Another fantastic day of geometry!
Monday’s loose ends
We started the day going over answers from the Quadrilateral Identification Task as a group, then each table compared notes on the Portfolio of counter-examples. Each table had fruitful discussions in separate directions. Andrew raised the issue of how the word “base” gets used differently depending on context: it could be the non-congruent side in an isosceles triangle, one of the two parallel sides in a trapezoid, or “a designated side” that the height is drawn from (Thanks, Jennifer Gilbert for bringing her classroom materials.).
Block 5: Transformations
Block 5 expands the view of “flip, turn slide” into a larger framework of transformations. We settled back down to reflections (flips), rotations (turns), and translations (slides) because they are nice: they do not change the length of segments and they do not change angles. We called such transformations rigid motions, since they treat the shape as unstretchable and unbendable. We then spent a lot of time using patty paper to explore reflections, rotations, and translations and what happens when you perform them in succession. In general, we found that the order in which rigid motions are applies matters. Doing A, then B will produce a different net result than doing B then A. There were special instances (such as rotating about the same pivot) where order did not matter, but the default is that order does matter.
Block 6: Symmetry
Block 6 expands our usual view of symmetry into a broader context through transformations. The usual mirror symmetry can be viewed as reflection symmetry: performing a reflection about the line of the mirror produces an image identical to the original figure (“you can’t tell anything happened”). In the end we said a figure could be called “symmetric” if there is some rigid motion that preserves the figure (i.e., the end image is the same as the original figure). This means a figure like a capital S has rotational symmetry about the center.
We had a good discussion about how many people (adults and students) want a [non-special] parallelogram to have mirror symmetry, either about a diagonal or some other line through the center. Andrew’s theory is that they see a “same-ness” on each end, but it is a “same-ness” based on rotation and not reflection. However, the untrained eye cannot tell the difference, and if you only know about reflection symmetry you assume that the same-ness you see can be explained by a mirror.
We did notice one important connection between reflection symmetry and rotation symmetry: if a figure has mirror lines, then it will also have rotational symmetry about the point where those mirror lines cross. Going further, the number of mirror lines gives the order of rotational symmetry (e.g., the regular pentagon has 5 mirrors and 5-fold rotational symmetry).
We skipped many handouts from this block, everything from 6-8 onward. If it looks interesting and you’d like more guidance on the matter, see the “EscherMath” link in Miscellany.
Shaun put together a nice exploration of where area formulas come from. We used the area of a rectangle as a baseline, which is a natural and easy to understand formula. We then explored other shapes’ area formulas, trying to connect them back to previously-discussed shapes. We found a way to “rectangle-ify” a parallelogram, “parallelogram-ify” a triangle, and so on. Our major tools were (1) cut off a piece of the region and move it via a rigid motion, and (2) duplicate the shape (or a part of the shape), move that clone to make a “big” shape whose area we know, and then take half of the area of the big shape. In this way we connected transformations with area formulas. Justifying area formulas is definitely a place where there is more than one right way to do it.
Patty paper can be purchased at Amazon (and elsewhere).
The PBS show with early-grades math content that Andrew mentioned is Peg+Cat. It is honestly Andrew’s favorite kids’ show. They have a whole song about how there is more than one way to solve a problem, and another about persisting in solving a problem. Episodes are available on pbskis.org, Netflix, Hulu+, and Amazon Prime.
There was a general education course offered at St. Louis University on the mathematics of M.C. Escher. There is a lot of material there on transformation, symmetry, and tessellations. Their course materials are online at http://mathcs.slu.edu/escher/index.php/Math_and_the_Art_of_M._C._Escher. They include a lot of activities, too.
[UPDATE 7/10/15]: Solutions for the portfolio of counter-examples, along with commentary, now available.
Email Andrew at firstname.lastname@example.org if there are other questions.