Thursday, August 23, 2012

Gummy Bear Revisited

The giant gummy bear problem has been floating around the blogosphere for a while. When I first saw it, I knew I wanted to use it. I finally have the perfect opportunity in Foundations of Mathematics 20 this year. (Saskatchewan Curriculum).

History of the Problem (As far as I know)
  • Originally presented by Dan Anderson here. Included original Vat19 video and driving question about scale.
  • Adapted by John Scammell here. Edited video and new driving question.
  • Dan Meyer provided a 3Act framework for the problem here.
  • Blair Miller adapted his own 3Act structure here.
My apologies go out to anyone else who played with or re-posted an original interpretation on the problem.

My vision for the problem (As far as I know)

Students will work on the problem as a task force. Whether or not I choose the groups will depend on the mix of students I get. The task will take place in a unit on rates, unit analysis, and scale factor. Students have been presented with mathematics that will help them through the problem. In that sense, this is not a "discovery" lesson, but I feel that that word attracts so much heat--I dare not use it.

The structure of the unit is highly influenced by Bowman Dickson's whiteboard structure. He uses it to induce wall-to-wall mathematics in a class period. That purpose remains here, but I also feel that a single medium to record their thinking (in a limited space) encourages efficiency and teamwork. With these two purposes in mind, students will enter the room to find tables equipped with the following supplies:
  1. A whiteboard. (Not too big, not too small)
  2. Two fine-tipped whiteboard markers
  3. A ruler
  4. A bag of gummy bears
  5. A folder containing documents
The folder will consist of pictures:

The first three are of the dimensions of the giant gummy bear from the original video.

Each group will also get screenshots from Meyer's weighing of the bears. (Act 3).

When students enter the room, they will take a seat and are free to explore the contents of their folder. After a few days of similar tasks, some will assume their job and get down to work. Others will dwell on the novelty of the situation.

When class officially begins, I will play Scammell's version of the video found here.

From there I pose the task:
How much does the giant gummy bear weigh?

Students are required to show all their work, reasoning, and calculations on the whiteboard. My role becomes one of transient learner. I move from table to table asking questions about their strategy and supplying my own questions for those who are stuck or finished. 

The group whiteboard task uses the mathematical ideas in a realistic setting. There are no prescribed methods or rigorous assessments. Students are simply required to use the mathematical information given to them, mix in a few mathematical tools, and provide me with a reasoned argument why they employed their school math in a correct manner. Kind of liberating--in a sense. 


Saturday, August 18, 2012

The Guess Who Conundrum

Every so often, an idea comes out of left field. I woke up with this on my mind--must have been a dream.

Back in the day, my family had a dilapidated copy of the game "Guess Who?" My siblings and I would take turns playing this game of deduction. You essentially narrowed a search for an opponent's person by picking out characteristics of their appearance.
I vividly remember playing with my younger sister one time at a family cottage. She--foolishly--chose a female person for me to identify. Anyone who has played the game before knows that the males far outnumber the females. I had her narrowed down to six or seven possibilities with one question:

"Is your person a male?"

Move the same principles into the math classroom. Numbers have characteristics much like Robert, Frans, and Bob. Some are large, negative, prime, abundant, triangular, square, etc. Would it be possible to play a game of "Guess Who?" with an array of numbers?

Create a five-by-five game board with numbers from 1-100 dispersed on it. Sample boards are available for download. Both players must have the same numbers on their boards for the game to work. They can be arranged differently to avoid peeking. It would be wise to choose a variety of numbers. Some even, some odd, some prime, some multiples of 3, etc. Try to avoid one dominating characteristic. (Like my sister's gender choice).

Have each player choose a number, and play the game. I would place restrictions on what they can and cannot ask:
  1. Must be a yes-or-no question
  2. Can't pertain to a physical characteristic of the symbol. (Does your number have curves?)
  3. Can't ask if the number contains specific digits (Does your number have an 8?)
  4. Can't ask over-under questions. (Is your number over 12?)
Encourage students to ask if numbers are prime, square, cubic, multiples of specific integers, triangular, odd, even, etc. Be sure they are dealing with the characteristics of the numbers.

After a round or two--depending on the group speed--get down to the real task. Have students pair up with their opponent and answer the following question:

Which number is the best to choose and why?

Students will begin brainstorming the most likely questions, and which characteristics pertain to which numbers. They will be practicing prime factoring within the framework of a larger task.

Students are essentially looking for the number that shares the most characteristics with the other numbers. Any unique characteristics run the risk of being singled out with a single question. This categorization of numbers fits very well in the Grade 10 Pre-calculus factors and products curriculum in my province. These topics are fairly universal across school mathematics.

Great question. I bet characteristic games can be applied to the rational numbers as well. I can see a less than--greater than approach working well for ordering of fractions. 

It might be interesting to have students play in pods of three. Each person has to deduce the other two's numbers by finding common characteristics. Every question is posed to the pair of opponents and each must give a yes-or-no response. Maybe they are both even, but only one is a multiple of 6? Maybe one is even and the other is odd but they are both perfect squares? The game ends when a player correctly guesses both opponent's numbers with a single guess. This throws an interesting wrench into the deduction gears. It also provides a nice extension for those students who are ready to move onward while others need more time with basic factoring. 

This is a great way of throwing mathematical deduction and problem solving alongside factoring. Students should see numbers like they see eccentric faces--as culminations of a series of defining characteristics.