# Math Circle Session 5: Powers, Limits, and Fractals

Yesterday saw the next installment of our math circle with primary school children. Again, following but extending Rozhkovskaya’s book, we used fractals as an interesting way in to consolidate (and in the case of the younger pupils, introduce) powers of numbers, to explore the children’s intuition about infinite series, and to make some pretty pictures!

We started by producing a fractal tree. The rule is simple: after Year 1, the trunk has grown. At the end of Year 2, two branches have grown from that, after Year 3, two further branches from each existing branch, and so on. Children were quickly able to tell us that the number of branches doubled each year, some took great pleasure in calculating or reciting powers of two. The template for this exercise was taken from Rozhkovskaya’s book, and is quite clever: lines are drawn across the page heights of 8 squares, 12 squares, 14 squares, 15 squares, and 15.5 squares, and each successive year’s branches should be drawn to reach the corresponding height. A bird is drawn flying around 18 squares up. The question is posed: will the tree ever reach the bird?

The children had mixed views on whether the tree will reach the bird. The most common view was that it must. One of the older children was able to provide a line of reasoning: “Each time, the height of the tree increases. Since it is always increasing, even by small amounts, it must eventually reach any height.” We then looked at how much gap there is between the height of the tree and a line of height 16 as the years progress: 16 at the beginning, 8 at the end of Year 1, 4 at the end of Year 2, and so on. Children could see that there was always a positive gap between the height of the tree and the height 16, for any finite number of years. However, this clashed with several children’s intuition, and it took quite a lot of discussion before it was generally accepted that an infinite series can sum to a finite limit. One of the youngest children in the group asked some very probing questions, such as “is infinity a number”, which led to some useful side discussions. Random banter between children later in the session about “hacking” their siblings’ passwords led onto a consolidation of this discussion by asking the question “If you had to press an infinite number of keys on a keyboard, could you do it in a finite amount of time? No? What if the first key took 8 seconds to press, the next 4, the next 2, and so on…”

We then moved onto a practical activity of constructing a sequence of approximations to the Sierpinski triangle by sticking little white triangles on coloured paper (pictured). One of the children noticed quickly, and was able to clearly explain, that the number of white triangles was increasing in powers of three. At this point we ran out of time, and our session ended. I intend to pick up and use the Sierpinski triangle again soon in math circle, through Pascal’s triangle modulo 2.