Category: lessons

Total 94 Posts

[Confab] Tiny Math Games

Jason Dyer writes a very important post highlighting Tiny Games, a listing of games you can play quickly, almost anywhere, with only limited materials. He then pivots to ask about tiny math games.

Could one make an all-mathematics variant — mathematical scrimmages, so to speak?

His post, and Tiny Games, are important because they reject an article of faith of the blended learning and flipped classroom movements, that students must learn and practice the basic skills of mathematics before they can do anything interesting with them.

For example, here’s John Sipe, senior vice president of sales at Houghton Mifflin Harcourt, talking about Fuse, their iPad textbook:

So teachers don’t have to “waste their time” on some of these things that they’ve always had to do. They can spend much more time on individualized learning, identifying specific student needs. Let students cover the basics, if you will, on their own, and let teachers delve into enrichment and individualized learning. That’s what the good teachers are telling me.

This is a bad idea. People don’t mind practicing a sport because playing the sport is fun. It’s easy to tell a tennis player to practice 100 serves from the ad side of the court, for instance, because the tennis player has mentally connected the acts of practicing tennis and playing tennis. The blended learning movement, at its worst, disconnects practice and play.

Take multiplication of one- and two-digit numbers for instance.

If you need to learn multiplication facts, one option is to watch a video and then drill away. Or we can queue up all that practice in a tiny math game that’ll have students playing as they practice:

Pick a number. Say 25. Now break it up into as many pieces as you want. 10, 10, and 5, maybe. Or 2 and 23. Twenty-five ones would work. Now multiply all those pieces together. What’s the biggest product you can make? Pick another. What’s your strategy? Will it always work? [Malcolm Swan]

Easy money says the student who’s practicing math while playing it will practice more multiplication and enjoy that practice more than the student who is assigned to drill practice alone.

Jason Dyer helpfully highlights two examples of tiny math games, Nim and Fizz-Buzz, but he and I are both struggling to define a “tiny math game.” The success of the Tiny Game Kickstarter project indicates serious interest in these tiny games. I’d like to see a similar collection of tiny math games. Here’s how you can help with that.

1. Offer Examples of Tiny Math Games

This may be tricky. We all have games we play in math class. What distinguishes those games from “tiny math games?”

2. Help Us Define “Tiny Math Games”

This may be a better starting point. I’ll add your suggestions to this list. Here are some seeds:

  • The point of the game should be concise and intuitive. You can summarize the point of these games in a few seconds or a couple of sentences. It may be complicated to continue playing the game or to win it, but it isn’t hard to start.
  • They require few materials. That’s part and parcel of being “tiny.” These games don’t require a laptop or iPhone.
  • They’re social, or at least they’re better when people play together.
  • They offer quick, useful feedback. With the multiplication game, you know you don’t have the highest product because someone else hollers out one that’s higher than yours. With Fizz-Buzz, your fellow players give you feedback when you blow it.
  • They benefit from repetition. You may access some kind of mathematical insight on individual turns but you access even greater insight over the course of the game. With Fizz-Buzz, for instance, players might count five turns and then say “Buzz,” but over time they may realize that you’ll always say “Buzz” on numbers that end in 5 or 0. That extra understanding (what we could call the “strategy” of these tiny math games) is important.
  • The math should only be incidental to the larger, more fun purpose of the game. I think this may be setting the bar higher than we need to, but Jason Dyer points out that people play Fizz-Buzz as a drinking game. [Jason Dyer]

What can you add to our understanding of tiny math games?

2013 Apr 17. Nobody wanted to tackle the qualities of tiny math games, which is fine since you all threw down a number of interesting games. I’ll be compiling those on a separate domain at some point soon.

Featured Tweets

Jason Dyer elaborates on his contribution above.

2013 Apr 24. Jason Dyer elaborates in another post.

2014 Oct 30. Eric Welch shares his thinking around tiny math games from his Masters thesis.

2014 Oct 30. Julie Reulbach lists several games in a Google doc.

Your Contributions

Number Sense & Operations

David Petro:

Basically you get groups of three. Two students grab a card from a deck and without looking at them put them on their foreheads facing out. The third student multiplies the two numbers and states the product. Those holding the cards then try to guess the two numbers.

YouCubed:

This game is played in partners. Two children share a blank 100 grid. The first partner rolls two number dice. The numbers that come up are the numbers the child uses to make an array on the 100 grid. They can put the array anywhere on the grid, but the goal is to fill up the grid to get it as full as possible. After the player draws the array on the grid, she writes in the number sentence that describes the grid. The game ends when both players have rolled the dice and cannot put any more arrays on the grid. How close to 100 can you get?

Nat Highstein:

Create a 5 x 5 grid of cards, in which every row and column adds to 31. We decided as a group that J, Q, and K were worth 10, Aces were worth 1, and all of the other cards were worth their face value.

Jonah:

A Tiny Math Game I used to play on the train: take the car number (usually 4 or 5 digits) and add operations between the digits and an equals sign (somewhere) to make a true equation. Try to come up with as many different solutions as possible.

William Carey:

You need a special deck of cards, but it’s an easy deck to make:

ten cards with a “2” on them
ten cards with a “3” on them
ten cards with a “4” on them
two cards with a “7” on them
two cards with a “11” on them

Each player draws four cards. They multiply their hand together, and announce the only the product (!) to the group. They then play go-fish.

Dan Anderson:

Write today’s date with just the number 4 and math operations.

Jeanne Bennett:

Krypto- given 5 random numbers (use a card ddeck, krypto deck or random numbers under 16) add, subtract, multiply and/ or divide to find a 6th random number. Good for strategy development as well as fact practice. I have used it in teams and as individuals.

Raj Shah:

Divisimainders. First player chooses a secret number. The other tries to guess the number by asking if it is divisible by a number, x. The first player only responds with the remainder when the secret number is divided by x. Play continues until the second player successfully guesses the secret number. Goal is to get it in the fewest number of guesses.

Marty Romero:

Taxman is one of my favorites – it can be played many times within some fraction of your class period.

The game consists of a set of numbers – I use 1 through 40. In the game, the numbers are represented as money. You and someone else take turns choosing numbers.

If you choose a number, you get the number of points equivilant to that number. Your opponent (the Taxman) gets the factors of that number. When the Taxman chooses, you get the factors of his choice. Numbers that are chosen go to each player’s respective side of the board. Once a number has been chosen or is a factor of a chosen number, it is removed from play.

Whoever has the most points (chosen numbers plus factors) after the last number is chosen, wins.

ecvluic:

My boys play a factoring game in the car on rides: Start with a number between 1-100, but not even. The other one can then pick a multiple of that number, or a factor. Continue, no repeating numbers, until one of them cannot take a turn. There is strategy, they do a lot of multiplying and dividing, and once they’ve done it a couple of times, they get bored with trying to win, so they help each other try to get the longest game possible.

Andrew Stadel:

Game 1:
Players: 3
Materials: one deck of cards.
Cards are assigned Black Jack values (aces=11).

Person1 (referee) hands one card to player1 and one card to player2 so each player can’t see the value of their own card. Without looking, each player puts the card on their forehead so their opponent can see the value. The referee adds the two visible cards and announces the sum to the two players. Each player has to figure out the value of the card on their forehead. First player to correctly state answer, wins. (*contestants can only say one number). Play a few rounds or best of three/five/seven. Switch roles and the winner is now the referee.

The game can also be used for multiplication where the referee announces the product of the two visible cards and each player has to deduct the value of their card.

Increase the level of challenge for both sum and product by making the black suits positive and the red suits negative.

Adam Poetzel:

Another one is called “strike”. Each player starts with a ten pin bowling set-up, each pin numbered 1 – 10. Then 4 dice are rolled (dice don’t have to be standard 6 sided ones). The player’s goal is to “knock down” as many pins as possible. You knock down a pin by using the 4 numbers shown on the die, in any order and with any operations and grouping symbols, to arrive at the number on a pin. For example, if you rolled a 4,5,7, and 8. You could knock down pin #1 by writing (8-7)/(5-4), and pin #2 by (8-7)+(5-4), … The winner is the one who knocks down the most pins.

Number Boggle.

Graph Theory

Erin Gilliam:

The Game of Sprouts! This is a quick game that requires nothing more than a pen, paper and a partner. Based on graph theory the game of sprouts is easy to learn and easy to play, but also has opportunities for higher level analysis.

If students don’t know graphy theory terminology, vertex can be replaces with “dot” and edge can be replaces with “line”. The word degree can be explained as the number of lines emanating from the dot.

The rules:

Start with a finite number of vertices.
Connect two vertices with an edge, and add a new vertex to the edge you just created.
An edge cannot cross over an existing edge.
A vertex is “dead” once it has a degree of three. You can no longer play off this vertex.
The winner is the person who makes the last possible move.

Inductive Reasoning

Jason Baldus:

With a deck of cards – Eleusis: http://en.wikipedia.org/wiki/Eleusis_%28card_game%29

One person thinks of a rule about what cards can be played. For example, “alternate even and odd numbers” or “only even numbers can follow a black suit and only odd numbers can follow a red suit”

Players try to discover the rule inductively, by trying to play a card. The rule thinker-upper tells them whether the card they tried to play is a legal move or not and the table keeps track of the attempts. If you make an incorrect guess, then you put your card in a row of bad guesses and draw another card. If you make a correct guess, your card is placed on the play pile, but you don’t draw another. The goal is to be the first to run out of cards.

It takes a while to explain at first, but once the game is in your repertoire it can be started and played pretty quickly. It also has the benefit of having students choose secret rules that are at a complexity level that they are comfortable with. It’s really fun.

Functions

Taylor:

1. Class splits into teams of four. (Or whatever. I liked four.)
2. A team gives me a number which I evaluate for some function either in my head or on my graphing calculator–depending on the complexity of the function.
3. I tell them the output. (I did not write it down to encourage participation.)
4. The team that gave me the number gets the first chance to predict the function. If they pass or are incorrect, all other teams may raise their hands and volunteer an answer.
5. Proceed clockwise from the first team.
6. Correct functions score points for a team. I began at 100 points for a correct guess after one output and decreased by 10 points after each output given. Incorrect answers took away 50 points to discourage random guessing.

Probability

J.D. Williams:

We play a quick game called “Never a Six”. You need one die. We usually play half the class against the other half, or boys v. girls or something like that. I’ve had students play individually, but they don’t get as involved in the games this way. We talk about probability when playing (sometimes).

Each person gets a turn rolling the die. If they roll a 1 – 5, they add that to their turn total sum and decided to continue rolling or end their turn. If they roll a 6, their turn total is 0 and it is the other teams turn. Players can choose to stop rolling at any time and let the other team begin their turn. When students decide to stop their turn, their team gets to keep the turn total sum and add it to the group total sum. The group with the highest total group sum after everyone has had a turn wins.

Memory

Raj Shah:

Blind tic-tac-toe. Fill in the boxes of the board with the digits 1-9 in any order. Spend 30 seconds memorizing the pattern. Never look at the board again. Call out the position you want to place your X or O by using the numbers. You must keep track of where all previous moves have been made. Not so much math, but it’s tiny, fun, and builds your working memory

Coordinate Plane

Phil:

VECTOR RACES. Have a basic track printed on grid paper. Pupils guess the vector they should translate their car by and then draw this vector and slide the car along it. If they hit the sides they miss this go. If they draw the vector wrong or try to cheat and their opponent notices then they also miss a go.

Estimation

Adam Poetzel:

Another game that can be applied to many math concepts is “Mental Math Golf”. You play 9 “holes”. Each “hole” gives you a problem (ex: What is 8% of 70?, what is square root of 90?). Using mental math/estimation strategies, each player gives their best guess. Their score on the hole is the percent error of their guess and the actual answer. Thus a perfect score on a problem would be a zero. After 9 holes, you average your percents, and the lowest score wins. You can play 9 hole courses that consist of concepts like percents, radicals, logs, missing side length of a right triangle, etc.

Factoring Trinomials

Bethany:

Ok, one that really is ‘tiny.’ I use it to practice skills for factoring a trinomial by grouping. Draw an X and put two numbers in the X. The rule is that the two side numbers have to add to = the bottom number and multiply to equal the top number. So, for example, -54 on top and 3 on the bottom… the students must figure out that the side numbers must be 9 and -6.

Miscellaneous

Jim Pardun:

CALENDAR GAME (HEAD TO HEAD)
Object: Force your opponent to say “December 31″
Rules:
– The game begins with a date in January.
– Players take turns increasing either the month or the number
but not both.
– You may skip over months or numbers however you cannot
move backwards or wrap around back to the number 1.
– The player who says “December 31″ loses the game.
– Can you figure out the strategy?

Geoff Krall Combs The Internet For Lesson Plans So You Don’t Have To

If the deluge of interesting problem-based material on the Internet overwhelms you, as it does Jonathan Claydon, Geoff Krall’s curriculum maps are a great place to start. He’s taken the Common Core’s scope and sequence documents and combed the Internet for items that fit. He’s included a few of my own items, some items from the Shell Centre, along with a lot of great lesson ideas I’d completely forgotten. Bookmark it. Throw him some love in the comments.

Teach The Controversy

I was walking with my wife along the River Corrib in Galway last weekend when we got into an argument that lasted the rest of the walk. I’ll present our two arguments and some illustrative video. Then I’d like you or your students to help sort us out.

Argument A: It would be much harder to swim to the other side of the river in the fast-moving water as in still water.

Argument B: It would be just as easy to swim to the other side of the river in the fast-moving water as in still water.

I hope this gets as out of hand for you and your students as it did for me and my wife.

Featured Comment

Scott Farrar:

This excellent question exhibits a quality that is not found often in math curricula: it has the “specificity sweet spot”: it is specific enough for a student to answer, but non-specific enough for every kid to agree on the answer. Students making different assumptions will have different responses, thus creating a real mathematical argument.

Odds / Ends

  1. Bill McCallum runs a “standard of the week” contest through his Illustrative Math Project, the goal of which is to illustrate what the different Common Core standards look like in student tasks. I submitted a task for 8.4.F, linear modeling, which was accepted. It’s called “Graduation” [pdf].
  2. Key Curriculum Press posted the Ignite talks from CMC-South. I did five minutes on the question, “When will I ever use this in the real world?

2012 Feb 18. Patti Smith used this task in class. Fun feedback:

My students finally understood the meaning of y-intercept as something more than “when it all began”. They also understood slope to mean rate – or how fast they read the names – rather than rise over run!

Shoulda Woulda Coulda

Two things I’d do if I were still doing the job instead of just talking about it:

Set Up The Expected Value Spinner

I don’t think people who understand expected value understand how hard it is for other people to understand expected value.

Let’s say I roll a die. I ask if you want to bet on an even number coming up or a five. You’re bright. You pick the even number. It has a 3/6 shot versus a 1/6 shot for the five. But what if I said I’d pay you $150 if the even number comes up and $600 for the five. What if I said I’d keep on giving you that same bet every day for the rest of your life? This is where expected value steps in and puts a number on the value of each bet, not its probability. The expected value of the even number bet is (3/6) * $150 or $75. The expected value of the five bet is (1/6) * $600 or $100. The five bet will score you more money over time.

This is tricky to fathom in gambling where superstition rules the day. (“Tails never fails,” betting your anniversary on the pick six, blowing on the dice, etc.) So one month before our formal discussion of expected value, I’d print out this image, tack a spinner to it, and ask every student to fix a bet on one region for the entire month. I’d seal my own bet in an envelope.

I’d ask a new student to spin it every day for a month. We’d tally up the cash at the end of the month as the introduction to our discussion of expected value.

So let them have their superstition. Let them take a wild bet on $12,000. How on Earth did the math teacher know the best bet in advance?

BTW: You could make an argument that a computer simulation of the spinner would be better since you could run it millions of times and all on the same day. My guess is that your simulation would be less convincing and less fun for your students than the daily spin, but you could definitely make that argument.

Host A Steepest / Shallowest Stairs Competition

Tonight’s homework: Find some stairs. Calculate their slope. Describe how you did it. Take a picture.

Your students should then determine whose stairs were the steepest and the shallowest and you’ll post those photos at the front of the classroom. You’ll make a big fuss over them. Then you’ll post a bounty for stairs that will knock them off their perch.

One interesting thing about slope is that it doesn’t have a unit, so you don’t need a measuring tape or a ruler to calculate it. Anything your students have on hand will work, including their hands.

Be prepared for a contentious discussion about the difference between the tallest steps and the steepest steps. It’s possible to design steps that are extremely shallow but too tall for anyone to climb up. Wrap your students’ heads around that one.

Be prepared also for students who can’t shake the sense that math is here every time they climb up a new set of stairs.

What a cool job the rest of y’all have.

[photo credits: moyogo, vulcho]

2012 Jan 17: Useful description and modifications from James Cleveland.