LESSON PLAN III

Increase students' knowledge of patterns in Fibonacci numbers and practice pattern recognition.

Students will be able to continue patterns found, generalize patterns they see and find new patterns.

(These numbers get big, students may wish to use calculators.)

1. Continue the table of Fibonacci numbers with at least 5 more entries.

2. Add one more term to the sum of alternate terms of the Fibonacci numbers illustrated. That is, find the sum

   f(1)+f(3) +f(5) + f(7) +f(9) + f(11) + f(13).

   a) Is this sum also a Fibonacci number?
   b) Do the same thing for f(1)+f(3) +f(5) + f(7) +f(9) + f(11) +
   f(13)+f(15).
   c) Write down a rule that expresses the pattern illustrated.
   d) Test your statement with 3 more examples.
   

3. Add one more term to the sum of alternate terms of the Fibonacci numbers illustrated. That is, find the sum

   f(1)+f(2) +f(3) + f(4) +f(5) + f(6).

   a) Are the sums related to Fibonacci numbers?
   b) Write down a rule that expresses the pattern illustrated.
   c) Test your statement with 3 more examples.
   

4. To write any number as the sum of Fibonacci numbers, follow this method. I'll use number 294, as an example:
   • Locate 294 between 2 Fibonacci numbers, f(13)= 233 and
     f(14) = 377.
   • Chose the smaller one, 233 and subtract it, 294 - 233 = 61.
   • Locate 61 between f(10)= 55 and f(11) = 89 and subtract it, 61 - 55 = 6.
   • Locate 6 between f(5)= 5 and f(6) = 8 and subtract it,
     6 - 5 = 1, which is f(2)

   The result is 294 = f(2) + f(5) + f(10) + f(13).

   a) Write 380 and 605 as sums of Fibonacci numbers.

5. Write the pattern found in these examples. Check with 2 more examples.

   a) f(1)*f(3)-f(2)2 = ?
   b) f(2)*f(4)-f(3)2 = ?
   c) f(3)*f(5)-f(4)2 = ?
   d) f(4)*f(6)-f(5)2 = ?
   

6. Show that the Lucas numbers l(5), l(6) and l(7) are each the sum of two Fibonacci numbers in the same way l(2), l(3) and l(4) were.

7. Another relationship: f(4)/f(2) = 3/1 = l(2); f(6)/f(3)= 8/2 = 4 = l(3)and
   f(8)/f(4) = 21/3= 7= l(4). Write 3 more examples of this pattern.