Lure of the Labyrinth and Mathpickle

I'm puzzled why it has taken so long for me to hear about these two resources. Seriously, folks, have I not been immersed in elementary-age math activities for the last several years, trawling through other people's favorites lists and doing my own searches? How could both of these have been (until now) undiscovered gems?

Mathpickle

Going beyond the silly name, this is a site with fantastic games, puzzles, and comments about learning and teaching math. The site is well structured and almost every link is a gem, so it will be easy to find worthwhile material. If you need any extra help, just try Games organized by Subject.

Note: at mathpickle, don't get caught worrying too much about "grade level" labels, especially don't skip things you think might be "too young" for your student. Once they have the (minimal) knowledge to get into an activity, there is usually no ceiling blocking further or deeper exploration or understanding.

Little Bit of Aggression: Part of J1's curriculum while at home sick last week

Some quotes/philosophy
Mathpickle is the work of Gordon Hamilton who aims to "get curricular unsolved problems into classrooms worldwide – one for each grade K-12."  This might remind you of the interesting post by Lior Practher from last september: Unsolved Problems with the Common Core.

Gordon's comments about board games strongly resonate with me:

A small fraction of games work well in the classroom. They must be resilient to damage and loss. They must be easy to teach and quick to play and put away. In the classroom budget, they must compete against cheap electronic games. 

A large fraction of games work well at home – that’s why parents must take the lead on establishing a culture of board gaming.  

Somewhere on the site, Gordon mentions that he is now developing some educational apps; I'm keen to see what he creates!

Lure of the Labyrinth

Gordon also has great taste in other resources, some of which are listed in his collection of Inspired People.  Within this sub-site, he has very high praise for Scot Osterweil's Lure of the Labyrinth.
The two older Js and I have been playing and are hooked.

Inside is a game world with a set of challenges related to a running story and a collection of (at least) nine different puzzle/game types that develop a range of mathematical ideas.

Some of the cool things about this game:

1. Fun challenges that get harder in interesting ways
2. Cartoon monsters and an interesting story context
3. The minimum of exposition. Challenges are not really explained, so you get to experiment to figure out what is going on. The designers found a great middle ground between "I know exactly what I need to do" and "I have no clue." Something on the order of "I know I can try these things, but am not totally sure what will happen."
4. Navigating around the game world is interesting, in and of itself. I was particularly delighted by the lay-out of the "triangle" wing (the last wing of the factory you unlock).
5. The game basics are high quality: good graphics, background music, interface
6. Free!

You have no idea what this is, but will be able to figure it out!

A classic puzzle with some gross-out twists

A minor, minor, nit:

In case the game designers ever read this.....

First, thanks so much! We love the game and really appreciate all the time, effort, and attention that must have gone into creating it.

Second, go back and read those previous two sentences again!

Third, well, since you persisted in reading this far . . . the number of actions allowed in the Managers' Cafeteria is a bit mean. As illustrated in the screen cap above, you have 12 actions to correctly specify and place 12 orders. That means no mistakes are tolerated, not if you mis-enter a serving size or accidentally forget to change the type of food. We understand that mistakes require a redo in other challenges (mineshaft, gardens, advanced testing labs) but:

1. none of those require the same shear number of clicks as the Managers' Cafeteria
2. It feels like a large number of clicks in MC are committed (they lock you into a path without possible change) while far fewer in the other challenges are so decisive
3. Many of the MC clicks are just mechanical, they don't bear on the level of understanding of the challenge.

Our suggestion is simple: add 3 more "lights" (allowed attempts) to MC.

How to do science (short story)

J1 has been at home sick the whole week. Each day, he writes a journal entry, usually a short story.

How do people jump?

One day, Ms. Jump got a call from Ms. Silly. She asked her to go to her home to jump for her
because she was doing research on how to jump.

When Ms. Jump got to Ms. Silly's home, she gave Ms. Jump 50 pounds as payment.
So Ms. Jump jumped and Ms. Silly recorded the jump.

Ms. Silly said, "Oh, now I understand. We make our head go up and down to jump!"

Little one counting

Late last year, Joe Schwartz wrote a very interesting post about the difficulties one student is having with counting and skip counting. I recall hearing a theory that many later math difficulties trace back to when a student missed solidifying the concept of one-to-one correspondence and some related concepts of counting (but don't have a citation or reference). As a result, I was thinking about how our little J's understand these ideas.

In particular, I wanted to try out Marilyn Burn's little game from one of her comments with J3:

Ask him to put out 8 cubes on a paper. [I chose 8 because when I remove one, the child won’t be able to know how many by subitizing.]
Ask: How many cubes did you put on the paper? (8) [Here I look for whether the child has to recount.]
Say: Watch as I take away one cube. Remove one cube and place it on the table.
Ask: How many cubes are there on the paper now? (7) [Does the child have to recount, or does the child just know.]
Say: Watch as I take away another cube. Remove one of the 7 cubes and place it on the table.
Ask: How many cubes are there on the paper now? (6) [This is the same as the previous question, a way to check if the child still needs to recount.]
Say: Watch as I put one cube back on the paper.
Ask: How many cubes are on the paper now? (7) [Similar, but adding 1.]
Sometimes I repeat again removing a cube and asking: Can you tell me how many there are without counting? Some kids shake their heads to indicate they can’t, others say they’ll give a guess, some are able to.

J2

Before I had a chance to try out the question sequence with J3, I had some time alone with J2. He was sorting colored pencils, so we used those as counters. Overall, he breezed through the questions, but there were two amusing points:

1. After separating out 8 colored pencils, I asked how many he had. His response showed that (a) he believes in conservation, so he knew there would be 8, but also (b) he is used to me doing something tricky, so he wanted to verify that there were still 8.
2. His method of verification: split them into two groups of 4, an amount he could recognize by subitizing, not counting.

He asked me why I was asking these questions and I told him it was related to his understanding of hierarchical inclusion. We talked briefly about what that means and he was delighted by the term, so ran off to ask J1, "how is your understanding of hierarchical inclusion?"

J3

My counting time with J3 came during dinner. She was eating cucumber slices, so we used these as counters. This turned out to be a mistake, since conservation doesn't work with edible counters! In other words, whenever I asked her how many slices were in our cluster, she would pop one in her mouth and smile, knowing that she was teasing me.

Mainly, though, I was able to verify that she doesn't yet have the concepts that allow short-cuts to the questions in Marilyn's sequence and needed to recount to get all the answers.

Incidentally, I started the activity by telling her that we were going to count something together. She immediately grabbed this coaster and then accurately counted the circles to 37.

For me, the entire experience was a really interesting illustration that counting actually requires a complex collection of sub-skills.

Polishing a ... (trying to fix a textbook lesson)

Graham Fletcher called out a bad textbook in his recent post, Placing a hit on Pseudo-Context.

He was inspired to create a related lesson. Here is another, super low effort attempt to salvage something from the original textbook. What do you think?

1. What is our scenario?

Okay, obviously, this sets us up to make boxes to hold DVDs. What do the kids think? Why is Mike doing this? What should he consider?

Seems natural that the teacher should have on-hand a bunch of DVDs and construction material.

2. So, Mike made some progress constructing his box:

Some questions for the kids to debate:

• What shape did Mike choose?
• What dimensions?

3. Here are the choices Mike made:

• What do you notice? 
• How do his choices compare with the ones you made?
• What do you wonder ?

*UPDATE* I talked through this post with J1, my 8 year old. Here are some of the things he explored and talked about:

• How would the DVDs inside the box be arranged? Stacked on top of each other, multiple stacks, with some type of dividers between them or not?
• How big are DVDs? He measured the diameter of one, then stacked 10 (in protective covers) to measure thickness.
• How many DVDs does Mike have? How does the desired structure of the box change if there are 5, 20, 100, 1000 DVDs? This also linked to practical considerations around finding and extracting a particular DVD from the box and what would be suitable materials for the box (paper, cardboard, wood).
• Since he measured our discs in cm and the book had inches, we had a conversation about conversions.
• As I'd guessed, getting him to think about the dimensions of his own box before seeing the textbook dimensions made him wonder about symmetry.
• Note: this prompt really didn't get him to think much about the volume of the box. I think this is because there isn't really a natural trade-off between the dimensions for DVDs, so it is natural to think of the base area compared with a single DVD and the height based on the number of DVDs. Graham's sugar cubes was much more direct for engaging the concept V = L x W x H.

Dots and Boxes variation

In grades 1, 2, 3, we played this variation of dots & boxes: Mathify the Squares Game.

I'm enthusiastic about this game, but can't resist a quick comment about the "mathification." Dots & boxes is already a mathematical activity, it doesn't need to be "mathified." This term implies confusing arithmetic and calculation with math, something I've written about elsewhere and, I hope, is clearly not implied by our blog.

In any case, I'll use the shorthand MD&B to refer to this dots and boxes variation.

Notes from playing in class

In class, we first introduced the kids to vanilla Dots & Boxes with a pre-printed grid of dots. We knew that it would be too much to play on a lattice covering the whole A4 sheet, but we thought a quarter of the grid would work. That turned out to be too big and the game started to seem monotonous to the kids as there was too much time spent on the opening (playing on squares that don't yet have any filled sides.)

We rectified this problem in 3rd grade and played on much smaller grids, with sizes between 7x7 lattices (which yield 6x6 squares) and 10x10.

After they were comfortable with the vanilla game, we introduced the product version with dice. In our case, we just had the players take turns and didn't give an extra turn when someone completed a square.

Some rule variations
There are some simple variations depending on how you deal with completed squares:

• no extra turn (this was the version we played in class)
• player adds another side to a different square with the same value. For example, say the dice are 2 and 4 (product 8) and the player fills a square. They also must add a side to another square with value 8.
• player rolls the dice and adds a new side (a full extra turn)
• player adds a side to any square (dice and square values ignored)

Of course, you could also make the extra turn optional instead of compulsory. You might also have some ideas about different ways to handle cases where there are no more free sides on squares with the required number value.

Probability questions

Dice games naturally lead to probability questions. Here were two that I really liked, based on scenarios from a recent game play:

What question are we asking?

What is the chance a player will get both the 20 and 36 boxes:

One great answer was 1/2. The reasoning: we are going to play until all boxes are filled and each of us have an equal chance to fill this box, so 1/2. This is not quite right, since the person who is about to roll has an advantage, but I thought it was an interesting interpretation of the question.

A 2x2 square

In this configuration, what is the probability that the next throw will allow the player to complete a box in the 4-30-25-15 zone?

  

A bigger D&B family

One of the reasons I thought this D&B variation was so cool is because of our games matrix. Whenever we play games, J1 and I talk about some key characteristics of the game, particularly the amount of randomness and the strategic complexity. These are not entirely independent dimensions, since a larger amount of randomness reduces the number or importance of each player decision, thus the strategic depth.

The mechanics of this game gave us some ideas about how to dial up or down the amount of randomness in this family of games. Here is our list of members of this family, roughly ordered from least random to most random:

1. vanilla dots and boxes: no random element
2. mash-up with product game: squares are still labeled, but players control the two factors using selectors (like in the product game) instead of using dice. This can be played with different collections of factors and different size boards (including board variations where a product appears multiple times or only a single time, where values are ordered or randomly distributed).
3. half-way house: one factor is chosen by players moving a selector, the other is determined by a dice roll (either before or after the "free" factor is selected).
4. MD&B game: as played in class and described in the first link
5. MD&B game where each factor appears only once. This is a case of dialing up the randomness by reducing the strategic options of the players. 

Ideas for other games

I'm excited to see what other games we can modify use the underlying idea from the MD&B variation. To be clear: use numbers to label parts of the game and then constrain the players' actions based on a die roll to involve either the pieces with corresponding labels or board positions with those labels.

Three specific examples:

1. Hackenbush variation where segments of the picture are numbered. This could nicely incorporate probabilities by putting values of the least likely dice rolls closer to the ground.
2. Ultimate tic-tac-toe meets the product game: from Art of Math. This is an old post, but I just happened to see it when preparing this post.
3. Dice chess. Here's the wikipedia article. For some reason, I often forget about this variation, even though it is a nice way to reduce the strategic complexity of vanilla chess for beginners and has some nice links with probability.

If you have some favorites, I would love to hear in the comments!

*Update*
Playing through the MD&B version several times, we came up with these rule variations that are worth your consideration:

1. Game stops as soon as someone rolls a value that can't be played (alternatives are to let that person roll again or have them pass their turn)
2. Remove some of the randomness: (a) on your turn, you roll and play a side with the required value, but they opponent also plays a side with that value. As we played it, that means moves (without filled boxes) go: A, B, B, A, A, B, B, etc. (b) When a player fills a box, they can choose to re-roll both, either, or none of the dice for their extra turn.

I think our favorite was a combination of all three of these components. Mixing 2a and b, you have to be careful to keep track of whose turn it is, but it lead helped bring out elements of strategy and more thinking about probability.

*Update 2*
Game phases
Above, we talked about how A4 (or even 1/4 of an A4, which I guess is equivalent to A6) is too big for beginning players of Dots & Boxes. They found the game "boring." J1 and I talked about this experience and it led us to considerations around game phases: opening, middle game, and end game. These are terms we first learned in chess, and we found it useful to contrast the two games.
Here were some observations:

• Opening: a lot of choices, not obvious how most of those choices link with "scoring" or the winning objective of the game. At this stage, there seems to be little interaction between the players (there is enough space that most of their actions either don't bring opposing pieces together or there is a lot of open territory).
• Middle game: still many playing choices, increasingly direct conflict between players, interim objectives within the game become more clear and there are some chances for plays that either score or more clearly move closer to the overall game objective.
• End game: significantly fewer choices for each turn than the other phases, either because there are fewer pieces (chess) or most territory has been claimed. At this stage, players are able to focus on the overall game objective, rather than interim objectives.

What we realized is that the larger playing area for D&B significantly increases the length of the opening. Because this phase is the least connected with capturing boxes, it is the hardest for beginning players to see how their choices ultimate lead to scoring and it is the phase with the most available choices on a turn.

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?

Consecutive capture and Multiplication zones (math games class notes)

The games last week were taken from Acing Math's collection of card games and John Golden's wonderful blog (here's a list of games).

Consecutive Capture

This game comes from John Golden. The idea is simple, but it is a fun game. We used this in the first grade class. We made some slight changes to his rules.
Materials: pack of playing cards, including jokers, a number line labelled -13 to +13
Players: Two to Four (though seems naturally a 2 person game)
In this games, red cards are negatives, jokers are zero, and black cards are positive. Players are dealt a hand, then take turns putting their cards on the number line. Whenever they form three (or more) in a row, they can collect the cards that form the run. The cards they collect from runs count as points toward winning. At the end of each turn, they draw a card to replenish their hand.
If a point on the number line is already covered by a card, a player can add another card with the same value on top. If that subsequently becomes part of a run, the player collecting the run only takes one card for each value.
Variations: as noted, you can play with different numbers of players. When there are multiple cards on a value, you could allow a run collector to take all of the stacked cards. In John's version, he lets black aces take the value of 1 or 14, up to the decision of the player who adds them to the number line (and red aces -1 or -14).

Multiplication zones

This is from Acing Math. For 2nd and 3rd grades, We modified the card values from their rules, keeping aces as 1, J = 11, Q = 12 and removed the kings.

Surreal numbers and whole body integers

Two unrelated activities to note:
(1) Exploring checker stacks and surreal numbers with J1 and J2
(2) A whole body numbers game with J3

Surreal Numbers and Kids

If, like me from one month ago, you don't know about surreal numbers, I think you'll find they can be a really engaging exploration with kids. The main attraction is the appearance of infinities and infinitesimals, both of which really seem to resonate with young mathematicians. In addition, there's fantastic icing on this cake, too: you can explore by playing a simple (to learn) game with a lot of depth.

Credits: this exploration is strongly inspired by Mike Lawler's recent posts about surreal numbers and the Jim Propp post that inspired him. If you are interested in doing this type of exploration with your kids, I strongly suggest going through all of their posts on the subject.

Note: since we used black and red checkers, while the convention in the other posts is blue and red, I will abbreviate B and R so you can naturally substitute your own preferred color scheme.

How we got started
Using a set of regular, stackable, checkers (black and red), I showed each of the older J's the position RB + BR (a stack with red on the bottom, black on the top and a stack with black on the bottom, red on top) and explained that the basic moves.

This was a good initial example because it let us talk about each of the major scenarios:

• We will investigate cases where B moves first and others where R moves first
• Each one can only take stacks above one of their own color checkers
• If the colors allow, they can take a top checker and leave the rest of a stack undisturbed
• If their color is the bottom of a stack, they can remove the whole stack
• Usually, they will have choices about which stacks to remove

Then, I explained the losing condition: if you don't have any more moves, you lose. They quickly realized this was the same as when they no longer had any checkers of their own color on the board.

Next, we quickly played a set of simple games:

1. Single B checker
2. Single R checker
3. B + R
4. RB + BR
5. RB + B
6. B+B+B+B +R+R (and similar)
7. BBB+R+R (and similar)

Connecting with numbers

I told them that one amazing thing is that we can give each game position a value. Then went through:

1. Single B checker is +1
2. What do they guess a single R checker is? explain -1
3. What about B+R? eventually get to 0
4. Explain the fundamental trichotomy: positive value means winning strategy for B exists, negative for R, what about 0?

Powers of 1/2

The first really juicy bit came when I asked what they thought the value of a BR stack would be. This established a common sequence of investigations:

1. If alone, can we see a winning strategy for either B or R? In this case, obviously B. Thus, the value is positive
2. Compare with 1 by playing the game BR + R. They were really fast about seeing this link, for others this is worth writing out and spending some time discussing. In this case, they saw BR + R must be negative, so the value of BR is between 0 and 1.
3. Guess a value; they've got enough experience to think that often the answers are "nice," so 1/2 was a natural guess.
4. Test. In this case, it meant checking BR + BR + R

Next stop was RB. I wanted to make sure negatives weren't left out and to reinforce the symmetry in the colors, so would ask them to swap the colors and get a value as a quick follow-up.

Next, I asked them to see if they could find a configuration with value 1/4. This took a long time and there were lots of false starts. I didn't think this would be easy and didn't help them shortcut the exploration.

Once they got 1/4, though, they had a fast guess about 1/8. We checked it and then they made a conjecture about further powers of 1/2.

Deep blue and red

Seeing powers of 2 in some form is always fun, but both knew that they had been promised infinity and wanted to see it. Using our plastic dinosaurs, I introduced the deep checkers deep blue (represented by a blue mini) and deep red (represented by an orange mini).

Our sequence for these was similar to the powers of 1/2. The value of deep blue is positive, so they compared to 1 by playing the game BBBB.....+R. Each had a sparkle of insight, but quickly played BBB.....+R+R+R just to check, then announced that the value would be bigger than any integer.

I gave them the name omega, and we checked the deep red: RRRR.....

At this point, one of them put the deep red on a B checker and asked what that would be. Again, followed the previous recipe to realize that it must be positive, but smaller than 1, then smaller than 1/2. At that point, one of them made this arrangement:

Ok, I know that mastadons aren't dinosaurs

If it isn't clear, the realization was that the game sequence B, BR, BRR, BRRR, ... would have values

1, 1/2, 1/4, 1/8, .... and BRRRR..... would be at the end of that sequence with value ... larger than 0, but smaller than every power of 1/2. You can see that they made a similar connection with the negative values.

A little bit about deep purple

The two kids were totally satisfied now, having gotten omega and epsilon (along with omega +1, epsilon - 1, omega + epsilon, etc). To give them something to chew on for later (and because we had purple dinosaurs in our set) I introduced the deep purple, BRBRBRBR.... Immediately, they designated the yellow dinosaurs as the inverse of deep purple (RBRBRBRB.....)

So, what's the value of deep purple? What they've gotten so far:

• positive
• smaller than 1 (by playing BRBRBRBR..... + R)
• Bigger than 1/2 (by playing BRBRBRBR..... + RB)
• guess 2/3. I don't know where this came from, but I confirmed that is the value
• working on finding winning strategies for the second player in R+R+3(BRBRBR.....)

Wrap up videos

As a round-up in the evening, we watched the videos from the first post in Mike Lawler's sequence. I paused frequently to let my two shout out their answers and explanations for where Mike's boys were in their exploration. This seemed to be a very effective way to underline their experience for the day.

Number match on the number stairs

A game for J3. This is a simple game with some variations that makes use of our stairway "number line" and a three year old's natural enthusiasm for running up and jumping down stairs.

As pictured previously, we have labeled the stairs in our house from 0 to 36 (more to come). We have a set of cards with numbers on them. P mixes them up, then gives them, one at a time, to J3 to put on the corresponding step, and they sing count up and down:

Some other variations:

• using playing cards instead of number cards
• Using cards with dot or shape patterns
• Using cards with number words ("one" instead of "1")
• child sends the parent to a particular step, checks if the parent got it right

Pillow forts

In case you missed it, pillow forts have been in fashion recently. Here is an example:

war variations

Most of you have probably seen how the standard card game "War" can be modified to make an arithmetic drill game. Denise Gaskins probably has the best description here: Game worth 1000 worksheets.

We have used three variations of this game a couple of times: straight War (J3 and J2 playing with greater than, equal to, less than), addition war (grade 1), and multiplication war (grades 2 and 3). Frankly, I am often surprised how enthusiastic the kids are to play, since there aren't any choices for them to make when they play. For those who are ready to move on from the basic game mechanic, here are some extensions and related explorations.

Extension games

Build your deck
Currently, Vanguard, a deck building game, is very popular amongst the Js. One possiblility for War is to let the players arrange their deck in advance. In a sense, this is like a more granular version of rock-paper-scissors. I particularly like this variation for the 2 (or more) card versions where the kids need to think about how to mix high and low value cards. Also, the number of cards burned on each War battle can upset the organization for the rest of the deck, so that adds a layer of complexity for them to consider.

Choose your cards
My favorite variation is to deal a hand (between 3 and 6 cards, replenished after each "trick") to each player and then let them choose which ones to play. You can either require simultaneous play or, as we prefer, have each person play one card at a time going around clockwise, like in Bridge.

Explorations

Some exploration questions:

1. In basic War (high card wins): will there always be a tie at some point during the first pass through the deck?
2. In basic War: can there be a complete game (one player loses all their cards) without a tie ever occuring?
3. Does basic War always end with one player losing all their cards or can there be cycles?
4. How many times do we expect a tie on the first pass through the deck?

All of these questions can be explored for the different variations. For elementary kids, these are very challenging questions and I don't expect many answers. Two recommended ways to explore:

• Play many games, record data and observations. Make conjectures and see if there are any counterexamples that disprove your ideas.
• Play a simpler version of the game by reducing the number of cards in the deck. For example, play a demonstration game with only 6 cards: A, 2, 3 for two suits.

All your base are belong to us (cryptarithm extension)

If you don't know Futility Closet, I suggest you take a look. It is a fun and quirky combination of math puzzles, chess puzzles, and historical anecdotes

This recent post had a nice puzzle, Hidden sum, that led to a fun conversation with J2 and J1.

This was a fun puzzle on its own that I knew would appeal to J2, since one of his familiar number friends, 111, is lurking in the solution.

The base

Before I got a chance to discuss with J2, however, I spent some time considering a small clause in the question: "in base 10." Strangely, if this clause hadn't been included, I probably would never have thought to investigate in other bases. This restriction, though, seemed like an invitation to go exploring in other bases. Since the older J's had recently done some work in non-decimal bases, I thought they would enjoy this extra exploration.

I told J2 this puzzle. First, he worked through the base 10 version, including seeing an old friend (and familiar factorization) along the way, I asked what he thought about doing it in other bases. He was interested, so we started with binary. Luckily, his first idea was to consider possibilities for TTT. In binary, the only three digit TTT is 111, aka 7 in decimal. He saw that was prime, so couldn't be factored into two 2-digit factors. That proved to be the first key insight of the exploration.

We moved on to base 3, 4, 5, 6, 7, 8, 9, 10, and 11. At some point, J1 joined the game. Along the way, they made the following observations and conjectures:

1. if 111 is prime, there is no solution. This is because TTT will have to have a 3 digit factor.
2. If 111 is not prime, it will have one 1-digit and one 2-digit factor (why?)
3. If 111 is not prime, neither factor will end with a 0 in the ones place (why?)
4. Given a 2-digit number (ME) with a non-zero ones digit in the ones place (E not 0), and a (non-zero) one digit number X, there is a single digit value T such that multiple of T x X is of the form YE (a 2-digit number sharing the earlier value in the ones place). 

For the first three conjectures, "(why?" means that most of you should be able to prove these. For the fourth, this conjecture isn't true! However, there is something extra that happens in the scenario for the puzzle that gives extra information and makes it true when ME x X = 111

The sum

One other point is lingering for me: why does the original puzzle ask for the sum of E, M, T, and Y? Sometimes, this form of question is a clue that there is some interesting relationship that allows us to calculate the answer without finding values for all the variables. Though it is common, I still get a kick out of this, probably because there is such a strong instinct to solve for all the variables.

In this case, I really don't see a way to get the sum directly, without finding values for E, M, T, and Y. Any ideas?

If there isn't a direct path, why did they phrase the question this way? Without seeing the exam, my guess is that this is an information reduction operation that allows this to be a multiple choice question.