# Geometric Sequences

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## Objective

SWBAT apply a geometric sequence to a real world scenario.

#### Big Idea

If you folded a large piece of paper 50 times its thickness would be about 77 million miles...why?

## Launch

20 minutes

If possible, just show students the question first, "If you folded a piece of paper in half 50 times how thick would it be?"  Let students try to understand this question first with a partner (MP1).  Walk around and listen to student responses.  Many students will say that it will be 50 times as thick as it was to begin with.  They may also reason about the thickness of a piece of paper (approximately 1/10 of a millimeter).  Once students have had a few minutes to reason about the problem put up the answer: about 70 million miles.  This answer is usually shocking to students and they can't believe it.  So their task is to begin to investigate why this is true.  Tell students to organize their thinking with their partner in a way that can be shown on the document camera later in the class.  Students should show their mathematics and be able to explain their reasoning to the rest of the class (MP3).  Tell students that not everyone will come up with an answer and that is ok.  You are more concerned about how they reason about the problem and begin to put together a solution (MP1).

Because the answer is so large, students need to begin thinking how the paper could become so thick by folding it in half.  Look for students who are starting to think about the effect of doubling a number 1, 2, 4, 8, 16, 32 (After only 5 folds the paper is already 32 times as thick).  If students are having difficulty reasoning about this concept (MP2) encourage them to actually fold a piece of paper in half several times.*

*As will be shown later in the lesson the world record for folding paper is 13 folds.  Most students will only be able to fold a regular piece of paper about 7 times.  Some students may know that there is a limit to the number of times a piece of paper can be folded.  If this comes up explain that this is a theoretical question.

## Discussion

10 minutes

Have several pairs of students show their work starting with students who only reasoned about a few folds to those who made more progress.  You can tell students the average thickness of a piece of paper but try to resist showing them the solution to the problem.  If no students are able to show why the answer comes out to 70 million miles that is fine.  After more instruction regarding exponential growth they may be able to revisit this question and determine the solution.

Through this discussion you want all students to see how a geometric sequence is developed.  Scaffold the student presentations to show at least the first several terms of the sequence and how there is not common difference.  In this type of sequence there is a common ratio.  Students will be able to see how this type of sequence grows slowly at first but then gets large very quickly (MP2).

Solution: There are approximately 254 pieces of paper in one inch.

(2^50)/254=A gives the number of inches

A/12=B gives the number of feet

B/5280=C gives the number of miles (69,960,176)

It is also interesting to see that this "paper folding" problem has been solved by others.  See the slides in this area.  The first shows Brittany Gallivan who was able to fold a large piece of paper 12 times (the 11th fold is shown in the picture).  She also wrote a mathematical formula to determine the number of folds.  Google Brittany to find out more information to share with your class if you have time.  The second shows a video of the record being broken by students in Massachusetts in 2012.  There is also more information about this feat available online if you have students that are interested.

## Closure

10 minutes

Have students do a think-pair-share around this picture.  Let them try to write down the number of squares in the first three drawings first.  Once they are able to reason about the pattern (MP7), they should be able to figure out how many squares are in the fourth and fifth drawing without counting.  When students share out about their thinking connect this problem back to the paper folding question.  Show them that if they list the number of squares in the 5 drawings it forms a sequence.  This sequence has a common ratio of 5.  Each drawing has 5 times as many squares as the picture before.