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!