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

Further generalize:

·         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

§  Problems

·         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

§  Problems:

·         what is 9**2/3?

·         What is rule for 9**a/3

·         What is rule for arbitrary bases?

o   a**b/c

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.