Find It!

Design
The call: a game for 5th graders just starting with fraction multiplication.

I look at my games. Fraction version of the Product Game... great fun, but more for practice than introduction. The crazy Ant Man game ... fun, good for calculator use, but also dividing fractions, so probably not time for that. Hmph.

Answer the question
(this was the first one)

Get it right to get a chance to
shoot past the goalie.

I look around on the web. Googled fraction multiplication game and got a lot of really awful drill "games." Glgkh - they left an awful taste. Some are obviously just quick flash mass production, but there are a couple that people really put time into looks and animation. For a quiz set to 8-bit music.


So, I'm on my own. Often with introduction time I try to think about representation. One of the things to love about fractions are all the many representations.  I think the discrete models are underused, so I thought about about students claiming fractions of a common pot (similar to the GeoGebra percent game I posted recently) - but it was difficult to figure out how to keep to intuitive numbers and overcome the disproportionate effect of going first. Also, I had trouble thinking of a game context that would get students to see it as a fraction of a fraction instead of a fraction of a whole number.

Then I thought about the area model. I imagined carving up a rectangle, having kids carve up rectangles. Scoring a total... connecting two points... then I had a connection. Cutting down bit by bit, it felt like searching for something. I tried a 12x12 grid, and my first pass at a mechanic worked pretty well: rolling a die to get halves, thirds, fourths. I thought of a context - searching for a lost hiker. Too scary if you've been lost? Finding a lost pet... maybe. It was a little too direct. Is it a competition? It was starting to feel like Battleship (a fine game), and that was good. I tried finding multiple objects; 2, 3, 4... and 4 was right. Oh! They could come up with the context - and that would give them the opportunity to add rules of their own. That's worth a try!

Here's the handout on Google docs: Find It!

Playing
I launched the game with my own context:

They managed to find all three, before... well before nothing. I was pleasantly surprised by how engaged they were just trying to find the rings. Like spontaneous applause when someone found one. (Playing with the whole class, I have them pass the die to someone who's ready of the other gender. Usually works.) Afterwards, I shared how maybe I needed more rules. Or the Mandarin's searching also. Or if you roll two 5's the Mandarin finds a ring. Or...

It was clear this was going to work because there was immediately a crowd of students trying to tell me their context, Minecraft, aliens, how it fit into the story she's writing about two wolves who turn into humans. It was exciting. They experimented with more than 3 objects and asked me why I had chosen three.

The wolfgirls.

Quite complex. This was played on two boards,
with interaction between the heroes and villains.

The minecraft game.
This had hazards as well as the goal.

The zombie game, which also had a hazard.
You had three lives, and had to find the zombie solution
before you lost all the people in your party.

The playing went well also. I was impressed by students ability to divide regions equally, and the many ways they found to do it. They started inventing their own terminology for how they were doing it, like the strips or plus method for dividing into four.  They used horizontal and vertical divides, and one group experimented with non rectangular regions. One group played like Battleship, competing to find all three before the other team did.

In feedback, everyone gave the game a thumbs up (mostly) or so-so. (Rare to have one that no one dislikes.) They liked the Battleship connection, the feeling of searching and the multiple objects to find. They were very excited to tell about their context and rules variations.


Game Evaluation

1. Goal(s) - good - experience with representation, dividing up rectangular pieces into equal parts. Plus a context for future questions and rephrasing.
2. Structure - works well.
3. Strategy - puzzle like. Choice in which region to divide up with which fraction. Choices for where you hide the objects. Not the strongest element of the game, though.
4. Interaction - good and so-so. One person/team being the mechanic for revealing spots and checking the other team's work on dividing was good mathematically. But Battleship isn't strong on player interaction.
5. Surprise - die roll, so okay.
6. Catch-Up ... depends on the variation. It's a bit methodical doing the search, but there's no time element in the basic version. The chance to get lucky with a search or a roll will help.
7. Inertia - works for this. Students were anxious to play more.
8. Rules - toughest element is the dividing up equally. Once you've got that idea, rest is simple.
9. Context - here's the winner. Students being able to set their own context was very engaging for a vast majority.

Percent Game

I was thinking about percents and ways to gain experience with them, in preparation for GeoGebra work with middle school students.

Found a couple of neat percent sketches on GeoGebra...

Nice visualization from jholcomb.
(By the way, GeoGebra has gotten input boxes to work in the HTML5/mobile device. Good going!)

Slick discount problem visualization from Anthony Or (orchiming).

Nice double numberline visualization from David Cox.

But as I looked, the idea for a game came to me, just to give percent experiences. There's no context, really, it's just a race game. No strategy, just rolling random percentages. But the mechanic of smaller roll goes first creates some nice percentage situations, and a lot of games wind up surprisingly tight.

I debated having the students find the percentages to subtract, but decided to make it optional.  There's a short video of how the game works below.

Here it is on GeoGebraTube, for download or for mobile applet.






The test game came out extremely close - most games will be shorter than that. It can be surprisingly suspenseful, though.  All students found it pretty playable, and some got very into it. I think the best benefit might be from playing and then using as a context for problems.

Searching through GeoGebraTube, students found two sketches particularly of interest.

Arrange Fractions, Decimals and Percents by dhabecker (who has quite a few rational number sketches), which lets students arrange form numbers of different form from least to greatest. A few students got quite engrossed in this sketch, and the feel of the sketch is terrific - very much like pieces snapping in place.

Estimating Percents by David Cox, which lets students make a first estimate and then a second estimate with the tens showing. Students were happy to see their % error improve from 1st to second guess, and got better quickly through playing. They also made an impromptu game out of it, and I think that would be fun.



Do you know a GeoGebra Percents sketch that you think supports students' understanding?

Here's the results collected so far. Thanks!

PS> there's a follow up post to this one with two more GeoGebra percent activities.

Number Addict

I have an addictive personality, so why would I even start a game called this?

Number Addict is a free iOS or Android game. (Also has a Facebook page. The sequel is a 99¢ app.[App store link]) It's a Tetrisish puzzle, where you are trying to amass groups of equal numbers to score them. Two nice wrinkles: it takes a group of at least as many as the number to score them, and you can combine adjacent numbers by addition up to the max number tile available. (Which increases as you level up.) You can see upcoming tiles, so this and the adding makes for many strategic decisions. I think it's good for reasoning as well as mental compositon of whole numbers.

The scoring is completely non-obvious. Scores increase both for number scored and how many in a group. This post is really just about sharing my data for a possible lesson.

In an effort to continue to seek the most boring possible video, here is a screen capture of me playing the game. I used screenr.com and the Air Server app to make the video. 


(It has a fun song 'Pocket Calculator' that I muted for the video. Most Boring Ever.)

But, I spent some time (too much) gathering screen caps of scores... (google pdf)



So I've even collected the data for Act 2... (google doc)


That's editable if you gather more data...

Some questions are obvious. What are the missing data points? What will the level 9 scores be like? (I've never gotten there!) What's the pattern to the increase in scores? Is it determinable or were the programmers just having fun?

If you give it a try, share what you find in the comments!

EDIT: collected a bit more data... great patterns. If you want it to verify student work, here's the spreadsheet, and here's the evidence.

Cat Chase

Play the game!

I've been filling up the blog with my class notes from Learning Creative Learning, but an assignment from there really turned into a fun project.

In Week 6 we were supposed to remix Scratch projects from other users. I - of course - looked for math games. First I found 15 Seconds by jOHEEN_c which was an integer game. I wanted more leveling, so found Maze Levels 1-5 by Cats_Are_Awesome. I didn't wind up using their code, but I did learn a lot about Scratch by working through their programs.

I wanted the levels because increasing challenge is key to engagement. I have about 10 screens of increase, after which it levels out. The movement is a little challenging for me as an old guy: the cat follows the mouse instead of being controlled directly. The game forces you to add and subtract positive and negative, but gives you choices on how to get the target. I did add a restart button for if you got stuck that lets you proceed without having to play all the levels over. Is it a problem that the game never really forces an end by becoming impossible?

The game is on the Scratch website, where you can play it or download the code. (Scratch is free to download, of course.) All Scratch submissions are cc 3.0, which is very nice. (Direct link to video, made with Screenr.com.)
(You can embed the Scratch program directly in a webpage, but it starts immediately, which can be aggravating.)  I would be very interested in your feedback on the game, and delighted if you or your students would be interested in remixing it.

Music credit: Upbeat Ukelele Song by Akashic Records, via Jamendo.

Mathematics Taboo and Mathematics Pictionary

Here are two sets of games that I've created based around popular board games Pictionary and Taboo.

The first is a Mathematics Taboo - great for getting students to explain a concept without using certain 'taboo' words. I have used this activity for a starter or plenary to a topic before
You can download some of these cards from my TES resources here...
http://www.tes.co.uk/teaching-resource/Mathematics-Taboo-Cards-6318892/ and this is easily adaptable for the topic you are teaching.
Display a certain card on the board and invite students to explain the concept/topic to their partner without saying any of the 'taboo' words

Alternatively, you could just give the cards out in groups for students to play the game for a given time. Perhaps as an end of term activity or even as a tutor time activity!?

The second is a Mathematics version of Pictionary. This activity is great for mini whiteboard work in groups of 4 with pairs competing against one another. Can be used in lessons for a starter activity to get students settled, or even as a tutor time activity?!

You can download this from my TES resources at...

http://www.tes.co.uk/teaching-resource/Mathematical-Pictionary-6167906/

Here's how my cards look (once laminated)...

There are 'category cards' and then 'topic cards' included

#globalmath Math Games

Tonight there's a #globalmath session on games where I'm one of the three panelists. I have loved the #globalmath sessions I've attended or watched afterwards so far, so I'm excited to be a panelist. EDIT here's the recording!

Here are my slides:


All the links are gathered together in a urli.st, Global Math Games. I tried to add other links that came up during the presentation. Unfortunately one of the panelists, Elizabeth, @cheesemonkeySF, couldn't join us because of school issues. She was going to talk about the Life on the Number Line game. But James Cleveland, frequent writer about games on his blog, was there.

James talked about his basic rule for analyzing a math game: does the math action have anything to do with the game action? He gave a couple of examples where the math is like a pause on the game, and you have to do some math to continue playing. He shared Ice Ice Maybe (MangaHigh) and Dragon Box as examples of good games. (I like one of those better than Dragon Box.) And then he talked about his game Totally Radical, which seems very cool, and is a great example of the math action being the game action. There's a homemade version to print up or you can even by a slick commercial version.

In planning my part of the presentation, I wanted to get across:

• what makes a good math game
• what I like about my best games

 Hey, how much should you shoot for in 20 min?

Slide:

2. I like games because they're fun and engaging. I love playing games myself, so it's not crazy to want to do them with students. But I also think that games are mathematical, as math is game like. The actual process of being a mathematician is inherently playful. Trying things out, intentional variation, strategizing. I am actively jealous of literacy teachers, who have all these stories about the book that turned this student onto reading (Potter, Bridge to Terabithia, Desperaux, etc.) and we math teachers never get that.  For me games are our best shot. Watching kids play Pokemon or Yu-Gi-Oh you see them doing amazing problem solving.
3. I'm in favor of gamification, but see that as being different from a math game. There's been some fun discussion of the Ninja board on Twitter with Jim Pai (his ninja board), Jeff Brenneman and @algebraniac1. That's just fun. I think how video games present challenges for which you need new skills and giving you notice for things you've accomplished (level up!) are worthwhile for teachers to think about. And review games are a good use of gamification as well. (Here's my list of other people's review games.)
4. The Product Game (NCTM online, my copy) is to what I aspire. It's a great game, with the strategy of a Connect Four (which is a real game, if by some chance you haven't played since a kid), but all the game actions are math actions. Furthermore, the encourages you to organize multiplication facts by family, and use patterns to find products you don't know. Evenfurthermore, it encourages reverse thinking preparing students to think about factoring. Plus it is just fun. The game structure is so good it lends itself to adaptations (decimals, fractions, integers, exponents...)

5. Decimal Point Pickle is one of my favorite games that I made. For students who have worked on decimal representation, it encourages better number sense and place value before moving on to decimal operations. It has a blackjack feel, and never fails to generate excitement. Mathematically, the comparisons of decimals in tenths, hundredths and thousandths have been invaluable.
6.  Eleusis Express. This is an adaptation I made of a deep and difficult game by Robert Abbott. He captured an essential mathematical process, though: making and detecting patterns. It is an amazing game to play that affords opportunities to reason, discuss arguments, and problem-solve. It really helps me communicate with students the value of capturing thinking. Because it doesn't feel like mathematical content, it is safer for them to struggle. It's just a hard game. Handouts for this are at the top of my games page.
   Plug: I also love Robert's endlessly clever mazes.
7. Where I'm heading now is trying to get the students to be the game designers. This is done over several sessions, releasing more and more of the design task to them, or by working through the different aspects of design and having them work on the piece of a game. Even students who don't get into the playing of games find this to be rewarding and engaging. Plus, I know from making games that a lot of the math is in the design and getting things to work out correctly.
8. The framework:
   
   
   Adapted from Magic: the Gathering's lead designer's thoughts on design, this framework has helped me to better understand what I'm doing and to make better games.

People were nice, but I did not make a good presentation. Good thing I had cartoons.

I came upstairs and my wife asked me how it went. "I was too vague," I said.

"I'm not sure what you mean," she  says.

Sigh.

In retrospect, I should have just talked about the three games, and given some of the play of them. Don't spend time talking about what you're not talking about! But it was a good learning experience, nonetheless, and I love being a part of "PD you like." If you haven't peeked in at #globalmath, give it a try. Most Tuesdays at 9 pm ET. Next week looks good already: Building Intellectual Need.

#globalmath Math Games

Tonight there's a #globalmath session on games where I'm one of the three panelists. I have loved the #globalmath sessions I've attended or watched afterwards so far, so I'm excited to be a panelist. EDIT here's the recording!

Here are my slides:


All the links are gathered together in a urli.st, Global Math Games. I tried to add other links that came up during the presentation. Unfortunately one of the panelists, Elizabeth, @cheesemonkeySF, couldn't join us because of school issues. She was going to talk about the Life on the Number Line game. But James Cleveland, frequent writer about games on his blog, was there.

James talked about his basic rule for analyzing a math game: does the math action have anything to do with the game action? He gave a couple of examples where the math is like a pause on the game, and you have to do some math to continue playing. He shared Ice Ice Maybe (MangaHigh) and Dragon Box as examples of good games. (I like one of those better than Dragon Box.) And then he talked about his game Totally Radical, which seems very cool, and is a great example of the math action being the game action. There's a homemade version to print up or you can even by a slick commercial version.

In planning my part of the presentation, I wanted to get across:

• what makes a good math game
• what I like about my best games

 Hey, how much should you shoot for in 20 min?

Slide:

2. I like games because they're fun and engaging. I love playing games myself, so it's not crazy to want to do them with students. But I also think that games are mathematical, as math is game like. The actual process of being a mathematician is inherently playful. Trying things out, intentional variation, strategizing. I am actively jealous of literacy teachers, who have all these stories about the book that turned this student onto reading (Potter, Bridge to Terabithia, Desperaux, etc.) and we math teachers never get that.  For me games are our best shot. Watching kids play Pokemon or Yu-Gi-Oh you see them doing amazing problem solving.
3. I'm in favor of gamification, but see that as being different from a math game. There's been some fun discussion of the Ninja board on Twitter with Jim Pai (his ninja board), Jeff Brenneman and @algebraniac1. That's just fun. I think how video games present challenges for which you need new skills and giving you notice for things you've accomplished (level up!) are worthwhile for teachers to think about. And review games are a good use of gamification as well. (Here's my list of other people's review games.)
4. The Product Game (NCTM online, my copy) is to what I aspire. It's a great game, with the strategy of a Connect Four (which is a real game, if by some chance you haven't played since a kid), but all the game actions are math actions. Furthermore, the encourages you to organize multiplication facts by family, and use patterns to find products you don't know. Evenfurthermore, it encourages reverse thinking preparing students to think about factoring. Plus it is just fun. The game structure is so good it lends itself to adaptations (decimals, fractions, integers, exponents...)

5. Decimal Point Pickle is one of my favorite games that I made. For students who have worked on decimal representation, it encourages better number sense and place value before moving on to decimal operations. It has a blackjack feel, and never fails to generate excitement. Mathematically, the comparisons of decimals in tenths, hundredths and thousandths have been invaluable.
6.  Eleusis Express. This is an adaptation I made of a deep and difficult game by Robert Abbott. He captured an essential mathematical process, though: making and detecting patterns. It is an amazing game to play that affords opportunities to reason, discuss arguments, and problem-solve. It really helps me communicate with students the value of capturing thinking. Because it doesn't feel like mathematical content, it is safer for them to struggle. It's just a hard game. Handouts for this are at the top of my games page.
   Plug: I also love Robert's endlessly clever mazes.
7. Where I'm heading now is trying to get the students to be the game designers. This is done over several sessions, releasing more and more of the design task to them, or by working through the different aspects of design and having them work on the piece of a game. Even students who don't get into the playing of games find this to be rewarding and engaging. Plus, I know from making games that a lot of the math is in the design and getting things to work out correctly.
8. The framework:
   
   
   Adapted from Magic: the Gathering's lead designer's thoughts on design, this framework has helped me to better understand what I'm doing and to make better games.

People were nice, but I did not make a good presentation. Good thing I had cartoons.

I came upstairs and my wife asked me how it went. "I was too vague," I said.

"I'm not sure what you mean," she  says.

Sigh.

In retrospect, I should have just talked about the three games, and given some of the play of them. Don't spend time talking about what you're not talking about! But it was a good learning experience, nonetheless, and I love be a part of "PD you like." If you haven't peeked in at #globalmath, give it a try. Most Tuesdays at 9 pm ET. Next week looks good already: Building Intellectual Need.

Mathematical DINGBATS

I'm a big fan of DINGBATS and have used a number of these in class before (especially on Pi day). Here are some of my latest Mathematical DINGBATS...

 These ones are all based around the topic 'Averages'

 

 



These ones are all based around the topic of 'Circle Theorems'









That's me in Q3 and Q1 above - one picture I am referred to as 'sir' the other I am referred to as 'me'!

'Math Magic' Board Game

I found these in our Mathematics department this week - the Math Magic Board Game - very much like a Mathematics version of scrabble where your points come from performing the 4 basic operations +, -, x, and divide!

I played this with my Year 11 class after they had just done their examinations this week - it worked really well although I feel I was more excited about it than they were!
Still, it was a chance to introduce a bit of competition to the class with them playing in teams of 4/5 and then the top 2/3 in each swapping to face the top 2/3 in another group, leader board on the IWB etc!

It was a welcome break for the students having just sat their Mathematics exams and it ensured they were still practising their basic numeracy. Well worth getting for a maths club or for tutor times?!

Angle Acquisition

Quick game idea. I've had a few bustling around, and I've got to get started writing them down.

Observing student teachers is a great job. I get to see and have in depth teaching discussions with lots of hard-working, talented teachers.  And see a broad range of content. We use a coaching model, which helps these be positive exchanges and ratchet up the interest level of the dialogue.

I was observing Terry Austen (the class he writes about here) and realized that once students knew the terminology well for parallel lines they were correctly identifying and applying the relevant properties.

That gave me the idea for a game. I thought it would be neat if students used the terminology to capture points. My first idea was to have a parallel line puzzle - I like those as practice, too - and have the players build it and then take the pieces. That's a lot of set up, though, so I decided on cards to make for less preparation. There's name cards to cut out, still.



Let me know what you think. I'll try it out with the PSTs this semester and post an update.

Greater Than

Previously: planning and coaching on inequalities.

The origin of this game was trying to think about a game that helped give students enough experience to intuit the rules for how operations with signed numbers affect inequalities. I think this is doubly hard for students because they don't have much intuition for signed numbers, and learn both integer operation and inequality rules by memorization rather than understanding.

I love playing cards as a material for signed numbers because the red and black make such a nice positive and negative visual. (For more integer games, see this collection elsewhere on the blog.) I thought about students somehow constructing expressions, but I couldn't think of anything not clunky. I knew we wanted comparison, and I like War as a comparison structure. That made me think of the exciting moment of war, playing down extra cards on a tie.

When I played around with the cards, though, it didn't get at inequalities because each player was changing the value of only one side of the comparison. A big no-no in the context we want. So the played cards had to effect both players' values. It seemed confusing to have both players reveal at the same time, so I decided that - at least to start - players should take turns, even though that makes the optimal strategy pretty determinable. (If it's not a word it should be.)



After students get the strategy, go to the variation where both players flip at the same time or the dealer has one less card. If playing with middle schoolers for integer operations practice, try the flipping off the top variation.

I think this is a good educational game, but only close to a good strategy game. I can't quite figure out what's missing, so if you have a variation or adaptation to try, please let me know.

Evaluating this on my game design framework:

1. Goal(s). Gain experience with effect of integer operations on inequalities. Works well. Also good for gaining experience with integer computation.
2. Structure. The game generates a lot of these situations and the cards guarantee a mix of operations and values.
3. Strategy. Pretty simple. On variation becomes a pretty nice bluffing game, but not intensely strategic.
4. Interaction. What you do completely depends on opponent.
5. Surprise. Hidden information and opponent plus randomization of cards helps here.
6. Catch-Up. Victory is almost always possible.
   
7. Inertia. Might be too simple for loads of play, but good enough for the objective.
8. Rules. The idea of applying your card to both is non-intuitive, and remembering suits-operations connections is hard. You might want specialty cards
9. Context. No context, but all the variations generated engagement from the preservice teachers.

One thing I want from this experience is a deck with operations for suits. Not sure if it should have all 4 operations, or multiplication and addition with positive and negative numbers or some other variation. What do you think?

Photo credit: Abulic Monkey @ Flickr

Pass the Chicken!

Whilst teaching at ISSOS I found a website that had a few teaching ideas on it, one of these ideas was 'Pass the Chicken'! Here's the website I found http://www.educationworld.com/a_lesson/lesson/field_day_games.shtml

For 'Pass the Chicken' you obviously need...wait for it...a chicken!

So, before I explain how 'Pass the Chicken' works, I needed to get a chicken. Luckily, I remembered that fancy dress shops sell them! So I looked up the local Cambridge fancy dress shops, looked through their websites, found one that sold them and then trekked into town to get one. Unfortunately, when I got their they didn't have any in stock (in hindsight I should have rang them 1st!), but being as kind as they were they put one on order for me and I went back at the end of my few weeks in Cambridge and got it!

It looks like this...

Anyway, this is how 'Pass the Chicken' works...

You 1st choose a random student (I'll do this by either using my random name generator or by simply throwing the chicken out to the class and seeing who gets it). This person is then the 'chicken'. You then give the 'chicken' a topic, say 'quadrilaterals'. They then state how many quadrilaterals they can name. Make sure at this point that it is a suitably high enough number, definitely over 5! Then, the 'chicken' passes the chicken to the next person in the class and then this person to the next and so on until the chicken naturally ends back up at the originally chosen student (the 'chicken'). However, if the 'chicken' manages to state the number of (in this case) quadrilaterals they said they could before the chicken gets back to them then the student currently holding the chicken becomes the new 'chicken' and the game starts over with a new topic. If the chicken does manage to get back to the original student (the 'chicken'), then they are the 'chicken' again and must start over with a new topic.

This should create a bit of intrigue when I get the chicken out for the first time, and hopefully engage the class in their learning. I can see this sort of activity working well at the start of a lesson to test how much they already know about a topic, or at the end of a lesson to see how much they have remembered!

Size the Day

It was time for my last game with the fifth graders, and the content was multiplication and division of fractions.  Having just been critical of a game that was very computation focused (see my Math Evolve review) I was very wary of doing the same thing.

I'm at the point now where I've designed more games than I can remember easily, so one of my first steps in game design is to search my own stuff. Failing finding a game to use, maybe there's one to revise. Failing that, maybe one to revise. I did stumble across my post on multiplying fractions. Oh.

I have many lessons that get at the meaning of the operations, can be used to start discovering, exploring and justifying the rules... but only the Product Game (adapted for fractions) for practice. But that post closes with an Ant Man and Wasp cartoon, and we'd been talking a lot about them because of the Avengers movie.  My son is a comic book fan (sounds better than monomaniacally obsessed, right?) so this was quite a debate. He loved the movie maximally, but would have loved it even more with the original comic book Avengers crew. (Joss Whedon wanted characters to whom non-super powered audiences could relate as well as the super powerful ones.)

By Xavier Golden, Super Hero Squad style.
"The hexagons are Pym Particles."

So I was struck with the idea of a size-changing game. But why would our heroes have to constantly change size?  To get past obstacles! Sometimes they'd have to grow, sometimes to shrink. I tried to think of a battle game because there were several boys always interested in that, but for battle it seemed like you'd almost always want to be giant-sized instead of ant-sized. So a kind of maze... so it could be a race game.

I tried to think of a way to turn dice rolling into fractions for multiplying or dividing, or to roll three and choose two, but that didn't feel appropriate for such new content. I wanted actual fractions to see and think about. 






If you have transparent spinners, this is a good place to use them; I just used bobby pins, which make excellent spinner needles.  I experimented with the spinner entries and maze heights to find settings that were not too immediate but not too difficult either. Thinking about the framework I've been using...

1. Goal(s) - solid. I wanted students to get the understanding of the effect of multiplying and dividing by fractions, so contrary to their expectations. I wanted to get some sense of estimation, and some experience with calculations that would lead to support the symbolic rule they'll learn later. I'd also noticed that they were very interested in calculators, but had little experience with using them. This put everyone on an equal footing, as the numbers were messy and required a calculator.
2. Structure - like the stretching/shrinking as a context for multiplication/division. The spinners allowed a lot of flexibility in getting values to be used. Makes the game highly adaptable. And the intention of having to choose a multiplying or dividing spinner helped get across the stretching or shrinking effect.
3. Strategy - weak as it was.
4. Interaction - typically weak in race games, though .
5. Surprise - spinners help here and with...
6. Catch-Up .
7. Inertia - not meant to be a game that requires a lot of replay.
8. Rules - basic premise, spin and change your height. Move forward when you fit.
9. Context - thought this was strong, plus pop culture tie ins to a heavily advertised movie. Kids were interested and engaged, though I sold it a bit explaining about Ant Man and the Avengers. There's a little suspension of disbelief, as Wasp could just shrink and fly through all the obstacles, and it's rare that she grows in the book.

I added the Spin Again option to help with catch-up, strategy and interaction. But most students were so immersed in their own spins that they rarely used them! The other idea that I like quite a bit was the customizable board. Most of the fifth graders were happy to use the board as printed, but a few experimented with rearranging the board.  Maybe with middle school students, more would be interested in giving it a go. Designing a board for your opponents is a great opportunity for some open-ended problem solving.  I picked up a couple packs of mini post-its, and they were perfect for keeping track of the players' heights.

It was a good last game of the year. In the debrief, they definitely got the point that multiplying and dividing by fractions did not just have the same effect as multiplying by whole numbers, and a few kids were noticing that dividing by unit fractions was like multiplying by the denominator. I also saw considerably increased skill with the calculators, and some sensible rounding of the decimals involved.  (Parentheses were almost entirely new to them.) They asked me to leave the supplies so they could play later, and it got almost 100% thumbs up for keep or dump - both good signs.

Hopefully you can get a chance to give this one a try. It has some interesting features, and I think the choice of spinners and rearrangeable board will show up again - good game mechanic features. I'm always interested in your feedback, if you have any ideas or get a chance to use it with students. One dramatic need: the name is a terrible joke, and of absolutely no use with middle school.  Ideas?

Math Evolve

Math Evolve - a basic operations fluency game for iOS. Free Lite version to try out, full version $1.99.

Somehow Math Evolve came across my radar again this week; it's an iOS app for elementary math. I'm always willing to give those things a try. Tired, frustrated with other stuff, I tweet:

And didn't think any more about it.

The next day, Adam tweets back, very politely. Immediately I feel like a heel. If I was speaking to someone in person, I wouldn't express myself that way. So I want to apologize to Adam here.

I had high expectations because the app store page has the following:

★ Winner of BEST EDUCATIONAL GAME OF 2011 (2nd Place) in the Best App Ever Awards.
★ “The holy grail of edutainment math apps.” Editors Choice, 5/5 Score -Best Apps for Kids 

In Math Evolve you are an alien organism, completing number sentences on a quest to grow and return to your people. At the beginning of a level you fill in the first blank, then the second blank after the operation, then finally the result. Later you pick the first blank and the computer provides the second. (I always get the first one of those wrong as I don't notice when the switch occurs.)  All the while you are dodging combatants and trying to shoot them out of the way. You control your position on the screen by moving your character with a touch. Finally the boss appears and asks you to compute particulars.

Another part of why I expected/hoped for more is that it is slick. Trying to develop ParabolaX (or rather, watch while Alejo and Kevin program ParabolaX) has given me an appreciation for the work that goes into slickness in iOS.  Is there a role for a practice app? I think so, but I don't value it anywhere nearly as highly as one that helps students learn. After all, if they need practice, typically there's room for greater understanding.

I'd love a practice game that gave more for students to notice. Like asking 7+3, 6+4, 5+5, ... or 4x5, 5x5, 6x5... or 5x6, 6x6, 6x7... Noticing and generalizing are things that I value more about math.

Given the context, the practice mode could have meaning of the operation.  Gathering or eliminating to add or subtract, repeated groups for multiplication... lots of possibilities. I do like how at first you're gathering all the equation elements, then the game fills in the second spot, then the boss poses a computation. The incorrect answers are not particularly problematic, and the boss doesn't seem to use missed problems to pose problems. Maybe at higher levels? Ways for added difficulty could be including choices that are not computable, such as choices between 3, 6 and 7 when you have 21÷___.  I did only play the Lite version, so possibly some of these things come later.

Boss level.

I love how open and interested Adam is in improving the game, the context and game mechanics are well developed and suitable, and the game is fun enough that students will be happy to play. And he is obviously a far better internet citizen than I am, but hopefully he's taught me to watch my oafish responses.

As I think about evaluating math games and apps
Objective: facts practice, four operations, including negative numbers.
Gameplay: fun enough. Think Galaga/shoot 'em up.
Aesthetic: great. Cute characters, good effects.
Pedagogy: beyond difficulty selection, no visible fact strategies or support for learning. Minimal math practices.

I'd love to hear what you think of the app, and what do you feel like would make for a good facts learning game.

Guess My Rule

Some activities seem ageless. They work in elementary, middle and beyond. Guess my rule I have played with very young students, and this week I got to try it with my summer intermediate algebra course.

Guess My Rule
Any number of players

A rulekeeper makes up a rule that gives a number output for a number input. (It should be a function.) Players take turns giving an input, and the rulekeeper tells them the output. If a player wants to guess the rule, they tell an input and what the output would be. If they're right, they guess the rule. Then that player makes the next rule.

I don't know who originated this game; it feels ancient and right, so kudos to whomever developed it. I let rulekeepers use a calculator, and I usually start out with a few rules until the guesser feels comfortable making up a rule.

Today I started with the number times two minus three.  (Highlight to see it, or guess from the table.) Inputs came from all over. 12,  5, 10, 1... some shock at a negative answer. I encourage the students to record the results, hoping that it will lead to some ideas about what inputs to suggest. Organizing data is not a natural tendency. At one point they started asking 30, 40, 50...

inputs	12	5	10	1	30	40	50
outputs	21	7	17	-1	57	77	97

Oooh! That's a pattern. It goes up 20 when the input goes up 10. So they could predict the answer for multiples of ten. Thinking about that, they suggested a rule that worked. We looked at the rule to see from where the up by 20 pattern came.

The solver was not comfortable coming up with his own rule, so I chose another: eight minus the number. 

inputs	7	1	10	12	20	30	40
outputs	1	7	-2	-4	-12	-22	-32

\(7 \to 1 \) then \(1 \to 7\) drew an audible gasp/hmm. There was one student who loved asking for 12. The third multiple of 10 got the rule again. Subtract eight and then take the opposite of your answer. One of the reasons I love this game is that it always brings up equivalent expressions. I shared my original rule and we talked out the equivalence.

 

Now they were ready to suggest a few rules. A couple like my rules, then input times 5 divided by 2, which got a nice check of "oh, that is the same as the number plus half of it," and "I had the number times one and a half." Then one student came up with a stumper. When guesses were not focusing, I started taking notes on the board.










I also asked for what patterns they noticed. Some good stuff.  We got to the idea of asking inputs in a pattern, and I asked them to predict the output from 2 before Edras gave us the actual. (There's some really nice research on the power of prediction in math, some by my colleague Lisa Kasmer.) We spent some time on different expressions of the rule, like ___ x ___ + \(\frac{1}{2}\) ___ x ___, (___)^2 + ( ___)^2/2 and \( 1.5 x^2\).


For a last question, I asked them to find what input would give 100, and got unexpected riches. Students really worked hard, consulting group members. Several tried to solve symbolically, but didn't know what to do with the \( x^2 \). One student had an excellent guess and check. We spent some time discussing that, including what an excellent problem solving method it is, and how when I use it, using numbers gives me a much better feel of what's going on. Maybe they've been discouraged to use it by previous teachers, but it's a great strategy. We did discuss how to make our guess more accurate and efficient. Like was 8 or 9 closer to 100 in output?

Another then shared a traditional symbolic solution. One student asked, "why did you divide by 1.5 before you took the square root?"



"Because you have to."
"It has something to do with the order of operations..."
"Doesn't PEMDAS tell you what to do?"
"But you have to reverse it maybe..."

"Let's try it." (OK that was me.) They told me the steps and I wrote it down. Amazed to find the same answer. A good moment to point out to them that are almost always multiple methods in math. Teachers may have told them that there is only one way to do things in the past, but that is bullsh*t.

I should not swear in class. (In response to my usual 'jot down one thing you want to remember' there were many repetitions of that statement. In my defense, it is a college class. And the comedic potential was ripe.)

The next activity is adapted from Pam Wells' adaptation of an activity from the excellent Mathscape curriculum.




(Here's the Word file if you want to edit - it wasn't appearing correctly in the embed.)

It dovetailed very well into the Guess My Rule, allowing us time and grist to find many connections amongst tabular, symbolic and visual representations, reiterating the equivalent expressions and multiple solutions themes, and giving us a launching point for a definition. Using the Y pattern rule, I made a table for S=1, 4, 7 and 11. The changes in output were 9, 9 and 12... what went wrong? They found that the difference in the inputs was not constant, and when we changed 10 to 11, the output pattern worked. To me, that's the moment for the generalization.

We went on to launch Linear War, which gave me a chance to share why we use the term linear and some of what this stuff has to do with lines. Next class, we play!

All in all, it was a fun class. I was impressed with their willingness to try non-traditional problems, and they're gaining rapidly in ability to work cooperatively and make conjectures. The meta-messages about mathematics seem to be gaining some traction, too.

Multiplying Game Possibilities

I saw this quick and clean multiplication game suggestion from the Math for Love blog, and really thought it had potential.

• Roll three dice
• Pick two to add
• multiply times the third

A little bit of choice, clean mechanic... great. But I thought that it could be jazzed up a bit. Then I thought that this was a great opportunity for the students to do game design.

The warm up problem that Mr. Schiller had suggested was pure serendipity: what is the largest area rectangle with whole length sides and a perimeter of 30 units. Maximizing a product with a constraint on the factors with multiple choices - perfect! I couldn't resist asking them after they found 7 x 8, "what if it didn't have to be whole numbers?" One student said - maybe 7\(\frac{1}{2}\)? They verified the perimeter, and I showed them a way to find the area. (Area model for multiplication is definitely one of my favorite representations.)

I shared the game and introduced the idea that we could rebuild it, make it better than before, with these prompts.  (Here is the handout I gave them.)

• Is there a context that would fit or a story to go with it? Climbing, racing, building, digging, fighting, shopping?
• Should it be a set number of rounds? How many?
• Or play to a total? What total?
• Any special actions or situations or rules? 
• Is there a way to get people to try and make something besides the biggest score? Like a bonus if you get to a total that ends in zero, or something that depends on the story.

They were bursting with ideas! We shared a couple and then they got working. They quickly decided 100 was too small. A few students went completely away from the idea. Rolling a die to shop from numbered stores, or just a roll and move that many spaces game. Still a lot of pride of ownership, and some good problem solving to make the game work. Others liked the game just fine the way it was, and figured out the right total to play to; as low as 200 and as high as 1000 depending on the group. Some instituted catch up rules for if you got too far behind. (Glad to see that come back from the Spiral Races.) Others added some player interaction by being able to buy out your opponent's roll.

One group made an Escape from Planet of the Apes game, where you race to 100 (pick the locks to escape the cages), then to 300 (escape the village), then to 600 (back to your space capsule for the final escape); this was humans escaping the apes. It's a madhouse! Actually another group also made a Planet of the Apes game, but they didn't share.

One group made a really complicated scheme where you start with 400 points, roll for more, and spend your points on chess pieces that represented bad things for your opponent. First player to zero loses. I would be very surprised if these guys are not future gamers.

Several players made gameboards for the race, with some special rules. One group's game that I got to play had a route through town with obstacles like a storm cloud, mud puddle, etc. that you had to use your points to buy, and you got points by rolling the dice. There was no really clear explanation on how you decided where you were on the path, other than gradually moving forward. These girls were less concerned with that, and it almost felt like a role-playing game. One of the designers repeatedly reassured me, "it's not rigged. At all!"



Another player made/recalled a game from her uncle, a press your luck game. I think you could make a multiplication game out of it.  Here are some slightly cleaned up rules. I love how she wrote an example.

Multiply or Bust!

Roll 5 dice. Scoring rolls are:

• each 1 = 100
• each 5 = 500
• 3 or more 2's = #x200
• 3 or more 3's = #x300
• 3 or more 4's = #x400
• 3 or more 6's = #x600

Set aside your scoring dice. You can reroll any remaining dice. If some of those score, add to your set aside dice. You can reroll non-scoring dice as much as you want, but if you ever roll no scoring dice, you end your turn with zero points. You can only keep scoring rolls, so you cannot set aside two 6's and hope to roll a third. Winner is first player to 5000, or the player with the most points if multiple players beat 5000.

A very really interesting idea came from a designer who wanted a guessing game. Her initial try involved turning out the lights, but after some discussion we got to a really fun little game. Not something either one of us would have thought of by ourselves.

Masquerade - 2 players
Roll three dice but keep them hidden. Add two then multiply by the third and tell your opponent the score. They get three guesses to try to win your dice. If they guess a number right, they get the die and score that many points. After the third guess, players switch who is the roller and the guesser. Each player gets five turns guessing. Highest point total wins.

We closed by students sharing their games. I often encouraged students to write out their rules, which was an interesting ELA activity.

I also had a couple ideas inspired by their warm up question. Here's what I would try.



The main benefit of the grid is to make non-maximal multiplications more interesting. Hopefully it adds a layer of strategy.

Of course, it would be a nice variation to play on the same grid. More interaction, more strategy, and I think the little products are bound to be better. I'll be trying these out!



The game design aspects of these 5th grade lessons has been pretty powerful. We lose a bit of focus on the mathematical objective compared to all playing a set game, but the engagement is high, and the mathematical practices are strongly present, as well as having more math done than in many traditional math lessons. Even comparing the energy the students invested in the warmup problem, which correlated roughly with their mathematical self-identity, with the very similar problem of figuring out the sum and product of the dice in the initial game was a stark contrast.