Beast Academy and Dreambox (reviews)

Conflict of interest statement: I do not have a current or pending financial relationship with Art of Problem Solving, but I have several friends on their board and have had direct contact with several other people there. We purchased and currently own all of the books I review below.

I have no relationship with Dreambox. We tested the program using their free trial and then paid for a 6 month subscription.

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Beast Academy

What is it?
Beast Academy (from Art of Problem Solving) is a book series with 10 "guidebooks" and 10 parallel "practice" books targeted to 3rd, 4th, and fifth graders. Note that the first of 4 books for fifth grade has only recently come out and they are planning to extend to 16 x 2 books covering 2nd to 5th. We do not have either 5A or 5B yet.

We have read all 8 books from 3a to 4d; J1 and J2 have gone through practice books 3A - 3C.

While these are not math exercise apps, I'm going to borrow some of the elements I've used in past app reviews. One key point I want to emphasize for both books and apps: the way you use them can also determine their benefits or costs.

TZB3
To drive this point home, let's start with Tracy Zager's Big Three criteria (see here):

1. No time pressure: Neutral since this is really up to you, parents.

Do you set a timer when they start a page of practice or a question? Do you require a certain amount of time spent on math practice? While the books do not suggest or impose a sense of time pressure, there are story segments involving math competitions that imply speed is important.

One time element that is and has always been great about physical books is that they sit around. This means they are available and tempting. Almost every day, there will be someone flipping open one of the BA guidebooks, even J3 for whom the material is too advanced right now.

2. Conceptual basis: yes (pass)

The books introduce models, contexts, and conceptual ways of considering problems and techniques. 

3. How are mistakes handled: again, this depends on you and your kids

My approach is to go through the problems and select ones to discuss. I don't use the answer key, so I do the problems myself. This means we have three categories of questions to discuss (a) answered correctly and I found interesting, (b) answered incorrectly, (c) answered correctly by the kid, but I made a mistake. 

Also, I am very positive in how I talk about mistakes. The key message is that these are actually the best learning opportunities and create a chance for us to understand our own thinking.

Preliminary summary: whether Beast Academy (or any printed material) passes the thresholds depends on how  you plan to use it. If you want to deviate from Tracy's guidelines, either adding time pressure or incentives based on minimizing mistakes, you probably should think carefully about whether that's wise.

The good

For my kids, the stories and themes in the guidebooks hit the right tone. They are engaging and funny, with a humor that is occasionally silly or corny. An extended quote from The Princess Bride certainly wins some extra points as well. More bonus points for becoming, via malapropism, the source of J3's current catch-phrase, "I get it: pointillism!"

For me, the organizing theme of the material seems to be "ideas you encounter when playing with math." In some cases, the exercises create "aha moments," like when J1 realized he didn't always have to calculate side lengths of a polygon to use his knowledge of its perimeter in a challenge. In other cases, like calculating (n+1) x (n-1) there are interesting patterns to notice and connections to make.

I'd note that the workbooks are absolutely essential as there is a lot of material that is introduced in the context of exercises. I think these books are excellent, well selected, well sequenced, with enough repetition to facilitate mastery and enough variation to avoid boredom. In fact, I really enjoy doing the problems myself.

Overall, we find the practice books an especially good source of cues for quick (5-15 minute) math conversations.

The Bad
Any worksheet-based system is weak in generating exploration and deeper investigation. Beast Academy partially addresses this by including open-ended games and an occasional investigation. While nice, this point remains a weakness. I don't want to belabor this point, since it is not a unique problem with Beast Academy. Indeed, I think it is a universal issue with static educational material.

Unfortunately, the only solution I know is to involve a human guide. Fortunately, I am able to play that role, asking their thoughts about interesting problems, helping them form connections with earlier or other material, getting them to follow useful side-branches or to continue more deeply into a particular area.

Eventually, of course, we hope to develop enough mathematical habits of mind that the kids will do these things on their own. Realistically, I don't think that will happen until they are well clear of any elementary age material!

The Ugly
I don't see any fatal flaws in Beast Academy.

Grand Summary
If you can use the material the way we do, I highly recommend Beast Academy.
If you can't or don't feel comfortable engaging as your kids' mathematical guide, these books are probably still one of the best options. Just don't set up a timer and demand perfect answers to all the questions!

Dreambox

Dreambox is a math facts, basic skills system. It has material from pre-school through high school. We have spent a lot of time with the elementary grade material and a little sampling of the high school content.

TZB3
Dreambox was one of Tracy Zager's positive examples in her app post, so we already expected it would pass these three criteria. After spending so much time with the system, though, we've seen that not all activities within DreamBox completely satisfy the checklist:

1. No time pressure
Some activities do include time pressure. For example, there are a family of "games" around multiplication automaticity where a collection of calculations stream across the screen. This really does raise the stress level for kids.

In a slightly different form, there are other activities involving virtual manipulatives that require the student to do something using the minimum number of moves. Like the time pressure, this seems to create confusion where the kids can get something right, but still get it wrong.

2. Conceptual Basis
I mostly concur with Tracy's original assessment. Almost all activities have a conceptual component. The timed calculations mentioned above don't, so those get a double demerit.

3. How errors are handled
Again, mostly agree with Tracy. However, there are some activities where, for a minor mistake, one is required to redo a number of manipulations, rather than fix the earlier work.

The good
The underlying math curriculum here is solid, if basic. The clear strength of this system is the pictorial representation of manipulatives offering models that build number sense, reflect operations, and show place value. In the early years section, where we have been spending most of our time, almost every activity is based around one of the manipulatives.

The other thing Dreambox does well is present a sensible progression for the different activity streams. I think this works especially well for J3 who is going through much of the material for the first time. As she encounters a new formulation, she will study it for a while and then there is a clear moment when she has figured out the new complication.

I'll give two examples. For J3, there is an activity to replicate a number bead pattern and then click the number of beads in the arrangement. Her primary tool is to count the beads one-by-one. In the most recent module, she gets a short view of the arrangement and then it is hidden (it can be revealed again, if you choose). This is forcing her to build new skills, either memorizing the arrangement to mentally count or a more advanced counting technique.

For J2, one of the place value exercises involves grouping items into pallets (1000s), cases (100s), boxes (10s) or loose items (1s). The current module asks him to consider multiple different ways to pack a given number. For example, 1385 items could be packed in 1 pallet, 3 cases, 8 boxes, and 5 loose items, or 13 cases and 85 loose items (among many other options).

One other strength of DreamBox is the email feedback to parents. Christopher Danielson recently noted this in a post: Parent Letters.


The Bad
I have seen three areas of weakness with Dreambox: the way mathematical tasks are presented, the pace of adaptive adjustment, and the absence of rich tasks. I'll talk about each of these in turn.

The theme gives an irritating appearance of choice. For example, in the early elementary section, the kids can play with dinosaurs, pirates, pixies, or animals. Under each of these, they have a further choice about what story to explore. Those choices, at least, lead them to different narratives and animated sequences.

At that point, all of the stories involve finding missing items. Users then see another choice asking where in 6 map regions they want to look for the missing items, but this isn't really a choice as there are no differences between regions and they will have to go through each region eventually.

Similar to Prodigy Game, the math tasks are presented as an annoyance to be overcome, the cost the student has to pay to move on with the story. Again, I find this creates unfortunate subtext to the mathematical experience.

Second, the adaptive adjustment is very slow, if it actually exists. In their FAQ, I see that they get questions about how to increase the challenge level, so this seems to be a common experience. Part of the problem is that they intentionally start students with material below their grade level.

Finally, the tasks in Dreambox are basic. While they may present a challenge for a new learner, as J3 is experiencing, they should eventually become so easy that they are boring. In some way, this feels like learning to solve math class tasks without having to develop or use any mathematical habits of mind.  Further, the thrill and fun of playing Dreambox lies in unlocking the animated stories and collecting tokens, not in doing math.

For J1 and J2, this thrill has worn off after about 2 months with the system.

The ugly
Nothing in Dreambox is a show-stopper.

Summary
Properly understood as a basic curriculum substitute or source of practice exercises, Dreambox is a solid application. Just don't make the mistake of thinking it will either foster a love of math nor deeper mental habits.

*Update* A quick comparison with ST Math
I was sitting on this review, partially written, for a long time. One thing that got me to finalize the review was going through the demo challenges on ST Math with J2. We had previously tested ST Math many years ago with J1 and it was really good. Once again, this is what I saw with J2: really cleverly presented scenarios that gave us good models for the math and a really fun user experience. After playing for about 30 minutes, J2 said: "this is a lot more fun than DreamBox."

If I can get a subscription, we'll test it more extensively and write a review to see whether that really holds up.

Wimbledon game

We have been playing the game Wimbledon from John Golden. J1 and J2 absolutely love the game and J3 has enjoyed pretending to play as well. Definitely try it out!

Below is a session report sharing some of our experiences with the game, including some alternative rules (aka mis-reading).

Our basic play
For the serve, we allow the following options:
(1) serve a single card from the top of the deck
(2) serve one or two cards from your own hand

We also allow returns that have the same value, if the largest card in the combination is higher. For example, 8 + 2 can be played on top of 7 + 3.

When playing doubles, we followed tennis conventions: one player serves the whole game, return of serve alternates. For the return of serve, the designated player must return on their own. For other returns, either of the partners alone or in combination can play cards for a return.

Having gone back to John's original post, I now see that we played with the inverse rules for aces. On the serve, we counted them as 11, all other times 1. We didn't distinguish between aces played from the hand or served from the deck, those were all 11s. That formed a strong advantage for the servers, while making aces essentially worthless for all other players.

With three players, I had the kids play as partners and I played with a ghost partner. The ghost partner would contribute cards randomly. When the ghost played cards that weren't large enough to be legal plays, we considered that an unforced error and awarded the point to the other team. While the ghost was able to hold serve for one game, it was a big disadvantage. An alternative for three players would be to have the ghost partner with the server and for everyone to take turns serving. This would put the server advantage (with our "house rules" for aces) against the ghost disadvantage.

A modification
In our play, we have found that the 10 value cards and aces (on serve) dominate game play. Here are two ideas to address that:

• Assign face cards values 11 for Jack, 12 for Queen, and 13 for King. Ace, on serve, can have a value 14.
• Allow players to combine as many cards as they like. This would probably work best with our "Further extension" rules below. A possible sub-variant is to only allow gradual escalation where the players can step from single card plays to 2 card plays, from 2 to 3, etc, but could not jump from a single card play to a 3 (or more) card play.

My instinct is that the variation we will like the most is to differentiate the face cards and allow gradual escalation for multi-card plays.

Further extension
Now that we've gotten comfortable with the basic and doubles games, we are considering a more complex version. The idea we are considering is to somehow limit the players' abilities to refresh their hands by redrawing so that burning a lot of cards will have a cost.

This is the rule modification, written for a 2 person game:

1. Create a draw pile for each player with 15 cards.
2. At the start of the game, each player draws 5 cards into their hand.
3. Points are played as in the normal rules
4. At the end of a point, the players refresh their hand up to 5 cards from their draw pile
5. If a player runs out of cards in their draw pile, they cannot draw additional cards to refresh their hand.
6. If both players run out of cards in their draw pile, then shuffle the pile of face-up played cards and give each player a new draw pile with 10 cards.
7. Repeat step 6 as often as needed

The idea of this variation is to thematically mimic the idea that one player could push too hard, too fast, and get tired out relative to the other player. Strategically, our idea is that this will also create a tension between dumping your own low cards and letting the opponent dump.

Evens/odds and a quick update

Early years math seems to put a strange emphasis on even and odd numbers. Recently, a friend asked whether there was a point to this. Maybe it is just one of those little bits of terminology that we are asked to memorize for no reason?

By chance, this was something I had started considering about a month ago. It did seem strange that we spend so much time on this simple way of splitting integers. I wondered if it was worth the attention. From that point, my awareness was raised and I started noticing where it occurs and ways it links with more advanced concepts and future learning. My conclusion is that even/odd is surprisingly deep.

First, it is a simple version of concepts that will be developed further. For example, the alternating (starting with 0) even, odd, even,odd, even... is an illustration of a pattern. They will soon see other alternating patterns, then more complicated patterns and 2d or 3d patterns.

For another example, evens are multiples of 2, odds are numbers with a non-zero remainder when dividing by 2. This leads to understanding other multiple families, division, and division with remainder.

Second, the even/odd distinction is helpful for improved understanding of different calculations. For example, the observations that even+even = even, while odd + odd = even, etc. These can be used to help self-check their calculation and also will form early experiences with algebra. Similarly,
even x odd vs odd x odd reinforce understanding of multiplication. Again, this gets broadened for multiples of 3, 4, 5, etc.

Third, there are a lot of more advanced results that are easiest to prove by parity arguments. Sometimes we are working with a set of things that are even and the key observation is we can pair them up. Other times, we have a set that is odd and the key insight is that, when we pair them, one must be left over.

Recently, with the J1 and J2, we were looking at some constrained ways to put the numbers 1 to 25 on a 5x5 checkerboard. They were able to prove that some versions were impossible simply because 25 is odd, so there are more odd integers in 1 to 25 than even integers.

Lastly, there are techniques in computer science that involve even vs odd. This comes up pretty naturally because of the essential use of binary.

Some games

Recently, J1 and J2 have gotten hooked on some classic card games. In particular, we've been playing a lot of 3-handed cribbage. It is a nice way to do some simple addition practice and build intuition about probability. Probability is now getting even more share of mind: in the last couple of days we started playing poker together. This was actually inspired by some of our reading together.

We are reading the Pushcart War.

In one scene, there is a poker game. Of course, the J's insisted that I explain the game and were eager to try it out. J3 was the huge winner tonight, while I busted out. Oh well.

Broken Ruler and Multiplication refresh

Ruler Explorations

We noticed that one of our tools, a ruler, had gotten broken.

Is it still useful? As a challenge, J2 looked at measuring a noodle from his soup.

There were two ideas:

1. the noodles were too long, so had to be broken in pieces to measure with the remaining ruler
2. Our ruler doesn't have to start at 0, we can use subtraction!

While we were talking about this, I recalled the idea of Golomb Rulers. We came up with a ruler that was marked only with 0, 1, 3, 7, 11, 12 cm. This lets us measure 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12cm distances.

What if this ruler gets broken? For example, we imagined cutting our ruler between the 1cm and 3cm markings. What measurements are still possible? Is there anything interesting in the relationships between how many ways there were to measure a distance before the break and how many ways to measure after?

Multiplication Refresh

We recently re-watched Graham Fletcher's Progression of Multiplication. Both J1 and J2 did some practice around this. The most interesting point was J2's reaction to Graham's comment at 4:54: "This sucks!"

"Why did kid's say that?"  "Hmm, let's try out a couple of examples..."

We rolled dice to randomly generate digits for an example and were lucky to get 35 x 34. J2 quickly saw this as 35 x 35 - 35 and knows a pattern that let him quickly calculate 35 x 35 = 1225. As a result, 35 x 34 was pretty easy for him to calculate.

Then, he worked through a graphical representation and a powers-of-ten version. At the end, we got to compare and contrast the different approaches.

Continuing to play with some old activities

Fold-and-punch
We did some more fold-and-punch activities. This time, we folded the paper, then drew a location for the punch, and tried to figure out how many holes would result and where they would be. We broke out our serious hole-puncher:

Unfortunately, must be operated by an adult

In this example, we got a small surprise that the result wasn't a power of 2:

Chairs (and tables)
Another round of building chairs, following the NRICH activity. This time, with J3:

We got to compare and contrast our designs:

• how many cubes were used for the legs? Which one had more and how many more?
• How many cubes were used for the whole chair? How did they compare?

Improv Math and Division Dice follow-up

We had a really good experience playing Division Dice, the game that we introduced a couple of posts ago.  Mainly, I want to illustrate something fun that came out of really listening and paying attention to what the kids are doing and saying. I like to think of this as "improv math," as a way to credit my improv comedy experiences for heightening my awareness of how important this is.

Division Dice for number sense

I was really pleased about the quality of thinking stimulated by the game. We played with the most loose rules (1s are wild, the components of the 2 digit value can be flipped to their 7s complement). That gave a lot of opportunity for the kids to think through options to (a) make whole number divisions and (b) maximize values.

For example, rolling 3, 4, 6:

• what are the allowed groupings that give a whole number division? Remember, in the 2 digit number, we can use any of the values 1, 3, 4, 6, and it is possible for us to use two 3s or two 4s in our calculation.
• What is the highest scoring choice?

Division Dice for arithmetic exercises

As a way to create virtual worksheets, this game is mediocre. The basic structure means that students are never dividing by a divisor larger than 6. This leave out a lot of fact families. However, because the kids are trying to maximize their scores, they quickly realize that they can almost always get away with division by 2, occasionally must divide by 3, and rarely get stuck dividing by 4 or 5. I haven't yet seen a case in a live game where division by 6 was necessary.

Fun exploration: what scenarios will require division by 6?

Using playing cards or other dice shapes allows us to extend the possible values and reduce the likelihood of dividing by 2 or 3. However, it also increases the number of cases that don't have a whole number division relationship. We are thinking about ways to incorporate division with remainder and will try out a variant tomorrow.

Improv Extension

Playing at home, the 3, 4, 6, case led J1 to consider: how do 63 ÷ 3 and 64 ÷ 4 compare?
As he contemplated that, I realized that we had a nice sequence of multiples, meaning all of these are whole numbers:

There were several cool things for J1 to observe here:

• 4 of the 6 quotients end in 1
• The quotients are all decreasing
• The drops between successive quotients are themselves decreasing
• the dividends are equal to the divisors + 60

We pursued this in two ways:
Extension 1: what if we add something else to the dividends?
We tried three versions.

1. starting with 60 and adding 6 at each step
2. Starting with 60 and adding 60 at each step.
3. starting wit 1 and adding 7 at each step

You can see our notes mid-discussion below:

Later, when J2 was also involved, I offered them another sequence: starting with 66 and adding 6 for each increment:

66 ÷ 1

 72 ÷ 2 

78 ÷ 3

84 ÷ 4

90 ÷ 5

96 ÷ 6

120 ÷ 10

132 ÷ 12

150 ÷ 15

180 ÷ 20

240 ÷ 30

420 ÷ 60

3660 ÷ 600

36060 ÷ 6000

We're breaking the rule about the dividends being multiples of the divisors, but the last two calculations are still easy and nicely illustrate the limiting behavior.

Extension 2: can we find other chains of whole number division equations?
We started this by thinking more simply: for chains shorter than 6. For example, what are the smallest K, L, M, N larger than 1 such that all of the following are whole numbers:

K ÷ 1

 (K+1) ÷ 2 

L ÷ 1

(L+1) ÷ 2

(L+2) ÷ 3

M ÷ 1

(M+1) ÷ 2

(M+2) ÷ 3

(M+3) ÷ 4

N ÷ 1

(N+1) ÷ 2

(N+2) ÷ 3

(N+3) ÷ 4

(N+4) ÷ 5

After getting the shorter cases under our belt, we then went for a chain of length 7. J2 worked by himself for a while, then came back and announced that no chain with dividends smaller than 100 would work.  He went away and then came back quickly with the idea that maybe we could add 7! to each divisor.

Logic Puzzle collection and Dropping Phones

Recently, Mathbabe put out a request for riddles. There are some good links in the comments:

• The Riddler on FiveThirtyEight
• Logic Puzzles on Math is Fun
• Alex Bellos at The Guardian
• Riddles.com (naturally)
• Puzzling.stackexchange.com (is there a stack exchange for everything now?!)
• Authors cited: Dudeney, Gardner, Polya, Smullyan, Vaderlind. 

We've played with puzzles from almost all of these sources in the past, but this was a good opportunity to put together a nice list.

The one that stood out to me was on FiveThirtyEight. That's a site I read frequently, especially during this US election season ... and I was totally unaware of The Riddler feature. We kicked off with the oldest puzzle from their archive: Best way to drop a smartphone.

Breaking stuff

Right away, J1 and J2 loved the theme and were into questions about whether they could somehow keep the phones, if they weren't broken, or utterly destroy them, if they were broken. We talked about setting an upper bound with a very simple strategy of starting on the 1st floor and working up each floor. That's not a great answer, but it got them into modifications and improving strategies.

Through the conversation, it was interesting to see them start with the idea that the drops for one of the phones would be de minimis and could be ignored, but then start to pay attention to that aspect. Also, they had to grapple with the idea of balancing the number of drops that would be required in different cases.

While we didn't get to the optimal strategy, but the kids managed to get a version that, at worst, would take 19 drops for the 100 story building. Their intuition was based around taking the square root of 100. They could see that this probably wasn't the best answer, since there were still cases that, at worst, would take 10 drops and others that, at worst, would take 19.

Can you do better?

Smaller before bigger

The puzzle page poses the same challenge for a 1000 story building. However, we found something interesting when working on the version for a 10 story building: there is  strategy where all worst cases take the same number of drops, but it is not the optimal strategy!

It is a little hard to write about this without disclosing the strategy, but here's a hint:

• 4 + 3 + 2 + 1 = 10 and 5 + 4 = 9
• We are always allowed to assume that the phones will break if dropped from the top floor of the building

This led to another extension: what size buildings will have the same issues as for a 10 story building?

Probability comes in

Another extension is to think about the expected number of drops required and strategies that minimize this. Crucially, this extension introduces the idea about our prior beliefs about the sturdiness of the phones: where do we think the phones are likely to break, what is our confidence?

We didn't pursue this extension very far, but it did lead to some interesting conversations about terminal velocities. For those who want to follow that thread, this (other) stack exchange thread might suit you: How to figure out height to achieve terminal velocity.

Build the chair (part 2)

In class, we played with the NRICH Chairs and Tables activity (our outline here). We came up with an extension that we explored at home: making a sequence with smaller and larger chairs.

Kick-off

I knew that J2 had incorrectly counted the cubes in the NRICH sample chair, so I kicked-off by asking him to show me how he counted them. This was a surprising, and unintentional, kick-off. His method was to decompose the chair into three sections: seat, back, and legs. This upper left 2/3rd of this picture show how he determined the number of cubes in each section.

First Sequence

The previous picture also shows notes about how J2 thought of making the chair bigger or smaller. His idea was to keep the same size seat, but make the legs and back of the chair longer or shorter. You can see our drawing of back for the two next smaller chairs.

Along the way, we looked at the total number of cubes in each chair. A simple pattern jumped out to him: each step is a difference of 6 cubes. He quickly realized this was because each leg required one more cube (4) and there are two on the sides of the back.

I asked him how many cubes would be in the 10th chair. When I asked him to explain his thinking, he said, well, going backwards, the 0th chair should have 10 cubes, each step is +6, so I need 10 + 6x10.

From algebra back to geometry

His idea of the 0th chair really excited me as this was one of the ideas I had been hoping we would uncover. We talked about how this chair would look (a 3x3 seat with one cube in the middle of a side as the back). This was not something we would naturally have created when asked to build a chair.

What we had done is gone through a sequence translate from geometry to algebra, naturally extend the algebra to a new case, translate the new algebraic case back to geometry.

Second Sequence

One other delight in this activity was that J2's sequence was not one I had in mind when outlining the activity. Of course,  wanted to share my version, as well.

I followed his decomposition into seat, back, and legs. See if you can understand my notes and picture how my chairs are growing through the sequence. Chair D₃ is the starting example from NRICH.

I asked J1 to fill in the D₁ chair to see if he got the pattern.

Not linear

J2 noticed that, this time, the gaps between cube totals were not the same, but the second difference is constant.

To round up the discussion, we wrote down an equation for the nth chair and calculated how many cubes would be in the 10th chair. Finally, we tried our trick of extending to the 0th chair. 

This time, we realized that there could be something interesting in the 1st chair, too. See if you can build a version of our D₁ chair, using whatever your favorite building material might be.

Build the chair spatial reasoning (Gr 1 and 2)

who: Baan Pathomtham grades 1 and 2
where: in school

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Here in Thailand, summer is over and we are back to school! We are kicking off the math games and exploration class today with an activity from NRich (chairs and tables) that has a surprising depth. Also, keep your eyes opened for the hidden reasons why we are starting with this activity.

Build a chair

The starting directive is simple: use unifix cubes to make a chair. Here's an example, from NRich:

To start, we ask the kids to get 15 cubes each. What does 15 mean? How do they know they've got 15? Do they think they will need more or less than 15 to make a chair?

After they have built their chairs, how many did they need? If it was less than 15, how many are left over? If it was more than 15, how many more did they need to add?

Chairs for bears

Once we've all got one chair, can we make two more for the three bears from Goldilocks and the 3 Bears? We need a small one for baby bear, a medium sized one for mama bear, and a large one for papa bear.

As the final construction challenge, we ask them to make a table sized to accompany their original chair.

Homework

1. Based on the pictures above:  (a) find out how many cubes would be needed to build these shapes, (b) draw a 2d perspective of one of the shapes from one direction.
2. For those who have construction sets at home, try making chairs of different sizes. What things were similar to using the cubes at school, what was different?

Extension

Building off the three bears activity is a nice extension:

• What is the smallest chair we could make? How many cubes do you use?
• How would you make the next larger chair? The next chair larger than that? How many cubes are used for those?
• What about the tenth chair in this sequence? What would it look like? How many cubes would we use to make it?
• Same questions for the 100th chair?
• What is an equation for the number of cubes in the nth chair?

I'd note that these are challenging questions which go well beyond first and second grade. Also, there is no single correct answer, particularly as different students will have different ideas about what is required to be a chair or how the form should grow through the sequence.

Note: these questions follow the thinking of Fawn Nguyen's Visual Patterns.

Hive at the beach (quickie)

Have been at the beach for the last couple of days before school starts. We got to play with one of my recent impulse buys: Hive (carbon).

Chance for discovery

As an experiment, I started by laying out the pieces with the blank sides up. One at a time, I asked the J's to come in and tell me what they noticed. For each of them, at some point, there was a moment when they tipped or turned a piece over and discovered the insects. Their reactions were a real delight and they realized it was special feeling, so quickly helped rearrange the pieces and get another sibling so that they could have the same experience.

Making patterns

Having started with the blank sides of the pieces, it felt natural to the kids to sometimes play with these tiles just to make patterns. Here is one example:

Playing the game

Of course, there is also delight in playing the game itself:

From this detail, you can see that J3 (playing black pieces) is following an unorthodox strategy. At this stage, she is really learning the rules and sees the whole activity as a strange way to play together to create unusual patterns and combinations of the bugs:

Reversed inequality

Recently, Mike Lawler posted a challenge base on the first question from the recent European Girls Math Olympiad (side note: what is "European" about this contest now that a US team participates?)

I enjoyed thinking about this problem and wanted to come up with something related to do with our kids. One way to see this problem is that it appears to reverse the direction of the inequality between the arithmetic mean and geometric mean. My version for younger students involves exploring this inequality first.

Discovering the Arithmetic Mean-Geometric Mean Inequality

I had J1 and J2 select two dice from our pound-o-dice. Initially, J1 chose a d20 and d6 while J2 chose a d12 and d6. I asked them to make a table with 6 columns and 7 rows (we ended up adding more, so probably better to ask for 8 columns and 10 rows). I had them label the columns:

1. A
2. B
3. AxA +  BxB
4. 2xAxB

They rolled their two dice and put the values into columns A and B, the calculated the other two columns as labeled.

J1's first results were 16 and 5. After filling in the rest of that row, he paused and then switched to 2d4. This wasn't a problem for our investigation, but he did miss some calculating practice with more difficult seed numbers.

After filling in 6 rows of data, I asked them what they noticed.

Here were some of the observations:

• when A and B are the same, our calculated values are the same
• when A and B are different, our calculated values are different
• When A and B differ by 1, the calculated values differ by 1 (also vice versa)
• AxA +  BxB > 2xAxB
• We are only using positive integers

J2 really got into the spirit and asked for some more suggested columns. I told him to add:

AxA - BxB and (A-B) x (A-B). J1 also added the square of the difference.

With that, they noticed two more things:

• AxA + BxB was larger than (A-B)x (A-B)
• AxA + Bx B = 2x Ax B + (A-B)x(A-B)

You can see that they also added some negative numbers on J1's sheet and challenged one of these observations.

A picture proof

To finalize our exploration of the inequality, we used tiles to build intuition for a picture proof of this identity:

AxA + Bx B = 2x Ax B + (A-B)x(A-B)

This is one of the pictures we liked the most. In this case, A = 5 and B = 2. Along the left side and the bottom are rectangles AxB (the green + yellow and the blue + yellow regions). These two rectangles overlap in a BxB square (the yellow region) which they also highlighted with the 4-color square. The remaining red square is (A-B)x(A-B).

Side note: In our discussion, I was delighted that J2 would correct me whenever I slipped and said "greater than" instead of "greater than or equal to" when discussing the inequality.

Two quick algebra extensions

For kids who have already done a little algebra, proving the identity using the distributive law should be fairly straightforward. The other little extension is to show that our expression A²+B²≥ 2AB is equivalent to the arithmetic mean-geometric mean inequality.

4 Coins challenge

Recently, we were at lunch and the kids asked for a challenge to entertain themselves. I had just read this Numberplay column and had Lora Saarnio's Four Coins Problem in mind.

The basic rules are to choose values for coins in a new system so that any value from 1 cent to 10 cents can be made with a single coin or with two coins.

For our older two, this was a great on-the-go puzzle without pen and paper. Later, we got into some extension questions that required more careful organization of their observations and attempts.

Mostly, I want to report some of the extension questions that we found interesting. The key point: this problem is very accessible (J3 could also work on a simplified version) but it is ripe for a Notice & Wonder discussion that will reveal increasingly complicated and challenging extensions.

Note: we used names of neighboring countries, but none of the claims in these scenarios are true (as far as we know).

Number fussiness
In Burma, the president really likes the sequence 1, 2, 3. Is there a coin system we can suggest that includes coins with all three of those values?

No duplicates
We noticed that, for all the coin systems we created, there were always some values that were duplicated. What we mean is that there are two ways of using the coins to make that value. For example, in the coin system 1345, the following values can be made two ways:

4 = 4 = 1+3
5 = 5 = 1+4
6 = 3+3 = 1+5
8 = 4+4 = 3+5

Is there a coin system where all the values can be made only one way? If not, why not?

Two rival countries
Cambodia and Laos really love to compete with each other. As a result, they want coin systems where there aren't any shared coin values. For example, if Cambodia has a coin with value 5, then Laos can't have a coin value 5.

Is it possible? If not, what is the smallest overlap possible and how many possible coin systems achieve this smallest overlap?

Note: when we started this investigation, we had already designed a new system for Cambodia, so our starting question was whether there was a new system for Laos that wouldn't overlap this particular system for Cambodia, how much overlap was unavoidable, and then how to change both systems to minimize the overlap.

More values!
If we want to extend to amounts up to 20, how many coins are needed to have a compliant system (any value possible with only 1 or 2 coins)?

We haven't pursued this to a final answer, but the J's quickly recognized that, for any compliant coin system covering up to 10, they could create an 8 coin system that would cover up to 20. However, they also realized that four coins would be too few. This means that the minimum required to cover up to 20 is 5, 6, 7, or 8.

Coin triples
In the More Values! extension, we allow more than four coins in the system. The other way to relax the original constraints is to allow more than 2 coins to build a value. What can we achieve if we allow three coins at a time? Could we use fewer than four values in the system and still cover all amounts from 1 to 10? What about 1 to 20?

Chocolate cake reference

This was Nigella Lawson's recipe, but that page currently doesn't include the ingredient amounts, so I'm reproducing it here. This was a test ahead of J3's birthday, so she helped. Since it seems successful, we will be making it again and I will include more pictures of the process.

Heat oven to 170 deg C

MIX:

• 3/4 cup + 1 tbsp flour
• 1/4 tsp salt
• 1/2 tsp baking soda

1/2 cup water (boiling)

6 tbsp cocoa powder

COMBINE into a thick paste and allow to cool (doesn't need to come fully to room temp)

2 tsp vanilla extract

ADD to cocoa paste and stir

2/3 cup olive oil

1 cup sugar

3 eggs

COMBINE in mixing bowl and mix on high (directions said 5 minutes, we mixed for 2-3)

COMBINE wet ingredients and mix thoroughly

COMBINE dry mix and wet just to incorporate together.

Use small amount of oil to grease a 9 inch baking pan.

Pour in batter and bake for 45 minutes (I rotated the cake every 10 minutes).

Another dessert reference to store, unfortunately dairy: Caramel Sauce

Ode to a bead string (a non-poem poem)

from The Math Maniac

What is a bead string?
100 beads
Whites and Reds
5+5+5+5+5+
5+5+5+5+5+
5+5+5+5+5+
5+5+5+5+5
From hand and eyes to brain
But also: a caterpillar, a snake,
dog collar and horse bridle
a gun

a gate or a fence

a telephone line
a part of a train
lock for your door
tiny eggs (maybe spiders?)
coins
jewelry (necklace, naturally, or earrings, bracelet, belt)
roller tracks for a made-up truck
beans cooking in a pan
and so much more!

---

While reviewing Dreambox Learning, I spent some time thinking about physical manipulatives compared to apps. Apps can certainly have nice features, but physical objects own my heart. I'm not even going to talk about the educational value (see this Hand2Mind note for a summary and further references). The thing we really like is the open-ended flexibility about how they can be used, both mathematically and for creative play.

As an example, the bead string is (maybe?) one of the more limited maipulatives because it is linear and one connected piece. We don't even have one. However, the ideas above were immediate ideas coming from observations of how my kids play and a quick brainstorming session with the little ones.
BTW, we find that a 6 to 9 year old can make any object into a gun, in case that idea didn't make sense to you.

Favorite category of math manipulatives: food! In this case, freshly baked chocolate cake.

Returning to math, one of the cool things about this open play with math manipulatives is that it provides a lot of easy entry points into short math chats. These are fun, in themselves, and also reinforce the observation that math is accessible and all around us.

Manipulatives in Apps
While better than nothing, virtual versions of physical manipulatives always seem to fall short, with the following limitations:

• only one "correct" way to interact with them
• no possibility of combining
• can't take them apart

Perhaps there are also some benefits of virtual objects. In particular, they don't need to obey the law of conservation of mass. Can that, or other advantages, be used cleverly to make up for the disadvantages?

Let me know what you think and tell me where you've seen the best use of virtual manipulatives.

Powers of a permutation and Santorini review

Two unrelated topics today. Or ... I guess one could argue that everything in math is related, but I don't see direct connections myself. If you spot some, let me know in the comments.

Rainbow permutations

We have a collection of crayons that can stack. Well, to be honest, mostly the kids break the tips off. The second most common use is to stick them on fingers as fancy fingernails. The fourth most common use is to actually draw or color with them.

J2 was engaged in the third most common use, stacking, when he noticed something new. He started with the colors stacked in rainbow order (R O Y G B Purple Pink). Then, he took 2 off the bottom and moved them to the top. Then, he took the bottom two and moved to the top and repeated. He noticed that, eventually, he got back to the starting order. Experimenting further, he tried the same process moving 3 from the bottom and repeating. Again, he eventually got back to the starting order.

Here is an example of the crayons stacked together:

In this case, he is moving blocks of 5:

After showing me, he suggested trying 4, 5, 6, then 1. He noticed a couple of things:

• moving 6 is like moving 1 backwards (from the top to the bottom). 
• Similarly, 5 and 2 are related, 4 and 3
• 7 is prime, so maybe that is the reason the arrangement repeats

To test his hypothesis, he added another crayon (for 8) and tested. Again, he got back to the original arrangement. Hmm, doesn't need to be a prime!

Where should we take this first foray into group theory?

Santorini

Gordon Hamilton of Mathpickle, one of our favorite game and puzzle resources, has a Kickstarter for the newest version of his game, Santorini.

I encourage you to check it out. (For what it is usual disclaimer applies, I'm not financially related to this game or producers in any way.)

For reference, this is what the previous version of the game looks like:

Our DIY board (version 1)
While this version of the game will have nice custom pieces, the underlying game can be played with almost any collection of stackable objects. For our introduction to the game, we tried using our TRIO blocks:

To explain what you're seeing:

• 1x1x1 cubes are used for building levels. I originally thought we would color coordinate (red for first level, pink second, etc), but J1 and J2 liked mixing up colors.
• 4 unit straight connectors for our builders: magenta versus green.
• Angle or arc connectors are radio antenae to serve as the fourth building level and block further building
• Flags and wings as boundaries of the 5x5 board

Reactions

Both older Js enjoyed the game play. We quickly played 5 or 6 times. This game clearly has a lot of depth. We will have to play a lot more to see what patterns we can identify and whether we can develop any opening strategies. They are also very eager to play with some god powers. Perfect way to spend the rest of the holiday this week!

The TRIO blocks have pros and cons for this game. One of the best features is that it makes the game set-up robust to tipping the board, knocking the pieces off, or otherwise unsettling the position. It was clear to see the sizes of the levels and also immediate to identify towers that had already gotten killed (with a fourth level antenna).

In our play, there were two drawbacks. First, the straight connectors are a bit hard to remove from their positions. Second, the higher towers (2 and 3 levels) sometimes obscured allowed diagonal moves in a way that the stacking tiles version didn't. I wonder if this will also be the case with the new version of the game as the building levels seem tall relative to the size of the builders.

Longer-term, I wonder about buying customized games vs playing with generic materials. It is easy for the kids to grab a box off the shelf and start playing. Somehow, I suspect they will be less likely to grab a building set and a notecard with rules.

More thoughts on DIY game versions

Before Santorini, I had gotten excited about making DIY versions of the GIPF project games. I was thinking:

1. these games will be fun to play, so great motivation to replicate them
2. interesting challenge to make our own triangular grid board (squares we already have in abundance)
3. good creativity prompt as we repurpose items
4. it would make the kids feel power over the game structure and rules, leading to exploration of variations and deeper thinking about structure.

However, the kids, particularly J1, were surprisingly unenthusiastic. With Santorini, again, they didn't take much initiative in developing our DIY version. However, I wonder if they will be more motivated to modify or replace the TRIO blocks board since they now see that the game is fun and have experienced some of the limitations of our current version.

On the other hand, maybe they'll just push me to buy the commercial version.

Start of vacation math

Here in Thailand, the kids are now on summer vacation, so I've been exceptionally busy and found it hard to write blog posts. We've still been doing a lot of math, so I wanted to at least note down some of the activities.

Sonobe Modular Origami

Credit for this one goes to another parent at J1's school. One day when school was still in session for J1, but J2 and J3 were already on vacation, we happened to see a group making sonobe modules at the elementary school. This is something my mother tried to introduce to the kids last year, but it didn't catch them at the time. Not sure what was different this time, but J2 was really intrigued.

Most of the work has been J2, but J3 also got into the act and folded about 9 units herself. I helped to sharpen the folds for 6 of them, which J2 and J3, working together, made into a cube.

Here are two of J2's stellated octahedra, made in between swimming sessions at the beach:

Video references we found useful for guiding our sonobe exploration:

Sonobe Cube: this is where we started.

Stellated Octahedron: shows how to make the basic module and how to assemble 12 into a stellated octahedron. This proved to be easier than I had anticipated, so J2 was able to do this all on his own.

Doubled tetrahedron: unfortunately, this doesn't give a sonobe module tetrahedron, but rather a 6 sided shape. We had actually made one of these ourselves when experimenting with 3 sonobe modules created from post-it notes. These are delightfully strong and feel really nice to toss from hand to hand.

Some advanced observations
Euler Characteristics
The model for the cube gives a triangulation for which it is particularly easy to count vertices, edges, and triangles. I talked about the Euler characteristic and we then did the counts for other sonobe models as well as other platonic solids to check. We also talked about the Euler characteristic for a cylinder (without end caps) and a torus.

Finally, I liked to describe our calculation as "0 dimensional things" - "1 dimensional things" + "2 dimensional things." This led J2 to ask what we would get if we included the space inside the shapes (extending to a "3 dimensional thing") and to get curious about 4 dimensional objects (which led to a lot of discussion, below).

Chirality
Sonobe modules come in two flavours, depending on which corners you fold into the center. All three of us (J2, me, grandpa) accidentally made the two kinds of sonobe modules without realizing it, only to struggle when we went to assemble a larger model. This started a brief conversation about chirality. While we didn't explore in depth, it did spark a later moment of recognition when Matt Parker mentioned the concept for mobius strips (see below).

Robustness
I was really surprised that the sonobe models we built are so robust. The completed shapes don't fall apart very easily (especially the doubled tetrahedron). Even better, for the models we have built so far, the sonobe modules themselves don't need to be perfect. This was crucial for the beginning origamists in our gang as it allows them to do almost all the work from start to finish.

It also made me think about my old idea of error in construction recipes (straight-edge and compass constructions as well as origami folding constructions). To illustrate this for my father (and J2 unintentionally) I showed them David Eisenbud/Numberphile's construction of the 17-gon. This, in turn, prompted J2 to get excited about straight-edge and compass constructions (below).

Constructions

As mentioned above, J2 was excited about investigating classical construction problems. This is something he has seen a couple of times in the Abacus Math curriculum that P does with the kids. However, in that, we haven't generally been strict about the tools they use and the collection includes some standard triangles (45-45-90, 30-60-90) and measuring tools (rulers, protractors). Now, J2 was curious to see what could be done with just a straight-edge and compass.

For physical construction, our tool of choice has been the Safe-T compass:

Obviously, this doesn't quite correspond to the ancient limitations, but we ignore the ruled markings and the various holes are always within our margin of error on radius lengths anyway. Other than the obvious (for a parent) advantage, I think this compass style makes it easier for 2 people to help each other when drawing the circular arcs.

J2 has also been playing Euclid the Game. The virtual constructions have added two interesting points:

• strict restriction to what is constructed; he eyeballed a pair of equal lengths at one point, then wanted to talk about why he hadn't gotten credit for the construction (for angle bisection). This actually helped him identify that he needed the lengths to be equal; once he articulated that, he knew to use the compass to draw a circle.
• Chunking into advanced tools. For example, the game gives you a mid-point tool once you demonstrate the construction of a midpoint of a given line segment. This has helped him start to work on the coding concepts of functions/procedures and also gives him a new visual reference for the problem solving strategy "try to use what you already know."

Fourth dimension

The big insight here is about the relationship between understanding four dimensional shapes and our own ability to perceive 2 and 3 dimensional shapes. Other than that tip, I'd direct you to these great resources:

Flatland: we don't actually have a copy, but I know it well and it is a foundation behind a lot of the discussions we had and videos we watched.

Matt Parker's Royal Institution talk: He covers a lot that isn't about four dimensional shapes, but I'm not going to point you to the relevant starting time because the whole talk is great!

XKCDhatGuy: This was actually one of the first videos J2 found and, hey, the kid does a pretty good job explaining. Don't, by any means, stop with this explanation, but it is a fine place to start.

Carl Sagan's Flatland: J2 liked it, but I still remember the old cartoon from the BU library that we would watch at PROMYS and wasn't so impressed.

4th dimension explained: [no comment]

Drawing 4th, 5th.. dimension:  Somehow, the simple points in this video really helped demystify higher dimensional shapes for J2. I think part of the point is that it encourages us to attend to aspects of the shapes that we can understand easily (for example, the number of vertices in an n-dimensional cube) rather than worrying about the macro shape.

Rotations of 4 dimensional shapes:  Similar message to the previous video

There is no 4th dimension: Nice, quick explanation living up to the high standards of other One Minute Physics videos. A very worthwhile series, in its own right.

Cricket

This isn't mathematical, but I wanted to record this for posterity as well. Through random chance, J2 asked, "How do you play cricket?" At this point, he is still 2/3 English, having spent most of his life in London, so I guess he has a right to some information about this crazy pastime. For better or worse, though, that's not what he got. Our house rules:

1. batsman/batsmen pretend to stand in a traditional cricket batting pose, mime holding a bat
2. I stand a couple feet away, mime delivery of a cricket pitch (straight arm overhead throw)
3. we all (including spectators) run together into a giant heap, tickling each other and saying vaguely English-y, vaugely cricket-y things: "good show old chap," "ooh, looks like rain," "how do  you take your tea?" "you were almost to a century, young chap" etc
4. I break off the tickling and everyone stands in line, waiting for their points and an explanation of why points were awarded. Not spilling the tea is the most common reason for someone getting points, though boundary hits are also frequent. Note that points can be awarded in non-integer amounts (1/3, sqrt(2), and pi have all been awarded at some point).
5. Finally, I declare the match over with a result of "no result." As you can tell, we take our cricket seriously and are committed to the "test match" format.

Some important summary points:

• J2 is the player who has managed to accumulate the largest score in a single match. He was awarded "one hundred million one thousand" points by J3 for not spilling the tea.
• We are coining a new English-y phrase: "young old chaps." Usage example: when J3 is setting up a chasing game outside, she will point to J1 and J2 saying "these young old chaps are the runners," then point to me "and that old young chap is the monster."

Assorted math conversations

A collection of little items from the last month.

Optimal time to eat

P was talking with the kids about what order they should eat their food, which led to lots of graphs showing flavor over time. I don't know if this is a typical conversation for other people, but pretty common in our house. Here are some of the charts they made: Naturally, this also led to other charts. In particular, a debate about how quickly someone could develop skills by practicing music. Note the lack of labeling on the y-axis.

Some challenges

At lunch today, everyone got a math question. Apologies, I don't remember what challenge we gave to J1. J3 was given what is 5 + 2? After we let her consider and answer, J2 got asked what is the cube root of 125? Before either older kid could answer, J3 yelled out: 5.
Hmm. No, we also don't think she really understood, but it is nice to dream!

A joke

J2: look, a prime! (pointing generally at the 100 chart on our wall)
Everyone look expectantly at the wall
J2: No, it isn't. 22 is a compositive
All little Js burst into hysterical laughter. P looks at me as if to say, "this is your fault."

I wonder

From nowhere: Daddy, what 3 digit number has the most factors?
Exploration to follow tomorrow.

Cooking

J3 and I made scrambled eggs today in two batches. First, with milk for J1, using 2 eggs. Then, a second batch without milk for J3 using another 2 eggs.

J0: how many eggs did we use
J3: For J1, we used 2 (holds up two fingers on right hand). Then, we used two for me (holds up two on left hand). Let's count them . . . 1, 2, 3, 4. Four eggs!

Contortionist cubes

We got a lot out of this video from Mathologer: Contortionist cubes.
An immediate idea was to see which of the objects we could construct using our materials at home. The best we managed was using polydron to build one of the contortionist cubes, with sides that are either all red or all green. This high-tech enterprise was made possible by scotch tape!

Infinity plus one and half Infinity (more checker stacks and surreal numbers)

J1, J2 and I spent more time talking about checker stacks and surreal numbers this evening. Most of the conversation was with J1, so J2 and I will have to catch up. I may have J1 lead the conversation and see how much he really understood and can explain.

Tweedledum/Tweedledee strategy

I had taken a look at Winning Ways for Your Mathematical Plays vol1 and thought it would be good to warm them up with the tweedledum-tweedledee strategy. I thought the pictures from hackenbush would be more inspiring, so drew these Tweedledum-Tweedledees. With checker stacks as recent background, J1 got the rules for the new game right away and he also saw the inverse symmetry of the colors (in fact, J3 happened to be there at the time and also noticed that symmetry).

Our wonky anti-twins

I asked which player has a winning strategy and gave him four options: red, blue, first player, second player. After a bit of thinking, he chose the second player and explained the copy strategy. We talked through a couple plays of the game just to see how it would work.

One interesting side conversation was the idea that the copying strategy isn't necessarily the most efficient way for the second player to play. This is in the sense of when the first player makes bad decisions, there could be ways for the second player to open up a huge advantage, while the copying strategy basically keeps returning the game to a 0 value.

On its own, this was quite a nice conversation, thinking through the pros and cons of continuing with the copying strategy. The ideas we discussed were:

1. does the margin of victory matter? For some games yes, for others no.
2. Is there a chance that we make a mistake if we stop copying and go for a "bigger" win? There might be some subtle strategic cunning that we are missing and any choice to stop copying could be irreversible.
3. Is there a chance that we've mis-identified the game and it isn't exactly symmetric? Woe to us if we copy the first player until the error becomes obvious and we're now in a position behind them.

For what it is worth, this links to a dialogue in the Fritz & Chesster series (quoted from memory, not verbatim):

Fritz: What if your opponent isn't very good. Should you just clobber them?
Chesster: What do you mean?
Fritz: What if they make bad moves?
Chesster: Focus on your own strong play and developing your position. Don't play bad moves that assume your opponent is weak.

We also rounded out this part of the discussion by looking at the deep purple and figuring out a stack that is the additive inverse of deep purple.

Some stack values

Omega + 1
Watching more Mike Lawler videos, I saw his kids enjoyed thinking about ω + 1. We have previously talked about a more vague form of infinity and had broadly agreed that infinity - 1 is still infinity and infinity +1 is still infinity. I asked J1 to see if he still thought that (he did) and then asked about omega. If omega is infinity, what about ω +1?

His intuition was that it would still be omega, so we though about how to test. First, we recalled that deep blue has the position value omega, deep red is negative omega. With some thinking, he realized we could look at the game: blue + deep blue + deep red.

What do we need to check? See if blue has a winning strategy as the first player (why is this sufficient)?

Blue does have a winning strategy and J1 saw it faster than I had. Thus, (ω + 1) - ω is positive. "Wow, the omegas can cancel here!" He didn't expect infinity to work like this (nor did it, in our earlier conversations.)

Omega/2
We talked about a couple of other stack values with ω (2 deep blues, three deep blues, etc). I made an incorrect (I think) comment about ω² and then said that the surreal numbers even have ω^ω and sqrt(ω) but that I wasn't sure what checker stacks would correspond with those values. He was pretty intrigued about the square root and asked what it would mean. This is all we got:

if A = sqrt(ω), then AxA = ω

Unfortunately, since we don't really know how to think of surreal multiplication in terms of checker stacks (yet?), this doesn't really help us so much.

However, this led J1 to ask, what about ω/2?

For regular 1/2, we had gotten lucky by adding a regular red on top of a regular blue. Maybe that would work here? Could we just add a regular red on top of a deep blue? We wrote down the position deep red + (deep blue)red + (deep blue)red and started checking whether blue would lose if playing first. We saw that this quickly gets to positions that obviously favor blue and concluded that our starting position had a positive value.

We still had our starting game position deep red + (deep blue)red + (deep blue)red on the white board. I figured out the fix and then told J1 that there was actually a simple way to modify what we'd written to correct it. I think he was still following the analogy to 1/2 and suggested making the top checkers deep reds. Talking through the new game for a while, we were convinced that we'd found a representation for ω/2 (wow!)

I thought this was really awesome. While we are still exploring something that other people have done before, he asked and answered a question that wasn't in the guidebook (so to speak).

Deep deep checkers
With the idea of stacking deep checkers on top of each other, we came up with the idea of deep-deep checkers. For example, remember that a deep blue can be taken off, removing itself and anything above it, and the player adds any non-negative finite number of regular blues to that stack. For a deep-deep blue, that checker gets removed along with any checkers above it, then the player adds any non-negative, finite number of deep blue checkers.

Using arguments nearly identical to the deep blue checkers, we figured out that a deep-deep blue + Nx(deep red) is still a winning position for blue, for any finite value N. That means the value of deep-deep blue is larger than Nω, for all finite N. This now seemed like a good candidate for a representation of ω x ω (aka ω²)

One final stack

Now that we've got deep deep checkers, it seemed natural to try something similar to the trick we'd learned earlier when we stacked a deep red on top of a regular blue. Remember, that gave us a stack with position value ε, positive, but smaller than any power of 1/2. Another way to see it was to play games with R + n (B deep Red), see that these are all Red wins, and conclude that B deep Red's position value is smaller than any fraction 1/n, for all positive integers n.

Using similar reasoning, we tested the games deep R + n (deep B deep-deep R). Following very similar reasoning, we think we see a pretty easy winning strategy for red, so the deep Blue deep deep Red has to be positive, but have value smaller than ω/n, for any positive integer n.

Hmm, maybe the position value of this guy is tiny? To be sure, we checked the game nR + (deep B deep-deep R). That is, n regular red checkers against a single copy of our suspect. Well, despite the awesome potential of a deep deep red, it falls whenever blue makes a move and these games are clearly blue wins.

So, what is the value of our new stack? Since it is smaller than ω/n, for any positive integer n, it feels like it could be ω * ε?

The problem: In Jim Propp's post, he says that ε and ω are multiplicative inverses (also, attacked in more detail in section 2.6 here), so their product is 1. However, our stack has value much greater than 1. We will have to think about ways to attack this new value.

Where is J2?

With J2, I teed up the question of how to find the additive inverse for a given checker stacks position. He instantly answered that you just need to swap all the colors. Seems like he is in good shape to verify this by working out the strategy as I did with J1.
On his own, he wanted to move onto the value of the deep purple stack, which he remembered as 2/3. I only had time to encourage him to think about two things:

1. What stack is the additive inverse of a deep purple checker? As with J1, linking this with a RP stack was interesting.
2. What could be the second player's winning strategy for 3P + 2R?