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.

Don't punt; take skills into the end zone

In 2011, the National Assessment of Educational Progress (otherwise known as ``The Nation’s Report Card’’) asked fourth graders to calculate the following:

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The results were 83%, 64% and 52% correct, respectively.  Why was performance on the last task, arguably the easiest of the three, the worst?  Answer: probably because it was the only one in which calculators were not allowed.

Read more »

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.