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 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. |