# An Old “Dog” Teaches New Tricks Presented by Cary and Jerry Chien

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```					An Old “Dog” Teaches
New Tricks
Presented by
Cary and Jerry Chien

British Columbia Association
of Mathematics Teachers
2001 Fall Mathematics Conference
Our Bag of Tricks
Tips for Teaching Transformations
The Development of Logarithms
Introducing the Concept of “e”
Experiments with Logistic Curves
Introducing Limits
Probability Puzzlers
Teaching Transformations
An explanation of f(2x)

f ( x)  4  x   2
Zoom “Decimal”
f ( x)  4  ( 2 x) 2   Double the domain
Teaching Transformations
An explanation of f(x+1)

f ( x)  4  x  2
Zoom “Decimal”
f ( x)  4  ( x  1) 2   Add 1 to domain
Teaching Transformations
An explanation of 2f(x)+1
Why are vertical
transformations
different from
horizontal ones?
They’re not!
y  1 x   2
y 1
y  2 4  ( x) 2  1
 f ( x)
2
Teaching Transformations
Here’s a bonus question:
 Imagine a function f(x)
What is the order of steps

in the transformation
2f(-x+3)+1?
a     b     c      d
-x    -x    3     3
3    3    -x     -x
1    2y    1     2y
2y    1    2y     1
Development of Logarithms
Using Powerpoint presentations for
more than just a high-tech
chalkboard
Mathematics has a rich history that
students can appreciate
Want to share presentations?
Email me at jjchien@yahoo.com
Virtual TI – a TI-83 graphing calculator
emulator www.zophar.net/ti.html
Javaslide – a working slide rule
www.taswegian.com/SRTP/javaslide/
javaslide.html
Jeopardy Powerpoint Template
jeopardy_template.htm
Introducing “e”
Once upon a time,
there was a very
nice bank (yes, it
is a fairy tale).
They decided one
day to introduce a
new interest rate
of 100% per year.
Introducing “e” =2.718281828…
\$1 with interest rate (100%)
compounded:
Yearly (1+1)=\$2
Monthly (1+1/12)12 = 2.613
Daily (1+1/365)365 = 2.715
Every Second
(1+1/31536000)31536000 = 2.717
Continuously
(1+1/109)10^9 = 2.718281827
Introducing “e”
Little did they know, they’d calculated
the mathematical constant e.
e = 2.718281828459…
For an investment of \$100 over 10
years, the bank has to pay 100 x e10
(\$2 202 600.47!!! Who want’s to be a
millionaire?). The bank realized it was
going to go bankrupt if this continued
so they went back to an interest rate
of 10%.
Introducing “e”
If we perform the same calculations
as we did before, we’ll end up with
\$1.11 after investing \$1 for one year.
(Which is the same as e0.1)
e0.1 = 1.10517.
According to the interest calculation,
this should equal 1.1 Since the
interest rate is low, there is very
little difference between ex and the
compound interest calculation
Introducing “e”
e is used to calculate many functions in
nature such as the growth of bacteria,
decay, which are growing or decaying
continuously, rather than once a month
or day like bank schedule
Experimental Logistic Curves
There’s a secret that only I know.
Using the randInt( function of the
calculator, we can model the spread
of this secret in our class
For the TI-83, choose:
MATH  PRB  randInt(1, n)
Experimental Logistic Curves
Record the number of people who
know the secret in each round
Enter data into lists:
   Round # into L1
   # of people who know secret into L2
Here is a sample set of data:
L1 : 1, 2, 3, 4, 5, 6, 7, 8, 9.
L2: 1, 2, 4, 7, 13, 20, 22, 23, 24
Experimental Logistic Curves
Use logistic regression to find the
equation:STAT  CALC  Logistic Y1
Experimental Logistic Curves
Let’s see the first and second
derivatives of our graph:
Introducing Limits
Let’s find the limit of a function:
x 1
2
f ( x) 
x 1       By Graphing…
Introducing Limits
Let’s find the limit of a function:
x2 1
f ( x)         x  1, x  1
x 1
By Table…
Probability Puzzlers
How many pathways are there
from A to B if you are only able
to travel right or down?
A

B
Probability Puzzlers
How many pathways are there
from A to B if you are only able
to travel right, diagonally down
and right, or diagonally up and
right?
A

B
Probability Puzzlers
You meet a gentleman and his
beautiful daughter. After some
conversation, the gentleman
tells you that he has two
children. What the probability
that this gentleman has a son
and a daughter?
Probability Puzzlers
You are in the final round of a
game show where there is a million
dollar prize behind one of three
doors.
Probability Puzzlers
You pick a door, but before the
host opens it, he opens one of the
two doors that you didn’t pick,
showing you that it is empty.
Probability Puzzlers

He offers you a change your