# Infinity and Beyond!

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## Objective

SWBAT apply the appropriate formula to solve problems of finite and infinite geometric sums.

#### Big Idea

What's beyond infinity?! Explore infinite geometric sums to get a mathematical perspective of infinity.

## Set the Stage

15 minutes

I start this lesson with an Infinite geometric series problem on my front board.  I discuss my reasons for choosing this plus standard in my Infinity and Beyond video.   Most students will already begin trying to solve the problem even before the bell, but for those who don't I challenge the whole class to solve the problem. (MP1)  I encourage any collaboration and/or appropriate discussion my students want to engage in as they work to find a solution.  I generally let them work for a 4-5 minutes before asking if anyone has been successful.  There are always a few students who are certain they either have found or can find the answer if they just have enough time with their calculators, but most students recognize the futility of using brute force to find this solution.  I ask them to articulate what they're having difficulty with, my expectation being that at least a few students will say that they can't solve the problem because the numbers never stop.  I ask if anyone has any suggestions or other options we might try and when it's clear we've exhausted our possibilities, I give a simple derivation of the formula for the sum of an infinite geometric series, using volunteers to help me through each step. I've made an Educreations video to explain this further.  Some students get stuck on the idea that the value of (r^n) gets very small as n gets very large (infinite), so that we can eliminate it from the formula.  I give some concrete examples like the mass of Earth compared to the mass of a marble, or the size of an ant compared to the size of an elephant to try to demonstrate that sometimes things are so small that they can be ignored!

## Put it into Action

35 minutes

You will need copies of the geometric series handout for this section of the lesson.

Independent work 10 minutes: For the first section I tell my students they will be working individually with several geometric series.  I distribute the geometric series handout and ask if there are any questions. I tell them that their assignment for now is simply to identify whether each series is finite or infinite and write a brief explanation of how they made their decision. (MP7)  While my students are working I walk around giving encouragement and redirection as needed.  When everyone is done or after about 15 minutes I call time and tell my students they will be working with their right-shoulder partner for the next part of the lesson.  (I make a note of those students who may need additional support because they were unable to finish the assignment.)

Teamwork 20 minutes: I explain that each team now has the privilege of solving for sum of each geometric series. I again walk around while my students are working and offer encouragement and assistance as needed. (MP1, MP2)

Whole class 10 minutes:  Instead of collecting these papers and grading them myself, I have the teams do a self-check as I go through the answers with the class.  This give them immediate feedback and puts the responsibility for accuracy and precision squarely on their shoulders.  (MP6)

## Wrap it Up

5 minutes

To close this lesson I ask my students to individually summarize the similarities and differences between the finite and infinite geometric sum formulas.  I allow them to use any of the graphic organizers I have available to organize their thoughts but ask that the summary be written on lined paper using complete sentences as well as proper grammar and vocabulary.  This is their ticket-out-the-door and gives me an opportunity to see if my students truly understand geometric series.