Wednesday, June 19, 2013

Dice Sums Task

Dice are familiar tools in most mathematics classrooms. Their use in primary school games allows students to build preliminary notions of number and autonomy. (see Kamii) As the grades progress, dice sums become too simple and the tool is pushed into the realm of probability and chance. There, alongside decks of cards and coloured spinners, it enjoys almost godly status; it seems that there is no better way to calculate odds than to role dice and spin spinners (in outrageous cases—simultaneously).

The greatest thing dice have going for them is familiarity. Teachers can use this to upset the thinking of students and encourage them to think mathematically to resolve their cognitive conflict. A great example of this is my Fair Dice Task. Here students encounter dice that act in—seemingly—strange ways. They grapple with the structure of the problem to restore a sense of homeostasis.

Here I want to explore how students use their number sense and logical thinking to piece together unfamiliar dice. I begin by introducing a chart that shows the sums of the faces when rolling two dice.
We discuss the possible patterns that emerge and any symmetry that exists. I then hand out a sheet of blank dice charts and reformulate the task.

What if I gave you the 36 possible sums? Could you then tell me what the dice look like?

I haven’t decided the best way to list out the 36 numbers. You could project them on the board, but then students may make counting errors that would derail the underlying focus of the task. I think each group should have a set of data, and mistakes should be erasable. I have settled on giving each group a plastic sheet cover and handing them a dry erase marker and a sheet of data. Each data sheet has a place for them to write their guiding rules. My goal is to compile these “rules” and have them fuel a class discussion. Each sheet also has a "final answer" grid.

I try and start with simple dice and move into more complex. The whole time I encourage them to write down any “advice” they have in the form of rules.

I visit each group and take note of any differing strategies. I pay special attention to those I would like to highlight. I am sure to discuss this example with the class to keep the task floor low. That is to say, I want every student to have a shot at achieving some sort of mathematical accomplishment.

I want to see if students can sense if the dice doubles. Maybe there is a rule that can be developed.

Here I am looking for logical partitioning of numbers. I want the students to see the number “eight” flexibly. Within this context, eight will have two parts. Each constituent represents a side of a dice. Can students use the structure of numbers to break down tasks?

Depending on the flow of the lesson, I like to throw out some other questions. These would also serve as great extensions.

What if every sum was even? What could you tell me about the sides of the dice?
What if every sum was odd? What could you tell me about the sides of the dice?
Can you design two dice (with numbers 1-6) that never sum to 8? to 1?
What if half of the sums were sixes? What could you tell me about the sides of the dice?

The task requires a large amount of logical thinking and numerical flexibility.

If you wanted to move into the topic of factors (or even greatest common factor), you could present the same structure but give them the product of the dice faces. Prime numbered faces would present an interesting twist under these parameters.

On the whole, the task takes a familiar object creates a cognitive conflict by skewing its attributes. I’m having trouble penciling it explicitly into a “unit” under a “topic”, but any suggestions are more than welcome.


Tuesday, June 11, 2013

Leaky Faucet Task

This idea is not my own. The only problem is, I don't exactly know who it belongs to. I remember tweeps talking about about a task where a leaky faucet's effect was analysed on a water bill. When I encountered the situation at my Uncle's house, I had to capture the modelling in action.

The best part was the conversation from intrigued (and weirded out) relatives as I ducked and dived around the tap to get a good angle. We got into a conversation about teaching, and they were happy to present any questions that came to their minds.

The premise of the task goes as follows:

After the video is watched, I would garner a list of questions that the students may have. My relatives provided the following:

How much does the leaky faucet cost you?

Is it worth it to fix the tap?

When will the sink fill with water? 
(quickly altered to append a "if the drain was clogged" clause)

How long would it take to drown in the water leaking onto the floor? 
(followed by the fact that you could drown in an inch of water.)

My personal question:

Would you be able to tell if the faucet was leaking just by looking at your water bill?
(Again, stolen from someone, somewhere.)

The questions generally fall in two categories. Those that involve unit analysis, and those that involve some measurement of volume. Although you can certainly entertain the volume oriented questions, my goal here is to focus on rates. The task is great because its difficulty can grow depending on the information you give to students.

Students make a "wish list" of pieces of information they need to know to solve the problem. This is where you need to make a curricular decision. My province focuses on conversions within a single system of measurement (either SI or Metric) at the Grade 10 level. The Grade 11 level inserts conversions between the two.

I ask for the answer in mL because that opens the interesting conversation of mL and cubic centimeters.

I play the following informational video to grant some sort of "rate" of water leakage:

I also have the two measuring cup pictures on hand so I don't have to search through the video to find the correct spot. 



I also give them the cost of water. If I want them to convert between systems, I give them a picture of my statement in $/cubic feet. If I don't, I will do the conversion before class and have it written on the board when asked for it. 

A snippet from my water bill

Then I let them work. At this point, they have developed methods to solve the problem that are not based solely on multiplicative structures. Some will get the conversion from previous experiences with ratios or rates. Others will group together "chunks" and use an additive strategy. How much does 100mL cost? How long will it take for 100mL to leak? Students often fall back on tables of values and linear relations. 

Because my curricular goal is unit analysis, I take mental notes of all strategies but highlight those that use a unit analysis framework. At the end of class, I use a discussion to highlight and connect ideas. We then can move into altering the problem and getting rapid results. 

Every time my uncle walked through the kitchen, he would mumble to himself that he should really get to fixing the faucet. Depending on cost of hardware and time, it may not be such an obvious decision. 


P.S. The work and answers to this task are purposely left out. As my Math Ed. prof @MatthewMaddux would say, "I guess I lost the answer key."