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Archive for the 'lessons' Category

In my modeling workshops this summer, we first modeled the money duck, asking ourselves, what would be a fair price for some money buried inside a soap shaped like a duck? We learned how to use the probability distribution model and define its expected value. We developed the question of expected value before answering it.

Then the blogosphere’s intrepid Clayton Edwards extracted an answer from the manufacturers of the duck, which gave us all some resolution. For every lot of 300 ducks, the Virginia Candle Company includes one $50, one $20, one $10, one $5, and the rest are all $1. That’s an expected value of $1.27, netting them a neat $9.72 profit per duck on average.

That’s a pretty favorable distribution:

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They’re only able to get away with that distribution because competition in the animal-shaped cash-containing soap marketplace is pretty thin.

So after developing the question and answering the question, we then extended the question. I had every group decide on a) an animal, b) a distribution of cash, c) a price, and put all that on the front wall of the classroom – our marketplace. They submitted all of that information into a Google form also, along with their rationale for their distribution.

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Then I told everybody they could buy any three animals they wanted. Or they could buy the same animal three times. (They couldn’t buy their own animals, though.) They wrote their names on each sheet to signal their purchase. Then they added that information to another Google form.

Given enough time, customers could presumably calculate the expected values of every product in the marketplace and make really informed decisions. But I only allowed a few minutes for the purchasing phase. This forced everyone to judge the distribution against price on the level of intuition only.

During the production and marketing phase, people were practicing with a purpose. Groups tweaked their probability distributions and recalculated expected value over and over again. The creativity of some groups blew my hair back. This one sticks out:

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Look at the price! Look at the distribution! You’ll walk away a winner over half the time, a fact that their marketing department makes sure you don’t miss. And yet their expected profit is positive. Over time, they’ll bleed you dry. Sneaky Panda!

I took both spreadsheets and carved them up. Here is a graph of the number of customers a store had against how much they marked up their animal.

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Look at that downward trend! Even though customers didn’t have enough time to calculate markup exactly, their intuition guided them fairly well. Question here: which point would you most like to be? (Realization here: a store’s profit is the area of the rectangle formed around the diagonal that runs from the origin to the store’s point. Sick.)

So in the mathematical world, because all the businesses had given themselves positive expected profit, the customers could all expect negative profit. The best purchase was no purchase. Javier won by losing the least. He was down only $1.17 all told.

But in the real world, chance plays its hand also. I asked Twitter to help me rig up a simulator (thanks, Ben Hicks) and we found the actual profit. Deborah walked away with $8.52 because she hit an outside chance just right.

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Profit Penguin was the winning store for both expected and actual profit.

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Their rationale:

Keep the concept simple and make winning $10s and $20s fairly regular to entice buyers. All bills – coins are for babies!

So there.

We’ve talked already about developing the question and answering the question. Daniel Willingham writes that we spend too little time on the former and too much time rushing to the latter. I illustrated those two phases previously. We could reasonably call this post: extending the question.

To extend a question, I find it generally helpful to a) flip a question around, swapping the knowns and unknowns, and b) ask students to create a question. I just hadn’t expected the combination of the two approaches to bear so much fruit.

I’ve probably left a lot of territory unexplored here. If you teach stats, you should double-team this one with the economics teacher and let me know how it goes.

This is a series about “developing the question” in math class.

With school starting up, I thought I’d share the most interesting icebreaker I found last year. Copy, cut, and pass out these half-sheets [pdf].

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Each person in a group picks a dot and writes her name next to it.

Now the group’s job is to label the axes. Physical attributes don’t require all that much thought and don’t reveal all that much, so don’t allow them.

That’s it. It requires a surprising amount of creativity and conversation. Happy first day of school, teachers.

Previously. This Who I Am sheet, which I adapted from my first student-teaching placement, has been popular.

[I got this particular idea from a workshop I led with Jo Boaler, Kathy Sun, Jennifer Ruef. It may be a Complex Instruction staple for all I know. I’m not claiming ownership, just passing along the fun.]

2013 Aug 7. I am informed by Marty Joyce this icebreaker is from the College Preparatory Mathematics series.

2013 Aug 9. Rachel Rosales used the Math Forum’s Noticing / Wondering framework on her first day.

Featured Comments

Laura Hawkins:

I learned this from Carlos Cabana, at the Creating Balance conference at Mission HS in SF. I’ve used it with much success for a couple years now.

Be aware, however, that it can surface issues you might not be prepared to deal with. I teach at a private school, and there is huge income inequality between my kids’ families. This year, a group labeled one axis “number of bathrooms in my house” with the two quantities being 4 and “more than 4″. In my surprise, I don’t think I handled it well to support the kids in the class who might have just 1 or 2 bathrooms in their house and suddenly had their lack of wealth put in their face in math class.

Cathy Yenca:

I’ve used your “Who I Am” sheet for several years – after reviewing them, I file them away, then surprise students on the last day of school by returning them. They giggle at how much they’ve changed in 9 short months, often barely remembering filling out the page in the first place. The “self-portraits” are always a favorite to revisit.

Nathan Amrine:

Examples:

Desire to read
# of hours spent swimming
# of summer vacations
Love of Ice Cream
# of pets
Distance from school

2013 August 13. First day activities around the web from:

Featured Email

Joey Warren, with a nice modification:

I thought I would share with you an activity that I stole from you and adjusted a bit which worked out very well for my first day freshmen Alg students. I read your Personality Coordinates Ice breaker activity and made a little twist. I put the x and y axis on the ground and used the classroom for students to move around. I gave them different labels for the x-axis and y-axis and let them move around the room and meet each other and see who was similar to them and who was different. I used labels like desire to read, distance traveled this summer, and # of hours you sleep each night. It lead into some good movement and good discussions of where students traveled over the summer, what they read, etc. The best graph was “love for math” vs “desire to get better at math.” After they moved around I categorized students into ones that were going to be my project to get them to like math, ones that would be easy to teach, and the rest I was going to push further away from the origin to get them to love math more.

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?

https://twitter.com/ultrarawr/status/322538190185578496

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.

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.

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