Secret Numbers (Addition Boomerang variants)

We have been enjoying Mathpickle activities in our math games classes recently. This week, the 3rd and 4th graders will be playing with some of the more advanced Addition Boomerang variations.

During our planning, we came up with one extra pointer to tie the activity more closely with multiplication. Also, we had ideas for variations and wanted to record our notes so we can use them again in the future.

Tie with multiplication

In Gord's video explanation, he sometimes records in the center of each loop how many times that loop has been used. We emphasize this and write some related equations to help draw out the connection between the repeated additions in this activity and multiplication.

The first way we do this is by making tally marks inside the loop every time that branch is chosen. At any time, you can pause and write down an equation for the current total in a form

AxN + BxM = Total

where A and B are the values of the loops, N and M are the number of times each loop has been used.

Alternatively, we can show a "completed" round of throws by simply writing the number of passes for each loop in the middle and, again, write out an equation showing the total as the sum of two products.

Secret Number Variations

We start with a basic addition boomerang lay-out, either with 2 or 4 branches, both players (or teams) share a common set of addends and take turns adding on to a common running total. In our variants, the players choose and write down a secret number that helps inform their target for the game:

• Version A: players pick a number between 70 and 100. This is their target for the game and they win if the common total hits that value, whether the target is reached on their turn or their opponents turn.
• Version B: players pick a number larger than 15. They win if the total hits a multiple of their secret number. For example, if they choose 17 and the running total hits 51 (aka 17 x 3) then they would win. If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.
• Version C: players pick any number. They win if the total hits a multiple of their secret number that is larger than 60 (not equal to 60). If the total is a multiple of both secret numbers, then the player who chose the larger secret number wins.

Version B is, I think, the most directly playable.

Possible issues
I'm not sure how to deal with the case where both players choose the same secret number.

For Version A, it will be interesting to see what modification kids can find that will deal with the fact that it is very easy to miss any particular target. In the basic game, once the total is larger than your target, there is no hope of recovery. There are several ways to address this. I would be eager to hear any rule sets that kids create and hear about the experiences.

In Version C, I wonder if choosing 2 as the secret number is too strong a move?

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.

math games class catch up

It has been a while since I posted a summary of our math games, so this is just a quick catch up to summarize what we've been doing:

Grade 1

Addition war

2 players

pack of playing cards (A to 10)

Deal out all cards to both players and keep the cards face down in a stack. Each round, both players turn over the top two cards and add their values. The player with the higher sum wins and collects the cards in their points pile.

If there is a tie, those 4 cards are kept to the side as a bonus for the winner of the next battle. Repeat this with ties until there is a winner for one round.

After playing through the original stack, look to see who has collected more cards in their points pile. That person is the winner.

For a more challenging version, use face cards and assign values J = 11, Q = 12, K = 15.

Solo addition bridge

2 - 4 players (we played with 3)

pack of playing cards (A to 10 for beginner game, add face cards for extra challenge)

Deal out 5 cards to all players. They pick up these cards to form their hand. Proceeding clockwise, each player lays down one card from their hand, going twice around the group. Each player adds together the value of the two cards they played. The highest sum wins and collects all the cards played. That is one "trick."

After each trick, deal out 2 cards to each player to refill the hands to 5 cards.

The player who won the last trick is the first to play a card on the new round.

When there aren't sufficient cards to deal equally to all players, deal the hands equally (all players start each round with the same number of cards) and keep the remaining cards as a bonus for the player who takes the last trick.

We played with 3 players and, for the advanced game added 2 jokers to make a deck of 54 cards. Based on popular consensus, the jokers were assigned a value of 1,000,000. Interestingly, the extremely large value meant that the players were reluctant to play their jokers and, twice, both players kept them to the final trick so that the winner was actually decided by the higher of the second card!

Group addition bridge

4 players working as pairs (partners) with the partners sitting opposite each other

pack of playing cards (A to 10 for beginner game, add face cards for extra challenge)

Generically, play is the same as solo addition bridge, but each round the partners each play one card and the team that has the higher sum wins the trick. While there are still reserve cards in the deck, hands get refreshed up to 5 by dealing a single card to each player. The leader for each trick rotates clockwise so that everyone gets a chance to be first (second, third, and last) to play.

Second and Third Grade

Multiplication Blind Man's Bluff

3 players

pack of playing cards using A to 10 (A counts as 11)

One player deals a single card to each of the other players. They hold that card up to their forehead. The dealer announces the product of the two cards. Then, the two players try to figure out the value of the card on their own forehead.

Role of the dealer rotates after each round.

We played this as a cooperative exercise. To make it competitive, you can award points to the first player to get their card value.

There are two ways to make this more difficult. Adding face cards with made up values is one way. Instead, we had the dealer give one player two cards, add those, then multiply that sum by the value of the other card. 

Yet another step is to deal each player two cards, then multiply the two sums.

When playing this version at home, J1 came up with the idea of giving clues to figure out the value of the two individual cards. This was a really interesting activity because it got him to think about what characteristics help specify the two cards and which clues actually don't provide new information.

For example, if I know the sum of my two cards is 11, does it help me to know that I have one odd and one even number?

To make a standardized version, the second round of clues is to tell each player the product of their values. 

Largest Difference/Smallest difference

many players (at most 9 per deck of cards, fewer with advanced versions)

pack of playing cards A to 9

Deal out 4 cards to each player. They then form two 2-digit numbers and subtract the smaller from the larger. The player with the greatest difference wins that round.

For slightly greater challenge, deal out 6 cards (for two 3-digit numbers) or more (forming 4 or 5 digit numbers). Again, the aim is to form two numbers with the same number of digits that have the greatest difference.

For a much more interesting game, we shift the goal: now, we try to find two numbers with the smallest difference (larger minus smaller). After playing a bit, we had some good conversations about what the students noticed, what strategies they used, and whether there was always a unique answer.

Multiplication Pig (variation of addition Pig)

2 dice (we used 2d6)

2-3 players (or more, grouped into teams)

Players start with 200 points and try to work to 0 (or below).

Each turn, the player rolls both dice. If neither is a 1, they multiply the two values and add this to their score for the round. They can either choose to roll again or take their score for the round and subtract that from their cumulative score.

If two 1's are rolled, then  their overall score is set back to 200. If one 1 is rolled, then their score for that round goes to 0 and they lose their turn.

Variations come from varying to characteristics of the game:

- Start with 0 overall points and, each round, add the points for your round to try to break a target (practices addition instead of subtraction in forming the overall target)

- Add the two dice instead of multiplying (shifts the practice to addition instead of multiplication)

- Use dice other than 2d6, possibly more dice or differently shaped dice (note: the overall target and/or penalty conditions might require some adjustment)

Some PIG observations

Dice games are loud games, compared with card games. I think this is because the value of the dice is revealed to everyone at the same time.

Based on expected values, the optimal decision whether to keep rolling to bank the points for that round depends on how many cumulative points you have and your score for that round. However, we observed that the students chose to bank their points very early, relative to an expected value maximizing strategy. I think this is because their experience with the game causes them to over estimate the likelihood of rolling a 1 (or two 1's) and/or to underestimate how many points they can earn on a single roll because of the multiplication.

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.

Deep Tic-Tac-Toe and tangram initials (1-3rd grade math games)

Note: we did 1st and 2nd grades in parallel this week.

First Grade

Warm-up

Our simple warm-up this week was a little before/after game with the days of the week and the months of the year. Starting out, we had a little discussion (debate) about what day it was (Tuesday, at the time). Then, what came immediately before and what came immediately after?

For months, no one was quite sure of the current month, so we talked about that for a moment, then talked about which months would be coming up. We should keep repeating and integrating into some other games so the students can get more comfortable with the sequence of months.

Clap and pat patterns

Moving on from the warm-up, I asked if they could guess some clapping patterns, where I would either clap my hands or pat my lap. Too easy, they both exclaimed. Okay . . .
Pattern 1: I clap 5 times and then ask them what comes next.  More claps! (admittedly, this was easy)
Pattern 2: Alternate clapping and patting (C-P-C-P-C-P) and then what comes next. Again, pretty easy for them to continue C-P-C-P. It was actually a step trickier for them to continue if I ended my sequence on a clap (so their continuation goes P-C-P-C etc)
Pattern 3: Same deal, but pattern is C-P-P- C- P-P etc. At this point, they were clapping and patting along with me to make sense of the patterns.
Pattern 4: C CP CPP CPPP CPPPP CPPPPP etc. Now, here's where it got complicated! We did this one several times.

Next time, we will do these patterns visually as well as aurally.

Tic-tac-toe Game

We played a couple rounds of classic tic-tac-toe (also called X-O here in Thailand). As we played, I asked them questions about what they noticed:

• how many spaces are there on the game board?
• how many lines do we draw to get those spaces?
• What shapes are the spaces?
• When we draw the board in the usual/lazy way (2 horizontal lines, 2 vertical) do all the spaces have the same number of border edges?
• If we made more lines, how many spaces would we get?
• At the end of the game, how many Xs and Os are there? Is it the same number for both?
• Do they prefer to go first or second?

We had several reasons for playing. The first is to encourage their habits of noticing and wondering. Even in such a simple game, there are a lot of mathematical things they can see and talk about. Our second reason is to prepare for the huge number of tic-tac-toe variations that we can play with number recognition, arithmetic operations, and more involved strategy.

A colorful pattern

We continued our exploration of the 100 board and the bead abacus with a coloring activity. This time, we wanted to color the even numbers (multiples of 2) and not color the odd numbers. For both of the kids, though, they preferred to use different colors for the two types of numbers rather than leave anything entirely plain. As they worked on each new number, they used a 100 bead abacus to figure out whether the number was an even or an odd.

Before beginning the activity, I asked if they had a guess about what pattern would result at the end. There seemed to be a consensus that we would get a checkerboard pattern. After working for a while on the first row, though, one student suddenly had the idea that the evens and odds would be in alternating columns. An interesting conjecture!

I promise pictures of the results next week.

Homework

Play X-O 3 times with a parent or older sibling and finish coloring their 100 boards.

Second and Third Grade

Warm-up

We used a warm-up game similar to last week. When someone has two secret numbers, can we figure out their values if we know the sum and the difference?

Tangram intro

In class, we worked on classic tangram rabbit and fox outlines. This was a really good exercise in paying attention to detail. For most of the class, students would claim to have a solution and we would point out something in their outline that didn't match the target (for example, a horizontal line where it should be on a diagonal or vice versa).

As with tic-tac-toe for the 1st graders, there is a lot to notice and wonder about in the simple tangram puzzles. For example:

• How many pieces are there? What are the shapes, what are the sizes?
• If we are using only some of the pieces, can we make the same shape in different ways? Trying to make different types of right isosceles triangles is one version of this.
• Which edges match exactly?
• What angles can we match up?

Homework

Make their initials with the 7 piece tangram. As many versions as possible and, if they want, try out other letters/other shapes.

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.

Math is all around (proof without words)

Ok, I can't resist writing a little explanation. These are pictures of activities from the last 2 months that never quite made it into blog posts or, if they did, I just like the pictures and wanted to show them again.

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A fractal tree made from rummy king tiles (part of our Natural Math/Moebius Noodles Multiplication explorers course):

Another fractal tree:

Is it an alien or a self-similar bug? How many body segments would we need to draw for the next smaller level?

Family Tree fractal:

I think this was a substitution fractal, again, Multiplication Explorers:

Halving sequence, another activity from the Multiplication Explorers course:

This little piece was 1/16536th of the starting square (2^-14)

How many animal hybrids can we make?

Many ways to make a whole. How many ...?

Even in the round:

A version of Pascal's triangle made into a puzzle. I think we made all the cells mod 10. Hand cut!

Some pictures we saw at an exhibit in one of the halls at NASA in Houston. We did our own calculations and didn't agree with their claims about how many stamps or how many coins were used.

Finally, a couple of nice magic squares, fraternal twins or identical? I can't take credit for this, but don't remember where I saw it (prob on twitter):

Examing a graph and 35 game (math class 12)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Weekend outing

Our school went to a play over the weekend, somewhat about chickens. Unfortunately, there were no mathematical chickens, but maybe someone will draw one this week?

Dice turn into a graph

The homework last week was to play damult dice with subtraction, targeting a score of exactly 100. We had discussed some observations about the game and had left open the observation that a lot of scores seemed to be multiples of 3. Over the week, I made a little pencilcode program to show the scoring distributions and thought this would be another good focus for the kids to make observations, conjectures, and pose questions. The distribution we discussed today was the max scores distribution, so you can play along here: Scoring histograms.

Once again, this turned into a good conversation:

• Why are some bars red and some blue?
• What is the meaning of the two rows of numbers on the bottom?
• How does this relate to our dice game?
• In the top row of numbers, why is 2 the smallest? Why is 72 the largest? Why is 3 missing?

For homework, we gave everyone a copy of the possible scores histogram and assigned them to write an observation or question about the graph.

35 Game

This is another card game. I'm not sure of the source, but will link back here when I find it.
Material: Pack of standard playing cards (J=11, Q=12, K=13), manipulatives for adding (we used 100 charts) are also helpful.
Set-up:Deal out all cards to the players. In the first version, it doesn't matter if people have different numbers of cards. Players don't look at their cards.
Play:Going clock-wise around the table, the player turns over one card and adds its value to the running total, if that result is less than 35. If the sum is 35, the player scores a point and the running total resets to 0. If the sum would be more than 35, then the player subtracts the value of their card from the total instead of adding.
Example: if we have gotten to a total of 26 and the next player puts down a 9, they would get a point. If they turn over a 10 or face card, then they subtract that value from 26.
Tips:

• When the card value is too large, encourage the kids to state the addition equation anyway. For example, "30+10 = 40"
• Before playing a card, ask what value would take you to 35.
• Occasionally ask for values larger than 13 and ask if there is a card that would let you get to 35 on the next play.
• For a more involved version, deal equal numbers of cards and let player's choose which card to play on their turn.

Homework

1. Write down at least one observation about the damult dice possible scores histogram
2. Play the 35 game with a parent or sibling until someone gets to 5 points and wins

We noticed . . . (math class 11)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Warm-up patterns Today, we started skip counting by 4 (first grade) and 6 (second grade). We are extending the damult dice game today and the kids benefit from as much practice as they can get.

What did we notice (dominoes)?

We split the class into pairs (the extra student went with P) with the following instructions: talk with your partner about what you noticed or discovered about the dominoes puzzle, write down one sentence and be prepared to share with the rest of the class. I assigned the partners and intentionally broke up traditional pairs. I made sure to have some boy-girl mixing and also paired students with different levels of historical engagement. Many kids had questions about what types of ideas they were meant to be discussing, so we gave them the following prompts:

• What were your strategies for solving the puzzle?
• What did you find for the smallest side sum?
• What did you find for the largest side sum?
• How many solutions did you find?
• How did this compare with the triangle puzzle? What was the same, what was different?

After 5 minutes (longer on request by the first graders who were still writing), we regrouped to discuss.
With this structure, we had a really good conversation. There were a lot of ideas shared and some good debate about each of the points. Some highlights:

1. Some students claimed to solve the puzzle by randomly ordering the paper dominoes we made last week. They were surprised when I praised this approach as they were expecting to be criticized. We discussed that the whole strategy was actually: make manipulatives, create a possible solution, check whether the solution is valid, reorganize into a different configuration. P emphasized that this is widely valid, if they are working on something and get stuck, one thing to try is to use a tool to help them, not to suffer in confusion.
2. For each answer we discussed, there were usually multiple ideas. This gave kids an opportunity to talk about what they found and how they found it.
3. Comparing with the triangle puzzles was a very rich vein for conversation. The contrast helped them identify structure in the puzzle that, otherwise, would have been either too subtle or too obvious to mention. For examples, the dominoes force the numbers to sit in pairs, the square arrangement has 4 sides vs 3 for the triangle, the domino puzzle had 8 numbers vs 9 (or 6) for the triangle puzzles, and there were some repeated numbers in the domino puzzle while the triangle puzzle had distinct digits.

What did we notice (damult dice)?

We repeated the discussion process again, this time based on playing damult dice. Again, this was a really good discussion, covering a range of ideas:

• the smallest result possible (2= (1+1) x 2)
• the largest result (72 = (6+6) x 6)
• strategies to make the largest result with any given roll
• strategies to make the smallest result with a given roll
• why do we get so many multiples of 3? (more on this below)

"New" Game: exact damult dice

We turn again to Math4Love's page on Damult Dice. In the comments, there were many suggestions for extensions to the basic game. The one we introduced today was simple: you have to hit the target number of points exactly and you are allowed to subtract the result of your dice roll instead of adding it. The basic calculation remains the same: roll 3 dice, add two, then multiple the sum by the third.

This is a very simple extension, but it makes the game much richer. Gone is the simple strategy of accumulating as many points as you can on each roll, replaced by something a lot more subtle. I'm looking forward to hearing what the kids find.

Food for thought: Multiples of 3

Playing damult dice last week with my own son, we noticed that maximizing the result often gives us a multiple of three. The challenge: why is this the case and how often do multiples of three actually occur?

Dominoes and Damult Dice (math class 10)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Warm-up patterns

Today, we started skip counting by 3, 4, and 6. This was because I wanted to play a game involving multiplication and I wasn't sure whether the kids were ready. They are familiar with skip counting and understand the linkage with multiplication, but clearly need more practice to become fully comfortable.

New Game: Damult Dice

The new game comes from Math4Love. Like NRICH, they are a consistent source of good material for our activities. I wanted this to be a more structured game than usual, so we established the following set-up:

• Class is split into two groups (3 and 4 students)
• One player in each group rolls three dice
• that student chooses two dice to add together. The third is for multiplying
• By skip counting, the team works together to multiply the two numbers.
• that result is their team score for that round and gets added to their total.

This game worked very well. It was especially encouraging to see them thinking about which combinations would yield the largest result and remembering prior results.

Old homework: The Triangle Puzzle

Homework from last time was to work on the triangle puzzles. I was curious to hear what they found. Nearly everyone found a solution to at least one of the sizes (6 gaps or 9 gaps), so they all got to share something about what they found.

• For the 3 space triangle, were there any solutions? No one had really thought about this, but during the discussion a couple of second grade students realized that all the numbers had to be the same for the sum of sides to be the same. I let them share their ideas about why.
• For the 6 space triangle, students found answers with sides summing to 9, 10, 11, and 12. 
• We talked about why 9 was the smallest (6 has to be linked with numbers at least as big as 1 and 2) and why 12 is the largest (1 has to be linked with numbers at most as large as 5 and 6).
• For the 9 space triangle, students claimed to find answers with sums 17, 20, 21 and 23. For an extra investigation, they can confirm these and try to find solutions with sums 18, 19, and 22.

For the 9 space triangle, I noticed a relationship between two of the students' answers, so drew these on the board:

Admittedly, I made some adjustments to highlight the relationship. You see that the numbers in corresponding slots add up to 10, right? In a sense, these are complements. If
you look closely, there is one other relationship, but I'll leave that for you to discover.

A dominoes version

I was lucky to find a direct complement to our triangle activity on NRICH: 4 Dom. This time, the challenge is to arrange 4 dominoes into a square with three numbers on each side where the sum on each side is the same. We got the kids to make their own dominoes by cutting out and coloring little strips of paper. That gave them a hands-on tool to explore three questions, all assigned for homework:

1. Allowing sides to have different sums, what is the smallest sum you can make with these 4 dominoes? How do you know it is the smallest?
2. Allowing sides to have different sums, what is the largest sum you can make with these 4 dominoes? How do you know it is the largest?
3. What arrangement makes all the sides have the same sum? What is the sum? How many solutions are there?

Homework

We gave them four pieces of homework. This seems like a lot, but the game is very short and most of the kids answered several of the dominoes questions already in class:

1. Damult dice: play the game with someone in your family, first to 200 wins. For each roll, write down the equation you are calculating, for example (6+1) x 2 = 14
2. Dominoes puzzle: answer the three questions listed above. In short, what is the smallest sum that a side of the square could have, what is the largest, what sum and arrangement works so that all sides are the same?

Congratulations for getting this far. Here is something pretty for your efforts. Please post in the comments any mathematical ideas this picture gives you!

Puzzling triangles and Pico/Fermi/Bagel (math games class 9)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Warm-up patterns

Some students requested we talk about the Fibonacci sequence, so I decided to start with some number patterns. If I were going to do these again, I would stop after the second and third terms and ask for a range of different ideas about how the patterns might continue.  Once we have 4 or 5 terms, though, it becomes pretty difficult to identify a non-obvious pattern that fits the available data.

The point we want to communicate is that there are many possible answers and the only condition is that they should fit the existing data.

Patterns we used today:

• 0, 1, 2, 3, 4, 5.... universally extrapolated as counting up by 1
• 3, 10, 17, 24, 31, .... best fit was counting up by 7 starting with 3
• 18, 14, 10, 6, 2, -2, -6, ... Subtracting 4, starting with 18
• 1, 2, 4, 8, 16, .... powers of 2 or doubling the prior element of the sequence
• 1, 1, 2, 3, 5, 8, 13, .... Fibonacci sequence

In the first grade class, we had an interesting conversation about possible continuations from 18, 14.
One idea was 18, 14, 12 which we explored a bit more and came up with the following:

• 18, 14, 12, 12, 14, 18, 24 .... the difference of the differences is +2
• 18, 14, 12, 12, 12, 12, 12, 12.... hits a floor and stays at 12
• 18, 14, 12, 11, 10.5, 10.25, .... the differences are getting halved at each step

Triangle Puzzles

Mathematics Mastery sent a Christmas card that inspired this exploration. As a basic set-up, you have a triangle with circles on the vertices and some number of circles on the sides. You have to place distinct digits in the circles so that the sum of the numbers on a side is the same for each side. You can see the pictures for triangles with 0 extra circles (only the circles on the 3 vertices), 1 extra on each side and, if  you look closely, 2 extra on each side:

The kids had a lot of fun with this puzzle. Here are some notes:

• 0 extra circles/3 total circles: Is it possible with three distinct numbers? If not, why not? Can you convince your friends you are right?
• 1 extra circle on each side/6 total circles: fill in with the integers 1 to 6. How many solutions can you find? How do you know you have found them all? Given a collection of 6 distinct integers, can you always find an arrangement that works? If not, are there any conditions that must be satisfied? Are there any conditions which are sufficient?
• 2 extra circles on each side/9 total circles: similar questions to the 1 extra circle
• non-equilateral triangles: with n-circles, when can you fit the numbers 1 to n into the triangle to meet the condition of equal side sums?

The non-equilateral version was suggested by Prim. She challenged me to work on putting 1 to 7 into a triangle with 7 circles: 3 on the vertices, 2 extra on one side, and one extra on two sides.

If you get tired of triangles, of course, you can explore square figures, polygons with more sides, and stars (put circles on the points and internal intersections).

Pico/Fermi/Bagel

This game was also a student request. I've talked about this game before (here), so won't bother to repeat the rules. Today, we did two rounds with 3 digit numbers, then let the kids play in pairs for a couple minutes. For playing in class, we found this a good game because:

1. It was fun!
2. Everyone could guess a number and stay involved in the game
3. Students got a chance to explain their thinking and try to reason logically through the evidence

We also experimented with having one teacher score guesses for different target numbers for individual students. This was workable, but it significantly reduces our ability to have a deeper conversation with the students about what they have learned from each clue and why they are choosing a particular guess.

Homework

The kids are assigned more exploration for the triangle puzzles:

• Find multiple solutions to the 6 space triangle
• Find a solution to the 9 space triangle
• Look for solutions to the 3 space triangle and talk about what you find

Advancing the calendar trick

I've written twice (first and second) about classroom experiences with a simple calendar trick that I originally got from Calendar Puzzles via Denise Gaskin's monthly newsletter. As happens so many times with these things, ideas from the kids make these activities into deeper and more interesting than I could have imagined on my own.

Tricked
On Tuesday, some of the first graders gave me their sums: 168 and 198. I immediately knew something was up. In the original calendar game, the square with the largest possible sum is the 23-24-30-31 square:

23	24
30	31

This has a sum of 108. I asked the students if they were sure of 168 and 198. They giggled, then the teacher smiled and told me she had checked it. What was going on?

My homework
I didn't have any immediate ideas, so I promised the kids that I would work on their puzzles. I told the kids that it was great to get my homework from them this time!

See through paper
One clue was that we were using a special calendar today and the paper was slightly see-through. This gave me an idea that the kids had turned the paper over and were seeing the numbers through the page with digits reversed. At first, I thought they were transforming 2s to 5s and vice versa, but was able to find 168 just by reversing digits.

How many carries?
For the original game, the crucial insight is simply that there are 7 days in a week and the calendar is organized into weeks. That means there is a simple relationship between each of the numbers in our 2x2 squares. Add a bit of simple algebra and you have an easy formula relating the upper left square of your 2x2 matrix to the sum (or, if you want to be fancy, a different formula relating whichever square you want to the sum).

For the reversed game, though, it isn't quite so easy. The relationship between the numbers can take one of several forms and is rather messy. I did manage to get 168. Can you?

But, I still couldn't get 198.

Two little helpers to the rescue
Last night, I "cheated" and asked for help. As J1 and J2 got ready to sleep, I asked what they thought their friend might have done to get 198. Their ideas from brainstorming:

• maybe the friend made an addition error
• maybe the friend also transformed 2s to 5s when reversing the paper
• maybe he summed a 3x3 square instead (which quickly gave rise to 4x4 and 5x5)

3x3 square? Interesting! Work through the algebra again and you can quickly see that there is (always!) a 3x3 square whose contents sum to 198.

Some further exploration, for you
More fun follow-on questions:

1. If the kids are allowed a choice of 2x2 or 3x3 section, but they still only tell you the sum and not the size of their square, can you still figure out which days they chose? Are there any conditions you might put on which month is chosen that allow you certainty in finding the square?
2. What if you allow 4x4, too?
3. Why stop at 4x4? What size squares are possible on a 1 month calendar?
4. If you make a year calendar instead, what sums are possible? If you are given the size and sum of a square, how close can you get to finding the source? In other words, how many squares have the same sums?

If you have other ideas, please let me know in the comments!

More calendar play and Nim variation (math games class 9)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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I can guess your numbers (again)

Since we had a long break, we decided to remind the kids about this calendar game. This time, we used a pretty calendar that was a gift from a friend recently visiting from Nepal.

We split the kids into 2 or 3 groups (varied by class) and had them choose a block of numbers to challenge me. As each announced their sum, I would write out the 2x2 square and then give them a sum to find. This has two purposes: first as practice repeatedly calculating and second to encourage them to look for patterns.

In the calculation practice, it often occurred that students would announce an impossible result (when all 4 squares are filled with dates, the sum has to be a multiple of 4). I think they were almost as surprised to be told that they had gotten the sum wrong as when I told them their squares!

Looking at patterns, we talked a bit about their searching strategies. If I told them 24, then they knew it they didn't need to examine any square that had a 20 or larger because that would be too big. One student made an observation that the diagonals of the 2x2 square have the same sum, so he started working on the idea that cutting the sum in half would be helpful.

We also explored this question: can 21 ever appear as the sum in one of our squares? Students were quick to see that it could be in a 2x1 square (days 10 and 11), but had to do a lot of searching to see that it wasn't possible on their particular month. For some students, we started looking at what numbers do appear as possible sums. Continuing this way will eventually help them realize how I can "magically" determine their starting grid.

Magic 1089

We showed them another surprising result. Each student writes down a 3 digit number with all distinct digits and digits in decreasing order. If you want to follow along at home, 211 doesn't work because the digits aren't distinct while 354 isn't allowed because the digits aren't in decreasing order, while 921 is ok. I will use that to illustrate the trick
Step 2, reverse the digits and subtract. For our illustration, 921 - 129 = 792
Step 3, reverse the digits of the result and add to the result. 792 + 297 = 1089.
Try it with your own number and . . . get 1089. Can you figure out why it works?
Some observations from the students: the middle digit after step 2 appears to always be a 9. Also, the numbers 792 or 297 seem to appear frequently in step 2.

Nim Variation: add to 100

Our new game this week, for the second graders, was to play a NIM variation on the 100 board. Two players take turns adding an integer from 1-20 to an accumulating sum. The player who gets exactly 100 wins.
For this game, we used 100 boards and buttons to record our sum.
We will play this game next week with the first graders who spent more time on the calendar exploration and didn't have enough time this week.

Homework

Try 5 different numbers and check to see if the 1089 trick always produces the same answer. Try with 2 digit numbers as well. What is the result then and is it always the same, no matter which numbers you choose to start? Parents: encourage your kids to also try out 4 digit or longer numbers if they want and to make conjectures about what is happening to make this work.

Calendar Tricks and Break the Bank (math games class 8)

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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As our warm-up today, we played calendar games (one for the 1st grade and 2 for the second grade).

I can guess your numbers

Source of this game is Calendar Puzzles which I read about in Denise Gaskin's monthly newsletter.

Choose one month on the calendar. In my examples, I'll use January this year:

Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday
				1	2	3
4	5	6	7	8	9	10
11	12	13	14	15	16	17
18	19	20	21	22	23	24
25	26	27	28	29	30	31

Without telling me their choice, the children select a 2x2 square with 4 numbers and add them all together. They tell me their sum and then I 'magically' know which square they chose and write it down.

For example, they might say 44 and then I write down:

7	8
14	15

Homework: to think about how I know which numbers they chose.
They should play with some examples and look for patterns. In particular, what is the smallest sum they can make? What is the largest sum? Do either of them depend on the month? In a given month, if they form as many sums as possible, do they notice something about all of the sums?
One point we did discuss was whether the 4 numbers are related. Because weeks are organized in rows, we know that any square has the following pattern:

N	N+1
N+7	N+8

Parents: using concrete examples, ask your children to explain this pattern. Why are the numbers on the lower row 7 greater than the upper row?

Determined Determinant

We only showed this next trick in 2nd grade. I'm not going to go through all the details since the kids wanted an opportunity to try to amaze their parents. Here's what you'll be asked to do:

1. As before, choose a month on the calendar
2. Without telling your child what you've selected, choose a 2x2 square with 4 adjacent numbers
3. Now, multiply the bottom left by the top right
4. Multiply the top left by the bottom right
5. Subtract your second product from the first
6. Don't tell your result, but tell your child that you are ready
7. They will tell you the answer you got

Just to make this a bit more clear, using our example from above:

7	8
14	15

You now calculate $14 \times 8 - 7\times 15$, but don't tell your child what you are doing. They will tell you the answer.
Homework: Again, try to figure out why this works. Play with patterns, look at parts of the calculation. Have fun practicing multiplication and think about different ways to do the calculations.

New Game: Break the Bank

For the first time, we are playing with dice. This game comes from math 4 love.

1. Roll your die and enter the number in your grid
2. Repeat 6 times for the 2-digit game, 9 times for the 3 digit game
3. Add up your 3 numbers
4. If your sum is 100 or greater (for the 2 digit game), you get 100 points. If it is 1000 or greater for the 3 digit game, you get 1000 points
5. If your sum is lower than the threshold, you calculate your points as Threshold - sum
6. Player with the least points wins

Homework: Play this game as often as you can. Think about these other questions: what is the largest sum you could get? What is the smallest sum you could get? If you knew all of the numbers before entering them into the grid, what is the best sum you could achieve? Can you always form numbers that stay under the limit?

Observations

The children really enjoyed these tricks and games. As we were playing with them, there were many opportunities for interesting little discussions and ideas from the children. As a result, these activities were much more engaging and stimulating than we might have guessed from the written explanation on the page.

Example: Nikki's idea for 7x15
During the Determined Determinants trick, Nikki needed to calculate 7x15. She got several pencils out of her pencil case and said:

• here are 7 pencils
• this pencil is 15
• two of them are 30
• Grouping them in pairs, I can skip count by 3, so 30, 60, 90
• I have one pencil left over, so I need to add another 15
• Together, I get 105

P asked whether someone taught her this technique. She said that she discovered it on her own.

Example: I know I've busted!
Playing Break the Bank, Tanya exclaimed: "Oh no, I know I've already busted!" Looking at her grid, she had five numbers filled:

6	4
2	5
	6

Jin and Ji Ping were next to her, so I asked the there of them: "Are you sure Tanya has busted already? How do you know?" Part way through the discussion, one of them said: "no, this just adds up to 95, so it isn't busted." They thought for a bit more and then realized that the next digit would add at least 10, so, busted indeed.

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

How do you know? (talking math with your kids)

who: J2
where: in the car
when: this evening, driving home from a picnic

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Yep, in this part of the world it is perfect picnic weather. I would have taken a picture, but

I was driving, so I didn't hear the full conversation, but what I caught was:

J2: Yes, 384 is a multiple of 12
P (mommy): How do you know?
J2: Because 384 is 192 + 192 and 192 is a multiple of 12.
P: How do you know?
J2: Becuase 192 is 96 + 96 and 96 is a multiple of 12.
P: How do you know?
J2: Well, it is 8 x 12 . . . also, it is 48 + 48 and 48 is a multiple of 12.
P: How do you know?
J2: 48 is 24 + 24 and 24 is a multiple of 12
J0: how do you know?
J2 (exasperated): Daddy, everyone knows 24 is equal to 12 x 2!

So, I'm still not sure how he knew that 384 was a multiple of 12. Surely it wasn't really through the decomposition in this conversation . . .

While "How do you know" is probably overused in this conversation, I think it is a good question to have as a parenting habit. You can see we fall back on it when very tired. Also, the kids expect to hear it, so they are immediately ready to enter into that type of discussion.  I'm looking forward to the times when (1) they turn it around and ask us "how do you know?" and (2) when they use it on other people (each other, friends, teachers).

Math games 7

Who: Baan Pathomtham 1st and 2nd grade classes
Where: In school
When: after science and before lunch

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Skip counting warm-up

We've talked about warm-ups in several previous posts. Today, it proved to be a more substantial discussion than expected for a warm-up.

For skip counting, we go around the room with each child saying the next number in the sequence. This time, we started with some easy skip counting (2 and 3, based on requests of the kids in the class) and then something a bit more difficult (6 or 7).

Along the way, we saw some people getting stuck and knew that they would benefit from seeing some new strategies. When one child appeared to do a calculation very quickly, we asked them to explain their thinking. It turned out to involve splitting and regrouping:

We asked why they decided to split 6 into 4 + 2 and this was the explanation:

This discussion happened in each class and, in each class, we gave it the name of the student who explained it (Minnie and Jiping). For the rest of the day, when someone got stuck, their friends would offer encouragement and say "try X's method."

By the end of this warm-up, the second graders were excited enough that wanted to show off their technique for multiples of 9, so they spontaneously launched into that.

Parents at home: you can do the skip-counting warm-up driving together or at a meal time. Ask your child to show you how to get started.

If they are having difficulty with a calculation, first give them time to think through it.  Next, suggest tools they can use: write it down on paper, draw a diagram, use some objects. Finally, ask if they can break it into pieces that they know.

Ring your neck

Since some kids weren't in class last week, we began our games with a review of the new game from last week. It was a bit messy in second grade, but in first grade we got them to take turns explaining rules (one child explained one rule) and then they split into two teams to play a demonstration round.

Our intention had been to discuss strategy for this game, but the group dynamics didn't work well. These are the types of leading questions that encourage them to think about the structure and strategy of the game:

• Do you want to go first or second?
• If there is only one card left, do you want it to be your turn or your opponent? What about 2 cards? What about 3 cards? What about 4 cards? ....
• Is it "good" to take a card (do you expect to get points or lose points when you take a card)? What is the most points you could get? What is the least? What are all the possibilities?  What is the average?
• Do these things change as the game is played?

Parents at home: when playing games with the kids, ask them to explain the rules. As they are explaining, encourage them to show examples.  When you are playing this game, encourage them to add up their score each time they choose cards. Be patient when they need to take time doing the calculations and use the suggestions above when they get stuck.  When they aren't stuck, ask them to explain how they were thinking.

Finally, ask them the strategy questions.  It can be a fun discussion, especially when you don't know the answers!

Strike it Out

Players: 2
Material: paper and pencil
Set-up: Draw a number line and tick for each whole number. We gave the kids printed pages with number lines up to 20, but you should feel free to make longer or shorter number lines.
Start of play: the first player chooses a number and circles it
Each round: using the latest circled number, the player chooses another unused number, and forms a number sentence where the result is a second unused number. Circle this result and cross out the other two numbers.
Winning: the last player with a legal move wins.

This is an NRICH game and these pages show examples: student page and teachers' note.

Parents at home: This appears to be a more complex game as there are so many choices at each step.  When there are a small number of choices left, ask them to predict whether they can win. Also, ask them how many ways there are to form the number sentence and whether it makes a difference.

Homework

1. Play strike it out with parents, friends, and siblings
2. Play at least one of the card games we have taught this term.

Observations

First, we had a lot of fun playing these games with the kids this term. I think we found a collection of games that were fun for the kids and reinforced mathematical concepts appropriate for their current understanding. When playing the games, the kids were focused and engaged. Also, at least one of these (ring your neck) also has potential as a more subtle strategy game, though you have to reduce the value of the final bonus card.
When we weren't playing the games, we often found it difficult to keep everyone together for a discussion. This is something we will discuss with the teachers and think about strategies for next term.