"Multiple" Pi Challenges for MS and HS

Consider the following table displaying the first four positive integer multiples of π rounded to 4 places:


So what's the challenge here? You will be a π-multiple investigator!

(1) Determine the EIGHT positive integer multiples of π, up to 1000π, which, when rounded to THREE places, produce an integer result. For example, the decimal 17.9996 when rounded to 3 places results in the integer 18.000, which we will regard as an 'integer'. Unfortunately, this decimal is not an integer multiple of π so have fun searching! Do you notice any pattern in your results? Describe it! Can you explain this pattern? [This requires more than a Yes/No response!]
(2) In fact, the "pattern" you may have found in (1) continues beyond 1000π. Show that you can go up to SIXTEEN multiples of π before the pattern breaks down. Why do you think the pattern eventually ends?

(3) Which of your results in (1) would produce an integer when rounded to FOUR places?

(4) Can you think of any practical application for finding multiples of π which are very close to an integer? Be creative! Responses may depend on how advanced your math background is.

Comments:

• Do your students know how to program their graphing calculator to do the search? OR in Java or C++ or Python or Perl?? This would certainly facilitate the search! I may display the code for the TI-83 or -84. However, students may also find a creative way to use the TABLE feature on their graphing calculators to save time without programming. Have fun!
• Please post feedback if you use this in your classes. I will not post answers ye! t in case students find this post in their 'searching'!
For mo st students, the full significance of this innocent looking search problem will not be apparent. You might want to give them a hint. Perhaps we should call this post:

IS π ALMOST RATIONAL?

rationalize the denominator calculator