The Beauty of Geometry

Every now and again, I take the opportunity to simply opine on the beauty of geometry. Math gets a bad rap because of it reputation of being cold, lifeless, functional and academic. (Some folks aren’t helping this by proclaiming that the arts are what we do to enjoy school and math is what we study to get paid later on in life.)

Don’t get me wrong, there certainly are academic ways of discussing art, music, iconography, fashion, design. There’s technique and vocab to all of these areas. Students of these disciplines are still students who must study, but their exploration isn’t saddled with the atmosphere that math is. In math, it is often believed, the box is set; the boundaries drawn. The math frontier is closed. There is no need for exploration when there is nothing to explore.

I’ve always felt like geometry has the capacity to challenge those notions. Kolams, quilts and origami help students understand the aesthetic value of straight lines, precise measurements, perfect circles and right angles. Sometimes, you have to build them to complete your understanding of them. That process alone can bring with it its own supply of feedback. When you are trying to create something visually appealing, often times, the eyeball becomes the expert in the room, not the teacher. Attention-to-detail and technique become valuable without encouragement.

At a recent professional learning opportunity, I was given some time to play with KEVA planks. So I did. The planks are all congruent rectangular prisms. So I placed one on the table. Then I placed a second on with a slight rotation (the diagonal intersection points were designed to sit right on top of each other, but the counter clockwise rotation was determined by the next block being placed so that the vertices were placed on the preceding blocks’ short-segment midpoints. It ended up being about 10 degrees.)

That all sounds pretty mathy (and probably somewhat unclear since I’ve never had to verbalize the process before). But the resulting tower is pretty cool-lookin’ (at least I think.) I simply love when objects with straight lines and right angles are arranged to look like curves. This can happen in algebra as well. As a teacher, of course I don’t know if my students will share my fascination, but fascination isn’t the goal. It’s tough to measure and, besides that, it’s fickle.

I’d encourage you to look for opportunities to change the cold, academic atmosphere surrounding the math. How can we warm this wonderful subject up? We used to take advantage of those tricky days right before a long break and do art projects. Thanksgiving Origami, or build a Christmas (or holiday… or wintertime) scene using nothing but triangles.

 

What ideas do you have?

Don’t forget Geometry when teaching Algebra

Right now, Michigan educators are trying to sort out the implications of the state switching to endorsing (and paying to provide) the SAT to all high school juniors statewide instead of the ACT as it had previously done. The SAT relies very heavily on measuring algebra and data analysis. This leaves plane and 3D geometry, Trig and transformational geometry under the label “additional topics in mathematics.” The new SAT includes 6 questions from this category. Compared to closed to 10 times that from more algebraic categories.

This comes up in every SAT Info session I lead. Should we just stop teaching geometry? All Algebra? Could geometry become a senior-level elective? If it’s not going to be on the SAT, then what?

These questions reflect a variety of misconceptions about the role of testing in curriculum decisions. In fact, these are the same misconceptions that are driving an awful lot of the decisions that are being made. And in this case, I think there’s more at risk than simply over-testing our students.

It would be a real shame to see Geometry become seen as an unnecessary math class. Because it’s not.

And to illustrate that, I’m going to tell you a story about a conversation I had with my daughter. She’s 6.

She wanted to try multiplication. She’d heard the older kids at school talking about it. So, I taught her about it and let her try some simple problems to see if she understood. And she did, for the most part.

So, I started to not only make the numbers bigger, but also reverse the numbers for some problems that she’d already written down. She had already computed 2 x 3, and 4 x 2, and 5 x 3, but what about 3 x 2, and 2 x 4, and 3 x 5. Were those going to get the same answers as their reversed counterparts?

She predicted no. So, I told her to figure them out to show me if her prediction was right or wrong. To her surprise, she found that switching the factors doesn’t change the product.

“Daddy… why does that happen? It should change, shouldn’t it?”

Translation: Daddy, how do you prove the Commutative Property of Multiplication?

How would you prove it?

The geometry teacher in me thinks about area when I see two numbers multiplied. We can model (and often do) multiplication as an array. It’s just a rectangle, right? A rectangle with an area that is calculated by it’s length and width being multiplied.

What happens if we rotate the rectangle 90 degrees about it’s center point? Now it’s length and width are switched, but it’s area isn’t. Because rotations are rigid motions. The preimage and image are congruent. Congruent figures have the same area.

If l x w = A, then w x l = A. 

Geometry helps prove this. Geometry also helps support a variety of other algebraic ideas like transformations of functions within the different function families. Connections between slope and parallel and perpendicular lines. There’s also the outstanding applications of algebraic concepts that geometric situations can provide. Right triangle trig, for example, is often a wonderful review of writing and solving three variable formulas involving division and multiplication. (A consistent sticking point for lots of math learners.)

As a teacher who spent years watching how Geometry presents such an environment for real, effective and powerful mathematical growth, eliminating it will leave a lot of holes that math departments are used to geometry content filling.

Gallons and Gallons of Pennies

Sometimes, “real-world” problems just go ahead and write themselves. And I say take advantage. Why be creative with the actual world can do the heavy lifting for you, right?

This floated across my Facebook feed. Pretty sure you’ll see where I’ve made some edits to the original texts.

Pennies Problem

Sequels could potentially include:

If Ortha wanted to exchange them for quarters, how many 5-gallon jugs would she need? You could do the same with nickels or dimes.

What would be the mass of each penny-filled jug?

What do you think? What other questions could come off this wonderful set up?

Exploring Reflections with Desmos Activity Builder

Geometric transformations take up a good chunk of the first quarter of Geometry (at least the way that I had it sequenced). The tricky part of teaching transformations in Geometry is the delicate balance between the non-algebraic techniques and understandings and their algebraic counterparts.

For example, consider the two following statements.

Two images are reflections if they are congruent, equidistant from a single line of reflection and oriented perpendicularly from said line of reflection.

The reflection of A(x, y) is A'(-x, y) if the line of reflection is the y-axis and A'(x, y’) if the line of reflection is the x-axis.

As a geometry teacher, am I to prefer one over the other? In my experience, they both present challenges.Students are often a little more enthusiastic about the first (which is why I start there), but can often be more precise with the second. The second requires the figure to have vertices with coordinates and the line of reflection be an axis. The first requires the figure be drawn (and fairly accurately, at that.)

So, in my attempt to learn how to use Desmos Activity Builder, I wanted to produce something useful. So, I made a Desmos Activity to bridge the transition from a visual, non-algebraic understanding of reflections to an algebraic one.

Full disclosure: This activity presupposes that students are familiar with reflections in a visual sense. It isn’t intended to be an introduction to reflections for students who are brand new to the topic.

Feedback, questions… all welcome.

Desmos-Enhanced: The remodeled Pencil Sharpener Problem

Lately, I’ve found it tremendously enjoyable to revisit some of my favorite homemade problems and use Desmos to model them.

I decided to remodel the Pencil Sharpener Problem this time. If you’re not familiar, go check it out. Here’s how it goes.

Three boys are held after class for detention. I told they have to stay for a half-hour, but they can leave earlier if they can grind down 100 pencils by hand in less than a half-hour.

So, the three of them decide to take the me up on my offer and begin cranking the pencil sharpeners as fast as they can. Each of their top speeds is recorded on video. If we assume that they keep their top speed up the whole time and don’t slow down, then how long with their detention last?

This problem has created some fantastic student work. Enough so that it is almost tempting to force pencil and paper work.

However, I couldn’t resist the temptation to create a Desmos worksheet for it.
Besides, by now, Pencil Sharpener Problem is ready for an extension. How’s this:

It seems safe to assume that the boys will tire as they crank the pencil sharpeners over and over and over. How about we say that the each subsequent pencil takes 5% longer than the previous pencil. So, if first pencil took 60 seconds, the second one took 63 seconds, and the third one would take 66.15 seconds, and so on.

Would it still make sense for the boys to grind away pencils? Or should they just sit quietly for 30 minutes?