Geometry – Day 55 – Translation & Tessellation

Why We’re Here:

  1. We’ll apply and derive translations from figures in the plane.
  2. We’ll discuss tessellations, which regular polygons tessellate, which don’t.
  3. We’ll get our hands dirty with triangles & quadrilaterals, devising a method for tessellating both of those.


  1. Scissors
  2. Triangles/Quadrilaterals Tessellation worksheet

The Breakdown

  1. Opener + Review (15 minutes)
  2. Translation Notes (15 minutes)
  3. Translation Classwork (30 minutes)
  4. Break (5 minutes)
  5. Show and Tell (1.5 minutes)
  6. Tesselation Notes (20 minutes)
  7. Tessellation Project (30 minutes)
  8. Clean Up (5 minutes)


Slide Deck

  1. 7.2 years (National Review)
  2. Ask them to describe it in words — it’s moved up and right — and then convert that to coordinates.
  3. No matter how ugly the figure, the translation is as simple.
  4. Black: just take every point one at a time and move them 7 left and 1 up.Red: Take every y and reverse the sign. What will it look like?
  5. Ask someone to draw it at the board.
  6. Ask someone to draw it at the board.
  7. Ask someone to draw it at the board.
  8. Ask someone to draw it at the board. Discuss why triangles, squares, and hexagons work but pentagons don’t.
  9. Tessellation handout. Devise a method for tessellating triangles and quadrilaterals.

Notes & Revisions:

  1. None.
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Hey Dan,

    Here are few q’s.

    – Why did you choose 1d?
    – Why did you choose examples in 3. the way you did? Did you ask them to tell you what the transformations were? Unfortunately, I don’t have your book, so I can’t tell what the relationship between classwork and these examples is. I’ll try to figure that out on my own.
    – You say: Ask someone to draw it at the board. Discuss why triangles, squares, and hexagons work but pentagons don’t. Does “discuss” mean you discuss or they tell you what they think? In other words, did they come up with an explanation? If not, would it be possible to give that as classwork?
    – In last two slides you have questions “Can any….?” Are they followed by questions “Why or why not?”

    Maybe some of this is answered in later lessons. If that is the case disregard the questions, I am catching up.

    About the previous lesson: you defined translation, but then talked about reflectional and rotational symmetry. When I think of symmetry I usually think of a shape having symmetry, that is it is the image of itself under an isometry. So a shape could have rotational or reflectional symmetry, but the actual transformations are called reflections and rotations. This is not a big deal by no means, I was just wondering if the book chose these words or are they what you usually use?

  2. On 1d: I do a very, very off-topic question at the end of each opener. Sometimes I can circle them back to math (particularly questions with obvious upper- and lower-bounds) but most of the time they’re just to keep an anything-is-possible vibe going.

    That particular question? I dunno. Pretty grim, I guess, but I found it interesting.

    On the examples in 3: the classwork hinged on translations, rotations, and reflections so I used at least one of each.

    On discussion: My involvement was to ask the class, why do these regular polygons tessellate and why doesn’t the pentagon? They took it from there. If they tossed out a reason I couldn’t contradict (like, multiples of 5 don’t tessellate) I would provide a counterexample. That’d could’ve been a nice hw assignment.

    On the quadrilaterals and triangles: our little workshop (and the slides) demonstrated that it was possible to tessellate any quadrilateral or triangle.

    On the language here: I admit I might’ve botched the language here. The book used symmetry as a noun so I stuck with it, but I’m not comfortable here.

    Thanks for the questions/comments.

  3. Thanks, Dan. I asked about 1d not because it was that particular questions, but because it was a non-math question. I’ve seen people do it elsewhere, and can’t figure out why.

    This: “(like, multiples of 5 don’t tessellate) I would provide a counterexample” I don’t understand. Did they come up with “no multiple of the angle of regular pentagon equals 2\Pi”?

    Our little workshop (and the slides) demonstrated that it was possible to tessellate any quadrilateral or triangle – My question was whether they actually articulated why it is possible to tessellate the plane with triangles or quadrilaterals.

  4. In the first case, they came up with, “the angles of a pentagon don’t divide evenly into 360°.” Elaboration, praise from teacher, etc. We don’t use radians in Geometry.

    In the second case, the articulation was a little less profound than I would’ve liked. They gave an algorithm for each. I talked with each group individually about how they would tessellate the next quadrilateral I gave them. Could they do it faster? Then how?

  5. Faster? I don’t know. I guess the question one can ask is that what it is that we’re teaching them. I had given this project to some freshmen in college and was disappointed by how little math they had done. They basically drew some pretty pictures and that was that. There are lots of things that they could say. Like in a triangle case they could notice that they can reduce a problem to a tiling with parallelograms, and it’s probably easier to see how that one works. And they could have also maybe noticed that if they looked in the quadrilateral picture, there is a part that gets translated around to get the whole tiling (in a triangle case as well). And so forth. Onto the next lesson :)