Systems of equations.  Finding the break-even point, or point of intersection.

Gym or pool membership plan.  


$80/mo + $5/visit


$50/mo + $10/visit



The Matrix vs. The Matrix.


1) Extremely small: nannoplankton.
2) One-billionth (10 to the -9th power):
eg: nanosecond.




Cooking a turkey.  

20 mins. per pound.   

Linear (line) relationship b/w time and weight

T = 20p



During the late 90’s bull market, shares of E*Trade brokerage explodes parabolicly / exponentially from $5 to $70 in 2 weeks)


“Parabolic” skis & "Parallel" skiing…


The parabolic ski has a dramatic sidecut, but the `extremeness` depends on the model.



Ratios Galore !


Fuel/oil gas mixture for 2 stroke race dirtbikes

Grocery store math.  Volume/quantity discount

Car’s rear end gear ratio (Driveshaft rotations: Wheel rotations)

Team’s Win/loss ratio

Ratios used in cooking recipes.

Currency exchange Rate


Car's Gear Ratio

The numerical ratio of the drive axle is the number of the teeth on the ring gear divided by the number of the teeth on the pinion gear.

A Shelby w/ 4.11 rear-end ratio.



Grading tests. 


If each question is worth 3 points, how can I model this situation?

Let w = # wrong

Let g = grade

G = 100 – 3w

Then use 2nd, Table to display the chart


Scale Models









Given a 50/50 chance to win each game, what are the odds of the Red Sox winning 4 games in a row?











  • Pentatonic minor scale has 5 notes

Scales · Pentatonic Scales · minor pentatonic



  • Hex keys/Allen keys


rhom·boid (n.) A parallelogram with unequal adjacent sides.

36,000, 50,000, 75,000 & 100,000 lbs.
Capacities Surface Mounted or Flush Mounted with Runway Lengths from 26’ to 48’ long


Some Gibson Les Paul guitars have “Trapezoid Pearl Inlays”


Gibson Les Paul Custom Shop
Gibson Les Paul Custom Shop Classic Mahogany

The Gibson Les Paul Custom is a classic carved from a classic wood. The guitar features a fast playing slim tapperd neck that is reminisent of the later 60's pauls. It is a long neck that produces a tight and resonant neck joint. There are also pearloid trapezoid inlays. This makes the guitar have that classic rock-n-roll sound and it very playable. You'll never know until you do.



  • One-piece mahogany back
  • Rosewood fretboard
  • Pearloid trapezoid inlays
  • Nickel hardware



POLYGON Graphics

Swing Plane in Golf



Symmetry and Line Reflection


In geometry, I've been teaching symmetry and line reflection.   At the end of the unit, I displayed a few palindromes, and kept my mouth shut until someone figured it out the relevance... The kids were intrigued by this, and even tried to make up their own..










Why is the Ambulance written "backwards" ?




5.0 liter engine block


460cc golf club head



Which is greater?  The volume of a Frisbee or a glass?







Algebra 2



When simplifying complex fractions, see the “fraction in a fraction” concept. 


How many triangles are there in the following figure?


How many fractions are in the following?



Inverse Variation:
The larger the shot, the smaller the gauge rating.





The standard deviation of the # of intentional walks given to Bobby Bonds


(excluding under 100 At bats)


Avg # of walks  = 37

Std. Dev = 26


Bonds # walks = 232


Bonds is almost EIGHT standard deviations above the average.

(Even 4 SD's means you are higher than 99.99% of the rest of the data points.)






Trig graphs.  Many guitar effects pedals are simply wave transformers. 

Phaser/Flanger pedals just transpose a shifted/delayed copy of the original sound wave onto the original.  Chorus pedals do the same, but they also modify the amplitude of the original sound wave.  (Led Zeppelin and Van Halen)



Sin wave.  Cyclic. 

Ups and downs.

Life is a sin wave. 


Composition of Functions


We start with 2 functions.  

1)     Paycheck is a function of hours worked. 

2)     Number of CD’s bought is a function of paycheck


Composition of functions is the process of writing a new function (or rule) that combines these multiple functions into one function.


The composition function allows us to determine Number of CD’s bought as a function of hours worked.  (in 1 step)


The amount you get paid is a function of hours worked. 

If you are paid $6/hr plus a flat $10 for gas, how do you represent ?  


Let’s say Jane spends her entire paycheck on CD’s.  The number of disks purchased is a function of the amount she was paid.     If each disk is $8 and the bus fare to the mall is $3, how do you represent the number of disks purchased? 


? How may CD’s can she buy if she works 3 hours?  5 hours?  10 hours?

? What if you want to determine the number of CD’s purchased directly from the number of hours worked?



As a check, let’s test a few values in the new equation.

They should match the results if the 2 functions were evaluated separately.






Conic Sections: Ellipses and Foci


St. Peter’s Square







Planetary Orbits are Elliptical






Elliptical machine at the gym.



Parabolic Satellite Dish



Logarithms   (Richter Scale)

A Richter scale 4 earthquake is TEN times more powerful than a scale 3 quake.


Richter Scale Graphic Representation



AOL Stock Chart.  (Linear vs Logarmthimic)


Notice how the price movement during 1994 to 1998 is almost flatlined b/c the Y-axis scale goes to 100.  However, the actual percentage move during 1994 to 1998 was huge.  (50cents to 4 dollars is an 8-fold increase.) 


Notice the “staggered” Y-axis units in the 2nd chart.  It more accurately reflect the price movement during 1994-1998. 







Exponential Growth


The risk of having a child with Down syndrome is 1/1,300 for a 25-year-old woman;

at age 35, the risk increases to 1/365.

At age 45, the risk of a having a child with Down syndrome increases to 1/30



Figure 1

Bell Curve (Average IQ)


80-90 (below average)
90-110 (average intelligence)
111-121 (above average)
122-130 (highly intelligent)
131-141 (gifted)
142+ (genius - top 1%)