalkPlanner.docx) to organize your notes.  Keep your goals for the class discussion in mind:

  • Highlight effective problem-solving strategies
  • Give students the opportunity to explain why they think a solution is in error or why they think it is correct.(MP3)
  • Help the class to realize that the area of a figure does not vary at a constant rate with the scale factor and invite students to suggest ways to discover what the true relationship between area and scale factor might be.

 

Have students share their work in pairs and teams (5 minutes).  To get students thinking before holding a class discussion, ask students to explain their solutions to a partner and with their teammates.(MP3

I used a Kagan Structure (Timed Pairs Share), with a few modifications.  The instructions are in the presentation MTPGeometry_Lesson6_0_4.pptx.


Conduct the classroom discussion (Strategy Talk) (20 minutes) (MP3).  Display student work using a document camera and overhead projector.  Plan the sequence in order to bring out key learning points.  Watch the time.  If necessary, reserve the last 5 minutes for a demonstration that will expose the misconception that the relationship between area and scale factor is linear.   Suggested questions:

How did this student find the answer?

What parts of the student’s solution do you think are correct?  Why?

What parts of the student’s solution do you think may be incorrect?  Why?

How could you check the answer?

If necessary, guide the discussion to focus on the erroneous assumption that area varies linearly with scale factor:

How do you know that the area of the enlarged sign is five times the area of the original?

To expose the misconception, ask students to draw the dilation of the block letter ‘O’ (as shown in the diagram that accompanies the problem) on graph paper using a scale factor of 5, then find the perimeter and area.  If time is short, ask students to consider a 2 x 2 square:

What is the area of the square?

Suppose the square is dilated with a scale factor of 5.  Now, what is the length of each side?  What is the area?

To encourage participation in the class discussion, my classes play a game we call Math Ball.  Instructions for this game are found in the document MathBall.docx.  

Suggested sequence for showing student work:

  • A solution with merit but with errors that can be revealed in discussion.
  • A solution that uses the approach taken by most students.
  • A solution that exposes the misconception that area varies linearly with scale factor.

 

I do not have students present their own work, because I want other students to explain the thinking behind the solution.
When displaying a student’s work, I make sure that the author’s name is not visible.  This allows the author to take ownership of the work or not, as he or she chooses.  

Lesson End

8 minutes

Have students reflect on the lesson in pairs and teams (5 minutes).  Students share in pairs, then in teams.  Teams report out by listing three important things they learned on the white board.

I used a Kagan Structure (Timed Pairs Share), with a few modifications.  The instructions are in the presentation MTPGeometry_Lesson6_0_4.pptx.

 

Assign Homework (3 minutes).  No homework was assigned at the end of this lesson, as students began standardized testing the following day.  I told them all to get a good night’s sleep and eat a healthy breakfast.