By Eric Olson, creator of PEMDice and Math Department Chair, Pingree School, S. Hamilton, MA

Mathematics educators have recently highlighted the need for “low floor, high ceiling” tasks that lead students into rich areas of inquiry. The “Four 4’s” problem is a fairly well-known example cited by Jo Boaler and others, but such challenges are hard to develop. It is often difficult to find an exercise, game, or a puzzle that is instantly accessible at a basic level, yet also leads to the exploration of higher-order thinking and deeper mathematical insights. The PEMDice game is designed to scratch this itch. Very simple in concept and, when played in its elementary form, it is ultimately as complex and as challenging as anyone cares to make it.

The PEMDice game includes two sets of 15 dice and can be played individually or in small groups. The dice feature numbers, operators, parentheses and the equal sign. Adjust the game for elementary students by removing the fraction cube and the two cubes with exponents, roots and factorial faces. Or use all 15 dice for upper grades. An example of a random toss of the complete set of 15 PEMDice is shown below with the dice separated into three groups: fixed dice, numbers, and operators. The dice have faces specially designed to give a variety of options, with just enough predictability.

The game needs little or no explanation; simply hand out the sets and ask the students to create the longest possible true equation. Equations must follow the standard order of operations, or “PEMDAS,” rules. Begin by letting students explore the many possibilities by saying that, “All dice are wild,” meaning that all the dice can be manipulated to show any face. This is the low-floor starting point upon which to build.

After making any long equation, students score their equation by finding the square of the total dice count. In order for parentheses to count in the scoring, they must change the equation when removed. This leads to student inquiry and debate about when parentheses are necessary, and provides a strong incentive to learn the skill thoroughly. Below are some possible student scoring equations for the dice that were tossed above. Can you spot the one that does not need parentheses? Do you see how that configuration is later used to an advantage at the bottom?

As the players adjust to the game and become more familiar with basic techniques and skills, teachers can raise the ceiling in real time through simple rule adjustments. PEMDice can evolve in several ways: the gradual elimination of wild-card dice, first numbers, then operators. Also, the use of the advanced operator dice create more, and more varied solutions--so are quickly embraced as options. They will quickly want to use 1! and 2! to their advantage, then realize that 3! and 4! are handy, too.

In the early stages it is likely best to play a collaborative version of PEMDice in which a teacher challenges the class to generate a certain amount: “1000 points in 10 minutes,” for example. Where appropriate, competition can be allowed, first between groups, then individuals--even to the point of playing the “duplicate” version, like Contract Bridge, in which each group has an identical starting configuration so that everyone plays with the same setup of dice.

## Insights

Playing PEMDice will provide an engaging, fun, and challenging way to add a little variety to the mastery of the order of operations and the creative manipulation of quantities.

• Even though there are no “1’s” or “0’s” in PEMDice, playing the game will spark a deeper understanding of the profound value of these identity operators. Many techniques in algebra use the addition of 0 or multiplication by 1 to achieve a goal. (For example, when rationalizing a denominator.)
• Additionally, the creation of the 1’s by multiplying dice that are reciprocals is also a valuable skill to reinforce, as are the other “tricks” of the game like raising numbers to the power of 1, or squaring or finding the square root of 1.
• Students who become proficient at PEMDice develop a quick facility with grouping numbers, looking for common factors, readily creating or identifying common powers of numbers, and understanding the many equivalent variations on equations.

## Games

There are numerous games that range from simple to complex; from collaborative to competitive. One of the strengths of PEMDice is that the teacher can adjust the challenge or game to suit a lesson. Here are a few options not found in the game instructions available online:

• A game in which you try to make an equation involving the largest possible number.
• Challenge the class to use PEMDice to make a specific number: 1024 for example on October 24th. Or maybe 22/7 on March 14th, or “Pi Day.”
• Require the class to use the open  “^” cubes to practice raising number to a higher power than just 2 or 3.
• Require the class to use mixed numbers in their equations.
• Require the class to combine two dice to make a 2- or 3-digit number.
• The teacher gives a configuration to the class and challenges them to find a 15-cube equation. Some hints can be provided. Below is a 15-cube equation to the original roll of the PEMDice.

The possibilities for PEMDice are endless, and with a little experience both students and teachers will become quite adept at playing. As students play, they will build their understanding of order of operations and their confidence in the language of math. They can post a selfie with the high score and let the world know!