Active+Learning+Investigation

Active Learning Investigation Introduction to Converging Infinite Geometric Series

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 * Overview: **
 * Name of Activity:** Tear It Up!! Introduction to Converging Infinite Geometric Series


 * Who is this for:** I would use this investigation for __Algebra 2__ and __Pre-Calculus__ students.The idea of a geometric series could easily be shown in earlier math classes but I would use this lesson as an intro into a deeper understanding of the formulas.


 * Time Frame:** This activity would take around 30 minutes.


 * Summary:** Students will use a piece a paper to help them find the summation of converging infinite geometric series. If they are given the geometric series 1/2 + 1/4 + 1/8 +1/16 +... they can use the piece of paper to model this series. They would first fold and tear the paper in half creating two halves. Then they would continue to fold the next piece creating a 1/4 piece of paper. They will continue doing this until they are no longer able to fold the paper. With this physical evidence in front of them students can get an idea that the summation of the series 1/2+1/4+1/8+... is 1. We would then do this activity with different fractions and different common ratios. The goal is for students to become familiar with the concept of convergence infinite geometric series and how they approach a sum. After working with this activity I would introduce the equation that is used to find the sum of a convergence series. It is a simple equation therefore this activity is intended to give the students a physical representation of what is actually happening.


 * Rationale and Relationship to Standards **
 * Important concepts:** Infinite Geometric Series, Common Ratio, Convergence


 * Real world importance of concepts:** Geometric series is a fancy term for pattern. Mathematics is used to help describe the world. Patterns are a straight forward example of a real world concept that are often modeled. In other words, I want students to make the connection between patterns we see everyday and how mathematics takes that pattern and make a formula to model it. Geometric series are common in our everyday lives and I want students to make those connections between what they witness (in finance, in nature, in computers, in social media) to mathematics.

[|CCSS.Math.Content.HSS-IC.A.2] Decide if a specified model is consistent with results from a given data-generating process
 * Relevant Standards: (There are not standards for Pre-Calculus. This standard is from statistics and probability)**

- 3 sheets of blank paper for each student - Handout investigation for each student (word attachment below)
 * Description **
 * Materials:**


 * Direction:** Students will be placed into pairs or groups of three depending how the numbers work out. Each student will be responsible to fill out the worksheet but they can share the responsibility of tear the paper. Directions should be read with students. Problem number one can be started as a class to get students a reference of what the investigation is asking. Therefore, give students thinking time for parts 'a' and 'b' from problem 1. Then go over those as a class.

Next, take a sheet of paper, fold it in half and tear it into two halves. On one of the pieces of paper write 1/2 on it. Then take the other 1/2 sheet and fold that in half, tear it, and label one of the pieces 1/4. With the remaining 1/4 repeat the process. Show the class the first three or four fold and tears to let them get the idea of what is happening. Then ask them to repeat this until they are no longer able to fold the piece of paper. Once students have complete problem 1 there could be a discussion about what happened. What was the sum of the series? How do you know? If there is a big discrepancy between students progress on problem 1 you could use problem 1 as a check point. The student have to get the teacher's approval before moving on.

Problems 2 and 3 ask the students to try their own fraction out with their own common ratio. Make sure the common ratio is less than 1 and has a numerator of 1. Beware!! When doing fractions that are less than 1/2 not all the paper is used. For instance, if I did a pattern starting at 1/3 and have a common ratio of 1/3 student need to understand that after getting 1/3 of the paper their next fold is 1/3 of 1/3 creating a piece that is 1/9 of the original paper. It is important they don't do 1/3 of the two thirds of paper that is still in their hand. They will need to make two pieces of 1/3. One piece to use as their collection of fractions and the other piece as their piece they will fold to make the next tear. Once students have completed 2 and 3 they will move on to the reflection.

The reflection should be answered on their own and quietly. This is for the students to reflect and it will be used to assessed to see if the learning targets were met.

Lastly, make sure the room is cleaned up!!

http://smartboardsmarty.wikispaces.com/file/view/TearItUP.pdf/385594488/TearItUP.pdf A pdf that provides the original lesson I got the idea from.
 * Reference:**

I will use problems 3 and 4 from the reflection as the assessment to check if students achieved the learning targets. Each question has 2 parts and I will give a check mark for each part met, with a total of 4 check marks. The check mark means they met the learning target. The assignment itself will get a certain amount of points for completion but questions 3 and 4 from the reflection is where I am assessing the students knowledge of the learning targets.
 * Assessment **



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