Posted in learning together

## learning together: a multi-level cooperative place value game

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!

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.

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.)

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 ðŸ™‚

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!

Posted in learning together

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

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.

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).

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.

Posted in wwlw

## what we’re learning wednesday, episode 10

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:

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 ðŸ™‚

Posted in wwlw

## what we’re learning wednesday, episode 9

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 ðŸ™‚ 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 ðŸ™‚

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.

Posted in family life, sqt, wwlw

## {sqt} – because limerick loves numbers

If you ask him, Limerick will tell you that his favorite thing in the whole world is numbers. More than milk, more than cookies, more than hugs – numbers are the best. So I thought I would capture seven ways he shows that love for this week’s {SQT}! Join Kelly for the rest of the linkup ðŸ™‚

1. Limerick’s favorite numbers of all are 1, 11, 111, and so on – anything that is all 1’s. So the other day as we were skip-counting back and forth together the way we do, he decided we should count by 11. When he got to 1111 (and he was the one who got to say it!), he was so happy that he stood up on his chair and clapped his hands together while laughing for joy.
2. This past week he’s been asking me to make number coloring pages for him, where I’ll draw the outlines of numbers on a page and he’ll color them in. Well, for one of those pages, he decided the best way to color them in would be to fill all the little spaces with smaller versions of the number he was coloring – very meta ðŸ™‚

3. In addition to coloring numbers, Limerick likes me to make skip-counting number boards for him – this week alone we’ve made one that counted by 499, one that counted by 999, one that counted by 4, and one that counted by 1 but had all the multiples of 3 drawn in a different color. After a board is made we’ll play a game with it once or twice but then it is on to the next one! I sometimes think he just likes watching the numbers appear on the paper as I write them…
4. Speaking of watching numbers, Limerick’s favorite book, You Can Count On Monsters by Richard Schwartz, gives him plenty of chances to do just that. He will sit for hours poring over every page of the book, noticing how the focal number of each page breaks down into its factors and figuring out how the accompanying monster illustration incorporates those factors (or the number itself, if it is prime). He’s been through it at least three times this week, taking 2-3 hours each time, and it doesn’t seem to be growing old yet.
5. I pulled out a math workbook for Limerick this week also, thinking he might be interested, and so far he has just been turning the pages looking at all the numbers and math problems and shapes. He isn’t interested in writing anything down, but when I ask him about any of the problems he knows the answer instantly, or knows how to figure it out. There are some fractions later in the book that would be more of a challenge for him, though, so maybe that will catch his attention eventually. It’s a bit of a tightrope balancing between guiding him towards new information and allowing him the joy of freely exploring numbers without pushing or interfering.
6. I did, however, get to explain different base systems to him this week! I just sat down at the table and started counting in hexadecimal on a piece of paper, and he glanced over and was immediately intrigued. We discussed what place value means in the context of the various base systems, and ended up writing out 1-32 in decimal, hexadecimal, binary, and base 6. I think binary was his favorite because there were so many 1’s and the numbers got long so quickly!
7. One other fun book we’ve read through a few times (though not as recently) is Bedtime Math by Laura Overdeck. It’s been a great introduction to the application of numbers, and a challenge for Limerick to translate the words into the more familiar arithmetic. He’s actually quite good at tracking along with the question as I read it, deciphering the logical connections, and doing the math in his head – he can for most of the stories do even the most difficult problem on the page already!

All in all, I just have to echo Limerick and say that he really does love numbers the best ðŸ™‚ And he has, honestly, since he was 18 months old and would sit on the driveway drawing them in wide circles around himself until he was familiar with each one, and since he was 2 years old and would count the bites remaining on his plate at dinner and practice subtracting them as he ate. I’m looking forward to watching this love continue to grow with him in the years to come!

Posted in wwlw

## what we’re learning wednesday: episode 5

This week, we’ve been playing math games!

While Aubade takes her naps, I’ve introduced the boys to dominoes (just the number matching beginner version – they aren’t quite ready for the multiples-of-five scoring version yet, but I wanted to lay the foundation for it as it is one of my family’s traditional games) and to war with playing cards. For war, we mess with the rules a bit by either playing addition war, where we pull two cards at a time and the person with the highest sum wins, or subtraction war, where we again pull two cards but the person with the smallest difference wins. (Multiplication war – highest product wins – and division war – smallest remainder wins – are other possible versions of the game but Rondel isn’t able to do that math quickly enough yet for the game to stay engaging and fun.)

I thought it might just be a fun novelty, but the boys have both really enjoyed it. Handling the cards, especially with our old sticky set, is good fine motor work, and the game itself is great math practice since the symbols on the card allow the boys to count to the answer if they need to. I also spent one game writing down the rounds in math language, to introduce the boys to symbols like “-“, “+”, “=”, “>”, and “<“; they liked the greater and lesser than signs the best, of course, because they are alligator mouths trying to eat the bigger number!

When we play games like these, Limerick gets super excited every time he figures out what his sum or difference is, and Rondel gets super excited every time he wins the round ðŸ˜› It’s interesting to see Rondel developing a sense of competitiveness, though I’m definitely grateful that he still has no negativity associated with losing. It’s also interesting to see how little Limerick seems to care about who wins or loses: he just loves the process of playing the game, and I wonder if that is connected more to his age/developmental stage or to his personality.

A note to add is that we don’t approach these games as a “math lesson” or as “doing math.” It’s just a way to have fun together – to incorporate Rondel’s newfound love of games (thanks to speech therapy) and Limerick’s passion for numbers. Especially at their age, it is far more productive to follow their interests than to try to force them into something they don’t want to do or aren’t ready to learn yet!

Posted in wwlw

## what we’re learning wednesday (episode 2)

Limerick is currently in love with numbers.

He counts everything he can, comparing amounts and sizes: he counted all the straps on the plastic pool chairs at the public pool Saturday afternoon, at least ten times in a row; he makes long chains of flakes and counts them over and over again as he builds to find out how long they are and which color is the longest; he counts the number of bites or slices on his plate and recounts each time he eats one. Essentially, he is spending time with the numbers and quantities, becoming friends with them, getting to know how they interact with each other and discovering their unique qualities.

Limerick figuring out how many sticks could fit on the length of our big wooden cutting board – I think it was somewhere around 40.

He will break numbers down into their component groups: if he builds a tower with fourteen wooden blocks, he will tell me proudly that it is two groups of seven. When he expanded his tower to eighteen blocks, I asked if it could split into two equal groups and he lined them up into two equal lines and counted each one to make sure that they both contained nine blocks. When I then asked if it could split into three groups, he made many little groups of three blocks and found out there were six of them – so, six groups of three and three groups of six. Since he’s been doing this on his own anyway, I introduced the vocabulary of “even” and “odd” to him.

“See this group of seven? I can’t break it into two groups of the same size – one of them is always bigger than the other. Numbers like that are called odd.”

Limerick pondered, then declared, “But eight is two groups of four!”

“Exactly! Numbers that can be split into two groups of the same size are called even!”

“Nine would be five and four,” Limerick told me with a slight frown, “but ten would be five and five!”

“So nine is odd, and ten is even!”

We’ll see if he remembers – he tends to hold a new word to himself for a week or so after he learns it, before he brings it out for everyday use.

Emboldened by all the math talk going on, I pulled out one of my favorite math tools from my own childhood (one that I believe my mom created, and that she now uses in her job as a remedial math professor at the community college):

It’s a physical representation of place value – the numbers on the cards in front show what the number would look like written down, while the buckets hold the appropriate number of craft sticks. In this picture, there are three single craft sticks in the box on the right, six bundles of ten craft sticks each in the middle box (the tens place), and nothing in the box on the left (the hundreds place).

We pulled out some huge foam dice and decided to take turns rolling and adding the number we rolled to the number already in the buckets:

Sometimes the boys knew right away what the answer would be; other times they would line up all the sticks to count them to make sure, and to bundle up the new ten-pack if needed.

Before this we hadn’t done much with written numbers – the boys know all the digits, but they didn’t understand double digit numbers. So it was a bit confusing at first for them, but by the time we stopped I think they were beginning to understand what the numbers meant and looked like, which is really cool!

Math tools and games like this aren’t necessary for learning math – Limerick has certainly been picking up on concepts ranging from quantitative comparison to division (with even a touch of fractions) just from everyday conversations about the numbers around us – but they are definitely helpful for illustrating a specific concept, challenging the mind to use a concept in a new way, or just having fun together with numbers! And since all you need for this particular tool are buckets, sticks, and paper, it doesn’t get much easier ðŸ™‚