Latest Updates to My Geometry Course

My geometry course… funny. It isn’t really my geometry course anymore. Little by little it’s becoming more of a curated set of links that make me wish I still had a classroom. That’s what community will do for you and the innovative math community more than takes care of people. I’ve noticed some excellent learning experiences that my geometry course has been missing. So, I’m adding them. All three are Desmos activities.

Parallelograms in the Coordinate Plane by Jared – This one asks some very interesting questions and has a nice format in which to do it. Due to the algebraic nature of proving this parallel, parallelograms are a nice target for a Desmos activity.

Transformation Polygraph by Julia Finneyfrock (@jfinneyfrock) – Seriously. Check this activity out and then daydream about all the outstanding vocabulary the students would be forced to use while competing with each other.

Working with Dilations by Mr. Rothe – This one is particularly handy because my unit on dilations and scale factor wasn’t my favorite. This is a very robust activity (36 slides), so time might become an issue, but depending on how you run it, I feel like there’s a ton of value embedded in the different learning activities.

If you give any of these a try, please let me know how they go.

 

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4 Desmos Activities I’m adding to My Geometry Course

It’s been about 14 months since I last taught Geometry, but that hasn’t kept me from keeping a keen eye out for high-quality activities to add to My Geometry Course.

Here are the goods that I’ve found looking through the searchable offerings on Desmos.

Ryan Brown offers a nice introduction to polygons using a do-it-yourself Polygraph activity. This strikes me as a nice kick-off to an early unit in Geometry. Serves as a good vocab review. For high schoolers, it would likely serve as a nice low-entry point to Polygons and wandering around listening would give a nice window into what your students could possibly missing as you are about to embark on a new unit.

Kate Nowak offers a nice algebraic introduction at circles in her Activity-Builder offering.  I like this because the common core circles standards tend to be a little more algebraic than some of the other units, so we put it toward the end of the year as we are ramping up the students for their next challenge (which is often Algebra II). Using specific vocabulary like “proportional” and having the student drag points around the graph sets a good tone for what students can expect in a second semester circles unit.

Pizza Delivery by Scott Miller is a very interesting take on reflections. I am actually bummed that I’m not teaching Geometry any more because I’d like to see what this one looks like. You’re going to want to look at this one.

Mathy Cathy offers a nice introduction to reflections as well. I really like the questions she chooses and the tasks seem approachable as reflections are explored for the first time.

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.

Redesigned Wedding Cake Problem with Desmos

It all started with this tweet:

I got a taste that Desmos was capable of conditional images and text and stuff. So, after I did a little of the ol’ one-man PD tinkering around with their link, I decided I wanted to try to model something. So, I figured why not Wedding Cake Problem? It’s gotten a few redesigns already, so what’s one more?

And so I created this on Desmos.

What I like about this one is that, after all this time, I believe that I have finally figured out a way to deal with the issue of the variable “frosting thickness”.

Also, I like that I was able to include the reality checking piece that the top had to be the smallest section, the middle had to be the second smallest, and the base had to be the biggest.

I encourage any feedback.

Also, I have a question for anyone who is able to answer it: Is there a way that I could make the function round up to the next whole value? I’d like the line to only snap to whole number values since the question is “how many jars of frosting will you need”? (That is, if you need 2.3 jars, you need to buy 3 jars.)

Update:

The last question received an answer by John Golden (@mathhombre) who taught me how to use the ceil() function. You can now check out the final Desmos worksheet.

Excellent Classroom Action – Art and Geometry in the Elementary Classroom

I’ve written about the connections in math and art before. The visual nature of Geometry lends itself quite nicely to this. I found that the right pairing could bring in the engagement of the visual arts while maintaining fidelity to the content.

Exhibit A: Sarah Laurens, 5th grade teacher at North Elementary in Lansing. Mrs. Laurens reached out to me excitedly while I was in her building to see a math activity that she was leading her students through. They involved quilts hand sewn by Sarah’s grandma.

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The activity went like this: Students were in groups of threes and fours gathered around one of grandma’s quilts. Each quilt was made of a series of geometric shapes. Sketch the primary “unit” shape of each quilt and identify each of the polygons that are contained within it. On the surface, it is a fairly simple activity, but listening to the students talk to each other.

[Students looking at the black and blue quilt above left]

Student 1: “Those are just a bunch of hexagons.”

Student 2: “Hexagon’s, no… no… those are octagons”

Student 1: “Yeah, yeah… same thing.”

Student 2: “They’re not the same, one’s got six sides and one’s got eight.”

Student 1: “Well… wait… one… two… three… four… five… six… six sides! See I told you!”

[Students looking at the purple and while quilt on the above left]

Student 1: “That’s an octagon with kites around the outside.”

Me: “Are they kites?”

Student 2: “They look like kites.”

Me: “They sure do. How many sides do they have?”

Student 1: “One… t, th, f… oh! five…. They’re pentagons!

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Ms. Laurens and her students were comfortably saying and hearing words like “regular”, “tessellation”, and using definitions to make sense of what these shapes are, and using the definitions to settle disagreements (the foundations of proof…)

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Interesting images like these were leading to some interesting conversations as well. I’m thinking of  a conversation I heard between two students who were trying to make sense of the shapes they were in the picture directly above to the right. They were trying to to determine if the purple section in the middle was one big shape or two smaller shapes put back-to-back.

Then after some discussion, they realized that their answer would be the same either way. (Quadrilateral was their choice for the shape name. The word “trapezoid” was getting thrown around, but the students were having to be prompted for it).

I very much enjoyed getting to see these fifth graders exploring. Ms. Laurens was excited, the students were engaged (and this was clearly not the first time they were expected to be a self-directed and collaborative).

I’m just bummed that my schedule forced me out the door before I got to see Ms. Laurens’ closure of the activity. The students were wrapping up their discussions as I had to head out the door.