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Squares of Differences: subtraction practice toward a greater purpose

Here is a phenomenal lesson, accessible to any child who knows how to subtract, and compelling to everyone, up to and including professional mathematicians. Get a kid engaged in it, and they’ll do hundreds of subtraction problems without complaint, because it’s helping them solve an honest mathematical mystery. I’ve seen this idea discussed in math education circles, but I haven’t found any references to it online, so I want to share it here.

Squares of Differences

Here’s the game: draw a square, and pick four numbers to go in each of the corners.

Once you’ve done that, put a dot on the midpoint of each side, and find the positive difference between the numbers on the closest corner. The picture is worth a thousand words:

Now connect the midpoints.

Lo and behold, you’ve got yourself another square! Which means you can repeat the process…

And then you keep going until…

there are no surprises ahead. In this case, we have zeroes showing up in the middle square, so there will continue to be zeroes forever.

At this point, this is purely a fun little exploration (that happens to involve about 20-30 one digit subtraction problems), and you and your child/students should try a few of these out. I recommend starting with one digit numbers while you’re getting the hang of it, but there are no real restrictions other than the instructions above. But once you do a few, you’ll start to notice something… you always seem to end up with a square of zeroes in the center.

And now we’re ready to ask the some serious questions, and pose some real challenges.

  • Will this game always end with a square of zeroes in the center?
  • What’s the most number of steps you can go before arriving at a square of zeroes?

My students generally count each square past the first one as being a “step.” The example above ends at step 7.

From a pedagogical perspective, here’s what’s cool about this problem: the challenge it is immediately engaging, and allows immediate entrance into the problem. Kids can get started right away, and by the time they’ve tried 4 or 5 squares, they’ve done a hundred or more subtraction problems. So they’re practicing a skill, but in a context where it’s motivated; they need subtraction to see if they can get more steps in their square of differences. But also, they can choose numbers that make this problem as easy or as hard as they need it to be. For example, you could put two digit, three digit, or 10 digit numbers in your starting square. You could also try fractions (or even irrational numbers!). So suddenly long subtraction, decimal subtraction, or differences of fractions becomes something a student needs to figure out (if they don’t already know it). It’s either great practice, or it’s motivation to learn something new.

They can also try out different strategies. What matters here? Some of my students thought that maybe using odd versus even numbers would make a difference (what happens if you use all odds? All evens?). Some others thought that using prime numbers would be the secret, while others had a hunch that using powers of 10 (1, 10, 100, 1000) would be the key. Should you try using a zero at the beginning?

Testing strategies also keys you in to important pieces of the mechanics at play. For example, one of my students noticed that if all the numbers on a square increase by one, you still get the same numbers at the next step. So the square with {1, 4, 6, 9} would give the same next step as the square with {2, 5, 7, 10}, or {3, 6, 8, 11} for that matter. Getting into the habit of noticing behavior like this is a critical habit to develop for the budding mathematician.

So that’s phase one of this lesson. What’s beautiful is that the kids will almost certainly usher in the next phases, because there are questions waiting to be asked, extensions to this problem just dying to be made. (See a good extension or question? Let me know in the comments.)

If you try this lesson out, tell me how it goes. Later, I’ll provide another post with my own favorite directions about where to go with it from here.

UPDATE: To read Squares of Differences II, check out the next blog post in this series.

Comments 6

  1. Elizabeth

    Thank you for this fantastic math conundrum. I work with 5th and 6th graders in at middle school and presented this problem with my 7th and 8th grade math/science teaching counterpart. We offered up the problem to all 5th through 8th graders, asking the questions of:

    Can you find different numbers that require four steps to complete the square?

    What is the minimum number of steps possible?

    What is the maximum number of steps possible?

    Can you find a “rule” to predict how many steps there will be?

    Because our school is technologically connected, many students asked for the name of these squares in hopes they might find answers online. We’ve kept the name a secret, and many of them have taken on the challenge. This coming Friday we will be coming together 5th-8th grade for breakout sessions to discuss findings or test rules and predictions.

    We are wondering what more you’ve done with this problem– in my experience thus far, the problem continues on and on as more questions pop up. I’ll let you know how the conversation goes on Friday!

  2. Marielle Wiesinger

    I just found your blog and I am very excited to try this with my kids at home and with my students in the Challenge Centre. I will let you know how it goes.
    I am interested in developing a workshop for gifted students looking at math concepts, so your blog will be great resource! Thank you!

  3. Pingback: Math Teachers at Play 38 | Mathematics and Multimedia

  4. Adam

    How does the process change if you ALWAYS have to move clockwise? For example, given a square with corners (starting at top-left, or northwest) 1, 2, 5, and 7:

    1-2 = -1
    2-5 = -3
    5-7 = -2
    7-1 = 6

    Your second square, then, would be:

    -1 – -3 = 2
    -3 – -2 = -1
    -2 – 6 = -8
    6 – -1 = 7

    …and so on. Will you ever reach an all-zero square?

    Adam

    1. Post
      Author
      Dan

      Adam– this idea came up in one of my classes, and we played with it for a bit, but I haven’t thought about it much. My guess is that virtually none of them resolve to an all-zero square. But I think there will be some other invariants worth noting.

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