His Treasure Seekers

Since Classical Conversations released The Math Map in 2023, I’ve gone through the 30 lessons at least 5 times in various forms, as a tutor, a teacher at home, and “lead learner” trying to understand concepts I never learned in school myself. This year, as I prep each week (taking my Challenge class through it on Community Day, in addition to using it at home with my four youngest children), I’m writing a post on each lesson, sharing things I’ve found helpful or offering some suggestions in hopes of making it less intimidating to those just getting started.

Tips for Lesson 7: Exponentiation

• Invention (i² page) This is a great time to discuss linear and logarithmic scales using the 5 Common Topics.
  • DEFINITION: Linear scales have a constant difference between intervals; Logarithmic scales have a constant ratio between intervals.
  • COMPARISON: How are these scales the same? How are they different?
  • Under what CIRCUMSTANCE might you use a linear scale? (ruler, measuring cup, thermometer…) When might you use a logarithmic scale? (Richter scale, pH, decibels… anything that covers large ranges of numbers. Ask your favorite AI site for examples and explanations if you’re struggling.)

• Memoria and Arrangement(i³page) The arrangement of this page is very similar to the one in the last booklet, so it should look familiar. Probably the most important concept of this lesson is the relationship between exponents, roots, and logarithms, so definitely spend some time discussing the three expressions of that relationship using base, index, and power as expressed in the top row: bⁿ=p, log_bP=n, ⁿ √p=b.

• How do we read these? Using 4³=64 as an example, we would say:
  • The 3^rd power of 4 is 64.
  • The 3^rd root of 64 is 4.
  • Log base 4 of 64 is 3.
  • NOTE: the index is not the power! The creators of The Math Map have tried to express the importance of being intentional about our words, so while it’s extremely common to say “4 to the 3rd power is 64,” that tends to make us (and our students) think that 3 is the power, rather than 64. I really like Dr. Gilpin’s explanation of this relationship. Even if you don’t want to watch a full episode of The Math Map Book Club, I recommend watching the part where she discusses it (cued here):

• Charts – Here are a few things that might be helpful to know before diving into the lesson pages:
  • C7:1 ATTEND to Notation and Standards (particularly Natural Log – ln=log_e)
  • C7:2 Looks at the use of exponents in different domains
  • C7:3 “Simplification Process” will come up in the lesson pages; the “Power Chart” in the bottom right hand corner will be handy for reference
  • C7:4 Similar to i³ page

Other help as you explore Lesson 7:

• I have found The Math Map Book Club to be a great resource, but there are so many episodes out there, it can be a bit overwhelming. If you just want to focus on this specific lesson, here’s my YouTube Playlist of Book Club videos about Lesson 7: Exponentiation (with episodes from 2023 to the present).
• Need a refresher on exponents? I really liked this Introduction to Exponents video for covering the basics.
• In some domains, page 1 shows a number line and mentions that “Powers start at 1, not 0.” This is why the zeroeth power of any number =1. I like to use paper folding to help explain exponents, and the video below does a pretty good job explaining it. My only complaint (and this is extremely minor, since it’s pretty universal), is that she uses “to the 4th power,” going against what I mentioned above, but unless it’s something TMM puts out, I think that’s what you’re going to find.