##
* *Reflection: Developing a Conceptual Understanding
Infinity and Beyond! - Section 3: Wrap it Up

Some of the compare/contrast summaries surprised me with their clear understanding about r^n approaching zero so long as the original ratio is between -1 and 1. I overheard one student explaining it to a classmate like this; "Just think about if you start with a cake and you take one fourth of it, then cut that piece into four and take one of those little pieces, then cut that into four...pretty soon you won't have a piece big enough to even cut!" I really liked the analogy and asked her to share it with the rest of the class. The only downside was some groaning from students that the cake story made them hungry!

*Ticket out the Door!*

*Developing a Conceptual Understanding: Ticket out the Door!*

# Infinity and Beyond!

Lesson 9 of 11

## 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.

*55 minutes*

#### Set the Stage

*15 min*

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!

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#### Put it into Action

*35 min*

*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)**

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#### Wrap it Up

*5 min*

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.

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- UNIT 1: First Week!
- UNIT 2: Algebraic Arithmetic
- UNIT 3: Algebraic Structure
- UNIT 4: Complex Numbers
- UNIT 5: Creating Algebraically
- UNIT 6: Algebraic Reasoning
- UNIT 7: Building Functions
- UNIT 8: Interpreting Functions
- UNIT 9: Intro to Trig
- UNIT 10: Trigonometric Functions
- UNIT 11: Statistics
- UNIT 12: Probability
- UNIT 13: Semester 2 Review
- UNIT 14: Games
- UNIT 15: Semester 1 Review

- LESSON 1: Whatchamacallit
- LESSON 2: Puzzle it Out
- LESSON 3: Polynomial Rewrite
- LESSON 4: More Puzzles
- LESSON 5: Rational Rewriting
- LESSON 6: Formula 1
- LESSON 7: Geometric Series Formula, Too
- LESSON 8: Working the Formula
- LESSON 9: Infinity and Beyond!
- LESSON 10: Algebraic Structures Review
- LESSON 11: Summative Assessment