Posts tagged early math
This week’s Math Goggles challenge is very simple – watch one of your child’s favorite cartoons and look for math in it. If it seems interesting, intriguing, strange, weird, and worth investigating, look further into it. Yep, that’s it! Here’s how it worked out for me.
My child’s most favorite cartoon of the moment is called Lego: The Adventures of Clutch Powers. I like it, but not enough to sit through all 88 minutes of it every other day. Still, I overhear snippets of dialogue as I go about doing my own things. One of the snippets that caught my attention was a trick question from the Troll Warrior to Clutch Powers:
Troll Warrior: Using nothing but standard eight-stud bricks, how many bricks would it take to build a spiral staircase as tall as three minifigs?
Clutch Powers: There are more than 915 million ways to combine standard LEGO bricks. But they will never connect on the diagonal.
Really? I mean, I understand the impossibility of connecting on the diagonal (had to explain it to my son a few times before), but 915 million combinations?! This sounds pretty amazing.
Turns out, the story is even more amazing than it sounds. The 915 million figure, or more precisely, 915 103 765, is the number of unique ways to combine just six two-by-four studded LEGO bricks of the same color. If this sounds unbelievable, check out the explanation of the LEGO Counting Problem. It is a great read and has just one formula that’s pretty easy to understand.
One of the best parts of the paper is in the “Why is this interesting?” section:
Such challenges are always important to drive mathematical research, and it often happens that methods developed to study a problem with no practical applications (like this one) are useful to study problems which do have an impact on everyday life.
Back to the original question of building a spiral staircase out of LEGOs… This is when the second part of my challenge started. And let me tell you, I totally cheated by looking at… my 6-year old’s recent builds.
He recently built a 3D treasure map using some unit blocks and LEGO bricks, complete with the “X marks the spot” signs which he made out of LEGOs, of course. That’s how I knew that, under some conditions, regular LEGO studs and plates can intersect at angles other than 90 degrees. So I used this idea and built a very basic spiral staircase in no time although it does not follow the ”use nothing but two-by-four studs” rule.
Now it is your turn to find math in your child’s favorite cartoon!
It’s time to put on the Math Goggles (not sure what these are? Head over here to find out). Last week I invited you to search for math at your local library. But this week I haven’t had much free time for impromptu field trips. So I’m donning my goggles and hunting for math around the house. To make things a bit more interesting, I decided to only look for one thing – patterns.
I found my first pattern walking down the stairs. Rise-run-spindle-spindle-rise-run-spindle-spindle… I only have one cat, so the pattern was broken there.
Next I had to put some books away and noticed this weaving pattern on a basket on my bookshelf.
And what about a bookshelf itself? Large shelf, two small shelves, large shelf, two small shelves. Moving to an even larger scale, the furniture arrangement pattern in the room (as noticed by my husband) is “furniture to sit on, furniture to pile books on, furniture to sit on, furniture to pile books on…“
But of all the patterns I found around the house today, this one had to be my favorite. I had to replace batteries in one of the toys and here it was, positive-negative-positive-negative terminals.
What patterns can you find in your house?
P.S. Once you start noticing patterns, it gets very addictive. So here’s a cool and very relaxing idea for a collaborative pattern hunt video I found on YouTube. Next time you have a few minutes to spare, doodle the patterns you notice. Here’re my doodles, including the furniture arrangement pattern, the stairs pattern (reflected and rotated), the batteries pattern and a few more.
Some time ago Maria and I had a chance to present the Moebius Noodles project at an Open Source/Creative Commons event hosted by Red Hat. As we tried to distill our big ideas into a 3-minute presentation, we had to choose the most important points to cover. They are
- In many ways, young kids already are mathematicians.
- Beautiful mathematical adventures are all around us.
- Math is not a worksheet.
- Freeing ideas and experiences (i.e. through Creative Commons licensing) is critical for success.
Want details? Check out a video of our presentation.
If you are developing, teaching, or playing rich early math games, we want to hear from you!
P.S. Can you tell this was my first public presentation?
This is the game my son and I are calling “Gummy Bear Go!” even though most of the time we play it with paperclips instead of treats. I got this game from the Russian mathematician Alexander Zvonkin’s book “Math from Three to Seven: The Story of a Mathematical Circle for Preschoolers“.
To play the game all you need is a piece of paper with a grid drawn on it, a pair of dice, and some counters. The grid has 7 rows and 15 columns, although you can certainly make more rows for a longer game. Each cell in the bottom row is numbered 1 through 15.
Decide how many counters each player will have. The first few rounds we played, we each had 3 counters (make sure you can tell your counters apart from the other player’s). Place the counters on the numbered cells. Make sure to not put more than one counter in each cell. Now, roll the dice, add up the dots, find the counter with the cell number corresponding to the rolled sum and move the counter up one row. Repeat until one of the counters reaches the top row. The first to reach the top row wins the game.
When we first played the game, my son placed his first counter on 6, but his other counter – on 1 and his third counter – on 15. If this happens when you try this game, don’t rush to correct your little one. Instead, play along and let him learn from experience. If your child, like mine, doesn’t do well with loosing a game, you can level the field a bit by placing your counters on “impossible” (1, 13, 14, 15) or unlikely to win numbers (2, 3, 11, 12).
Rolling two dice means that there’s a lot of addition work in this game. Which is great, but keep in mind that practicing additions is not the main goal of the game. I didn’t want to give the answers to my son, but he did need help with the larger sums. So I gave him a ruler that he used as a number line.
Of course, since there were just the two of us playing with a total of 6 counters in the game, most of the numbers were left open. When we happened to roll a sum equal to an unoccupied number, I’d make sure to say things like “ah, too bad neither one of us had a counter on 7″ or “hey, 9 again?! I just might play it in the next round!”
At the end of each round we’d stop to survey the game board and note which counters didn’t move at all and which ones “put up a good fight”. I had the idea to mark each round’s winning number.
After the first round, my son’s choice of numbers to place counters on became much more interesting. He clearly understood that his best chance at winning was on the middle numbers – 5, 6, 7, 8, 9. He abandoned the impossibles and after one or two games moved away from the marginal 2, 11 and 12.
Finally, after playing this game for a few days on and off, we played one last round that I called “the grand parade”. We put one counter in EACH cell in the bottom row. This way, every time we rolled the dice, something moved. And once the game was over, we surveyed the battle field.
And then we filled out this little table of all the possible roll combinations. That’s a whole lot of additions which gets pretty boring. So instead I suggested to look for patterns. My son quickly noticed the horizontal and vertical patterns.
Finally, I suggested we try to figure out what’s the most likely winning number. To do that, I asked my son to find the smallest sum in the table, 2. Which explained why placing a counter on 1 was a waste of time. Then I asked him to find the largest sum, 12. Which ruled out 13, 14 and 15 once and forever.
Next, we started counting how many of each sum we had in the table. The 2 and the 12 were easy-peasy. It was interesting to see that although he noticed the horizontal and vertical patterns right away, he failed to see the diagonals. But after finding and counting all the 3s and 4s, he noticed the diagonals and after that counting was a breeze. But the best part was that once we were done counting all the different outcomes, he knew right away which three numbers were the likeliest to win. It was so awesome to see him go through the “Aha!” moment! Plus we got to have gummy bears and mini-marshmallows to celebrate!
My daughter and I have learned so much math by finding it wherever we are and in whatever we’re doing. For the last year we have been paying attention to the physical world around us and finding as many different examples of math in our lives as we can. It’s quite stunning how beautiful and full of math even a city sidewalk can be if you have your math glasses on.
Back in May, for example, I wanted to start looking for spirals but only found two examples, one in a garden and one in our local playground. Long story short, at some point my daughter picked up on the spiral thing and started pointing them out, only to have me say, “No, those are actually concentric circles,” which then lead to a few days of clarification about what a spiral is and isn’t. Now she sees them everywhere!
We’re a team, her and I. It’s really fun that things we have taken for granted all our lives suddenly have a new dimension. This is why, I think, that a recent return to reading familiar picture books from our home library made me notice math in books that are not obviously math readers.
My very favorite almost-hidden math story book is Five Creatures, by Emily Jenkins. It’s about the similarities and differences (attributes!) between the members of a lovely little family.
“Five creatures live in our house,” it begins, “Three humans and two cats. Three short, and two tall….Three with orange hair, and two with gray.” We read this book when my daughter was in preschool and it was fun for both of us to look at the pictures to see who matched each description. The categories of family attributes are not always straightforward, which makes this a wonderfully interactive read.
In Ezra Jack Keats’ The Snowy Day, cut paper illustrations show math from the very first pages. In addition to great spatial vocabulary (up and down the hills, tracks in the snow, on top of, snowballs flying over the boy’s head) patterns abound. Check out this wallpaper — I love how the pattern units are so different from each other, and yet the overall pattern is so regular:
Parallel lines made by sticks and feet and gates:
The foot prints alternate, making a kind of frieze pattern:
I love this grid pattern in the mother’s dress, and it’s not just a color pattern. If you look closely there’s another attribute of shading (solid and striped):
This background is a great example of ‘scattered’ like in a scatter plot. Which section has more dots, and which has less? How do you know?
In nature, every snowflake has the same structure yet each one is different from every other snowflake. That’s not exactly the case here. How many different kinds of snowflakes can you find? How are they different and how are they the same?
So, now I’m curious what other books are out there that have this kind of ‘hidden’ math? I just thought of one more book: My daughter listened to the novel Half Magic on CD back in the fall. In the story, the kids find a charm that gives them half their wish and they quickly learn to wish for twice as much as what they really want. It’s fabulous.
What other kinds of books have you found that have this kind of hidden math? I’d love to hear your ideas!
Yelena recently reviewed the book “1+1=5″ and shared the game of “I spy” she plays with her son.
The book inspires kids (and adults) to see everyday objects as sets, or collections of other objects. For example, a triangle can be viewed as a set of 3 sides while a rectangle is a set of 4 sides. An octopus is an example of a set of 8 (arms) while a starfish hides a set of 5 (arms) in plain sight. If one set has 8 elements and another set has 5 elements, then when added, the two sets have 13 elements total. Hooray!
I thought it could be fun to invite readers of this blog to play a round of the game. Here is the big question I am contemplating: “How can we make our descriptions of games we design so interactive that they become, literally, playable games?”
Add your own example! Of course, this is the ocean, for our Moby Snoodles.
This is what people added so far! It takes about five minutes for your answer to appear here. Wait and then reload the page to see!
Days are flying by and everyone is busy preparing for the holidays. While you might not know what gift(s) you will be getting this year or whether the kids will catch the sniffles right before the party, one thing you can count on is tangled lights. Seriously, no matter how carefully we pack them, when we open the boxes next year, here they are, all knotted up. Turns out, it’s all about math and physics as opposed to our carelessness. Check out this NPR article for details. Here’s the best part – next time you have knotted up strings of lights, give them to your little one to undo and proudly tell your friends he was busy solving a knot theory problem.
In addition to tangled lights, another thing you can count on is a broken ornament or two or three. If your tree looks a bit bare, don’t rush to the store. Instead, consider some really cool last-minute DIY ornaments. Bonus is they are all about math. The easiest ones are paper chain garlands. This year, add a twist, (ok, half a twist) and turn them into Moebius chain garlands. Or bend pipe cleaners into shapes and grow Borax crystals on them. What shapes you choose is totally up to you. Inspired by Vi Hart’s Borromean Onion Rings video, we made this Borromean Rings ornament.
Finally, we’d like to share a holiday printable we found thanks to the link from one of the readers, Elena T. She printed it for her daughter to color days ago, but we just got around to it. This Christmas tree might seem like just a giant cute coloring page, but don’t let it fool you. It’s got lots of things going on, including some great math. Can you find examples of gradients, tessellations, pattern, rotational symmetry, radial symmetry, iconic quantities and a lovely Sierpinski triangle?
Share your holiday math with us on the blog and on our Facebook page.