# [SCBG] Concept Exploration: “What’s Infinity Times Infinity”

This year I’m going with a Skills and Concept based grading system.  The skills will be like your typical SBG system.  The concept portion will allow us to dig a little deeper into the conceptual side of algebra.  Here’s the first concept lab/project/exploration/whateva you wanna call it.

## Overview

This will be in the second week of the school year for algebra I.  After we have talked about operations and evaluating expressions.  The exploration’s main focus is the real number system and the number line.  This exploration will take two class periods, plus a short period of time for a small individual summative assessment.

## 2. Individual Reflection

Give the students 10 minutes to reflect on the video in writing.  They may answer the prompts or write their own thoughts, but they must write in complete sentences.  Prompts include:

• Is infinity a number?
• Can we count to infinity?
• How big is infinity?
• Does “infinity plus infinity” make sense?
• Which group of numbers is bigger: whole numbers or natural numbers?

## 3. Socratic-ish Discussion

At this point, we should have about 30 minutes left in our 45-minute period.  I would aim to spend 10-15 minutes discussing what students wrote.  Let them guide the discussion.  Eventually, steer it towards that last question so we can talk about the real number system.

## 4. Vocab & Day One Wrap-Up

Flip to page two of the exploration and define the various real number classifications for the students.  Ideally, the structure would have been brought up in our discussion.  The cool thing here is we’re talking about cardinality in algebra as a way to set up the real number system.  That’s some high-level stuff!

## 5. Day Two: Individual Work

Day two is a chance for students to work through some exercises based on the real number system, the number line as well as basic integer and fraction operations.  Really, this is a chance for me to get around to each student and see where they are with their abilities to perform arithmetic with negative numbers, multiply and divide fractions, etc.  However, there will be some gems thrown in there for the students who do have their skills down.  This one is my favorite:

True or False: When you multiply two irrational numbers, the result is always an irrational number.

I am really looking forward to seeing how my Algebra I students grapple with that.

## 6. Day Three: Short Assessment

On the third day, I anticipate a short, low pressure assessment to follow up the lab.  Something like 2-4 typical problems related to classification of numbers and the number line, perhaps 1-2 arithmetic problems and maybe an essay question just to get the kids used to the idea of writing in math class.  This would be worth no more than 20% of the entire lab.