### First proofs: engaging math activities for the summer break

Mathematics is certainly not (only) about counting, graphing and solving equations. I do not believe that every child can reach beyond those. I do not believe that a child who does not show an inclination to dig deeper into math mysteries lacks in intellect or creativity. I do think that it is worth trying to find out. I child who gets excited on a discovery of uncommon patterns will have enriched his/her life experiences.

So do try.

### Counting heads and tails

**What do you need?** Three elongated objects I'll call *pins*, with two distinct ends called, say, a *head* and a *tail*.

**What is the task?** Form a triangle with the three pins and observe the three configurations of heads and tails at the three vertices. Count and record the number of head/head, head/tail, and tail/tail configurations. Repeat the experiment several times and try to make a general observation of what's happening.

**What to expect?** A child can arrive at several conclusions. For example,

- If there is a head/head vertex there is also a tail/tail vertex, and vice versa.
- There is always a head/tail vertex.
- The three configurations are either all different or all the same, in which case all are head/tail.

**Now a proof** Make child to argue in support of any of the possible conclusions. For example, how is it possible to prove #2: there is always a head/tail vertex. Here's how I would go about doing that.

Put together any two pins. If their joint is head/tail, we are done. Make it then head/head, say. This leaves two exposed tails to be joined by the the third pin. However this pin is placed, the two new vertices are always tail/tail and head/tail - and we are done in this case also.

**Where to go from here?** Generalize! See, for example, an older and a more voluble page

[...] Bogomolny* presents First proofs: engaging math activities for the summer break posted at CTK [...]

June 28th, 2011 at 9:37 am[...] who gets excited on a discovery of uncommon patterns will have enriched his/her life experiences. [ Full article [...]

July 10th, 2011 at 9:48 pm[...] in an effort to better engage math students while Alexander Bogomolny brings us a whole host of engaging math activities for the summer break and Pat Ballew introduces a Sweet Geometry [...]

August 13th, 2011 at 7:42 pm