here there is an image of a 555 timer chip.

http://www.evilmadscientist.com/images/articles/CylonOLantern_1.jpg

my question is to find out -

What will happen to the “clock speed” (the output of pin 3) if I changed the values of the resistors from 180k ohms to 1k ohms and the value of the capacitor from 0.22 µF to .10 µF? Will the “ticks” from the clock speed up or slow down?

I dont understand how to form this into a math question to understand it. Nor do I know if I need to have it in a math form to fine me answer.

 

The oscillator period consists of two phases; charging and discharging the 0.22uF capacitor. It charges through both resistors in series or 2X180KΩ, or 360KΩ. It discharges only through the lower resistor (180K), so the total period is proportional to 3X100KΩ*0.22uF.

If you halve the capacitor, the oscillator will run twice as fast. If you halve the lower resistor, it effects both the charge time, and the discharge time. If you halve the upper resistor, that effects only the charge time without effecting the discharge time. I will let you figure that out...

There are many online calculators that predict the frequency (1/period) of a 555 astable as a function of R1, R2, and Ct., including this one or this one.

 

Almost all there is to know for the 555 here.
Free E-book.http://www.talkingelectronics.com/projects/50 - 555 Circuits/50 - 555 Circuits.html
http://home.cogeco.ca/~rpaisley4/LM555.html
http://williamson-labs.com/pu-aa-555-timer_med.htm
Max.