His Treasure Seekers

Our Classical Conversations year is starting, and as I take my Challenge class through The Math Map (TMM), as well as using it at home with my four youngest children, I’m sharing some tips each week from things I’ve learned over the last 3 years of using TMM.

Tips for Lesson 2: Number Notation

• Euler diagrams are introduced on the i² page. You’ll see them pop up throughout TMM, as well as Euler’s number. Euler was German, so his name is pronounced like “Oiler,” not “You-ler.”
• Fun fact: Speaking of Germans, the reason Z is used for the domain Integers is because it comes from the German word “Zahlen,” meaning “numbers.” (If you want to learn more about domains, you might find this article interesting.)
• On the i² and i³ pages there are lots of symbols or concepts that may be new to kids who haven’t done TMM before. Be sure to start by asking them to look for something familiar. It helps these pages feel less overwhelming!
• If you’re overwhelmed, remember you don’t have to master everything on your first trip through TMM. Remember Andrew Pudewa’s EZ+1 concept from IEW? Pick one new thing to try to understand this time. I promise, next time through, more things will feel familiar. (My own children and the Challenge students I’ve tutored who’ve done two or more tours of TMM can all attest to that!)
• If you’re not familiar with systems outside of base 10, take some time to attend to the top box on the i³ page before looking at the other base systems. Find as many patterns as you can. then compare what you’ve discovered to one of the other base systems on the page.

• Regarding the Charts. If you’re overwhelmed by the i² and i³ pages, proceed to the Charts with caution. They are detailed and thorough, as their purpose is to explain all the concepts related to each lesson that one would hope a Challenge IV student would be able to understand and articulate after years of going through TMM. Because of that, they can be a bit intimidating. While sometimes they are necessary for filling in the lesson pages (look at the title bar across the top of the lesson page and then find the corresponding chart), I would estimate about 90% of what I’ve needed to complete lesson pages can be found on the i² and i³ pages.

• Remember the Flashcards! Even if you and your students just learn 1-2 a week, that’s more than you would know if you did none at all! There’s a box to check them off on the first page of each set of four lesson pages (1, 5, 9, and 13).

• Don’t forget about The Math Map Companion on CC Connected. You can pull up the pages from each lesson in any available domain and click on the different sections of each page to get more explanation about the concepts. (I’ve blurred these screenshots to protect copyright but still give you an idea of what I’m talking about.)

Other helps: When I first started to dive into TMM, I jotted down notes of things that might be helpful as I taught each week’s concepts. Here’s what I found for Lesson 2:

• Here’s my playlist of all the Book Club episodes that are about Lesson 2: Number Notation: YouTube Playlist of Book Club videos about Lesson 2
• Want to explore different base systems? Try these videos: “introduction to number systems and different bases” and Base numbers, Place Value, Number Notation on Math Map
• Confused about imaginary numbers (these were totally new to me when we first started TMM!) I found these videos helpful: Why is the square root of -1 i? and Imaginary Numbers Deep Dive (specifically references TMM)

• On p. 1 the students are told to study and memorize the pattern for imaginary numbers. I find it really helpful to walk through each step and help them understand the reasoning behind each step. (A Challenge A tutor shared something similar to this on Facebook, and I’ve tweaked it a little as it made sense to my kids).
  • i⁰ = 1 (because the “zeroth” power of anything = 1)
  • i¹ = i (because the first power of anything = itself)
  • i² = -1 (because the whole basis for imaginary numbers is that we have to go beyond Real numbers and imagine the square root of -1)
  • i³ = -i (because i³ = i²  x i¹ =, or -1 x i)
  • i⁴ = 1 (because i⁴ = i² x i² , or -1 x -1) 
  • i⁵ = i (because i⁵ = i⁴  x i¹ =, or 1 x i)
  • And the pattern just continues…
• Finally, I love this visual for Euler diagram of domains. I’m collecting boxes as we speak to try to create this for my kids at home. I want to write some numbers of scraps of paper and have them figure out which box to put them in.