Some simple dice games

J3 and I played several simple games recently that I want to record. One of them, race to the top, is a variation of a more sophisticated game that can be used more generally.

Digits three in a row

Materials: a 100-board, 2d10, colored tiles
Players: 2
Goal: mark three spaces in a row
Basic play:
Two players take turns rolling the two dice. On a player's turn, they form a 2 digit number using the two digits and then claim that space on the 100 board. If the space has previously been claimed, they lose their turn.

The first player to claim three adjacent spaces in a line (horizontal, vertical, or diagonal) is the winner.

Variations:
(1) we used one dice marked 0-9 and another marked 00 - 90 (all multiples of 10) and then added the values to get our two digit numbers. This eliminated any player choice, but helped reinforce the idea that the value in the 10's digit is the number of tens.

(2) the winning condition can be increased to require a line of 4 spaces

(3) the winning condition can be changed to allow any three (or four) spaces that are colinear; these spaces would not have to be adjacent.

Notes:
This is a very simple game, especially the variation we played, but J3 found it fun. It was a useful exercise to practice locating the numbers on the 100 board.

Race to the top

Materials: An 11x 6 grid with one side labelled 2 to 12, 4d6, 3 tokens. Optional: 11 distinct tokens/small objects per player.
Players: 2 - 4
Goal: capture 3 columns.
Basic play:
This game is fairly simple, but has resisted our attempts to succinctly summarize the rules. Here is an explanation as we play two turns.

Here's our playing material, the three green squares are temporary markers:

First player, J1 rolls two ones and two fours. With this roll, J1 could group them into two fives or a two and an eight:

J1 decides on two fives and puts a temporary marker on the second level of the 5's column. After you understand the rules consider whether this choice is better or worse than the 2 and 8.

J1 chooses to continue rolling and gets 1, 1, 4, and 6. J1 groups these as a 5 and a 7, then moves the temporary marker in the 5's column up one level and adds a marker at the bottom of the 7's column.

J1 rolls a third time, getting 1, 1, 2, and 6. The only option is to group these as 3 and 7, so J1 places a temporary marker in the 3's column and advances the marker in the 7's column.

At this point, J1 ends his turn and marks his progress. On his next turn, if he gets a 5, for example, the temporary marker will start at the fourth level of the 5's column (building on his consolidated progress).

D has the next turn. He gets 1, 4, 4, and 5, which he chooses to group as 5 and 9:

D chooses to roll again, getting 4, 5, 5, and 6. This has to be grouped as 9 (advancing in that column) and 11:

D chooses to roll again and gets 2, 3, 5, and 6. This is a lucky roll that can be grouped as 5 and 11, allowing two tokens to advance:

D presses his luck and rolls a fourth time, getting 1, 3, 3 and 5. The dice can't be paired to get a 5, 9 or 11 and there are no more temporary markers available to place, so D loses his progress. J1 will have the next turn.

To be clear about the failure condition: the player must be able to place or advance a temporary token for both pairs of dice. For example, if D had rolled 1, 3, 3, and 6, he still would have lost his progress.

Further rules:

• the first person to end their turn on the sixth level of a column "claims" that column.
• the first person to claim 3 columns wins the game.
• columns that have been claimed by any player are safe values for all players. Players do not need to allocate a temporary token to those columns.
• Players can occupy a square that an other player has marked.

Variations:
(1) Change the winning condition so that the first player to capture a column wins
(2) Change the height of the columns, either fewer than 6 for a faster game or more than 6 for a slower game
(3) Change the failure condition so that only one pair of dice needs to be playable and reduce the temporary tokens to 2.
(4) only allow each player to roll one time. This eliminates the "press-your-luck" aspect of the game and is much more basic.
(5) allow players to jump over a square that has been occupied by another player. This rule particularly fits well if you use objects to record your consolidated progress (which also makes the grid re-usable).

Notes:

I was originally taught this game by Mark Nowacki of Logic Mills.

J3 and I played the variation where each player only rolled one time on their turn.

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.

Division Dice (math games class)

Who: grades 3 and 4
Where: in school

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A dice division game

We created a simple dice game to practice division. Here's a description of the basic element of play:

1. roll three dice: for example, 3, 4, 5
2. group two of them into a two digit number: for example, 45
3. Divide the two digit number by the remaining single digit: for example, 45 / 3 = 9
4. This value is your score for the round
5. First player to 200 or more points wins (we used 100 for the initial game)

Key constraints

• You can only score points if the single digit is a factor of the two digit number (remainder must be 0)
• Where there are multiple options, the player can choose the combination that gives them the maximum score

This pencilcode program (see code) analyzes this basic game structure, identifying how often there will be no legal scoring arrangement and showing a histogram of the largest scores.

Modifications/Extensions

I wasn't satisfied with three elements of this game: (a) any time a 1 occurs, the division calculation is too easy, (b) too many combinations don't allow a score (about 15%) and (c) there aren't many decisions for the students to make (just six combinations to investigate).

We addressed these by adding two extra rules:

• 1 is a wild that must be replaced by a value from 2 to 6 (cannot be left as a 1)
• On your turn, you can flip the over the dice in the two digit number. For example, a 6 can be flipped to a 1, 5 to 2, 4 to 3, etc.

The first point removes the division by 1 cases, the second one allows more choices and reduces the number of non-scoring cases.

*UPDATE*
Allowing the dice flip and wild 1s seems to make the game too loose. Instead we dropped the wild 1s rule and added these two:

• Division by 1 is not allowed in the game
• If you roll triple 1, re-roll

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.

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.

Random 100 board and some subtraction review (math games)

In grades 1 and 2, we (re)introduced the card subtraction game from one of our earliest classes.

Since this is our first class of the new (calendar) year, we also did one of the calendar tricks again in all of the classes (the first one on this page).

Repeating activities is something that we rarely do. Usually, we have been introducing a new game or activity each class, possibly following up on that activity in the following week. However, we realized that it takes more time for the kids to extract all the ideas or practice. Also, since these things are often still challenging, their interest level is high.

Four in a row

PK came up with this game, but it is based on an idea she saw elsewhere. If we recall the original source, I'll link to give it credit.
Players: 2
Material: randomized 100 board (we used 0-99 randomly arranged on a 10x10 grid); 4 dice (we used 4d6)
Turn: Player throws the dice, then combines the values using arithmetic operations to create a final result. They mark the corresponding cell in the grid with their color.
Winning: The first player to get four adjacent cells in a straight line wins (NB: diagonals also allowed)

We also asked the kids to write down their number sentences for each move, below the playing grid.

Here are the 4 grids we used (random order created using pencilcode similar to the 6th graders final assignments last term!) There's nothing magical about these, but we wanted to mix up the "easy" and "hard" numbers as well as give some additional variation for each play.

Homework

• Grades 1 and 2 should play the subtraction card game at home
• Grade 3 should play the 4-in-a-row game and explore the calendar trick. Can they figure out a way to find the square their friend chose? 

another race to 100 game

Today's game at the math classes was not particularly well liked, but we are including this note for completeness and future reference.

Race to 100

how many players: 2-5
material: 1d6, 100 board, position markers (the kids made their own out of play-dough)
start: all players start on 1 on the 100 board
turns: each player's takes a separate turn. They roll the dice, then move their piece up the 100 board some multiple of the dice value (up to 10x).
winning: first player to get exactly to 100 wins

This game practices multiplication, skip counting, and factoring. Here are some example questions to stimulate thinking about game strategy:

• Would you rather have your piece on 99, 98, or 96?
• What about 71 and 70?
• If you are on 88, what are your chances of winning on the next roll?

Game reception
The kids found this game fairly easy. In retrospect, perhaps we should have played this game before the Times Square factors game.

Potential extension
The game is nicely suited to analysis by working back from the higher positions and/or analyzing a simpler version of the game. This may be a nice exercise for our programming classes, especially as we have recently been working with arrays.

Damult dice extension

Apologies that it has taken me so long to post. We are on break between terms right now, which actually means I have less time than usual!

We played an extension of damult dice with the third grade class during the last week of classes that was very successful. Here is the quick story:

First, we played a round of the standard game, with a point target of 200. Every student had a chance to roll 3 dice, add two of them, then multiply the result by the third. At one point, on my turn, I had 124 points and the kids had 199 points. We stopped and I asked what they thought would happen. They quickly saw that I had no chance to win.

Next, we drew a card from a playing deck. In this case, it was a 5. Based on that draw, we then played damult dice again, but players are only allowed to count points that are multiples of 5. To soften this additional constraint, we added a fourth dice. In other words, players throw four dice, choose two values to add together, and choose a third value to multiply by that sum. If the result is a multiple of 5, they can add those points to their running total.

For the next game, we chose a different card to determine the multiplication family for that round.

Multiplication families practice

At the basic level, this variation made the game a really good way for the kids to practice their multiplication families. To focus the practice, I had actually doctored the playing cards ahead of time and took out many of the cards (face cards, 10, aces, and 2s).  For other student groups, I might take out the fives or the threes, if they were already very comfortable with those families, or I might include the twos and tens if they needed more work there.

Their observations were great

During play, the students quickly launched into observations and hypotheses about how to make use of their new degree of freedom (the extra dice) to overcome the constraint. For example, they realized that the only multiples of 5 possible are where one of the factors is already a multiple of 5. That means (a+b) * 5, a+b = 5, or a+b = 5. They had different observations when looking for multiples of 6 and again for multiples of 8.

These conversations became a natural step into thinking about the factors of our constraint and, even better, the prime factorization.

Another possible twist

For the class, we chose one multiplication family for each complete play of the game.  In subsequent play at home, we've tested two other versions. In one version, the player starts her turn by drawing a card, then needs to score points that are a multiple of that card's value. This adds a further element of randomness to the game as each player potentially faces different constraints on their turn.

Alternatively, we drew one card to constrain the next two turns. In other words, each player faces the same constraint as the other, but the multiplication family can change between a complete set of turns. 

My recommendation is to be flexible and try the version that suits your players the best. I think the first version is my favorite for encouraging the players to really think about what is required to make a point value that is a multiple of their specific constraint, to make observations and hypotheses.

math games class: 24 game and connect 4 dice

First grade

As a warm-up, the first grade played tic-tac-toe. This was to prepare them for the main game of they day: dice connect 4.

Dice Connect 4
Two players alternate occupying spaces on this grid:

The constraint is that, each turn, the player first throws two dice and then can only select a space in the column that corresponds to the sum. The winner is the first to get four in a row (horizontally, vertically, diagonally.) In our version, the winning spaces need to be a chain of neighbors, in the sense that each space has 8 neighbors.

Second and third grade

In both older grades, we played the 24 game. I saw this mentioned recently on Benjamin Leis's blog about his own math club (maybe just in passing here, I thought he had a more extensive post about the game elsewhere, too).

We played the "war" variation:
players: 4 (when necessary, three players can have a ghost player, or two players can modify each round slightly)
material: deck of playing cards with face cards removed
set-up: deal out all cards to the four players, cards stay in piles face down.
each round: players all turn over the top card in their pile and then race to get as close to 24 using the values of the four cards and standard arithmetic operations. In our version, we allowed addition, subtraction, multiplication, and division, but we didn't insist on using all four cards. When we played, someone would call out a value they could make and the number of cards, say "21 with 3 cards." The other players would then have a minute to try to improve that, either getting closer to 24 or making the same value with more cards.

The player who wins each round collects the four cards and puts them on the bottom of their pile, face down.

Ending the game: when any player runs out of cards, the game is over. The player with the most cards wins. Note: you can also keep going until only one player remains, but this wasn't suited for classroom play.

Playing at home: As mentioned above, you can play the game with 2-4 players. With two players, just have each player turn over their top two cards. With three, we used a dummy/ghost pile of cards to make sure each turn had four open cards. Another variation that is flexible is to keep all cards in one pile, turn over four each round, and then the player who is fastest or has the best result collects them. After the main pile is exhausted, the winner is the player who collected the most cards.

If no one else is available, this can also become a practice or puzzle session. Just turn over 4 cards and see how close you can get to 24. Write down your equations!

math games class

First Grade

Warm-up
As a very quick warm-up, I asked the kids about days of the week: for example, what day follows Monday? what day precedes Friday? This was more challenging than I expected and I think need to practice.

First Game: number line squeeze
We made a number line with integers marked from 0 to 20. To start the game, I think of an integer in this range. Then, the kids take turns guessing. With each incorrect guess, I will give them an extra clue: "my number is smaller than [the number they just guessed]" or "[guess] is larger than my number," etc. When they get this clue, I had them draw a bracket on their number line, ( when their guess was too small and ) when their guess was too large.

This is a quick game to help encourage their familiarity with the number line and some logical inferences. For example: if they already know that 5 is too small, what do they know about 3?

We also took turns where they picked the secret number.

There are a couple of extensions we can play in the future:

1. Allow them to ask about ranges, for example: is the number between 5 and 11?
2. Ask "or" questions: is the number smaller than 3 or larger than 15?
3. Ask calculations: is the number equal to 3+4? 

Don't make a triangle
Our second game was a fun one from Math4Love. Don't make a triangle. The idea is simple: draw a bunch of starting dots in black (we used 6). Each player has a chosen color takes turns connecting the original dots. You lose if you form a triangle with all sides of your own color.

This was a fun game which, as you can see, leads to some pretty pictures. Especially when we play this back to back with tic-tac-toe, it leads to an interesting question: can the game end in a tie? Why or why not? Subtle stuff for first grade.

2nd and 3rd grades

Our game this week was related to Dice Miner from last week and comes from friends at Logic Mills in Singapore. As with Dice Miner, players start with 11 tokens of the same color and we play on a grid with columns labeled 2 through 12. This time, though, the grid has 8 rows, 3 players use the same grid at one time, and we had three extra markers (we used loom bands).

On each player's turn, they throw 4 dice, then choose to group them into two sets of two. They take the sum of each set and place a temporary/common marker (the loom bands) in the columns. After marking their territory with the temporary markers, the player has a choice: throw again or claim their progress by placing one of their tokens on the spaces they've reached. If they throw and can't place a temporary marker (there are only 3 at a time!) then they lose all of their progress for the turn.

When a player has reached the top row of a column, they claim that number. The first player to claim 3 numbers wins.

For those who like this type of classification, the game has a territory conquest objective played through a race-against-each-other mechanic.

Here are some pictures, mid-play.

In this picture, orange has capture column 7. Now, any two dice that sum to 7 are safe for all the players and orange is 1/3rd of the way to victory.

A bit later, we see that blue is close to claiming 9 and pink is close to claiming 6, but orange is close behind there.

In a different group, white is close to claiming column 8, but a good turn from yellow could disrupt that plan.

Dice mining (grades 2 and 3 math)

Notes for Grade 1 will come later

This week, we continued our run of dice games. This is a simple game we found in Marilyn Burn's About Teaching Mathematics. She calls it Two Dice Sum Game, but I like Dice Miner better (to pair with Dice Farmer, of course). Here's how to play:

1. Students make a number line with slots for each integer 2 to 12
2. Everyone gets 11 blocks or other counters (we used unifix cubes which were nice for stacking).
3. Players put their counters on the numbers, distributing them in whatever way they want. In particular, they can put more than one counter on a number or none on a number.
4. Everyone takes turns rolling two dice (normal 6-sided). Players can take one counter off their board for the sum of the two dice. For example, if I have 3 counters on 7 and roll 6+1, then I can take one counter off and now have 2 counters remaining on 7. If I had no counters on 7, then I would not take any action.
5. The first player to have all counters removed is the winner.

Any activity with construction cubes is bound to create opportunities for other mini-conversations:

• using the cubes as a ruler to make the number lines
• comparing how many cubes one person has to another by measuring lengths (before we made sure everyone had exactly 11)
• using the cubes to form a ruler to measure other things (our whiteboard, a table leg, a friend)
• making a pattern with light and dark colors from the cubes

As to the main game, there is some basic arithmetic practice through all the addition, but the real interest lies in trying to figure out how best to arrange your counters at the start. All the kids had their own hypotheses, but it was interesting to see these highlights:

• One student realized that 7 was the single most likely result and put all his cubes on 7 (this did not win that round)
• Most students started with a uniform distribution, one cube on all numbers
• There was a surprising amount of enthusiasm for 2 and 12, with many students initially playing multiple cubes on those extremes
• One student tried to be tricky and put cubes half on 6 and 7, presumably planning to take them off if either number came up.

Toward the end of the class, there were two comments that were really interesting. First, one student suggested rolling the dice many times, recording the results, and seeing how often all the numbers came up. This would then inform his strategy for how to distribute the dice.

We asked the students how many times they would have to roll. Would 1 or 2 rolls be sufficient? No, everyone was sure that wasn't enough. What about 25 times? One student pointed out that there are 11 slots, so 25 times is only an average of 2 per slot, so that didn't feel like enough to get a good sense of the "true" answer. Most students were eager to see for themselves, so this became their homework.

Homework

1. test the distribution by rolling two dice 100 times (or more) and tallying how often each number comes up as the sum of the two dice.
2. Play Dice Miner 5 times with your friends/parents at home. Record your initial starting position for the cubes, how many rolls to take them all off and who won each round.

Seasonal math and dice, dice, dice

who: J1, J2, J3
what did we use: mangosteen(!), dice, a checkers set.

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Queen of mathematical fruits

My condolences to everyone outside of Thailand, you are current missing some delicious pre-school math: Mangosteen!

The mathematical secrete is on the bottom of the fruit. The flower shape on the bottom has petals that correspond to sections of the fruit inside. This one will have 7 sections

Just to trick you, this is a view from a different one, with 6 fruit sections.

J3 has been having a lot of fun counting the "flower" petals, opening the fruit, then counting the segments to check.

J2 has been using these to practice subitizing (recognizing number groups without counting them). J1 has just been gobbling down the mangosteen.

Dice Games

A dotty dice puzzle
In school for the 2nd and 3rd grades, we introduced a tic-tac-toe variation using dice where the winner is the first player to make a straight line that is a multiple of 5 (more notes here). For homework, the kids played this game with other multiples, in our case 6. During one game with J1, there was an interesting scenario that I saved to discuss with J2. The board has the numbers as marked and the player has just rolled a 4. Where should you play (you can add 4 to any cell of the grid, remember that you don't want your opponent to be able to make a multiple of 6 on their turn):

Race to 20
For J3, we modified one of the games that we had played in the 1st grade class last week. In this game, two players race to be the first to 20 on a 100 board. In this version, both player has a token/marker of their own. For each move, the players roll 2 dice and choose which value to move, then count up their position on the board. First player to 20 or more wins.

The strategy here is very simple, but it was a good game for J3 to start learning about dice and the 100 board. Once she got the point of the strategy, though, she had her own idea:"I'm going to add both dice together and move that amount!"

Snakes and Ladders experiment
J1 and I tried an experiment with snakes and ladders: a race between two different teams.
Team 1: a single marker that has to move according to the roll of one hexahedral die.
Team 2: two markers, the player can choose which one to move on each turn.
The winner is the player who gets a marker to 100.

Which team has an advantage? How big is that advantage? What if you play so that the winning marker has to land exactly on 100 (otherwise bounces back)?

Knights Move

SolveMyMaths recently tweeted about Knights Move (aka Razzle Dazzle.) Since we are always looking out for more/new games to play, it sounded interesting. Also, we could repurpose existing material to play. The game is a cross between chess and basketball. Below, see J1's set-up:

Yes, we are using icosahedral dice as the balls. Also, the chess pieces on the side are spectators, they aren't part of the action (since this is played on a 7x8 board).

One of the most interesting parts was our ability to experiment with rules. Since we didn't have a definitive version, we felt free to try slight modifications. In particular, we experimented with:

• Passing: allowing multiple (chain) passes or just a single pass on a turn.
• Defense: moving into the 8 squares around the ball holder forces them to pass on their turn
• back-and-forth passes: player who passed the ball has to move again before they can receive the ball (we turned the checkers upside down to indicate they couldn't receive a pass). 

Mystery Numbers and a Tricky Pattern (math games grades 1-3)

who: Baan PathomTham grades 1, 2, 3
when: Tuesday morning
where: in school
what did we use: a pack of playing cards, some cube dice

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While our second time with the older kids, this was actually our first time with the new 1st grade students since both were out sick last week.

Grade 1

Warm-up
This time, we wanted to play a tricky pattern game involving various ways to write numbers. To start, we all practiced writing the numerals 0-9 using Thai and arabic numbers. To help show the commonality of value, I also included little dot pictures for each. After that, we played a game where I started a sequence, then each child tried to guess and draw a number that would continue our sequence. Here is an example. The guesses that are crossed out are values that don't fit my rule:

This is a re-creation of the actual play. Obviously, the kids have better handwriting than I do

This is a strange game where wrong answers give you more information than correct answers. See if you can figure out what is allowed to come next.

Our friend: The 100 board
We had used 100 boards last week, but I realized that the kids would need a chance to really examine them, notice, and ask questions about what is there. Here are the types of things we discussed:

• what is the largest number?
• what is the smallest number?
• how many spaces are there across the top? what about down the side? what about the long diagonal?
• when we go down, do the numbers get smaller or larger?
• when we go to the left, do the numbers get smaller or larger?

Another dice challenge: Race to 20
Armed with our 100 board, we were ready for a new dice game. I think this was my own invention, but I spend so much time reading about games and mathematical activities, that it might come from someone else. If you know who I should credit, please tell me in the comments.

# of players: 2
tools: 2 dice, a 100 board, a small object to mark our position
Game play: first player rolls both dice, chooses one value and adds that to the accumulated sum, moving the marker up to our new position. The next player takes a turn in the same way.
Winning: the first player at or over 20 wins .

Obviously, this gives some adding practice, along with reinforcing the 100 board as a tool for visualizing. After playing a couple of rounds, though, some strategy emerged when we got over 10 as we started to notice that adding smaller values was better than adding larger values. This was a slightly tricky observation since strategic errors don't necessarily lead to a loss, they just increase the probability that the opponent will win.

Most components can be varied: increase the number of dice, use non-cube dice, change the target from 20, increase the number of players. The last one, however, is a bit dangerous as it makes the game even less strategic, though I think that could also be fixed by creating partnerships for a 4 player game.

Homework
Homework this week is to play the Race to 20 (or a higher target) 5 times with a parent or friend and to test their parents with the original pattern rule.

Grades 2 and 3

Warm-up
We started with a couple of mystery number puzzles:

• Mystery couple 1: two numbers that add to 10 and have a difference of 2.
• Mystery couple 2: two numbers that add to 12 and have a difference of 4

Several students created their own puzzles to challenge me:

• Student mystery a: add to 16 and difference of 7. I told them to figure out the product of their two numbers, I would write down the product as well, and they could come back and check me.
• Student mystery b: add to 20 and difference of 5. Good to see some fractions come into play!

We didn't prescribe any particular way to attack this warm-up challenge. For students who were guessing-and-checking and got a bit stuck, we did help them organize their information into tables to make things clearer.

Challenge Josh: make 31 strategies
From last week's homework, kids came prepared with their variations of the make 31 game. For simplified games with either 1s and 2s or 1s, 2s, and 3s, they got to choose the target value and whether to go first or second. To my delight, there were conflicting views in each class, where kids had the same target value, but disagreed about whether to go first or second. I paired up those groups and we got to test it out. In a couple of cases for each class, I accepted the challenge and played against the kids. The results were mixed, I scored my share of victories, but was defeated twice.

More dotty dice
Remember, this family of games is itself a variation of tic-tac-toe, so we have a 3x3 grid, roll a cube dice, and add that number of spots to a square in our grid.  The first version we played was for each square to be filled when it got 6 dots, then the winner was the first to get 3 filled squares in a row. As an alternative, we tried playing so that the objective was to get a straight line that added up to 20.

This version proved unpopular for two reasons. First, squares would get filled with so many dots that it was hard to tell how many were there. Unsurprisingly, that was exacerbated because not every player was very neat about their dots. Second, it was possible to overfill a row so that it already had more than 20.  This week, we introduced a couple of fixes.

First, instead of using dots, we raised the idea of using tally marks. As far as I can tell, this is the whole point of tally marks, so not a surprise that it comes in handy for this problem.

Second, we changed the objective so that the target was any multiple of five, with the restriction that all squares in the line had to be greater than 0 (none could be empty). This variation proved quite popular, so it became the standard for the rest of the class. When played to win, this provides a lot of practice adding since there are so many lines that need to be checked.

Challenge: what square was last played in the RHS version?

Homework
Play 5 rounds of dotty dice, either with the target being a multiple of 5 or, for a more challenging game, multiple of 6, 7, 8, or 9.

We learn to simplify and Dotty Dice (math class 2 Y2)

who: Baan PathomTham grades 1, 2, 3
when: Tuesday morning
where: in school
what did we use: a pack of playing cards, some cube dice

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While our second time with the older kids, this was actually our first time with the new 1st grade students since both were out sick last week.

Grade 1

Warm-up
To get started, we did some skip counting by 2 and counting backward from 20. For counting backward, we got a funny reaction when it came time to say 0. At this stage, neither student was willing/able to go below 0, but we did get to talk a little about whether there are numbers below zero. Next time, we will make this a more explicit part of the class.

Dice sums and re-roll game
Our first game is a simple one. All players have 2 dice (regular, 6-sided, for now). Players roll their dice, identify the numbers that are showing and say the addition number sentence. For example, if someone rolls a 2 and 3, they would say "this is 2, this is 3, 2+3 is 5." Next, they choose how many dice to re-roll, either all, one, or none of their dice. Once everyone has re-rolled, they again talk about the numbers showing, the addition fact, and then figure out which player got the largest sum.

We played a sample game to show them how it works and then split up, one student playing with Pooh and another playing with Josh. As we played, we would ask various questions about what they noticed and why they made certain choices. For example:

• How many sides are on the dice
• What numbers are on the dice? What is the largest, what is the smallest?
• What is the largest sum they could get? What is the smallest?
• What do they think of our choices? Did we play well or badly?

Variation
While we were playing, we noticed there are actually two slightly different versions. In one game (Version A), all players choose which dice to re-roll and then roll at the same time (or, equivalently, in secret). In Version B, players take turns deciding what to re-roll.

We only played as a 2 player game, but there are additional variations if we include more players. Again, they can re-roll simultaneously or in sequence. However, you could also add a points system based on which player has the largest sum, the second largest, etc.

Homework
We asked the students to play this game at home. They should play Version A and Version B at least 5 times (play at least 10 total rounds). Are the games the same or different? Do they use different strategies? Why or why not?

Grades 2 and 3

Warm-ups
As with grade 1, we began the older kids with skip counting warm-ups. We did two things slightly differently that I wanted to note. First, when we said that we would skip count by 7, several of the kids grabbed their notebooks with the times table on it. Instead of starting with 0, though, we started with 2. Second, I sat next to one of the students and had him secretly point to the numbers of the sequence on our 100 board while waiting for his turn. That also made it possible for him to predict what was coming next and to figure out the number he would say, which he thought was really cool.

31 strategy
The homework from last week involved playing a variation of the 31 game and investigating strategy. Almost none of the students ended up playing the original 31 version. Instead they generally played with 1 (ace) through 10 with a target of 71. When we talked about strategy, almost everyone had figured out that there was something interesting happening in the 60s, with several ideas that you wanted to make 60 on your turn, though there were some thinking 61 was an interim target. Everyone acknowledged there was complexity if you had already used up all of a particular number and there were some ideas about explicitly forcing that outcome, mostly looking at using all of the aces or other small values.

No one felt they had a clear strategy and no one was sure whether it was best to be first player or second.

I told them this wasn't surprising because the game is actually pretty complex. One of the ways we can get a better understanding is to make it simpler. We then talked about ways to make the game simpler (use fewer card values, have a smaller target) and then began to investigate.

• Ace only: the simplest game we wanted to study was to use only 4 aces. What is the largest target we could use? I joked in each class about playing to 1000. For all the reasonable targets (1 to 4) who wins, first player or second player? This was very easy, but I wanted to emphasize that it is great to start with something we really understand well as preparation for jumping into a more complex version.
• Ace and 2: an obvious next step is to add the next value card. Again, we talked about what the largest reasonable target is and then stepped through strategy for different targets. First player has a winning strategy if we target 1, 2, or 4, while the second player can win targeting 3. Are there any values larger than 3 for which the second player has a winning strategy? We didn't talk through all strategies for all targets, so this is part of the homework.
• Ace, 2, and 3: stepping up again. This time, we played a couple rounds with a target of 24. I was slightly surprised that this held their interest, even on the second round. Again, we worked a bit on strategies for different targets, but we didn't talk through it completely, so this is part of their homework. One student guessed that 17 might have a different strategy to smaller targets. 

Dotty Six (another dice game)
The basic idea for this game family comes from NRich, so I won't repeat the basic rules. In addition to the dotty six version, we played a dotty ten version where each cell of the board is only filled once it has ten dots (or tally marks) in it.

Homework (grades 2 and 3)

1. For the Aces+2s variation of 31, find a target larger than 3 for which the second player has a winning strategy.
2. Prepare to challenge Josh: for the Aces+2s+3s variation of 31, choose a target (larger than 4) and whether you want to be first or second player against Josh.
3. Keep making observations about the strategy for these games.

Dice Farmer (game)

who: J1 and J2
where: reception floor (having been deported from the kitchen doorway)
when: after lunch

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We recently got a pound of dice:

From the toy category: "dad uses kids as an excuse to buy something for himself"

Today was our first round of dice farmer (from Leftover Soup, rules below) and the three of us had a lot of fun. Actually, J3 stole a handful of dice from our reserves and had fun, too.

This is a fast game that both J1 and J2 found pretty compelling. In fact, J1 and I were in the middle of playing Munchkin, J2 was getting antsy, so I told him the rules and started playing on the side. J1 got so interested, he abandoned Munchkin mid-combat!

J2 kept winning, so that helped make sure everything was fun. They were both surprisingly tolerant when their dice "died" and pretty friendly about sharing favorite dice (as you can see, almost all have a unique color). One parent warning: on a hard surface floor, dice will bounce everywhere.

The basic rules are easy enough so this game is a low threshold activity, but the number of combinations made it computationally challenging for our players. During play, we started having conversations about the shapes, probabilities, and how to assess who was ahead. All of these will take some time for us to really develop.

Just before going to sleep, J1 asked me to promise to add this to the blog tonight.  That's how much he enjoyed it!

Rules

Equipment: A "herd" of dice of different shapes. We used standard D&D platonic solids + 10-sided.
Set-up: all players start with three 6-sided dice
Play: on each turn, roll all your dice. Any dice that come up 1 are dead and go back to the reserve. From the remaining dice collect sets that add up to sizes of dice shapes. You then add these to your stable.
Who wins: the first person to collect three 20-sided dice as part of their herd
Variations:

• Change the starting number and/or composition of dice. For example, I often started with 3d4 to reduce my chances of winning. Usually, though, I suggest all players start with the same configuration or establish a budget for # of sides.
• Change the winning condition. J1 got excited about requiring the winner to have 1d8 + 1d10 + 2d12 + 3d20. I wasn't around to see how this worked out.
• Use different interpretations of what it means to form sets that add up to a target number of sides. For example, you could require hitting the target exactly or, as we did today, set that as a minimum. Perhaps you could also get some amount of excess back as a rebate, though I usually considered those lost.
• Change the condition to die or add other scenarios (e.g., 1 = that one dies, 2 = asleep, so doesn't count for that round). 
• Add unusual dice configurations