Mad Math, or Math Libs

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car

Did you ever play Mad Libs? I loved to play this game on long car rides when I was a kid. You could get books of them in the drug store, and best of all, your parents didn’t mind spending the money to get you a whole package, because it was “educational”!

Now the game has a new online incarnation: http://www.madlibs.com/, and you can even find an app to play it.

In Mad Libs there is a leader, who asks everyone else to give them words to fill in the blanks — but the leader does not tell the rest of the group the story until all the blanks have been filled in! Once the blanks are all filled in, the leader reads the story to much hilarity.

I created my own story, with a twist — it has numbers at the end that students also have to fill in. When my students finish reading out the story, they also read out and do the problems they have created. The particular problems you’ll see below involve factoring, but could be changed to suit any topic. The great thing about this game is that it brings in topics from English (interdisciplinary!) and story telling. It gets students laughing and more ready to do the problems, and it allows students to create their own problems.

Mad Math: Factoring Frenzy

 Directions

  • The group leader does not show the group this piece of paper!
  • The leader asks each person in the group in turn to contribute a word, letter or number until all the blanks are filled in, including the number blanks for the factoring problems.
  • If a person gets stuck on a word, they can use one of the ones on the board.
  • Then the leader reads the story and the group works out the problems.

My ___________ subway ride started when a giant  ___________   _____________ up from the subway               adjective                                                                    animal         verb ending in –ed               

and into the ____ train.  People were  ___________, but I got a ___________, so I was ___________.

                  letter                                    verb ending in –ing               noun                                adjective

When I got to school, my ___________ professor would not ___________my excuse and said that if

                                                       adjective                                        verb

was late one more time, I would get a ____. What a ___________ day! Luckily, I found out that if I could

                                                               letter                    adjective

do these ___________ factoring problems, everything would be ___________!

                   adjective                                                                                 adjective

Factor:                             Caution: one of the problems is not factorable!

 

  1. x2 + 3x – ___                                             2.  x2 –  ___x + 25                                 3.  x2 + 12x +  ___

an integer between 3 and 5          an  integer between 9 and 11      a perfect square betw 30 &40

 4. x2 – ___                                                16x2 –  ___                                6. x2 +  ___

any perfect square                              an odd perfect square                              any perfect square 

Bonus: change the problem that is not factorable into one that is.

The word file here: mad-math-example gives you a better copy, plus some signs I made up to put around the room so that students would know what an adjective, adverb and noun were.

I invented this game at a What’s Your Game Plan workshop, with the help of Joe Bisz, Carlos Hernandez and Francesco Crocc. Much thanks, you guys!

Barely a game

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carddeck

I’ve been playing around lately (pun intended) with something I’ll call, “barely a game.” It’s closely related to a similar concept which I’ll call, “A really dumb game that’s fun anyway.” More on that second idea in my next blog post.

Barely a game is an activity that seems game-like because it has some game-like components but that’s missing something essential to make it a true game. For example, it might have randomness or game tokens or play money or scoring (all great signifiers that a Game is Happening), but no clear winners and losers.

My latest foray into “barely a game” with my algebra students had two great game elements:
1. Randomness (through randomly dealt cards) and
2. Competing teams.

Game play, round 1

I put the expression  expr1 up on the board and asked my students what they thought it would simplify to.

They all pretty much agreed on 1, with some great reasoning as to why. (We started with “they cancel” and managed to get to the much more mathematically sophisticated “the same thing over the same thing is 1.”)

Then I showed my students some cards I had hastily created on half sheets of paper.

cards

I asked them which values of x would be easiest to plug in, and we all agreed that 0 would be best, with 1 as a close second. I shuffled the cards and gave each team a card, wishing them luck in getting the “best” one. Each team had to evaluate the expression  for their value of x. We noticed that all the expressions came out to 1, as predicted by what we thought would simplify to.

Game play, round 2

Next, I told students to quickly trade their card with another team – maybe they would get an easier number this time!

Each group then evaluated the new expression expr2. Students were surprised that the expression simplified to 2 for every single group! We went over how to factor this expression to expr3, which, since it was 2 times our previous expression, made sense would simplify to 2.

 

Game play, round 3

One more shuffle and trading of cards! Now each group evaluated the expression expr4

We discovered that each group got a different answer! This lead us to the conclusion that this expression would not simplify. Discussion then turned to whether you could cancel over addition, which in turn lead to my favorite meme:

Every time you do this a kitten dies

I have three cats, so I had to assure everyone I was not planning on sacrificing a kitten any time soon. But I do love this meme so much, and it made my student laugh.

So that was my “barely a game”! Why wasn’t it really a game? Because, although it was sort of pitched as a contest (which team will get the easiest number?), there was no actual competition, nor even any scoring.

Could it be changed into a “real” game? Probably, and with not much tweaking. But I have to say that I really liked it this way. It was fast, it was fun, I didn’t have to prep that much to play it, and I’m not really fond of the winner/loser aspect of games in an educational setting anyway.

In addition, it’s versatile — I could see modifying this for calculus or for arithmetic.

For calculus, we might try plugging various similar expressions into the definition of a derivative formula.

In arithmetic, we might try the following cards, to explore what happens when we multiply or divide by powers of 10:

10cards

That concluded this blog post! Stay tuned for the next one, when I talk about how an awful game can be really fun. smiley

 

Calculus: Art on the Wall Game (Teena Carroll, Saint Norbert College)

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This is a great calculus game that I saw demonstrated at the MAA/AMS joint conference in Boston in January. It was created by Teena Carroll of Saint Norbert College.

Students are in groups of 4, each with a post-it note. On the post it note, each student draws an arc that goes from one corner of the post-it to the opposite corner:

Student 1 is then asked to position their post it so that is is concave up and increasing, student 2 so it is concave down and decreasing, student three so that it is concave up and decreasing, and student 4 so that it is concrete down and decreasing.

The group then links their post-its together on the wall, in any order, and identifies points of discontinuity and inflection points.

I’m not teaching calculus this semester, so I played this game with a student I am tutoring. I was wowed at the way the game teases out the difference between concave up (positive second derivative) and increasing (positive first derivative). I’m looking forward to playing it with a whole calculus class!

Finally, I wonder if this is a game, really, or is it art? Or is it not art, but just great math? Whatever it is, it’s certainly a lot of fun and a great learning tool.

Bizz Buzz for Base Systems

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numbers

A simple game for learning base systems illustrates many of the connections between game based learning and other pedagogies. This game can be played in a liberal arts or mathematics for elementary education class. The game is a variant of Bizz Buzz, often played as a drinking game.

Students sit in a circle and count off – one, two three, four. The fifth person, instead of saying five, says “bizz.” The count continues – one, two, three, four, bizz-bizz, one, two, three, four, bizz-bizz-bizz, one, two, three, four, bizz-bizz-bizz-bizz. After this (four bizzes), the count changes — one, two, three, four, buzz.

This is a base 5 counting game, with 105, or 5, represented by bizz, and 1005, or 25, represented by buzz. The game typically engenders much laughter as students who are not quite paying attention say 5 instead of bizz, or bizz instead of buzz. Students help each other to say the right word, “Say bizz!” they call out to the confused fifth person. But the game is not too hard, and soon everyone gets the hang of it.

Explicit connections can then be made between the game and the notation for base 5. For example, the seventh person is bizz + two = 125 in base 5. The connection can also be made to base 5 manipulatives — units, 5-unit rods, and 25-unit squares.

The game can later be played in a different base, to extend the difficulty level and to deepen understanding. I like to ask my students “how would you play this in base 7?” and they can quickly come up with the new rules.

The Spread of a Rumor or Virus

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rumor

This game introduces students to the concept of exponential growth. It can be played as the spread of a rumor, or the spread of a virus, and works well in an algebra or modeling course, in a quantitative reasoning course, or a liberal arts mathematics class.

Each student gets a card, labeled “Round 0 ____, Round 1 ____, etc.” On one student’s card, there is a yes next to round 0, while on the rest of the cards, there is a no. The student with a yes is the student who “knows” the rumor or who has the virus.

Students are instructed to stand up and mill around. In each round, they must look at one other person’s card. If that person’s card has a yes, the student who did not have a yes now has one, while everyone else writes no – without saying anything about which they have on their card. After enough rounds so that everyone has a yes (for a class of 35, this is usually about 6 rounds), students sit down and a chart is made of how many had a yes at each round. Connections are then made to doubling, and to powers of 2, which then leads to a discussion of exponential growth.

Note that the growth modeled here is actually logistic, since there is a limit to the number who will have the rumor or virus, but if the game is played only up to a certain number of rounds, it mimics plain exponential growth nicely – as does the spread of a rumor or virus in a large population. The game can later be played with different growth factors, such as introducing some amount of immunity (a person only gets the virus after being exposed twice, or three times) or increased virulence (each person shows two or three people their card, on each round).

The Spread of a Rumor can be seen as a simulation, rather than a game, although the distinction between a simulation and a game is often only a matter of semantics. However, for serious 18-year olds, it can be problematic to be seen “playing” – whereas older students and future teachers do not seem to mind as much.  I usually introduce this one without saying the word “game.”

NIU-Torcs for Numerical Methods

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car

See the game video: https://www.youtube.com/watch?v=LYGwaI-haOM

NIU-Torcs is an example of a college-level mathematics game that allows for deeper learning within the game. Brianno Coller and colleagues developed the game through an NSF grant to help their mechanical engineering students learn numerical methods (Coller & Scott 2009). Students begin the game by learning how to code acceleration and steering using the programming language C++.  They then move to making the car move fast without skidding off the road, by calculating numerical roots, solving systems of linear equations, and doing curve fitting and simple optimization. The authors report that students are motivated to keep trying far more than when given these types of problems as meaningless homework exercises. Concept maps produced by the students in both the game-based and traditional classes showed that although measures of low-level knowledge were statistically identical, students in the game-based class had much greater levels of deep thinking, which included being able to compare and contrast methods and link concepts together.  In addition, student attitudes about the class had changed – they were more engaged, and more able to recognize the value of the mathematics they were doing.

 

Japanese Ladder Games for Combinatorics

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Three fascinating games from Steven T. Dougherty and Jennifer Franko Vasquez at The University of Scranton appear as “Ladder Games” in MAA Focus, June-July 2011. Mathematicians will enjoy the group theory and combinatorics used to explain the way the games work, while their students will have fun with these games as an intuitive introduction to the concepts.

The authors write that the three games, “…can be played by anyone regardless of their level of mathematical sophistication. When you are done playing these highly addictive games you will have a deeper understanding of permutations and group actions and you will learn a very interesting connection to coding theory. We have found that the games are very enjoyable to play and that players end up understanding complex mathematical ideas without realizing they have done so.”

https://sites.google.com/site/professorstevendougherty/games

Bingo in a Math Lecture

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bingo

This game could work in any definition-heavy math or science class lecture. In statistics class, hypothesis testing is traditionally one of the most difficult units, filled with new vocabulary and difficult concepts. To alleviate tension introducing this topic, I used bingo cards, with the vocabulary words randomized on it.

See the full story in the Faculty Focus.