Often times, it is accompanied with questions concerning the sums of the faces that appear on each dice. I think the obsession with this specific subdomain of probability questions comes from the elegant way in which a table of outcomes (pictured below) leads to a counting of favourable and total events.
|The sums of two dice neatly organized in a table|
Shifting the problem ever so slightly offers a new subdomain of question concerning the product of the two dice. A typical problem might look like:
Roll two fair, 6-sided dice. What possible products can be made by multiplying the faces together?
What is the probability that:
a) the product is 12
b) the product is a multiple of 4
c) the product is greater than 15?
My assumption is that most math teachers could construct numerous more subquestions of the same variety.
In an effort to make the problem more interesting, I flipped* it.
I rolled a pair of dice 100 times. I multiplied the result from each dice after every roll and recorded them in this table.
What numbers appeared on the faces of each dice?
It should be noted that opening up a problem to divergent and emergent action does not devalue the elegant organizations afforded to mathematics. Many think that rich, open tasks are just a rejection of neat and orderly work. Open tasks signal a rejection of the notion that interaction with a problem should be neat and orderly, and, by extension, that learning mathematics is the process of achieving lockstep with an ideal way of acting. The learning of mathematics is a holistic, messy, and convoluted experience filled with triumphs, dead ends, and ways of representing action. In order to harness these movements, the problems need to allow space for them.
As a frame of reference for the students, I offer them the following image of 100 products from two regular 6-sided dice.
The problem is posed intentionally vague. I want students to develop the culture of precision, and demand that the problems I present meet the same requirements. For instance, are both dice identical? Can a number appear on a dice more than once? How many faces are on each dice? etc.
In this case, I refine my parameters to two dice each with six sides. The dice do not have to be identical, and have whole number sides only. A number can appear more than once on a single dice, and that will be reflected in how many times certain products are created.
To structure their work, I provide them with two handouts:
The first is a sheet with a completed table of products from two normal dice (similar to the first image in this post) and the corresponding nets of the dice that would create the distribution. (download .pdf here).
The second is an identical sheet with the table and dice nets left blank for them to work with when creating the dice that resulted in this new distribution. (download .pdf here).
I give one of each handout to a group (to hand in) and have them do the majority of their work on a shared whiteboard space. I find the communal (and non-permanent) workspace to encourage a greater density of neighbour interactions.
Anticipated student action:
I've chosen to organize anticipations of student action around possible questions that 1) groups may pose during the course of action or 2) teachers may pose to trigger student action or perturb student reasoning.
(1) Will all possible combinations of rolls occur in the 100 trials?
This question gets at the very heart of probability. I guess, in theory, the answer is: No, not every possible pairing must have been rolled in the 100 trials. But there are three caveats to this: First, every product must be possible, so that means that all products must have at least one pair of factors exist on opposite dice. Second, how many products are there? If there are less than 36, that doesn't mean that some did not occur. It could mean that some have multiple pairings that produce them. Third, each outcome has a 1 in 36 chance in happening. Even if each product were unique, there would still be a good chance that each occurred in 100 rolls.
(2) Does where you place the factors make any difference?
We assumed the dice were fair so where you place the numbers on a single dice will not matter. However, switching numbers between dice will change the possible outcomes. Keep in mind that each product needs to have one factor from each dice.
(3) How often will certain outcomes occur?
My guess is students will set up some type of table of results and compare the experimental results with a dynamic theoretical probability as they build their dice. They will feel comfortable when their theoretical calculations closely match the experimental results. If they set up a system where "6" should be rolled 5 times in 36, but only appears 7 times in the 100 trials, are they wrong? What is the margin for error? What if theoretically rare outcomes actually appear more often than theoretically common ones in the trial? Should we be switching numbers, or assuming that the trial resulted in a deviance from expectation?
I think the beauty of the problem is in this question. Arguments are rooted in theoretical mathematics (which are probably the curricular outcomes at hand) yet guided by a mathematical intuition.
(4) What do prime numbers tell us?
Well, a prime number has two factors, so we know those two factors must exist on opposite dice. Also, one of those factors needs to be a "1". The case of "1" is interesting. How can we know that a "1" won't exist on both dice? If we know a single prime product exists, can we be sure that others must exist?
(5) What do products with only four factors tell us?
Every product will have "1" and itself as factors, but what if there are only two other options? Take the example of "27". We are fairly certain that "1" and "27" didn't multiply to achieve "27" because if "27" existed on a face of a dice, we would see other multiples of "27" in the results. There is one (54), but it seems unlikely that either the other dice is dominated by 2s and 1s or the "27" just wasn't rolled in conjunction with the other options. It leaves "9" and "3" as excellent options for dice sides.
(6) What do perfect squares tell us?
The knee jerk reaction here is to assume that a perfect square is created by the multiplication of identical factors. Something is attractive about that symmetry. This, of course, ignores the possibility of pairing the perfect square with a "1". It is not true in the case of "4" and "9", and gets even more convoluted in the case of "36". It is interesting to note that a "4" must be constructed with two 2s if there is no "1" / "4" pairing possible. The same goes for the product of "9". The existence and placement of a "1" gains importance yet again in this case.
If your goal is to study factors and products, allow the probability to slink into the background and remain governed by intuition. Focus on the composition of the numbers and the characteristics of primes, perfect squares, and common factors.
If your goal is to study probability, use the factors as a vehicle to have conversations of likelihood. Encourage formality to justify conjectures of probability, and live in the very real tension between experimental and theoretical probability.
The task doesn't throw out the representations of classical probability or the theorems of factors and products. Rather, it opens a space for students to act on both topics in ways that emerge as necessary at the time.
While I don't tend to give out solutions to problems, I feel like teachers are more likely to use this task if they know the composition of the dice that created the 100 trials. (Yes, I created the dice and actually rolled them 100 times).
Dice 1: 2, 3, 1, 6, 4, 6
Dice 2: 9, 10, 5, 3, 2, 2
My suggestion is to never reveal this composition to your students. I know you, as a teacher, probably have this weird "master of math" vibe and most likely feel incredibly insecure in the uncertainty of all of this, but your students haven't earned the right to be this unnaturally uptight about their work yet.
I like to collect viable solutions complete with justification. I attempt to focus on the process of establishing the mathematics, and a grand reveal of the right answer bankrupts that immediately.
*tongue-in-cheek reference to the recent fad of "flipped classrooms" of which I am still extremely skeptical of any claimed pedagogical innovation.