## Wednesday, May 20, 2020

### Playful Mathematics Carnival 138

Welcome to the Playful Mathematics Carnival 138, the edition with its choice of theme song? (Techno or punk. I vote for the Misfits, I guess.)

I was frankly a bit surprised at how interesting 138 is. 138 is a sphenic number(the product of three distinct primes from the Greek for "wedge shaped"), but is special among the sphenics for being the smallest product of 3 primes, such that in base 10, the third prime is a concatenation of the other two: (2)(3)(23). What is the next such special sphenic? (Some of the pictures here were made with some Brent Yorgey inspired GeoGebra or David Mrugala inspired GeoGebra.

138 is the sum of four consecutive primes (29 + 31 + 37 + 41); which is the previous and the next? 138 the sum of 2 successive primes; which? And not only is 138 the average of twin primes, it is a number such that 6 times 138, is the center of twin primes, 827 and 829. Is there another number like that?

I had never heard of Ulam numbers: a(1) = 1; a(2) = 2; for n>2, a(n) = least number > a(n-1) which is a unique sum of two distinct earlier terms in exactly one way. 138 is an Ulam number... what are the two terms which make it? Which is the first counting number not an Ulam number?

Obviously, this month's edition is in the time of social distancing. So it has produced some stand out creative home mathematics. Nikky Case made an interactive explanation how the epidemiological models work. Eva Thanheiser wrote a post about numeracy in the pandemic time. Here's an applet if you're trying to model a gathering... or a classroom?

Denise Gaskins, the initiator of this here carnival here, has a post for those new to it on how to homeschool math and Peter Rowlett at the Aperiodical wrote about playful math at home

Whilst there you might check out the 2020 Lockdown Math-Off, with some really accessible entries this year. And you can submit until they run out of entries or they're allowed outside.

 Lily Cole
Math Art
Evelyn Lamb collected some of my favoritest math away from school resources. Cited there, Annie Perkins' #mathartchallenge (Twitter) is maybe my favorite thing ever. Here's the home (blog) for it. Brings together so many amazing projects from so many amazing artists and mathers. Like, Paula Krieg's origami firework on Day 53. Several have compulsed me to try to make things, like the Hitomezahi stitching from Day 14. Annie's going until she hits the magical 100!

Clarissa Grandi's #Maydala challenge has filled Twitter with lovely images.
 Clarissa herself
 Janet Annetts

 Japleen Kaur

 Miss Bowkett

 Paula Krieg

More mathy math
Henri Picciotto put together two amazing posts looking at wallpaper symmetries, and a catalog of pattern block Wallpaper symmetries; part I and part II. (Challenging, I think, because the blocks themselves are so symmetric.)

Luca Moroni pointed out Raffaella Mulas' self-illustrated article on Wild Mathematics. Quick mind bending read or just ponder the pictures.

Paula Krieg also pointed out this classic puzzle that Dr Olsen points out can be made from origami.

Karen Campe covers a wide variety of puzzles, from the jigsaw to the Catriona Shearer.

Simon Gregg started this thread about duck tiles, which continue to delight. He also published a sweet book on pattern blocks. If you just want the pdf, let them know and they'll donate the book to a local school.

Pat Bellew (of On This Day in Math) chases a white rabbit, running around My avorite Theorem and the arithmetic triangle. (I'm hoping that catches on over Pascal's someday.)

Justin Time
Not sure how to categorize this, ironically, but Justin Aion's post on alignment charts and teaching is amazing.

Games
Denise Gaskins, the initiator of this here carnival here, has a War meta-post with variations from preK through HS. I use several of these in a variety of classes.

Sam Shah collected many activities and games you can do over videochat with a class. Nia (@ihartnia) put together a Google doc of quarantine games: tinyurl.com/quarantinegamez

Dan Finkel updated his Horseshoe Math game with a theme song!

Ben Orlin shared 6 intriguing pen and paper strategy games.

Carole Fullerton covers a silly game called Penguins. Kind of a computation bingo variant.

Kent Haines' kids are chips off the ol' block, having invented Math Ball.

My retired colleague Char Beckmann developed many math games with our students and they've all been collected here. Materials, instructions, and video demonstrations.

This past semester I got to supervise three preservice elementary teacher/math majors in their senior projects to make math games. Char Beckmann's project! Our emphasis was to make instructional games that you could play with materials on hand or with minimal printing. Sam Bosma made a fun Guess Who variant for multiplication and division, Multiply Who. Maggie Eisenga developed Choose Your Path, a hit the target game with playing cards with some very cool discrete elements. Grace Gay increased the math content in Quixx, a dice game with some subtle strategy, to make Rolie Polie Operation Olie. All are upper elementary-middle school, and have how-to-play videos.

In Memoriam
Playful math lost two giants recently and I wanted to close with some remembrances.

John Conway was an all time great mathematician. Siobhan Roberts, his biographer, wrote two pieces for the New York Times, a memorial and a personal memoir. (Her book is excellent.) Sunil Singh said if you watch just one bit of Conway talking, watch this. James Propp is an unabashed groupie. Ivars Petersen shared a couple recollections. Pat Bellew remembered his impossible knot and Quanta covered its solution. Matt Baker remembered some of his lesser known results. I can't pick a favorite, but some of the most fun I've had with students is the rational tangle.

Don Steward was an amazingly creative and generous middle school maths teacher in England. Brilliant problem poser and visualizer and entirely low tech. Colleen Young wrote about some of her favorites. Jo Morgan wrote about her collaborations. Steven Cavadino shares an irrational triangle. Many people shared their favorite Don Steward task on Twitter. He did a sweet analysis of a Keith Richardson-Jones drawing that inspired some GeoGebra from me. Don's local paper wrote up a note on his importance to the community, and of course you should check the blog of the man himself, which he made arrangements to keep available and free.

On that somber note, we close the carnival.  Be safe, be kind!

But on your way out... maybe check last month's Playful Math at Life Through a Mathematician’s Eyes or their recent post, 10 math movie recommendations, and look for next month's at Math Mama Writes, or check her post about Pythagorean Triples for an online math circle.  The carnival's homepage is at the site of Denise Gaskins, the initiator of this here carnival here. Contact her for your chance to host!

### Rolie Polie Operation Olie, a Math Game

This past semester I got to teach a senior project class. Four preservice elementary teachers working on understanding math games, game design and making their own. Grace was fascinated by several games, but especially Quixx, a dice game. All the games were tested with kids, and went through multiple revisions and I'm really proud of their work and the games they made.

GUEST POST by Grace Gay

Rolie Polie Operation-Olie is a quick-playing mathematics game played with dice! It is a spin-off of the family dice game Quixx, in which it is a simple game to play but each decision is crucial. There is no downtime in between your turns, so there is a lot of catch-up as you always have a chance to gain from each and every player’s roll.

If you have kids or students who are working with their basic operations of addition, subtraction, multiplication, and division, then this game is a fun way to test their computations! The object of the game is to score the most points by filling in as many boxes in the three columns as possible while avoiding penalty points. Each player takes a score sheet and something to write with. Before playing, there is one basic rule that is similar to Quixx, which is boxes must be filled out from top to bottom in each of the columns. If you choose to skip any boxes, they cannot be filled out afterward.

Take a quick look at this scoresheet. The score sheet has three columns. One “up” column, in which the values that you will fill in the box will be in increasing order; one “down” column, in which the values that you will fill in the box will be in decreasing order; and one “choice” column, in which you are able to choose whether you want an increasing or decreasing column. In the sixth row of the columns, there is a 12 that is filled in. In order to fill in any boxes past the 12, you must fill in the 12. Lastly, in the last row of the columns, there is a lock symbol. In order to lock a row, you must have filled in at least 6 boxes and rolled either the attached 36 or 0. The “Operation column” is a place for you to write the operation that you computed in order to derive the number in the box.

A how-to-play: the first player to roll a 6 takes on the role of the “active player.” The active player would then roll all four dice. They now have two play options:

• The active player announces the first two dice that were rolled. All players may then (but are not required to) use any operation in order to combine the two dice to find a value that they can fill out in one of their columns.
• The active player (but not the others) may then (but it is not required to) take one of the other dice together with one of the first rolled dice and combine them, using any operation in order to fill in a box with the number corresponding to this found value in one of their columns.

Similarly to Quixx, there are penalty boxes, which must be crossed out if, after the two actions, the active player doesn’t fill out a box of at least one number. Each penalty box is worth -5 points at the end of the game. The non-active players do not take a penalty if they choose not to cross out a number.

Once all players are ready, the player to the left becomes the new active player and re-rolls all four dice. Then the two actions described above are carried out again, one after the other.

Now, If you wish to cross out the number at the very bottom of a column (up 36, down 0, and choice either 0 or 36), you must have first filled in at least six numbers in that column above, including the 12. If you cross out the number on the bottom, then also cross off the lock symbol directly next to it. This indicates that the column is now locked for all players and numbers of this column cannot be crossed out in the future rounds.

If a column is locked during the first action, it is possible that other players may, at the same time, also cross out the number on the bottom of the column and lock the same column. These players must also have previously filled in at least six numbers in that column. Also, the cross on the lock counts toward the total number of crosses marked in that column when you are scoring.

The game ends immediately as soon as either someone has marked a cross in their fourth penalty box or as soon as someone has locked two columns. Beneath the three columns is a table indicating how many points are awarded for how many crosses within each column (including any locks marked with a cross). Each crossed out penalty is worth five minus points. Enter your points for the three columns and the minus points for any penalties in the appropriate fields at the bottom of the scoresheet. The player with the highest total score is the winner.

I hope you enjoy this fast-paced mathematics game!

Handouts: gdoc & pdf

John's Postscript: This game is pretty complex on first approach. But the strategy is subtle, and repeated playings
have a good variety. So don't give up with this one! I actually like Grace's over the original. More streamlined, and
the extra operations add interest.

In addition to their own games, the teachers selected an already made math game to promote for classroom use.
Grace selected one of my all time favorites, Fill the Stairs. I have a post on it here, and Joe Schwartz has

### Choose Your Path, a Math Game

This past semester I got to teach a senior project class. Four preservice elementary teachers working on understanding math games, game design and making their own. Maggie wanted a game that used a graph/network as a playing board, and tried several options until coming up with this. All the games were tested with kids, and went through multiple revisions and I'm really proud of their work and the games they made.
GUEST POST by Maggie Eisenga

In my capstone with Professor John Golden I got to play and research math games. The big project of this class was to create our own math game that teachers would be able to use in their classrooms and/or parents could use at home! I created a game called “Choose Your Path.” This game is designed to help elementary students (focused more on 3rd grade but can work for other grades as well) work on their fluency in integer equations. I wanted to add strategy to my game as well, which you will see as I explain further.

The materials for this game are simple. You will need a deck of standard playing cards, small household/classroom items, and a scratch piece of paper and pencil if students need to check their answers. This is a two player game, but students can also work in teams against each other. I found it to work better as two individual players, but either works! The goal of the game is to make an equation with the cards you choose to equal the designated number in that round.

To set up the game, shuffle the cards and lay 12 cards, face up, in a 3 X 4 fashion and lay the rest of the deck face down in the middle of the two players. This picture is how it will look.

Have the students pick their household item, this will be their game piece.

To determine who goes first each player will draw a card from the deck and whoever has the bigger number will go first. In this game K=13, Q=12, J=11 and Ace=1.

The players will start at a corner of the “board”,” but they cannot start at the same corner. The player who goes first will pick up 2 cards from the deck and they get to decide what operation they want to use to get the first designated number and this can be any operation. That player then gets to decide where they want to go on the board. They only move one card at a time and can go up, down, left, and right, but not diagonal. They then pick up the card they were just on and keep it and replace it with another card from the deck.

The next player then gets to move their piece and do the same thing. This continues until one of the players believes they can make an equation with their cards to get the designated number. If they can, they discard all the cards they used in that operation and pick up the two cards that were used to make the designated number. They may keep any cards they didn’t use. The player who won that round then gets to pick up another two cards from the deck and choose any operation to get the next designated number. Whoever wins 3 out of 5 rounds wins the game.

The strategy behind the game is choosing which cards the students want to use for their equations. Having them use any operation is a good strategy as well.

In addition to all of that, if a player draws 2 cards of the same suit they may have an extra turn in that round and may use it whenever they would like but it has to be in
that round and if you land on the same card another player is on, you can kick the other player off and they will have to start their next turn at a corner of your choosing. For more difficulty, have the players win a round by using 3 or more cards and using 2 or more operations when getting the designated number. For example, they may use 3 cards but then it has to have 2 different operations. Not just adding for instance to make it simpler, use only adding and subtracting and take out the bigger numbers in the deck meaning the jack, queen, and king.

Whenever I played this with students they really enjoyed it and loved the challenge it brings! Being fluent in integer equations, I believe, is very important for students to have as they continue in their math education and this game is a good way to practice just that!
Here are the standards that involved within the game:

• CCSS.MATH.CONTENT.2.OA.B.2 Fluently add and subtract within 20 using mental strategies
• CCSS.MATH.CONTENT.3.OA.A.1 Interpret products of whole numbers
• CCSS.MATH.CONTENT.3.OA.B.5 Apply properties of operations as strategies to multiply and divide.
• CCSS.MATH.CONTENT.3.OA.C.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division

John's postscript: while this is similar to many computation games, there are several interesting elements that make the play more fun. One, it naturally levels as the operations the players use to make the target connect to the operations they'll use with the cards they pick up. It's turn-based, taking out the speed element to a lot of these games. And being limited to the cards they can get to helps even the playing field, with some chance involved in those replacement cards. I think there is a lot of opportunity for strategic thinking.

In addition to their own game, each teacher chose an already created math game to promote for classroom use.

Integer Solitaire

We looked at so many math games as a class and individually, but I found that the one I am about to talk about, is very beneficial for students who love a challenge and need a little more integer practice. It works for middle school, possibly late elementary (if they are ahead), and even high school students who want to be more fluent in solving integer equations. This game was created by Kent Haines back in February of 2016.

All you need to play this game is a deck of cards and a small white board and marker. If you do not have a white board, a piece of paper and pencil would work just as well! The board will look like the picture to the right.

Students can work by themselves or in pairs. I recommend using pairs because it helps build teamwork and in this game it is nice to have a partner you can bounce ideas off of. The student will draw 18 cards at random. The black cards will be positive integers and the red cards will be negative. In this particular game Ace=1, Jack=11, Queen =12, and King=13.

The goal of the game is to have the students use their 18 cards to somehow fill in
the 14 blanks on their board to make 4 correct equations. If the students finish early have them start over and pick 17 cards to make it more challenging.

Overall this game can be a challenge because they could get 3 correct equations but not be able to make a fourth. However, students are persistent to win so they will keep trying. For this reason, the fact that you don’t need a lot of supplies, the range of students who can play this is large, and because it is really good practice and fun for students who need to work with integers are all of the reasons I believe this game is a great math game for students.
A couple standards that are involved within this game are:
• CCSS.MATH.CONTENT.2.OA.B.2 Fluently add and subtract within 20 using mental strategies.
• CCSS.MATH.CONTENT.2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

Cf. Kent's original post on Integer Solitaire

## Tuesday, May 19, 2020

### Multiply Who, a math game

This past semester I got to teach a senior project class. Four preservice elementary teachers working on understanding math games, game design and making their own. Sam came into class with the idea for her game. All the games were tested with kids, and went through multiple revisions and I'm really proud of their work and the games they made.

GUEST POST by Sam Bosma.

Learning to Go Forwards to Go Backwards

Many of us remember ‘learning’ multiplication in second or third grade. I remember ‘learning’ multiplication in 3rd grade through the use of flashcards. It didn’t matter what was happening behind the scenes just as long as I got the right answer/ could memorize the answer.
Too often with multiplication, we see children simply memorizing that, for example, 3 x 8 is 24 and not really understanding the ‘why’ behind how it works. The problem with this memorization is that once students move on to division, we see them getting very quickly. This makes sense since how can we expect students to go backwards (the undoing of multiplication) if they can’t even move forwards (multiplication).
In the game I have invented, Multiply Who?, my attempt is to have student thinking more deeply about multiplication instead of simply memorizing. In this game, students will have to find patterns and think about the components of each number in order to ask the most strategic questions in order to guess an opponent’s number.

If you are interested in learning more about this game, watch the video down below or click here for the instructions and rules.

John's postscript: I got to see one of the playtests of this game with a class of 5th graders and was impressed. The format of the game works really well for 2 on 2 playing. And though Sam put the sheets in sleeves, you could play directly on a printup as well. And the game is highly adaptable to level, by modifying the sheet. But the best thing to me is that this could definitely lead to students making up a sheet for playing and really getting to do the math.

Handout: gdoc, pdf

Make It, Take It
In addition to making their own game, the teachers made a video for a game of their choice that they'd like to see in a classroom. Sam chose Make It, Take It, a money game I made up a long time ago for a teacher that wanted a money game for 2nd year.

Again, highly adaptable. It's nice how just the progression of play gets players to think of new combinations.

Handout: docx, pdf