Inductive intro to exponents and logs:
The following is a condensation of my understanding for our inductive intro to exponentials. I am inventing notation here.
First: teach some properties of composition for linears – the concept here is the generalization of multiplication from multiple addition to scaling: that students will immediately understand but never realized:
Using a concrete example (draw graph), argue that If L is linear, then:
Whatever operation computes L(x+1) from L(x) is identical for all x. (including fractional x) Call this Lu(x) // u for unit
· u for unit
· L(x+1) = Lu(L(x)
· notice that Lu(x) = x+c for some c.
Whatever operation computes L(x+0.5) from L(x) is identical for all x. Call this Lh(x)
· h for half
· L(x+0.5) = Lh(L(x))
· notice that Lh(x) = x + c/2
Observe composition property: Ln(n+1) = Lh(Lh(L(n)))
· work through details with class
· The composition principle applies to inverses: L(n) = Lh-(Lh-(L(n+1))
· the same principles applies to thirds, or any other fraction.
· The same principles applies to units larger than 1
· The same principles applies to inverses
Now examine if it makes sense for exponentials to be continuous – suggesting that it would be convenient if the same properties applied..
draw graph of first 3 exponents of 9 (1,3), . (2,9), (3,81). Don’t draw (0,1) yet because, while students may know it, they won’t know why. We’ll derive It from basic principles shortly.
· Define E(x) to be exponential. Say 9**x
· Note to instructor: Why 9? It’s the smallest square of an integer that is not 2. Will minimize risk that students will confuse factors and multiples of 2 with roots products of 9. .
· Draw reasonable exponential curve – and indicate that its properties must make sense – just like for linears
Now examine indicate convenient properties that we would like to hold for E(x) fractional x:
· There should be a unit transform such that E(x+1) = Eu(E(x))
o E(x+1) = Eu(E(x))
o Eu(n) = n * 9. Discuss how it works for all bases
· Consider inverse: Eu-(n) = n / 9
o Thus E(0) must be 1. E(-1) must be 1/9. Etc. Draw them..
· Consider implications for fractional x
o Draw dots on exponential curve where x = 0.5, 1.5
o Observe that 1.5-0.5 = 1
o Suggest that it would be convenient if unit transform properties from linears applied
§ E(1.5) = Eu(E(0.5)); E(1.5) = 9*E(0.5)
§ Same for computing E(2.5) from E(1.5)
· Now consider half-step transform Eh(x)
o E(x+0.5) = Eh(x)
o It ought to compose
§ E(x+1) = Eh(Eh(E(x))
§ Eu(n) = Eh(Eh(n)) = n * 9 = n * (3**2)
§ Obviously Eh(n) = n * 3
§ Why 3? Because 3 is square root of 9, which is the base of E()
§ Observe that 9**1/2 = sqrt(9)
o Observe intuitive inverse property
§ Eu-(n) = n / 3. Observe that it works .
o Now consider for quarter steps.
§ Students should suggest that Eq(n) = n * sqrt(3)
§ Eq-(n) = n / sqrt(3)
· Now generalize to other fractional steps
o If x step is 1.5
§ Eh(Eh(Eh(n))) = 3**3 = sqrt(9)**3
§ Observe that 9**3/2 = sqrt(9)**3
· figure out 9**5/2
· figure out rules for 9**a/2 for any a
o if thirds rather than halves
§ Fractional step is cube root of 9
o Observe that 9**1/3 = cube root of 9
· what is 9**2/3?
· What is rule for 9**a/3
· What is rule for arbitrary bases?
Notice how the students should be well prepared to figure out the important algebraic concept themselves. We just review it. Would not work if we told them first.
© 2011 Eric Freudenthal and collaborators. All rights reserved.