Lately I've been looking for different ways for my seven year old and I to conceptualize multiplication. As has happened many times before on our math journey, this graphic showed up at just the right time (albeit somewhat circuitously through the excellent influence of the Math Munch blog).

My favorite thing about it is that it's not about numerals; when I look at factoring trees I can make some surface sense of them, but my mind goes numb pretty quickly. In this visualization, however, there is an incredible connection to shapes and grouping. I find this visual especially well-suited for kids in general and at least this adult specifically.

Last night I printed out the graphic and left it advantageously on the kitchen counter. I thought maybe my kid

*might*be interested but was truly surprised by her reaction when she found it this morning. It is probably the first piece of math my daughter has ever admitted she was excited to know more about, which is saying a lot.
She wondered what it was about so we looked it over together. At first it was basically 'count the dots' and notice that each configuration was one more dot than the one before. Then, in the same way we tackled the 100's chart last winter, we started looking around and noticing things: The ring of seven dots on the far right column has multiples of seven underneath it. The 6 shape shows up two more times on a descending diagonal. It's fun just to look and talk about what you see.

It's the geometry of the design that really shows the relationships between numbers. And, even though this was not meant to be a multiplication chart, it's probably the best one I've ever seen.

All our talking and looking got my mind spinning.

**What if...what if I made little playing cards out of each factorized number?**What kind of game would it be?
I was about halfway through constructing the cards when my big AHA! moment hit. As I made and sorted them one by one it became completely clear to me that the integers 1 through 7 formed shapes that were echoed in the other factorizations. As an attempt to organize my growing pile of cards I laid out a top row of 1 through 7. But where to put the other cards? For example, 5 is a pentagon made out of single dots and 10 is a pentagon group of two dot groupings. Where does it belong? The 2's column or the 5's column? This kind of question is at the heart of the new game.

Here's how my daughter decided to sort them in a 'get acquainted' activity before we started playing:

As we went along I refined the language she needed to help her make her choices. Was she going to place a particular card based on its large grouping (outer shape) or the smaller groups? As you can see above, there's a 5 shape of 3s in the 3 column, because the smaller group is a match to that number. But, every other 5 shape is in the 5's column. She's also got a 7 shape in the 3's column for the same reason -- the smaller grouping matched and, ultimately, the whole 3's column is consistent on that criterion.

For some comparison, here is how I sorted the cards, earlier in the day. I was trying to match to the category of 'outer shape':

I'm not sure I got it the way I wanted it, but no worries. There is probably no one right way to sort these cards and the activity in itself makes for some really interesting thinking and conversation.

After she familiarized herself with the cards we started in on the new game which I'm calling

**Factor Dominoes (with a side of Scrabble)**. The title alone should give you clues as to the game's aesthetic and procedure, but here's how to play:
Split the deck equally between two players. Player 1 puts down the opening card. Player 2 tries to find a match. If Player 2 has no match the card is put aside face up for future use and play returns to Player 1. You can find a match either by outer grouping/shape (triangle, square, pentagon, weird six shape and seven ring) or by similarity between the small dot groupings. In our game we also matched 'echoes' -- small groupings that are the same shape as another number's outer shape.

For example, in the picture below the first card is a 5 shape with small groups of 2. The 6 shape next to it works because even though it's a different shape it also is comprised of 2s. And, the card directly below the first card also works because the smaller groupings of 3 match the 5 shape of the larger grouping. Make sense?

Here's another example: The top line of matches have the 3 shape in common. The bottom row connects to the top with small groupings of 4.

And, here's a picture of a couple more interesting matches. See if you can figure out our reasoning on this section of the game:

Play the game until there are no more cards. This is a cooperative/conversational game but feel free to give it a point structure if you like. You can also make the game bigger and more complex for older students -- just cut out more factors and make more cards! That's what I'm going to do for our next round of play.

Here is our completed first game:

Based the exponential growth of my personal understanding of primes and factors, gained in just one short day, I am firmly convinced that a wide range of ages, experiences and abilities can get something of value out of this game.

My seven year old was perfectly challenged as we focused on groupings, but what if you added the prime numbers beyond 7 into the mix? How would that deepen or change things? What about adding exponents as a match category? What if you figured the value of each card and matched them in sequences (like {25, 26, 27, 28...} or {4, 8, 12, 16...} or even a sequence of primes, in order)?

If you do play this game PLEASE let me know how it went and what other ideas you have for it. And, please do consider joining us on the Math in Your Feet Facebook page. We're having a good time over there!

p.s. I cut up the chart of 49 integers and made my own version. What do you think?

Great game idea! I'm going to have to bookmark this for the next time I teach a co-op math class.

ReplyDeleteGreat! Let me know how it goes. I'd love to hear about how you adapt it for your learners. :-)

ReplyDeleteMy son might be willing to try this. I'll have to stop by a copy shop to get some color copies. Nice!

ReplyDeleteAwesome! It actually reminds me of the board game Qwirkle which I've been meaning to buy, but haven't so far. Maybe I'll just use your game idea instead.

ReplyDeleteYeah, it is like Quirkle, isn't it? I find the factor version more engaging in terms of really looking and enjoying similarities/differences. Anna from Math Munch pointed out that it's sort of like Bananagrams. I think a good game structure is something that you can adapt endlessly -- crossword puzzles, scrabble, dominoes... whoever invented those was a genius! :-)

ReplyDeleteVery interesting. Are you sharing the document? I tried to load it, and it wouldn't save as a jpeg. Many thanks, Glory

ReplyDeleteWe made a PDF suitable for printing on 11x17 card stock, if that's still useful.

Deletehttp://rwoodley.org/?p=427

Hi Glory -- What happens if you 'right click' on the image? (I used the first picture in this post to make my game.) When I do that I get 'save image as' and also another option to 'print picture'. I think either will work.

ReplyDeleteIf that doesn't work for you, perhaps follow this link to the original image: http://mike-naylor.blogspot.com/2012/11/factor-visualization.html and try right clicking again. I included links in this post but when I went to check just now, the links had disappeared! They should all be good now.

Let me know how it goes, and thanks for reading! :-)

I love this! Pinned it, shared it, posted about it. What a brilliant way to play with math and see factor patterns. :)

ReplyDelete~Alicia

Thanks Alicia! :-)

ReplyDeleteThis is a revolutionary moment! When I showed this animation to my third graders: http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/ , they automatically and intuitively understood how numbers go together. They'd see a circle made up of dots, and they would chant, "PRIME!" It can take years to teach the concept of prime and composite with good ol' Arabic numbers. Thank you for showing us this, and our deep gratitude goes to Ph.D. student Brent Yorgey for first publishing this online, only last October! http://mathlesstraveled.com/2012/10/05/factorization-diagrams/

ReplyDeleteHi Laurie -- I agree, the visual graphic is truly revolutionary on so many levels. For people like me, like your students, and also for math education in general. I think it also shows, once again, how the power of the internet can be used for good! There was the original design by Brent and then other people took it places and refined it (I think that's the right progression) and then people started *using* it and *applying* it and we all benefit.

ReplyDeleteI was going to say that your game reminds me of Quirkle, but someone beat me to it! I actually created my own version of Quirkle to be played in small groups with my 7th and 8th graders, but I LOVE your version better because factoring is such a tedious concept for my struggling students! Thank you!

ReplyDeleteCompletely blown away. Printed, pinned, emailed, shared. LOVE the factor-visual. So intuitive, and pretty. Thanks!

ReplyDeleteBrilliant! (Again!)

ReplyDeleteI'd seen the wonderful animation of this -

http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

but I love the idea of this game.

I checked it out, and I like Brent Yorgey's latest colourful version of it -

http://mathlesstraveled.com/2012/11/05/more-factorization-diagrams/

I'll probably just print his pdf onto card and cut it up.

Yeah, I liked the 'factor conga' dance, but it moved too fast for my eyes and brain to comprehend. I wanted to slow it down a little and have more control over finding relationships between shapes/numbers which is why I thought to print it out in the first place. I'd love to see how your students play the game and what interesting things they find. :-)

ReplyDeleteThis post really sparked a lot of comment in our household. We spent some time coming up with a set of rules that would work and could be understood by our 6 year old.

ReplyDeleteAlso we attach a PDF suitable for printing out on card stock so you can make your own set of cards.

http://rwoodley.org/?p=427

Excellent, Robert! Thank you for sharing!

ReplyDeleteI don't understand what makes your mind go numb with factor trees? Also, why does it take years to "understand the concepts of prime and composite with good ol' Arabic numbers?"

ReplyDeleteI'm honestly kind of baffled as to why those concepts are difficult.

Kathy W.

Since my comment above, I dabbled with the javascript animations by Sean Seefried to create 2 related products:

ReplyDelete1. a calculator, and

2. a factorization game.

The calculator helps you explore factors in multiplication and division. The game was just my attempt to inject some competition to incentivize kids to think visually when factoring numbers.

http://rwoodley.org/?p=492

I've left a reply on your blog, Robert, but in general very cool! I hope others will check it out and give you feedback!

ReplyDelete