As
the school year seemed to be dragging to a close, I was looking for an active,
engaging lesson for a small group of fifth and sixth graders. These
students choose to come and participate in a challenge group during their
Friday lunch hour, so I feel the pressure to make our time
worthwhile. They had already demonstrated their ability to solve complex
number puzzles, and they loved the Pan Balance activities we had done earlier
in the school year, so I looked again to the Illuminations website. We had
done a Pi Day activity, measuring circular objects and calculating the ratio of
circumference to diameter, and so when I found the Golden Ratio lesson, I
knew it was going to be perfect. What I didn’t expect was how the students
would take the problem in their own direction.

During
the first session I handed out the Fibonacci puzzle. The students were
immediately intrigued by the idea that they were working on a puzzle that had
been posed over 800 years ago, especially when I mentioned that mathematicians
are still discussing it today. There was a lively discussion as they
started sketching the rabbits, but it was not long before they had discovered
the number pattern and began extending it. I showed them the Number
Patterns in Nature page and they were amazed. That was when I pulled out
the calculators and let them explore what happened when they divided each
number by the previous number. Again, they could hardly believe it when
they continued to get the same answer. I told them that this was a ratio
that like pi had its own name, phi, but was commonly called the Golden
Ratio. We spent our whole next session measuring and calculating the
Golden Ratios of our bodies (an idea from another Illuminations lesson) and of
items we had found that fit the Golden Ratio (including the body of a violin,
which was exciting to several musical students).

For
our third session, I had planned to have the students create a presentation to
show their classmates what they had discovered. That was when one of the
students proposed, “What if the question was changed and the rabbits could only
have babies every third month? Would the pattern change?” I had to
honestly say, “I don’t know. Let see.” The students spent the next
two sessions hard at work modeling and discussing how the pattern would change
and grow. They looked for patterns within the numbers and discovered that,
rather than the a+b=c, b+c=d progression of the Fibonacci sequence, their
sequence, which they named SMEEC (to include all of their initials), had an
a+c=d, b+d=e sequence. The process itself was so exciting to them, as they
truly felt they were “real mathematicians.”

The
school year is almost coming to a close, but I could easily see this activity
continuing into next year as the students try to graph their results and look
for patterns within the pattern. It was a breakthrough moment for me as a
teacher to throw out the lesson plan and let the students guide the activity
entirely--and a moment I hope happens again!