# Co-curricular lessons between geometry and chemistry?

My school is hyped about the promise of co-curricular education and they are giving the math and science teachers paid days off to develop lesson plans that synergize our learning goals. I'm on board with this.

The problem is that I'm teaching geometry and my science grade-level peer is teaching chemistry. I'm confident that she could use my help reminding students about the kind of algebra that could help with balancing equations or doing stochiometry, and statistics refreshers would probably also be awesome. But I can't think of much of anything that would use plane geometry on the Euclidean or Cartesian plane. And while my experience of chemistry is several paradigm shifts ago, I can't imagine what she would teach that would help my students get better at proofs or constructions.

Does anyone have experience or references that could help me to bring some productive ideas to our collaborations?

• If you had a well-defined curriculum you would not have to worry about this, would you? There is much tighter integration between algebra and chem and physics. Geometry and physics too. Geometry and chem not so much besides some pretty models of molecules and crystal structure. The angles between the bonds can be explained with chemical properties, but aside of that there is no application of geometry to chemistry. Time can be better spent on geometry or chemistry separately. – Rusty Core Nov 14 '19 at 19:06
• @RustyCore Yeah, we both do have approved curricula for our individual classes, but cross-curricular learning seems like the flavor of the month around here so we'll probably have to find some way to look busy. But, IKR, I could probably get more use out of a social studies collaboration than chemistry! – Matthew Daly Nov 14 '19 at 19:10
• Some of the ideas, and maybe even methods (but I realize vectors and trigonometry are beyond your course) discussed in How can I prove that the angles of the tetrahedral structure is $109.5^\circ$ with calculus. I could do it with geometry might be of use. (I have .pdf files of the papers I cited, if you're interested.) – Dave L Renfro Nov 15 '19 at 9:37
• Entire theses have been written on the geometry of the Periodic Table of Elements, and of course salt crystals, graphite, etc all have interesting shapes that are surely worthy of investigation. (allperiodictables.com/AAEpages/aae3d.html) – Strawberry Nov 15 '19 at 15:58

## 4 Answers

One angle you could look at is molecular geometry. Not really my subject area but a couple of examples:

• Organic molecules can have different chiralities. That means that while one is the mirror image of another you cant rotate one molecule to the other. The reasons for this are pretty deep mathematically, but chemically give rise to interesting things as well: for example the smell of lemon and orange are chiral opposites. https://en.wikipedia.org/wiki/Chirality

• Bond structures restrict the geometry of molecules. As an example, can get them to make Buckyballs and prove that carbon structures have to take certain forms https://en.wikipedia.org/wiki/Buckminsterfullerene

• Geometry of crystalline structures. Combining the two above, did you know that we've classified all of the regular tessellations? What does that say about solid structures like crystals made out of different compounds? Of course, the 2D and 3D versions are different but you can start with 2D (we have vertices that can attach to 3 edges and vertices that can attach to 4, what kinds of regular crystals can we make?) and then extend to 3d. On the chemistry side, the and specific molecules determines the material properties of the solid.

Finally, there's something about bond angles, but that's almost more physicsy. This sounds like a really interesting area to mine though!

• But this just involves changing the chemistry curriculum, doesn't it? If the math class is supposed to teach geometry and the chem class is supposed to teach stoichiometry, adding stuff about the geometry of molecules is just getting off-syllabus. – Jack M Nov 15 '19 at 10:22
• @JackM I think Mathew was using as examples of where math is currently used but that are not related to geometry; I interpreted his question as soliciting topics that could connect the two. Usually when you do these kinds of collaborations its okay to get off syllabus a bit. – Nate Bade Nov 15 '19 at 14:41
• +1 for the geometry of crystalline structures. This is a standard first-year college chemistry topic, and I think it also shows up in AP chemistry. In particular, computing the size of a single crystal cell based on the sizes of the atoms it is composed of is common, and a great exercise in the 3-D Pythagorean Theorem. – Opal E Nov 15 '19 at 17:52
• If you just want interesting Maths questions in molecules, you can also ask yourself whether every molecule can be drawn as a planar graph. I've asked myself this and had to quickly retreat from a deep rabbit hole of interesting papers and articles. – Fabian Röling Nov 16 '19 at 11:56

Pick your battles. Don't expect to have synergy in every place. But where you do have synergy, exploit that, call it a win, and move on. Concentrate on the partial fullness of the glass, not the partial emptyness. For that matter, you don't have time to totally redesign each course from the ground up in a way new to man. Nor do you want to screw up the basic stuff (in each course) that they need to learn no matter what.

1. But certainly geometry of molecules is a NORMAL, MAJOR part of the chemistry course. See chapter 10 ("Covalent bonding") in 1980 Masterton, Stanitski, Slowinski. Or the equivalent chapter in whichever text your chem colleague is using. [It's about halfway throught the year, but so what. Give you time to concentrate on your classes themselves, not try to start complicated joint immediately.]

2. I suggest to add a small joint segment on point groups. A small addition of content. I remember doing this as a lab in my AP chem class (without geometry collaboration) as the chem instructor didn't think our normal chem text covered it enough, but she liked it because she was an inorganic structural chemist. Fortunately our lab text had a nice lab that covered the topic.

Note that this topic will be descriptional (you're not going to have character tables for IR stretching modes...and nyet on teaching group theory). But you can learn the basic symmetry operations (planes, rotations, inversions, etc.) and can build a few models with toothpicks and foam and/or diagnose a few common objects and molecules. At that point, you declare victory. It's a solid exposure which was taught a lot about an important topic to people unaware of the topic (steep part of the learning curve). The kids will get it heavier, better in college inorganic chem class (if chem majors). But it's enough for now.

If you did an hour of lecture (probably joint) and then an hour of lab (joint if you want or just let the chem guy supervise the toothpickinig), that would cover it fine. Kids would hand it written lab report with a few questions to answer, as normal for chem labs.

These two should be enough to declare victory, not screw up your classes, not confound the kids, and keep the admins happy. And you and the lab geek can go have a beer afterwards and watch the footie.

• The 3D modeling sounds like something with a lot of potential. My colleague was discussing an octahedral compound that I forget the name of, and that's definitely achievable with toothpicks and marshmallows! – Matthew Daly Nov 16 '19 at 19:51

This is perhaps more molecular biology than it is chemistry.

There are some accessible planar geometric questions suggested by the H-P (hydrophobic-hydrophilic) model of protein (amino acid) folding, which could be explored with simple manipulatives (such as K'nex). For example, which proteins in this model have a unique minimum energy folding?

Figure from Geometric Folding Algorithms: Linkages, Origami, Polyhedra.

Are proofs still part of the geometry curriculum? (Some of my math colleagues have mentioned they've been downplayed over the last decade or so.) A good Chemistry answer looks an awful lot like a good proof; same sort of logic flow. (As does a good programming solution, if you're looking for another connection.)

It might be a bit simple, but you could talk about why volumetric measurements get turned into linear ones. (E.g. a graduated cylinder is designed to keep volume proportional to the height of the liquid.) The measurement difficulties of accurately measuring spherical volumes compound errors. (I.e. a 5% error on 100 mL measured in a G.C. means something way different than a 5% error on 100 mL on sphere's radius.) So an activity could be to measure a small ball's volume (maybe by using string, maybe calipers), and then submerging it (if your colleagues have a large enough graduated cylinder) and discussing any discrepancies.

Long shot, but are logarithms part of your curriculum? It might be a stretch for the Chem teacher (depending on how deep they get into acids and bases), but pH is all about logs. (In case you don't know, pH = -Log([H+]), where [H+] means the concentration of Hydrogen ions in solution.) (I would juuuust got to this in my Honors Chemistry classes, so I don't know if your colleague will.) Plus, there is the pOH scale --> pOH = -Log([OH-]). For acids and bases near 25C, pH + pOH = 14. and [H+] * [OH-] = 10^-14. (This might be really pushing for the Chem teacher, but who knows.)