Dusty Grundmeier, of the Harvard Math Department, hosted the session on what we can learn from pictures. Although many students think of math as a series of equations, and very text-based, Dusty led us through an exploration of what we can “see” mathematically in some favorite picture-proofs, for the Pythagorean theorem, Integration by parts, Fibonacci identities, and more. Prior to the session, Dusty whet our appetites with this problem:
What (if anything!) does the following picture help you figure out about sums of cubes?
(i.e. 1 + 8 + 27 + 64 + …)
Maksym Fedorchuk, from Boston College, hosted the session for us. We considered ideas involving mathematical invariants – things such as no matter what type of triangle you start with, and how you change it around, its angles will still sum to 180 degrees. The Law of Conservation of Energy is another such invariant (that the total energy of all objects in an isolated system remains constant no matter what happens to the objects). Other invariants might not be particularly obvious, and studying what quantities stay the same under certain operations can be a fun and profitable (mathematically, at least!) activity. Invariants can also be used to show that certain results are impossible – and using invariants is one of the key tools for doing this.
To get our thoughts flowing in this direction, Maksym asked us to imagine an 8 by 8 chessboard with two opposite corners cut out.
Is it possible to cover the chessboard (which now has 64 – 2 = 62 black and white squares on it), with a set of 31 dominoes so that every square is covered? The dominoes are each 2 by 1 rectangles as shown, and they can be placed either horizontally or vertically as you wish, but they have to align with the chessboard square – i.e. each domino should cover exactly two squares. Good luck!
Rosalie Belanger-Rioux from the Harvard Math Department hosted a chance to explore some intriguing issues to do with what it means to be “average”! We looked at various ways of describing data (using means, medians, modes and standard deviations), at ways one can be fooled by simplistic descriptions of data, and explore better ways to do this! As a teaser problem to get us thinking about this, Rosalie had passed along the following to take a look at ahead of time:
Dan Goldner asked, where should you put 6 beads on a rectangular wire so that they are as far apart as possible? Most folks settled on interpreting the problem to mean, maximize the minimum straight-line distance between two beads, and different solutions were found in different regimes of aspect ratio of the rectangle.
Led by Jameel Al-Aidroos, we explored some of the mathematics (graph-theory, geometry and combinatorics) of paper-folding, and also explored the fold-and-cut theorem, which says it’s possible to cut any polygonal shape (or even lots of them at once!) out of a piece of (sufficiently large and thin!) paper by folding the paper flat and making a single straight cut (with a very sharp pair of scissors!).
This morning, Joel Patterson led us in exploring the question:
Given a polygon, is it cyclic (all vertices land on a circle), or not?
Early related conjectures:
all regular polygons are cyclic
all squares are cyclic
all triangles are cyclic
some quadrilaterals are non-cyclic
all cyclic polygons are convex
A question arose about interior angle relationships and cyclic-ness (cyclicality?), and we were off and running:
Joel checking in on our progress
Ellen relates regular polygons to our problem
CiCi connects arc angles to polygons
Tom: a helpful pentagon?
Nicole and Bryce: yes, the points are all on a circle – is it the same circle?
Prior to the session, Joel had offered the following “Top Ten” list of possible titles for the session:
10. A roundtable on polygons
9. Wheel In The Sky Keeps On Turning–actually it’s not in sky just one some paper on the table.
8. Fitting square pegs in round holes
7. A roundabout conversation on polygons
6. DISCover something about polygons
5. R.E.M.’s first album, track 6
4. A circular discussion of polygons
3. A roundabout look touching on polygons
2. Joel rolls out an exploration of polygons of all types
And the number 1 title for next week’s session is…
Turning the MTC into an MTP (Math Teachers’ Polygon!)
For those of you who haven’t joined us for a session, please do come by – all the sessions are independent, and we love having new folks join in!
Here’s a sample of what we worked on at a spring Math Teachers’ Circle meeting with Justin Lanier. If the green pool ball is sent off in the 45 degree angle shown, then it will bounce off the opposite wall, one unit below point B.
If it continues to bounce around the table, will it eventually go into one of the pockets marked A, B, or C, and if so, which one will it reach first?
For a neat Geogebra applet that shows off this and other pool bouncing arrangements please check out: https://tube.geogebra.org/student/m93224
We are excited about creating a math circle for Boston area teachers: a community where teachers and mathematicians can celebrate the creative spirit of math together, and explore ways to bring the joy of mathematics to our students. We are putting together the 2014-2015 program now, which will focus on middle school teaching. We will start the year off with a summer retreat (August 4-8) followed by monthly Saturday sessions at Harvard through the school year. In addition to earning professional development points, participating teachers will gain a deeper sense of the joy, beauty and excitement of mathematics, and a network of teachers and mathematicians to draw on through the year.