math – little thoughts

learning together: counting in binary

Posted on 2021-02-062021-02-04 by kristinabean

Limerick has been interested in different base systems for several months now; he’s enjoyed playing place value games in bases other than 10, converting between bases, counting in alternate bases, and so on.

The other day he came up to me and told me that he’d figured out a new way to count on his fingers. Holding up the pinky finger on one hand, he told me that was 1. Just the ring finger extended was 2; the ring and pinky together was 3. The middle finger on its own was 4, middle and pinky together was 5, middle and ring together was 6, and middle, ring and pinky all at once was 7.

It may be easier to see his logic with 0’s and 1’s, where a finger curled down represents a 0 and a finger extended represents a 1.

00000 00001 (one pinky extended)
00000 00010 (ring extended)
00000 00011 (pinky and ring extended)
00000 00100 (yeah, this one looks wrong, but he’s six so he has no clue)
00000 00101 (middle and pinky)
00000 00111 (middle, ring, and pinky)

And so on.

It’s binary! Assigning each finger one place in the binary place value system, he figured out how to count on his fingers to 1023 (which would be all fingers extended). In addition to showing how natural this aspect of mathematics is to him, it also shows a good foundational understanding of computer science – because electric signals are either on or off, just like Limerick’s fingers could either be extended or curled down, and therefore represent data in this same bitwise manner. It is so amazing what kids can think of and create when they are given the chance to deeply explore something they love.

(We tried developing a base 3 counting system but found it was too difficult to keep some fingers folded only halfway down when the fingers next to them needed to be extended or completely curled – someone with more fine motor control might have more luck though!)

sqt

snippets of life

Posted on 2020-05-072020-05-07 by kristinabean

We are now flooded with blackberries! I have made 4 jars of jam, a crisp, and the most wonderful simple sauce to pour over crepes filled with tangy whipped cream – not counting all the berries we’ve simply eaten straight off the bushes! And honestly this is just the beginning of the season; we probably have another few weeks of fruit.

I’ve been texting my sister all about my blackberry goodness and I think she’s a bit jealous, as she lives in a place where blackberries ripen much later. It might even be one of those places where you can legitimately eat the last of the blackberry harvest on September 29th for Michaelmas (which just boggles my mind as that seems so late for berries). But her turn will come in a couple months and then she can make me wistfully recall these blackberry days.
My other sister, who lives locally, made me a very nice cloth mask to wear when I need to go out! She’s actually been making a lot of them and donating them to one of the local grocery stores where she knows some of the employees, and she even offered to donate some to the security personnel where I work after they commented on mine and on their anxiety about not having enough (the man I spoke to was going to be on patrol on one of the floors scheduled for deep cleaning after a confirmed covid-19 diagnosis and didn’t have a mask, so there’s clearly a problem here for people’s state of mind at the very least, even though the masks aren’t as crucial as keeping your hands clean and off your face).

Sporting my mask! It’s like the opposite of one of those superhero masks that just go around their eyes.

I also got to wear the mask down to Aubade’s dermatology follow-up this week. Of all the places I’ve been, they are definitely taking the virus most seriously. Two people wearing masks stood outside the waiting room to take our temperatures before letting us in; the chairs were mostly removed with just a few left in isolated locations; and everyone there was masked and gloved. And while Aubade’s eczema (about which we were following up) is well under control currently, it was still a useful visit with regards to the molluscum epidemic among the kids (a possible treatment at last! Hurrah!) as well as Aubade’s nevi spilus birthmark (which apparently slightly increases her risk of skin cancer over her lifetime and is something to watch as she grows). Also Aubade got to show a bunch of new people her sparkly light-up sneakers, and we got to chat just the two of us for the whole drive down and back, which is always nice.
Limerick is still loving numbers over here. He really just needs a whiteboard and a marker and a new concept to sink his (figurative) teeth into and he is so happy! This week he spent quite a bit of time calculating powers of different numbers until they got to be 6-7 digits long; he made lists showing the conversions between fractions and decimals for 1 down to 1/12; I introduced him to long division; and we had a fun afternoon counting in different base systems. (It always seems like we aren’t doing enough school, but I think that’s actually a lot for one week now that I list it out like that).

We did base 10, 16, 2, and 3 together; then he did base 4 on his own and here he’s working on base 5. He ended up making lists for bases 6-9 as well, counting as high as he could fit on the whiteboard for each base.

In general, Limerick has a good mind for solving problems of the quantitative sort. Paul’s been setting up the boys’ loft beds so that one of the ladders and the two railings lie across the space between (with Aubade’s mattress on the floor below) so that the kids can crawl across and hang and swing and jump off them. Well, they decided they didn’t want to wait for him to be done working, so Limerick and Rondel figured out how to do it together. It’s not easy! The ladder is heavy and difficult to position even for me, as is the mattress they have to drag in from the other room (through two doors and a narrow hallway). But Limerick has an eye for angles and positions and possibilities, and was able to figure out how they could set it up without the advantage of adult height. It was really awesome listening to him direct the whole project, with Rondel providing good input and feedback as they worked together.

And… that’s all I’ve got for quick takes this week! (Well, I suppose I could add that Rondel and I have almost finished The Horse and His Boy since last week’s takes. I’m not always sure just how much he understands, especially in places with lots of names and places, but he keeps wanting me to read more.) Visit the link-up at This Ain’t the Lyceum! I hope you all have an excellent week filled with unexpected pockets of joy 🙂

sqt

finding joy in the little things

Posted on 2020-05-01 by kristinabean

Aubade has been so into princesses lately that I decided to make a couple little dresses for some of her tiny dolls, to match the princess costumes one of her aunts and uncles gave her for Christmas. I guess one silver lining of the quarantine is that I have a little extra time at home for little crafty projects like this! (Also, her pink nails are Crayola marker… she’s been coloring them to match her outfits!)

For Limerick, I wrote a few quick Python scripts to let him see some of his favorite number sequences up to whatever parameter he wants – Fibonacci numbers, triangle numbers, square numbers, powers of any base, and reciprocals of integers. He loves being able to see those numbers in more detail (and more quickly!) than he could with a calculator. So far I haven’t gotten him interested in trying to write his own code, although since this is his first time using a computer he does have the whole learning curve of the keyboard and trackpad to deal with first!

With Rondel I’ve just been reading and reading and reading. We started The Lion, The Witch, and The Wardrobe towards the end of Lent and have now finished it, as well as Prince Caspian, The Voyage of the Dawn Treader, and The Silver Chair. He absolutely loves worlds of fantastical beasts, bold adventures, kings and queens, and magic.
One of the hardest things about the quarantine is making sure we all get enough physical activity. The other day we set off on what I thought would be a short exploration through the neighborhood and ended up being a 3.2 mile trek (Aubade in the stroller, Limerick on his bike, Rondel walking)… but now the temperatures are hitting 100 every day and those long walks are a lot less enjoyable. Also I’ve been sick for the past week and really didn’t have energy to do anything active with the kids. My parents’ pool has been such a blessing, as their house is the one place we are still taking the kids, but for the first time I’m wishing we had our own! I’m sure when things are back to normal this desire will fade, though.

My other personal challenge is maintaining a sense of rhythm and structure when all the milestones and pivot points of a regular week are gone. I think especially as an autistic person, I struggle significantly with having an uncertain routine. So far I’ve been doing alright with bookending the day – prayer in the morning, reading to the kids while they eat breakfast, and exercising on the stationary bike in the evening while listening to podcasts – but the middle of the day is a great gaping void. And when I think the day is going to have a certain structure but then it doesn’t end up working out, it’s really bad. I suppose if I had to find a silver lining here, it would be both the confirmation that I am autistic and didn’t somehow trick the psychologist as well as the reassurance of God’s faithfulness and grace as I find myself needing Him more.
Related to that last point, the Easter season has been such a gift right now. The daily reminder that Christ is risen, the reaffirmation of the hope and joy to be found in Him, even just the singing of the alleluias – those things help me stave off negative emotions and unhelpful thought patterns. They give me a starting point for seeing joy in each day, for learning to be thankful, and for abiding in hope.
Finally, the sudden burst of warmth has made the garden flourish! The last of the winter beets are rounding out under their thin blanket of soil and the herbs are thick and bushy. The blackberries are ripe, the peaches are blushing, the corn is shooting up, and the beans are filling in around the trellis. I even have some sweet potato and purple basil sprouting up on their own from last summer! This is definitely a joy-bringing aspect of this time as well.

I hope you are all doing well, staying healthy and finding joy, and that you have the support you need right now! I am linking up with Kelly today so head over to check out the rest of the link up!

family life, learning together

learning together: infinite series

Posted on 2020-03-302020-03-28 by kristinabean

I asked Limerick the other day what was, in his opinion, one of the neatest things he knew about math and numbers.

In response he told me he thought the neatest thing was that if you start with the powers of 2 (say, 128 for example) and kept dividing by 2 (32, 16, 8, 4, 2, 1, 1/2, 1/4, and so on, he said), you would go on forever and never actually reach 0.

Basically my five year old uncovered on his own the concept of an infinite series approaching a limit and (very naturally!) decided it was just about the coolest thing numbers do. I love how his brain processes numbers and analyzes the world in their light!

learning together

learning together: a multi-level cooperative place value game

Posted on 2020-03-112020-03-02 by kristinabean

We’ve been working on place value for a while. Rondel unfortunately decided that my default place value game was his least favorite thing ever, probably primarily because Limerick utterly loves it and finds it intuitive and easy while Rondel has struggled more with the concept. Fortunately, however, we were able to adapt it using place value blocks (wooden blocks in denominations of one’s, ten’s, hundred’s, and a huge thousand cube) into a game that let each kid operate on their own level of mathematical ability while working together to earn chocolate chips!

757 represented by a stack of place value blocks, with three dice and a pile of one’s cubes on the side

Our goal as a team was to reach 1000, rolling dice to add to our total on each turn. Along the way, we could get chocolate chips: one for each person every time we added a new hundred square, and ten for each person if we made it to the thousand cube. On Aubade’s turn, she would roll just one die and practice counting the dots to find how many she had rolled, then practice counting again as she put the right number of cubes onto our combined tower in the center.

Aubade carefully placing her one’s cubes on top of our combined tower.

On Limerick’s turn, he rolled two dice, multiplied them together, and then added his total to the combined tower. (Yes, this is easy for him. Next time I’ll have to come up with something more challenging for him to do! He also tends to supervise everyone else, however.)

Limerick straightening out the blocks on the combined tower after adding his total

On Rondel’s turn, he rolled four dice and added them all together (which was perfect for him! Adding two dice is easy for him at this point, but four lined up with the addition we’d been encountering in Life of Fred and he remembered and mentioned that.) Seeing how his one’s cubes lined up to form a group of ten, and how his ten’s lines added up to form a hundred square, the concepts of place value finally started to make sense to him! These blocks are such a nice visual/tactile representation of that 🙂

Rondel adding a one’s cube to the top of the tower

By working together, we eliminated both the stress of competition and the need for everyone’s individual rolls to come out to, on average, comparable amounts. Because we were working together, it didn’t matter if Aubade was rolling much smaller numbers than Limerick or Rondel, or that Limerick’s highest possible total was higher than Rondel’s – everyone just contributed towards our shared goal in their own way. It also didn’t matter who was fastest or reached a goal first, and the shared celebration every time an intermediate goal was met (i.e., the chocolate chip for each hundred) prevented anyone from becoming jealous or discouraged. And finally, because none of those things were important to the game, we could tailor it to each participant’s math level, allowing all three kids to play together despite ranging from counting to multiplication/division with their math skills (which I’ve found surprisingly difficult, mostly when it comes to including Aubade in the game.)

Now I suppose I just need to come up with a name!

learning together

learning together: a “3R’s” treasure hunt

Posted on 2020-01-152020-03-09 by kristinabean

Rondel really enjoys looking for treasure (thing-finding in the tradition of Pippi Longstocking, where almost anything can be considered treasure), and as I pondered what to do with our morning (unexpectedly open since Aubade had too bad of a cold to handle the hike we’d planned), I thought he might enjoy following a series of clues to find a treasure at the end.

Because there is always some way to incorporate math, reading, and handwriting into life’s activities (please take that with some humor!), I decided that each clue would be a numerical cipher but that the numbers in the encoded message would have to be determined by calculating a series of math problems. I made short messages like “under the desk” and “lego box”, converted them to series of numbers using a key, and then came up with arithmetic problems at Rondel’s level. He started out with the key and the first clue, which led him to the second clue, then to the third, and all the way to the treasure after six clues in total (probably about 50 math problems all together, which is a lot more than he’d normally do otherwise!). In addition to the math, he got a lot of handwriting practice in from writing down the numerical answer to each problem as well as the corresponding letter value from the cipher, and then even got to do a bit of reading to put all those letters together for each clue.

Above are two examples of keys and clues – the orange set was for Limerick and the red set was for Rondel. Changing the key values to larger numbers would let you create even more difficult math problems without needing to alter the method of encryption. Limerick kept commenting on how the problems corresponding to the letter “A” – where the answer was 1 – were too easy, and a different key value would have eliminated that issue.

Rondel did a whole treasure hunt, despite the difficulty of focusing with two younger siblings running around at high volume and also being very interested in everything he was doing and not giving him any quiet or space! I was really impressed with his determination and motivation, because nothing about this was easy for him but he didn’t give up.

Rondel sitting at the kitchen table with sound-reduction headphones, writing out the answer to one of his clues. — The sound-reduction headphones were a huge help! I could see him relax and focus more easily the instant he put them on.

Limerick wasn’t interested at first, because he isn’t all that into finding treasure, but he like the idea of following a path of math clues so I made a set for him later and he finished his as well! He does not like to write or draw often so pulling him into an activity where he writes this much is a rare and pleasant thing (I think he couldn’t resist the math).

Limerick standing by the kitchen table writing on his clues

One thing I did notice from the activity was that both boys have legibility issues, and I’m going to have to find a way to work with them on pen grip and letter formation that hopefully doesn’t result in daily fights. Rondel’s letters in particular are like people, each with their own personality and opinions, and they dance around the page and swing by their toes and jump on each other’s heads and sometimes sword fight – and they are highly offended by the idea that they should arrange themselves in a neat orderly line! So if you have any ideas or suggestions I would love to hear them.

wwlw

learning to love math

Posted on 2019-12-302019-12-28 by kristinabean

Rondel never took to the “3 R’s” of education quite as naturally as Limerick (most people don’t, honestly), though he is an information sponge for the things that interest him, which have throughout his life been mostly science-related. We’d tried a couple of different approaches to learning and practicing math before I came across the Life of Fred curriculum (and I don’t even remember where, or I would definitely send them my thanks!).

Life of Fred is very, very different from any other math curriculum I’ve used. Every math concept is introduced in the context of a story about Fred Gauss, a five-year-old math professor (yes, it’s strange, but you have to just roll with it), and the end of each short (ridiculous, hilarious, bizarre) chapter has a few practice questions that work in both new and old concepts. So kids reading through the stories begin to see math as something useful, lovable, and even beautiful as it snakes its way through Fred’s everyday life (and his very odd adventures). And the stories will have kids laughing out loud along the way, if they are at all like Rondel (and myself!).

The elementary series for Life of Fred starts with Apples and Butterflies, which are both kindergarten level, and goes from there – through decimals, fractions and percents, through pre-algebra and algebra, and all the way through calculus (which was actually the first book the author wrote, oddly enough). While the books are published by Polkadot Publishing, I couldn’t find a way to purchase directly through them and ordered from ZTwist Books instead (free shipping!).

We are just beginning Cats now, having spent an average of 3-4 weeks each on the first two books. Rondel has been asking me to read Life of Fred all the time – more than I can right now with my lingering sore throat, and more than Aubade has patience for at times – and I can see his confidence with math growing week by week (we’ve been using the books for over two months by now). He will now tell people that math is his favorite because of Fred; he doesn’t get overwhelmed by basic addition and subtraction problems; he is starting to understand analog time-telling; he is getting better at remembering the days of the week and months of the year; he will skip-count for fun; and he is learning to notice patterns and sets in the things around him. The practice problems force him to focus as he has to recall information and use concepts in new contexts, but there are never so many in a set that he can’t make his way through them all.

In short, I am so glad we found these books and highly recommend them for anyone else, particularly those kids who are struggling with a traditional approach to mathematics.

wwlw

what we’re learning wednesday, episode 10

Posted on 2019-09-042019-09-04 by kristinabean

To incorporate some math into our current space-themed enthusiasm, I came up with a new game that we have been calling Space Race. All you need are three dice, a deck of cards, and some sort of token for each player, so it’s pretty easy to set up.

Using all 13 cards from one suit, we have enough cards to represent the sun, all 8 planets, and 4 dwarf planets (Limerick’s favorites). While you could line up the cards from 1-13 in order from the Sun outwards, Limerick prefers to assign each card to the appropriate planet according to its ranking by mass, as follows:

Sun	13 (King)
Mercury	5
Venus	7
Earth	8
Mars	6
Ceres	1
Jupiter	12 (Queen)
Saturn	11 (Jack)
Uranus	9
Neptune	10
Pluto	4
Haumea	3
Makemake	2

In the table above note that the dwarf planets are italicized and that Uranus, despite having a larger diameter than Neptune, has less mass and is therefore given a smaller number. If you had cards going up higher than 13 you could incorporate more dwarf planets, but that would also make the game longer.

On a player’s turn, they roll the three dice and attempt to use the three numbers rolled to make the number of the planet to which they wish to launch a probe. Once they show how they can reach a planet’s number using basic arithmetic (addition, subtraction, multiplication, and division), they can place a token on that planet’s card to symbolize the successful mission of their spacecraft. If the player is unable to make the number of any planet to which they have not already launched a probe, that turn is considered a “failed launch” and play proceeds to the next person. The goal is to be the first player to send spacecraft to the Sun and all the planets.

We have these awesome cards with non-standard suits, one of which is little rocket ships! And the face cards, while still called J, Q, and K, have the correct number of rocket ships on them which is helpful for this sort of game. We use brain flakes for our tokens since we have a lot of them in multiple colors, but Legos or coins would also work well.

If the game is too easy or goes by too fast with open-ended launch destinations, a more challenging variation is to select a specific order in which the planets need to be reached – smallest to largest or center of the Solar System outward, for example. Limerick prefers this variation; it forces him to be more creative with his use of the numbers while also removing the need to decide which planet to choose when there are multiple options. Rondel prefers the original more flexible version, however, since it allows him to launch to the first planet he is able to find a solution for and lets him play with the math skills he is more comfortable using. I would encourage the more flexible version if you have players who are primarily relying on addition and subtraction, and the ordered variant for players who are comfortable with simple multiplication and division as well. Either way, it’s fun and it’s good math practice – always a winning combination 🙂

family life, sqt

{sqt} – just living life

Posted on 2019-05-232019-05-23 by kristinabean

This is a real {sqt} post this week: just seven updates from our life 🙂 Visit This Ain’t the Lyceum for the rest of the linkup!

I now officially have my autism diagnosis! So if you read my series for Autism Acceptance Month, which I wrote during the diagnosis process, you can now be comfortable in the knowledge that it comes from a “real” autistic person instead of an imposter. Not that I think most self-diagnosed individuals are – but it was how I was afraid I would be perceived (and honestly, I was deeply afraid that it was true of myself). It was a lot easier than I thought it would be, and also a lot more uncomfortable. I was so afraid, the whole time, that the psychologist would tell me I was just intelligent with typical gifted quirkiness – and then I would be left wondering, if that were so, why I seemed to struggle so much with things that came naturally to the gifted friends I grew up with? But fortunately for my peace of mind, I can now say I’m autistic with confidence, and I say it to myself a lot when I need to advocate for myself or address areas of weaknesses in my life, and it helps to stop the perfectionist depressive thinking patterns from asserting themselves.
I have realized how much game play helps with the development of strategic thinking and forethought, by watching Rondel grow in those skills. I’ve seen him take the initiative to plan a course of play at the beginning of a game; stay aware of the events of the game so that opportune moments for deviating from that plan can be seized; look ahead at his opponents’ possible moves to make the optimal choice for his own; and see several steps ahead on the pathway to his desired end – in several different game settings. These are really valuable skills for life, not just for games! This is all about considering options, observing the environment, planning for the future, and making decisions in the moment that affect long-term goals. When I write up his kindergarten year summary, I may include some of these games in a SPED section under executive functions…
Teaching something that I don’t remember learning is challenging. In other words, while we are all into math and science over here (definitely at least a grade ahead in math, and more for Limerick), we’ve barely done more than the alphabet and letter sounds when it comes to reading, and I’m struggling to know where to go next. I have a few ideas from my sister-in-law and I looked up some phonics/beginning reader games online that look fun (my kids are always up for a new game) – but to me, reading is like breathing. I can’t imagine (or remember) life without it. And how would you go about teaching someone to breathe?
I may have a new favorite food, and I think Aubade would agree. I whipped up some heavy cream, added some yogurt and maple syrup, and discovered paradisiacal creaminess with just the right balance of airiness and weight, sweetness and tang. We’re calling it “breakfast cream”, over here.
Picture is of Aubade in a black and white striped polo shirt at a kitchen table with a bowl of whipped yogurt, eating a spoon of it, with white smears all around her mouth and nose.

The recipe is very straightforward: two parts heavy cream, whipped until very stiff; beat in three parts plain Greek yogurt (I used full fat); sweeten with one tablespoon maple syrup for each quarter cup of yogurt. Last time I made it, I rolled it up inside fresh crepes with diced peaches; Aubade just ate three bowls of it unadorned 🙂
The cantaloupe vines have reached the top of the trellis (8 feet high!) and are beginning to claim the other side. It makes for a beautiful shady green retreat from the world, tucked under the trellis on a camp chair, looking out at the sunflowers starting to bloom. The fruits themselves are not overwhelming in number (which could be because I planted too many too close together), but they are massive. Paul keeps asking me if I’m sure they aren’t actually watermelons and I can’t really blame him because I have never seen cantaloupes this size in my life…
Every few months for the past couple years, I’ve pulled out my old pattern blocks to see if the kids are interested in them – and now at last their interest and their fine motor skills are there! Limerick and I make patterns (he prefers to work with me rather than on his own, even if he’s making all the decisions), and Rondel tends to build animals. Aubade isn’t really ready – but she has fun playing along with the boys 🙂

It is such a great foundation for an understanding of geometry and the more mathematically abstract styles of art, and having the hexagonal base is a nice contrast to our other building toys which are either octagonal (Brain Flakes) or rectangular (Legos). And it’s just so much fun… I could make patterns for hours.
This past week was rather interesting for me in terms of theological discussion. My sister-in-law and I had a discussion about Protestant/Catholic differences that spilled over onto Facebook (where actual Catholics got involved, to my delight) and many text messages days later. Then, I spent a morning with two Protestant missionaries on home assignment, and finally was accosted by two Mormon missionaries that same afternoon. These are all concepts and divisions I have thought about and researched a lot, but I don’t often have the opportunity to actually discuss them in real life very frequently. And I realized that while I still am officially Protestant, I was arguing the Catholic side and thinking in Catholic terms more often than not during all of these interactions. So, having surmounted the autism diagnosis hurdle, addressing this theological hurdle is next on my list of Important But Uncomfortable Things To Address. I’d be interested in any resources, thoughts, or experiential wisdom you have to offer here!

Again, don’t forget to visit the linkup today! If you share your own blog there let me know and I’ll make sure to read it, or I’d love to hear some of the highlights of your week in the comments as well 🙂

wwlw

what we’re learning wednesday, episode 9

Posted on 2019-05-072019-05-07 by kristinabean

Every spring, as part of her job as a math professor at the local community college, my mom goes to a math education conference and brings home all sorts of interesting books, games, and ideas. One of these was a set of playing cards containing the integers from -12 to 12 (two sets of -12 to 0 and two sets of 0 to 12), from Eureka Math. I can only find the cards sold in packs of 12, which isn’t very useful for a single family in terms of either quantity or price, but if you have an interested support group or co-op I’d definitely recommend them.

So far we’ve come up with two different games to play (neither of which is the game for which instructions are provided with the cards, though that also looks fun). First, we’ve been playing addition war, where whoever’s sum is closest to zero wins the round. Limerick picked up the concept of negative numbers really quickly – I introduced them with a number line while we were waiting for Rondel’s speech therapy, and he understood them and was picking up speed in figuring out the sums in our game of war by the end of the 45-minute session. Rondel is more visual and tactile than Limerick, and was struggling with the concept and calculations when they were presented that abstractly, so we made a large helpful number line that has really helped.

Unlike a traditional number line that runs horizontally, this one extends vertically, with negative numbers at the bottom and positive numbers at the top. So positive numbers are like steps up a mountain, while negative numbers are steps down into a hole underground. When we make the sums during war, Rondel knows that he can start with either number and count the number of steps either up (for a positive number) or down (for a negative number) to calculate the result. After a week or so of playing with this board every day, he’s now grasped the concept well enough to be able to figure out most of the sums completely in his head, just like Limerick.

The board also gave us an opportunity to develop another game using the Eureka Math integer cards, this one cooperative instead of competitive. Each of us starts out with one playing piece (a small animal counter, usually) on the zero tick in the center of the board. Taking turns, we draw one card at a time from the pile and move that number of steps from our current position. Every time someone goes off the board (28 or -28 – my tick marks are centimeters on standard printer paper) or lands exactly on 0, everyone gets a chocolate chip 🙂 The board is large enough that the amount of chocolate earned is fairly small, but small enough to keep everyone from getting frustrated. I usually end this game (to loud protests) when we’ve gone through the entire deck three times… more than that could lead to excessive amounts of chocolate and the concomitant hyper silliness 🙂

From what my mom says, negative numbers were apparently one of the more difficult mathematical concepts for me to grasp (along with distance-rate-time problems), though I don’t remember learning (or struggling with) them. So I’m glad that these two fairly simple tools have made them intuitive and fun for my kids.