Factorization diagram cards are here!

It’s been a long process, but factorization diagram cards are finally available for purchase!

If you just want to purchase a set right this minute, then click the above link! If you want to learn more, keep reading.

History

As explained in this original blog post from 2012 and this follow-up post, the basic idea behind factorization diagrams is to visualize the prime factorization of a positive integer $n$ by taking $n$ dots and recursively grouping them according to the prime factors. For example, $30 = 2 \times 3 \times 5$ can be visualized by making two groups of three groups of five dots, as seen in one of the cards above. You can find a lot more information about factorization diagrams here, including links to related things people have made, posters for sale, and so on.

Very early on I heard from teachers who had printed the diagrams, cut them out into cards, and used them successfully in their classrooms. After hearing that, I decided that there really ought to exist a high-quality deck of factorization diagram cards for purchase. It’s taken four years for that idea to come to fruition, but they are finally here!

The deck

So, what’s in a deck, you ask? Each deck contains 54 large (3.5 inch) square cards. The front of each card has a factorization diagram, and the back has the corresponding number and factorization written out. Numbers with multiple distinct prime factors have multiple cards with different diagrams, one for each distinct permutation of the prime factors. (For example, $12 = 2 \times 2 \times 3 = 2 \times 3 \times 2 = 3 \times 2 \times 2$ has three different cards, as illustrated below.)

You can buy your very own deck through The Game Crafter for $13.99 (see below for an explanation of the price). The images used to make the cards are freely available here (in case you’d rather just print them yourself, or do something else with them), and the source code is on github. Everything is released under a Creative Commons Attribution 3.0 license, which basically means you can do whatever you want with the source code, images, design, etc., as long as you credit me as the source (preferably by linking to https://mathlesstraveled.com/factorization). If you end up using these cards in a classroom and come up with any fun activities/games/puzzles using the cards, please let me know! I would like to collect a big list of suggested activities for different ages and eventually be able to publish the list along with the cards. Pricing and print-on-demand As mentioned above, I have published the deck through The Game Crafter, who have done a great job. The process was easy and professional, and I am pleased with the final product—and I hope you will be too. You might think$13.99 is steep for a deck of cards, but (a) the print-on-demand model means there are no economies of scale to be had, and (b) these are definitely high-quality cards (3.5" square, high-quality card stock, with a linen texture and UV coating that reduces glare). I think a deck should last you a while, even if young kids are handling it.

Speaking of economies of scale though, if you are—or know—an educational/game publisher who would be interested in publishing this deck at a lower price point, please contact me! Until something like that happens though, I don’t have the resources—temporal or financial—to be able to coordinate a larger print run. The print-on-demand model means that I can get these cards out without a huge commitment.

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Post without words #11

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A few words about PWW #10

If you still want to think more about the picture in my previous post, stop reading now!

Here’s a simple way to think about how the picture is made, as noted by Fergal Daly. The $n$th circle (starting with $n = 1$) has $n$ evenly spaced radial spokes, which we think of as being numbered clockwise from $k=0$ through $k=n-1$, with spoke $0$ always lying on the positive x-axis. Then spoke $k$ has a blue dot at its end if and only if $k$ is relatively prime to $n$, that is, $\gcd(k,n) = 1$. So, for example, the tenth circle has dots on spokes $1$, $3$, $7$, and $9$, since every other number shares a factor with $10$. Note in particular that $\gcd(10,0) = 10$, and in general $\gcd(n,0) = n$ (this is not a special case, it’s just a natural consequence of the definition of \$). This is why no circle except the first has a dot on the positive $x$-axis. (Though as noted by Macbi, we can also think of the blue dots on circle $n$ as highlighting the generators of the cyclic group $\mathbb{Z}_n$—this is almost the same as the definition in terms of $\gcd$, but means that possibly we should not put any dot on the first circle.)

It’s worth pointing out a few consequences of this definition in terms of $\gcd$:

• As noted by Naren Sundar, prime-numbered circles always have $(n-1)$ blue dots: one on every spoke except the $0$th.

• The dots always have reflection symmetry across the $x$-axis, since if $k$ is relatively prime to $n$, then so is $n-k$.

• As noted by Dan Kearney, once a blue dot has appeared in a specific location on a circle, no blue dot will ever appear there again. For suppose a blue dot appeared on circle $n$, spoke $k$. Then if a spoke ever appears in exactly the same spot later, then it must be spoke $ik$ on circle $in$ for some multiple $i > 1$—but then $\gcd(ik,in) = i \neq 1$. For example, note the blue dots on spokes $1$ and $2$ on the third circle. Those same spokes show up on circle $2 \times 3 = 6$—but now they are spokes $2 \times 1 = 2$ and $2 \times 2 = 4$, so they have no dots. Likewise, the same spokes show up again on circle $3 \times 3 = 9$, as spokes $3 \times 1 = 3$ and $3 \times 2 = 6$.

We can use this last fact to make some cool pictures: for example, we can give a different color to each $n$ and then superimpose all the circles on top of each other. No dot will ever overlap with another dot (well, at least they wouldn’t if the dots were infinitely small). To make it more visually obvious what’s going on, I’ve also scaled circle $n$ to have radius $n$:

This kind of reminds me of Paul Salomon’s Stars of the Mind’s Sky, except where we have deleted any star that can’t “see” the origin when it is exactly blocked by another star.

Now, Naren Sundar also made another conjecture: he observed that in addition to the reflection symmetry, the blue dots seem to have rotational symmetry whenever $n$ is divisible by some $k^m$ with $m > 1$, that is, whenever it is divisibe by the square of a prime. This turns out to be true, but unlike the other properties mentioned above, it is not at all obvious just from the definition! I hope to talk more about this in some future posts.

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Post without words #10

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Factorization diagram card redesign: feedback welcome!

After getting a printed set of factorization diagram cards, I decided there were a few design tweaks I wanted to make. I’ve gone through a few iterations and I think they are definitely better now. Here are some representative samples (namely, 6, 13, 21, 29, and 30):

• Better color scheme (at least I think so!)
• Primes now have a visual representation that does not depend on color (though the color is still meaningful). For example, 29 is represented by an outer shell with two half-circles (representing the 2) and a trio of triangles (representing 9, that is, three threes).
• The triangle representing 3 is flipped upside down so it never intersects with anything.

I’d love to hear any and all feedback! Modulo any final tweaks I plan to make sets available for purchase soon.

Posted in arithmetic, counting, pattern, pictures, primes, teaching | Tagged , , | 6 Comments