Multiplication Mazes - a puzzle for fact practice

 All this month I'll be posting games from the Fall 2023 GAMES seminar at GVSU. This senior capstone was begun by Char Beckmann. See many of the games from her seminar in this YouTube playlist. Many of the games completed in my seminar are in this playlist. In the seminar, we play lots of games and math games, the future teachers make a first video to promote a class math game that already exists, we develop a group game (a monster-themed middle school Desmos escape room and Math Heads, a number mystery game this year), and they develop a game of their own.

Keri Herman chose Tens Go Fish, a classic addition fluency game. As an extra feature, she demonstrates the game with Tiny Polka Dot cards. (Find them here at Math for Love.)

Keri's original game was a new idea for the seminar: she was interested in making a puzzle. I recently saw a description of a puzzle as a game for one person.  That certainly fits here. The puzzles are available on a Google doc here. What follows is Keri's story of making the game, and her ideas on why we should play games in math class.

Story of my Game

I knew when I got the opportunity to create my own game, I wanted to develop something that was related to quick multiplication facts. The reason being that my memory of learning my multiplication tables was always timed and quite stressful for me as a young student. I wanted to create a game where students could get great practice of their multiplication facts, and build in aspects of a good game; strategy, any player can win, etc. 

My first idea was a game board, moving the amount of spaces of the product. However, I was then drawn to the idea of more of a maze. I started with a small grid and filled in very small multiplication facts, students would have to find their way to the end. This turned into the development of three mazes, 5 x 6, 7 x 8, and 9 x 10, all with their own unique solution. To figure out how to design these mazes took a lot of different approaches, starting from scratch, and overall just thinking about how to make them work. I believe that the final product of these mazes will provide students with a very fun way to practice their multiplication, while being able to try to solve the maze. 

The goal was to have a large mathematical objective for the game. Students will be focused on trying to find the solution, even if they are going the wrong way, or have to start over, they are still constantly doing the math and getting practice of their multiplication facts. I think this game would be something that teachers should play with their learners because you can never have enough practice with multiplication. Especially in the 9 x 10 maze, all multiplication facts are used from 1 through 9 (not including zero). These mazes will also help students recognize patterns between multiples, factors, and products. 

These games could be used within a lesson, if students finish early, or simply just given as an opportunity for more practice, without time constraints. I also share within my video the development process of these mazes. With students who have learned multiplication facts,  I think it would be a great idea to turn this into a project or performance assessment. Students can work to develop their own maze. Not only does it take strategy, but at the same time students are able to continue working with the facts themselves and continue to recognize patterns. Overall, I am very proud of the way these mazes have turned out. I want to continue to show these to math educators and I hope that students will enjoy solving them as much as I had hoped. 

Why Play Games in the Math Classroom?

As a future math educator, incorporating games into the classroom is something that I want to use and will continue to encourage others to consider as well. It is often looked over to play games in the classroom, but the reasons as to why they are beneficial to student education should be considered. There are few specific reasons that are important to point out, including; building mathematical knowledge and skills, collaboration with peers, student engagement, critical thinking skills, and more. Each one of these reasons in its own makes games in a math classroom worthwhile. 

Building Mathematical Knowledge

Math games all are built upon their own goals and mathematical objectives. Teachers have the option to choose a game that targets the content that is being focused on. To find a game that can build mathematical knowledge, choosing a game that is relevant to your current learning goals within a classroom can help students extend their skills. There are so many aspects built within games that students can pick up on mathematically, without noticing. This can be beneficial to students because they are still learning, but without the title of class, homework, or assessments. 

Collaboration with Peers

It is important for a classroom to have communication among students that can lead to quality discussions. Discussions can uncover so many helpful aspects to student learning. In a game setting, a lot of times students will play with each other in teams, or against each other. In both cases, students are able to communicate and learn from each other. Students are able to pick up on each other’s strategies and build off of them. When playing with each other, this can help build a more positive classroom environment. This is because this type of communication is not usually seen in a regular lecture or discussion. 

Student Engagement

Oftentimes we hear negative assumptions about math and negative attitudes are common when stepping into a classroom for some students. It is important as teachers that we are able to increase student interest by engagement and participation. Incorporating math games into the classroom is a great way to develop student engagement. A lot of times, the mathematical objective of games are mixed in with aspects of interaction, surprises, and fun. A game can also change the view of many students. All students can participate and it is important to use games where any student can win. In math class, students can often point out the “smartest” students and become discouraged. When using games that are designed that anyone can win, not just based on skill, this can build a lot of confidence in students. 

Critical Thinking Skills

In many situations, students become disengaged after they reach the level of knowledge and understanding. However, it is things like analysis, critical thinking, and application that get students to really push past that level of reasoning for the content that they are learning. Math games provide a different way to push students to build upon their critical thinking skills. Having to figure out a strategy to finish or win the game is a very important tool when it comes to building these skills. With that being said, games that are chosen to play in a class should have aspects that involve strategy. 

Overall, math games have so many advantages when it comes to incorporating them into the classroom. Being able to play different types of games this semester has taught me so much about what a good math game should look like. Being able to develop and create our own group game, and my own game has changed my perspective on math games. Math games can help students learn in a unique, fun, and interactive way. 

Boxzee - Flexible Computation Game

All this month I'll be posting games from the Fall 2023 GAMES seminar at GVSU. This senior capstone was begun by Char Beckmann. See many of the games in this YouTube playlist. Many of the games completed in my seminar are in this playlist. In the seminar, we play lots of games and math games, the future teachers make a first video to promote a class math game that already exists, we develop a group game (a monster-themed middle school Desmos escape room and Math Heads, a number mystery game), and they develop a game of their own.

Jordan Burnham selected Close to Zero, and integer addition game for her first video. Handout and original blogpost.

Jordan's original game Boxzee crosses one of my favorite classroom games, Number Boxes by Jenna Laib, with the classic Yahtzee. What follows is Jordan's explanation of the game and thoughts on why play games in math class.

Boxzee

When I was first brainstorming games, I had absolutely no idea what kind of game I wanted to make. It wasn’t until one day when I was sitting on my bedroom floor that the starting ideas of Boxzee came to me.

Originally I imagined the game to have more moving parts. I first had players each being dealt 4 cards. From there they would roll a dice twice to determine a specific operation they would be using (odds = subtract, evens = add). Then after finding out those operations you would choose 3 cards from your hand to find a largest total value for that specific round. I found that this became a little confusing and players wouldn’t necessarily be able to truly “compete” if all of their rounds operations were different than each other. If one player only rolled odd values then they would be predetermined to loose solely because the other players would have a better chance of having larger numbers if they rolled more even values. 

Moving on from here, I decided to instead come up with the number box sets. Rather than using the dice to determine operations I decided this was a more structured way that players could still affect the total value by the cards they put in without having so many moving parts. I first came up with the idea to have four different rounds. The players would both have 4 cards in their hands and needed 3 to fill into the number box sets. I also decided that they would both fill in the top box row, then move downward. After playing this a couple of times I realized it could be very common to tie. So then I chose to create a number box set that would be the final round and would use all of the cards in the players hand. I liked this much more. 

Then to incorporate more of a feel of Yahtzee, I decided that players should be able to substitute their cards into any of the top 4 number box sets of their choice in any order. This gives them more of a chance to use higher cards and lower cards when they have them for specific rows that those cards would be more valuable for each round. 

Some final touches were made after play testing with Professor Golden and my classmates. These included allowing players to chance any of the cards they have in their hand. I really enjoyed this change because it gives players more risk opportunities. The queen card was introduced as being a wild card during this time as well. I appreciated this idea because I feel like it allows players to more strategic and intentional about where they substitute certain card values into the number boxes. Finally I made a coupe of variations. I came originally came up with the addition and subtraction version of the game. I then decided to toy around with the idea of multiplication and division and made the multiplication and fractions versions.

I think that teachers should play this with their students because it makes basic operations more exciting. I think that allowing students to have so much control over placing values into expressions and solving these is something they will enjoy. I also believe that it allows students to grasp where they may rather place a larger value versus a smaller value. Since the goal is to have the largest total value for each number box set, it will look different for each set. Placing a 9 in the same value that you place a 1 or a 0 has much different affects. 

I believe that this game can be adapted and used for so many reasons. The framework of the rules and rounds is something that creates such a great skeleton to then use with multiple content areas. I have thought about creating a Binomial Boxzee and think that this would be a great next step as well.

Why Play Math Games?

Math can sometimes be a very intimidating subject area for some students. Because of this, I believe that it is important to keep the classroom environment exciting and reassuring that every student has the ability to be a mathematician no matter what level of skills they may think they have. To do this, incorporating games into the classroom can be very beneficial.

Math games are a great resource for teachers to use to introduce and practice content. When playing games in the classroom in allows students to learn content in a more relaxed environment. This allows students to feel less pressure when making mistakes. This is important because students will be more likely to try and continue trying even after making mistakes which will help them master content areas. Similarly, playing these games allows students to build their strategic and problem solving skills. They want to perform their best and win, so they are able to develop strategies that can help them succeed throughout the game.

I also believe math games are beneficial in the classroom because they can be interactive. This allows students to also help each other in teaching the math skills. By not only performing the skills needed for the game, but also using their skills to help teach their classmates they develop a deeper understanding for the content. 

Finally, playing math games allow students to build a love of math. When students are engaged and having fun playing these games, this is when they will be doing the most learning. Exposing students to games that are centered around math subjects, they will be able to see that math is more than just what they may be learning to compute in class.

Now seeing some of the benefits associated with math games, it is also important to identify what makes a good game. One of the biggest things that I believe makes a good math game is having minimal time constraints. When students are practicing their math skills within a certain amount of time some may start to feel discouraged if they are not as fast as their other classmates. With this in mind, choosing games that give students the same opportunity to be successful at completing the game whether they are fast thinkers or need some extra time is very important. 

I also believe that a good math game allows for catch up. This means that even if a student is “down” in a game or is behind, there are aspects of the game that allow the players to quickly catch up and still have an opportunity to win. Since some students may not succeed right away, offering an opportunity for them to catch up and still have a chance to win this makes the game more fun for all players. This also makes students more likely to want to play and in turn allows them to practice and learn without the fear of losing. 

In conclusion, math games being incorporated into the classroom that I urge many educators to try. Not only to practice content, but also to help build up students’ love for the subject and confidence in their own skills.

Escape Dr. Latham's Laboratory

The final game from the 2020 math capstone GAMES class. Begun by Char Beckmann, I was thrilled to teach this course and will soon have my second group. Here's the final entry, an escape room. Ashly was committed early on to make an escape room and I am impressed at the work as well as the perseverance required to finish while student teaching during the pandemic under the conditions .

GUEST POST by Ashly Latham

Escape Dr Latham’s Laboratory is a game I created for my capstone project for MTH 496. When I first heard that we were creating a game I thought of the escape room me and my friends played before the class where we sat down and started talking about creating our game. This inspired me to create an escape-room inspired math game that allowed students to have fun while also doing math!

This game originally started as a fifth grade game but as I got into an accident and injured my hand, I had to postpone the idea. Finally, when I received enough function to get back to work I was a Teacher Assistant in a third grade class classroom. These third graders inspired me to adjust the game so they could play it. 

The puzzles students will complete in this game will focus mostly on multiplication skills as this is what my students were working on at the time I created this. There are ten different puzzles but some puzzles have two answers which allow students to only play six puzzles each play. 

To create this game I searched the internet for fun multiplication puzzles. This included literal puzzles but also some riddles. After I found enough puzzles for what I wanted, I began to construct the scene cards. These tell a story as students work from puzzle to puzzle. The hardest part was constructing the solution wheel so that it was hard for students to guess answers and each answer had a different code. This took me spending lots of time creating my own symbols.

When I tried it out with my third grade students, they LOVED it. I tested out the first couple puzzles and the students were constantly asking where the rest were. After finishing it, I had those same students try it. It took two 30 minute sessions to complete but they sacrificed their recess time just to finish it! They even want to do it again to find the other solutions! 

In the attachment for the game, all of the topics and standards are explained along with how to begin the game and set everything up. Though it’s in color, you can absolutely print it out in black in white. I hope you give it a try as it’s very fun and rewarding! Though carving out an hours time of your classroom might be hard, you could have students do one puzzle a day for a morning or ending activity for the day. 

Ashly's extensive materials and instructions, including for her supper cool solution wheel are here.

In addition, each capstone student picked a good math game to promote with a video. Ashly picked Michael Pershan's Baldermath.

Wimbledon

Quick game idea dreamed up while watching tennis. I'm excited about it, but it's untested. Maybe someone will playtest with me at TMC16?

Wimbledon
2 or 4 players (doubles, naturally)

Materials: deck of playing cards, score sheet.

Set up: deal each player 5 cards. Randomly determine who is serving first.

Tennis scoring:

• In a game, you score love (0), 15, 30, 40, game. But games have to be won by two points. If you get to 40-40 it is deuce (tie). One point from there is Ad (advantage). If the player with ad wins, it's game. If the other player wins it's back to deuce. 
• It takes 6 games to win a set, but you have to win by two games. 
• If you get to 6-6, there's a tiebreaker. The server serves once, then players take turns serving twice. First to 7 points wins the tiebreaker and the set, but, everyone say it, you have to win by two points.

(Game design aside: having to win by two points is one of the best remove first player advantage mechanics ever. The tennis tiebreaker was also a great innovation.)

A match can be one set, best of three or best of five. I recommend playing one set to start.

Playing a point: players draw up to 5 cards. The server can play a card or flip over the top card. Each card played has to be higher than the card before. Players can play two cards together as a sum. If a player can't play or chooses to pass, the other player wins the point. The same player or team serves for an entire game, then the other player or team serves the next game. Continue alternating serve throughout the match. When the deck runs out of cards, shuffle the played cards to continue.

Special cards:
Ace: 1 on a serve (flipped from the deck) 11 any other time.
Face cards: count as 10, but can only be played as part of a sum. However, King beats Queen beats Jack beats 10, so Queen+2 beats 10+2.

Variation: card pairs are multiplied rather than added.

Doubles: in a doubles match, either player on a team can play a card, or play a sum, or a sum can be made with one card from each player.

Notes: The weird scoring is the downside, but kids learning tennis scoring is not a bad thing. And it's a classic for a reason. I'm really excited by the strategy here. When to give up on a point, how to get rid of low cards, which pairs to form and play.

Product Placement

The discussion from two days ago led to another nice observation from Marilyn Burns.

Michael Jerrell suggesting thinking about it as an array. That makes it pretty clear that the products can only be different by 11 if the factors add up to 10.

I had never thought of that before... but that meant that if you decomposed the other way, then you could turn the product into a series.

So 4x6 = 3x5+3+5+1 = 3x5+9 = 2x4+7+9 = 1x3+5+7+9 = 3+5+7+9.

Hmmm, so 5x8 = 4x7+12 = 3x6+10+12 = ... = 1x4 +6+8+10+12.

Which means
MxN = (M-1)x(N-1) +M+N+1 = 1x(difference of M &N+1)+ (that +2)+...+(M+N-1)

But that's an arithmetic sequence.

So something like 23x28=(28-23+1=6)+8+10+...+46+48+50. These are easy to sum - ask any 5 year old Gauss. 23x28 = (50+6)x(# of terms)/2. So how many terms? Whichever is smaller determines, since you're going from 28-23+1 to 28+23-1. So 28-22 to 28+22, difference of 44 by 2s. 22 times up plus the first term... that's ... 23 terms. Hmm. Of course, since we're saying 23x28 = (56)x(something)/2.

Of course! First + last = (N-M+1)+(N+M-1) = 2N. So (first+last)*(# of terms)/2 = (2 x larger) (smaller)/2.

No new product, which I was hoping for when I started, but a nice connection from products to series.

Note if M=N, this is the classic squares are the sum of odds. Since N-N+1=1, N^2= 1+3+5+...+(2N+1), making this a good generalization, too.

Thanks, Marilyn and Michael!

Decimal Games: Burger Time

 My last three games for Mr. Schiller's 5th graders have all been about decimals. They worked pretty well, and I definitely tried new things for myself as an educational game designer.

The first game came when they were first digging into decimal multiplication, just doing whole number times the decimals.  I went through a large number of gyrations about a good context, but I kept coming back to measurement. I thought about a race game, where students measured out multiple decimal portions of a meter or centimeter - and I still think that has some potential. If it was warm, especially, I'd love to see them out with meter sticks on a course they made around the school. Of course, we were in the midst of the harshest winter in 70 years. No one even remembered what the sidewalks looked like. I also thought about physically stacking things, but I wasn't sure about what materials we had to stack. I'd like to see more of what game designers call dexterity games in math.

I finally went with burgers - I suppose with Robert Kaplinsky's In-n-Out burger activity rattling around my skull. The game itself is almost more of an activity. Some students built their burgers and didn't care about the game layer on top of that. Choosing ingredients, making the picture - that was very engaging for most of them.

Build a Burger
Who makes the best burger in town?
 
Materials: 5 dice (or 5 dice per team/player)

Idea: Roll 5 dice to get ingredients for your burger. The numbers correspond to how many mm tall each part is. Three 5s means 3 all beef patties. Two 6s means two layers of bun. You get to reroll one time. Pick the dice you want to reroll.

1 – sauce, .1 cm. Choose from ketchup, mustard, mayo, hot sauce, , barbecue sauce, secret sauce.
2 – cheese, .2 cm.
3 – bacon OR onions (Mix if you have 2 or more)
4 – tomato OR lettuce, .4 cm. (Mix if you have 2 or more)
5 – patty, .5 cm. Maximum: 3 patties, UNLESS you have 3 buns, then you can have 4 patties.
6 – bun layer, .6 cm. Maximum: 3 layers. Every burger needs at least 1 layer of bun!

The choice on the different rolls seemed crucial - their customization increased variety and student ownership. The rerolling mechanic is the child of Yahtzee, of course, but a great way to add choice and chance for redemption.

We launched the game with stories of great burgers with excessive description. I rolled up a burger with help from the class and demonstrated the multiplication. As the students played, they ignored some of the rules - which I see as making the game their own. In the end of class discussion, that gave them plenty of fodder for suggestions for the game. Add another ingredient, let people go meatless or bunless, and - certainly - we should actually make the burgers.

The game was an undeniable hit, and seemed to provide some good experience for multiplication of decimals as groups of decimal quantities. As you can see in the student work below, there was lots of material generated for discussion between student and teacher and students with each other.

And finally, a student who clearly has a future in fast food marketing:

"See, it has meat instead of a bun, but still the regular meat, with the bacon and cheese in the middle."

Here's the form as a pdf if you're interested. Email me for the Word doc if you want to modify.

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.

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?

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.

Multiplying Fractions, Times Three

In a gig with Mr. Schiller's 5th grade last week, I was overcome with indecision.  They'd been working on fraction multiplication and I had three related activities, and asked the teacher to pick based on what he wanted for the class.
The first was just skill practice.  (This is the option Mr. Schiller chose.)   I had never made it, but was confident that you could make a good fraction version of the Product Game.  This is almost what I made:

The only change is that I originally left off 5/12, but the fifth graders convinced me that it should be on.  My thinking was that it would be nice if one unsimplified product was not available, creating a situation where players had to consider equivalent fractions.  But the game itself brought it up enough that I think it's unnecessary.  Click on the image for the full size image which should print properly.

These students have been practicing fraction multiplication, and simplifying and 'unsimplifying' the result.  They have played the Product Game, which is the greatest math practice game ever.  (In the Connected Mathematics Project Prime Time module now, may be from the Middle Grades Mathematics Project before that.)

I launched the game by playing me vs. the class.  Reemphasizing that you only get to change one factor at a time, the goal is to get four in a row, and the new idea that there are equivalent fractions.  If you multiply and get 6/12, you can cover 1/2, and vice versa.  Then the students played pair vs. pair.  At the end we summarized by discussing what they noticed about the game, and what they thought made for a good strategy.  I did point out to them that someone would tell them fraction division was hard, but they've already done it when they're figuring what to multiply 3/4 by to get 6/12.

The second idea was to have the students develop the ability to make sense of their answers through a constructive representation.  Right now, I think the students are mechanically carrying out the multiplication, without much intuition to inform them if their answers are sensible.  These questions are adapted from an activity I do with my preservice elementary teachers.



I like playing the video before the activity, but that is obviously optional.

Potatoes
"Potatoes, mash em, boil em, stick em in a stew.” – Samwise Gamgee.

Things are _______ (awful, bad, okay, good, great) because you have potatoes!  Draw a picture to justify each answer.  Write an equation or number sentence for each story, if you can.

Find how many pounds of potatoes you’ve got if you search the cupboards and find…
1)  1/2 a bag of potatoes, which started with 2 pounds of potatoes.

2)  1/2 a bag of potatoes, which started with 2/3 pound of potatoes. 

3)  3/4 a bag of potatoes, which started with 2/3 pound of potatoes. 

4)  1 ½ bags of potatoes, which each started with 2/3 pound of potatoes.

5)  1 ½ bags of potatoes, which each started with 3/4 pound of potatoes.

6)  4 bags of potatoes, which each started with 1 1/3 pound of potatoes.

7)  2 2/3  bags of potatoes, which each started with 3 ¼ pound of potatoes.

8)  ____ bags of potatoes, which each started with ____ pounds of potatoes.
(You make the problem!)

The numbers are chosen pretty intentionally to allow for some connections and the possibility of relating the quantities to each other.  I like potatoes (of course) because they can be used for a discrete or an area model or a nice casserole.  My plan was to start with problem 2, demonstrating for the class a couple different models, and then have them start on number 1.

The third option was to get at a new context for multiplication.  As anyone following the Keith Devlin multiplication fiasco knows, the prevalent contexts for multiplication involve repeated groups.  One of the other contexts that is often nice for rational numbers is the idea of stretching and shrinking.  That always puts me in mind of Alice, and how her terrific adventures began.

 Go Ask Alice
“One pill makes you larger, And one pill makes you small
And the ones that mother gives you, Don't do anything at all
Go ask Alice, When she's ten feet tall” – Jefferson Airplane

Sort of from “Using Alice in Wonderland to teach Multiplication of Fractions,” Susan Taber, MTMS, Dec 2006

“There seemed to be no use in waiting by the little door, so she went back to the table, half hoping she might find another key on it, or at time she found a little bottle on it, ('which certainly was not here before,' said Alice,) and round the neck of the bottle was a paper label, with the words 'DRINK ME' beautifully printed on it in large letters.”  Alice in Wonderland, Lewis Carroll.

It turns out that she drinks it, and shrinks to 1/6th her former size.  Now she later finds a cake…

“She ate a little bit, and said anxiously to herself, 'Which way? Which way?', holding her hand on the top of her head to feel which way it was growing, and she was quite surprised to find that she remained the same size: to be sure, this generally happens when one eats cake, but Alice had got so much into the way of expecting nothing but out-of-the-way things to happen, that it seemed quite dull and stupid for life to go on in the common way.”  But soon, “Just then her head struck against the roof of the hall: in fact she was now more than nine feet high, and she at once took up the little golden key and hurried off to the garden door.”

It made her grow almost 12 times larger.

1)    What height would she be if at 10 ft tall she took a sip of on-sixth potion?


2)    If she started at 5 ft tall, and then took a sip of one-sixth potion, how tall would she be?  In feet?  In inches?


3)    If she took a bite of ten times cake and then a sip of potion, would she be the same height as what she did, which was take a sip and then take a bite?


4)    Having had one sip and then one bite, how close can she get back to her original size?


Let’s add to the story shall we?  Suppose she finds a times-three cookie, and a one-fourth soda.
5)    Starting at 5 feet, what height does the one-fourth soda make her?


6)    What effect would the times three cookie have, followed by the one sixth potion?


7)    If she starts at 5 feet and wants to be 6 feet tall at the end of it, what should she eat?


8)    In the story she wishes to pass through a 15” door.  If she had the choice of all four magic items, what should she do, starting off at 5 feet tall?


9)    What other mixtures are possible with all four items?


What does this have to do with fraction multiplying? 

What did you learn from these problems?

It's just such an amazing context, and that's without getting into the mushroom, which she uses for more controlled growing and shrinking later on.  Of course when my son is ready for these problems (soon, I think) we'll have to switch the context.

He shrinks, too.  Perfect!

Multiple+Representation=Multiplication

Working with the 4th graders last week, the objective was just to develop multiplication facts, as they are struggling with the multi-digit multiplication.

I'm a big believer in automacity vs strict memorization, as I believe it leads to fluency and solid pre-algebraic thinking, as well as deepening operation understanding. The lesson was pretty simple, but a good place to start.

Objective: TLW see connections between adjacent multiplication facts, and use those connections to help computation.

Materials: unifix cubes, graph paper with a 5x5 structure (link goes to a 2 page pdf graph paper, so it can be printed both sides easily), blank multiplication chart.

Lesson:

Cubes 25-30 min

Start out with a small set of cubes, such as two stacks of three cubes. What multiplication problem is this? (You might choose if you're going to make an issue of order or not. To me, this is 2 of 3, making it 2x3.) This is 2x3 and 2x3 is 6. We're going to pass the cubes around our group.

When you have the cubes, each person can either add a cube to each stack, or add a stack of the same height. Then you say the new multiplication and what the answer is. I add a cube to each stack and say it is 2x4, which is 8.

As the stacks went around, I saw students slowly gaining an idea of figuring out the next problem by adding on to what they knew before. It took a little bit to get the idea of what multiplication computation it was, but they got the idea of what moves were allowable immediately. Soon, several of the students were adding to get the next fact.

We restarted with 3 stacks of 1. The students were much more fluid. There was a bit of an issue with the cubes being distracting with them. If I had thought about working with students who hadn't used the cubes much, I would have given them time to play first, setting up multiplication problems of their choice.

Graph Paper 15 min

The next phase of the lesson was to move to graph paper. We drew a 2x3 rectangle, and the students were comfortable with thinking about that as 2x3. We did one together, where the group decided which side to add squares to. 2x4, 2x5, 3x5, ...

Then each student got their own graph paper and started building a chain of rectangles, with the new dimensions filled in and the result. I saw several students using the adding strategy. One student didn't get the idea of what the connection was, and just drew rectangles and filled in the area. But maybe that was what she needed to attend to.

Multiplication Chart 5-10min

As it was time for students to go, we summarized by looking at a multiplication chart. Filled in a fact they agreed on, 6x5. I led them through how to use that to go on, by adding to get to 6x6 or 7x5. They went back to class with their own chart and a page of graph paper. As I saw them working on the charts in their free time, some were using patterns as they had seen them before, some were using them for the first time, and one student asked for how that worked. A couple of examples got her started.

The next week: The week after this I tried to get the students to help me develop a game. A couple of them found it less than engaging, but Mrs. B mentioned that all the students were antsy. Day before a long weekend? Cabin fever? The game is designed to become obsolete, but I don't think that's the issue. I picked it because I've wanted to work this out, and the other thing they've been working on in class is area and perimeter of compund rectangular shapes.

Break Up (In development)

Two players or teams.
5-structure graph paper, pen, optional dice.

Game play: Determine the size of a starting rectangle. This can be done through choice, each team choosing a side length, or rolling three dice, or rolling four dice, or rolling 2 dice plus 10. If dice rolling, each team should roll one side.

On your team's turn, you either divide a rectangle, or calculate an area, or do both. Your team gets a point whenever an area is filled in. After all the areas are filled in, the team's whose turn it is next gets to try to find the total area. To emphasize using known facts, you can only fill in a rectangle if you know the area as a fact.

Examples: you determine a 12x15 rectangle. The first time divides the 12 into 10 and 2, and fills in 10x15=150. The second team divides 15 into 10 and 5, and fills in 2x10=20. The first team fills in 2x5=10. The second team finds the total, 150+20+10 and gets 210. So 12x15=180.

You determine a 9x12 rectangle. The first team sections off a 6x9, and doesn't know that as a fact. (There was one student who loved dividing in half.) The second team split off 5x9 and filled in 45. The first team filled in 1x9. The second team filled in 6x6 as 36. The first team (fudging a bit) figured 3x6 with 12+6. The second team mis-added 45+9+36+18 (hard sum!) and the other team got 108.

Notes: I thought the game would be better as a cooperative game, but the kids wanted to try it with points. They thought about scoring the area (as I have) but that gives the first team too big an advantage. They looked forward to scoring points, but didn't seem to care much about winning. They thought maybe you should keep track of points across multiple games. It did strongly encourage mental computation.
There's not much strategy to this game. It's about tic-tac-toe level that way. I could see this turning into kids designing their own board of compounded rectangles, that might be interesting. But it definitely encourages relational thinking for multiplication facts, which is worthwhile. If anyone has ideas for improving the gameplay, I'd love to hear them.

EDIT:
Sue VanHattum, from Math Mama Writes, was reminded of a game called Raging Rectangles from a North Carolina instructional resource packet. See the comments for details.