[3ACTS] Pixel Pattern

You’ve heard of pile patterns? There are variations but generally you have three snapshots of a growing shape like this:

Questions follow regarding future piles, past piles, and a general form for any pile.

I wanted to know what this old classic would sound like with newer equipment. Would video add anything here, for instance? Here is the result of my tinkering:

Video adds the passage of time. I added a red bounding box to the video, which was an attempt to make the question, “Where will the pattern break through the box, and when?” perplexing to students.

I also added different colors, which allows students to track different things or ask themselves, “What color will be the first color to break through the box?” Different questions require different abstractions. If you care about total tiles, you’ll model the total. If you care about the breakout, you’ll model the width and height. Each one will require linear equations, which is nice.

Other notes:

  • The sequel asks about the “aspect ratio” of the growing shape which is a useful way to dig a little at limits.
  • Real-world math. Here again I’m thumbing my nose at our conviction that math should be real. This isn’t real in the sense we usually mean. If it interests your students, it will interest them because it asks questions that rarely get asked in a math classroom, questions from the bottom of the ladder of abstraction:
    • What questions do you have?
    • What’s your guess?
    • What would a wrong answer look like?
    • What information do you need to know?
I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.


  1. I’ve used a tile pattern lesson before (that I stole from someone else) that told a story and had students building their own patterns to hook them.

    It starts off with telling a story about the pet worm I used to have, his name was Blinky (or something) and isn’t he cute? Here’s what he looked like at age one (show three-tiles) and here he is at age two (show five tiles) and age three (seven tiles). Isn’t he adorable?

    Then, okay kids, make your own tile creatures – first make one up for age 1, then show how they grow for the next two years.

    Then extend to “okay so how big would it be at age 7?” … show how to make a table, then “okay but will I have room for this thing when it’s age 50??!??!” to motivate finding an equation. (Some kids won’t have made a linear pattern; lets you discuss what the difference is, although then those kids aren’t sure what to do for the equation, so I’m not sure how much I like that aspect of it.)

    The original was pitched to me using pattern blocks, which I wish I could’ve actually tried, but both times I’ve used this they weren’t available so I swapped in graph paper and filling in grid blocks / hexes. (Worked okay, having the blocks would’ve been almost guaranteed better though.)

  2. ps. Mentioning this lesson partly to randomly share and make the world a better place, but also because I’m not sure I see much of a hook in this 1st act. But a video intro combined with something hands-on for kids to create would be possible (and possibly awesome).

    pps. Did you mean “tiles” when you said “piles” or are these actually called pile patterns sometimes?

  3. Despite Dan’s bounding box, I wasn’t asking “Where/when will it break through?” Maybe I’m just conditioned as a math teacher to ask “How many tiles are there in the 100th (or nth) figure?” Now, if I had just watched that scene from The Office in which the employees are glued to a screen wondering if a pixel will hit the corner, I might have. (Dan, was it you who shared this?)

    I’ve attached context to these growing patterns (number of people seated around tables, etc.) which kids quickly ignore b/c the math itself– determining an expression– is what engages them.

    Here again Dan reminds me I too often skip past asking “What’s your guess?”

    PS – If I can speak for Dan, he meant pile patterns. I first came across this term in Jo Boaler’s excellent What’s Math Got to Do with it?

  4. Well thank God I’m not soon to be replaced by a computer…

    Funny thing is the same publishers who push computer based thins and that also publish authors whose books say the key to reaching students is making a connection. Wow what a concept

    We start at a disadvantage in math. Kids start off afraid of, intimidated by or significantly behind where they should be in order to succeed in Algebra and beyond.

    Still I like graphing calculators and the promise of what computers could possibly do for us and our students. (And I’m old enough to remember learning logs, trig and square roots without a calculator).

    I just wish I could go 1 parent teacher conference without a parent excusing their child’s math laziness, insecurity and deficiencies by saying ” I wasn’t any good in math either”. To which I usually ask how good they (the parent) was at text messaging back when they were young….

    Scott Hlls

    See you tomorrow at the Macomb ISD, Dan