Striking a Balance



by Roger McElmell

Unemployed and itchy (at press time).

Instructional Objective * Players will be able to analyze physical quantities into their fundamental units of measurement.

* Using dimensional analysis techniques, players will be able to develop balanced equations involving physical quantities expressed in terms of the fundamental units of measurement (distance, mass, time, angle, and electric current).


Learners/Context The target learners for this card game are students in high school- or college-level physics. The instructional purpose of the game is for remediation, enrichment (if the players are alert) and instructional punctuation. Playing the game will give the learners practice in balancing physical units in equations, an important abstraction of formulas in physics and many other scientific fields for that matter. It will also familiarize the students with the fundamental relationships between mechanics and electromagnetism.


Rationale In several scientific disciplines it is necessary that, when equating two quantities in a physical relationship, both dimensional units and numerical values must match. However, students easily become overwhelmed when dealing with physical quantities such as, velocity, acceleration, force, energy, power, voltage, electrical resistance, and so on. These quantities are simply constructs of the fundamental units of physical measurement: distance, mass, time, angle, and electric current.

To bring these quantities under control, a simple technique called dimensional analysis can be employed. Dimensional analysis involves three steps: (1) breaking down a complicated quantity into its fundamental units, (2) factoring common fundamental units, and (3) balancing them through products, quotients and conversion factors.

This proposed gaming activity will give students an opportunity to practice their dimensional analysis skills through balancing equations which they construct with the cards in their hand. It will also (hopefully) encourage discovery learning of relationships among physical quantities, which can easily be verified with the provided supplemental chart (the MKSA Derived-Unit Table) and/or classroom instruction. The game is adaptable to a wide variety of learners and disciplines. Although the game is for students of mechanical physics and electromagnetism, it can easily be modified to incorporate fields such as optics, thermodynamics, acoustics, and quantum physics.

One final benefit from playing the game is the opportunity to work extensively with the metric system. Our culture has taught us to describe distances in terms of feet or miles, and weights in terms of ounces and pounds. Most of the rest of the world has adopted the Système International d'Unités (SI) measurement system. This is a collection of measurement standards based on the MKSA (meter-kilogram-second-ampere) system of units. Most courses in science nowadays use the MKSA (or CGSA, centimeter-gram-second-ampere) metric system, but the shift has not taken effect in most other disciplines in the United States.


Rules The object of Striking a Balance is to form balanced equations of physical units that appear in many physics formulas. Equations are formed with cards in various combinations, each card bearing a physical quantity which is broken down into its fundamental units. The winner of a round is the first person to "go out" as in Rummy, which means that the player can use all the cards in his/her hand to form complete equations. Points may be accumulated at the end of each round and the grand wazoo is the player who reaches 100 points first.

Striking a Balance can be played by two to four players. If there are two players, each player is dealt 12 cards. If there are three players, each player is dealt 10 cards. If there are four players, each player is dealt 8 cards.

The deck consists of 48 cards. Also available is the MKSA Derived-Unit Table, which lists the several derived quantities used in the MKSA system, their symbols, a short description of how the unit is derived (where appropriate) and the number of cards in the deck which represent the quantities.

1. The dealer deals the appropriate number of cards to the players from the shuffled deck. The remainder of the deck (the draw stack) is placed face-down in the center of the table and the top card of the deck is placed face-up (the discard stack) next to the deck.

2. The person to the left of the dealer starts the play. S/he may select the face-up card from the discard stack or the top card of the draw stack. After selecting the card, s/he must discard one card from his/her hand, placing it face-up on the discard stack.

3. The play continues as described in step 2 in a clockwise direction. If the draw stack is depleted before any player "goes out", the discard stack is shuffled and placed face-down as a new draw stack. The top card of the new draw stack is turned over to form the new discard stack. From this point the play continues as normal.

4. To "go out," a player must use all of the cards in his/her hand to form balanced equations (see step 5). Players may refer to the MKSA Derived-Unit Table during play. The player who "goes out" must discard one card to the discard stack. That is, the player must go out with the same number of cards as s/he started with in the beginning of the game.

5. Balanced equations are formed as follows.

a. The minimum number of cards to form an equation is three. The maximum number of cards to form an equation is the number of cards in a players hand.

b. An equation is considered balanced when combinations of cards in a player's hand are arranged correctly to form a valid equation.

c. Additive and subtractive equations are not allowed. Only one term may be equated with another.

d. Two of the same card may not be used in a single equation.

e. A single card may not be shared by two equations.

See the Sample Equation for further explanation.

6. When a player "goes out," the round is terminated and points are then calculated. Each player who has a balanced equation in his/her hand receives points for that equation. The point value assigned to each card is simply the number of units required to form the quantity. The FORCE card has a point value of 4 since the quantity of force is constructed from one base unit of mass, one base unit of distance, and two base units of time (FORCE = kg*m/sec2). Players receive points only for the cards that are involved with a balanced equation.

7. The player who "goes out" receives 25 extra bonus points. The first player to reach 100 points is the winner...the grand wazoo.


Sample Equation Suppose a player has the cards AREA, TIME, ANGULAR MOMENTUM and MASS DENSITY. These may be arranged as

(AREA)(MASS DENSITY)/(TIME) = (ANGULAR MOMENTUM)

This relationship is built by breaking the quantities down to their base units and factoring out the appropriate units. In this case, the two factors of distance (meters) from the AREA quantity cancel out with two of the three factors in the MASS DENSITY quantity, leaving the appropriate combination:

(m2)(kg/m3)/(sec) = (kg*m/sec)

When a player goes out, s/he must place his/her cards face-up on the playing surface and arrange them correctly to form valid equations. Quantities which are treated as factors (i.e., quantities which are multiplied other quantities) should be laid next to each other. To denote that a quantity is a divisor (denominator), rather than a factor, the card will be arranged below the numerator card(s). For the above equation, the cards should be placed on the table as shown here.

By separating the cards as shown above, the equals sign is implied.

Note: an equally valid equation could be formed with the same cards by moving the TIME card to the right side of the equals sign. The arrangement of cards would then become

(AREA)(MASS DENSITY) = (ANGULAR MOMENTUM)(TIME)

where both sides of the equation are simply products of quantities.

This sample equation is worth a total of 10 points: AREA is worth 2 points, MASS DENSITY is worth 4 points, TIME is worth 1 point, and LINEAR MOMENTUM is worth 3 points.


Card and Deck Design

The deck consists of 48 cards, 2.5 inches x 4.0 inches (that's 0.6350 meters x 0.1016 meters!). Each card has a quantity on its face, along with the MKSA symbol(s) which represent(s) the quantity when it is used in notation form. The upper left corner of the card has the same MKSA symbol, represented in terms of the base units of distance, mass, time, angle, and electric current. This allows the players to hold their cards in a fan arrangement while still being able to see the various fundamental units for each of the cards. Much like regular playing cards, these cards are symmetrical so that players don't have to turn the cards upright to be able to read them.

Some cards occur in the deck more than others. Those of which there are multiples are physical quantities of lower order than the other more complicated quantities. The number of cards for each of the quantities is listed in the MKSA Derived-Unit Table, which is to be used during play as a reference guide, and a dispute settler in the event of a challenge.

The distribution of cards is based on the number of times the base units are used to form the more complex quantities. For example, the base unit second is found in nearly all of the derived units (there are 34 occurrences in all), so there are eight SECOND cards in the deck. Likewise, the unit radian is found only twice in the derived units, so there is only one RADIAN card. The higher order quantities occur only once in the deck.


Design Process Having studied physics for several years, I chose a topic with which I am familiar. Dimensional analysis seems appropriate for a card game format since it lends itself to building patterns based on component units, much like cards do. This pattern-building aspect in turn called for a Rummy-type game.

As opposed to Rummy, however, Striking a Balance has instructional value which goes beyond merely "looking for the card that will complete a run or a set of three." Through managing several types of quantities in a single hand, players may develop specific and interesting insights as to how physical quantities interact and build with others to form new constructs. As with a regular deck of playing cards, the number of possible combinations of patterns is (almost) limitless. This was a design feature I felt was necessary. With a virtual infinity of valid patterns (equations), both learning and the longevity of the game have a better chance.

My original game plan required players to conduct two processes at once: performing dimensional analysis and balancing out the powers-of-10 prefixes of the metric system. I decided that it would be too much to handle in a friendly card game. Therefore, I limited the equation-building process to a single, standardized measurement system, the MKSA (meter-kilogram-second-ampere) system.


MKSA Derived-Unit Table

    Quantity              MKSA Symbol(s)                     Definition             Cards   
BASE UNITS                                                                                  
Mass               kg                             kilogram: equal to 1000 grams.      5     
                                                  A gram is the mass of 1 cubic             
                                                  centimeter of water at                    
                                                  4[[ring]]C and 1.00 atmosphere            
                                                  pressure.                                 
Distance           m                              meter: the length equal to          7     
                                                  1,650,763.73 wavelengths in a             
                                                  vacuum of the radiation                   
                                                  corresponding to the transition           
                                                  between the levels 2p10 and 5d5           
                                                  of the krypton-86 atom.                   
Time               sec                            second: the duration of             8     
                                                  9,192,631,770 periods of                  
                                                  radiation corresponding to the            
                                                  transition between the two                
                                                  hyperfine levels of the ground            
                                                  state of the cesium-133 atom.             
Angle              rad                            radian: the plane angle between     1     
                                                  two radii of a circle which               
                                                  subtend an arc on the                     
                                                  circumference of the circle               
                                                  equal to the length of the                
                                                  radius.  There are exactly                
                                                  2[[pi]] radians in a complete             
                                                  circle.                                   
Electric Current   A                              ampere: the amount of electric      4     
                                                  current which, when driven                
                                                  through two straight parallel             
                                                  conductors of infinite length,            
                                                  negligible cross section, and             
                                                  separated by one meter in a               
                                                  vacuum, will produce a force              
                                                  equal to 0.0000002 Newtons for            
                                                  each meter of length of the               
                                                  conductors.                               
DERIVED UNITS                                                                               
Area               m2                             square meter                        2     
Volume             m3                             cubic meter                         1     
Linear Velocity    m/sec                          meter per second                    2     
Angular Velocity   rad/sec                        radian per second                   1     
Linear             m/sec2                         meter per second squared            1     
Acceleration                                                                                
Angular            rad/sec2                       radian per second squared           1     
Acceleration                                                                                
Linear Momentum    kg*m/sec                       kilogram-meter per second           1     
Angular Momentum   kg*m2/sec                      kilogram-meter squared per          1     
                                                  second                                    
Force              kg*m/sec2 N                    Newton: the amount of force         1     
                                                  necessary to accelerate a                 
                                                  one-kilogram mass at a rate of            
                                                  one meter-per-second per second.          
Pressure           kg/m*sec2  N/m2  Pa            Pascal: the amount of pressure      1     
(Mechanical                                       applied by one Newton                     
Stress)                                           distributed over an area of one           
                                                  square meter.                             
Torque (Moment     kg*m2/sec2 N*m                 kilogram-meter squared per          1     
of Force)                                         second squared                            
Energy (Work)      kg*m2/sec2 N*m J               Joule: the amount of work done      1     
                                                  to displace a body by one meter           
                                                  using a force of one Newton               
                                                  while moving the body in the              
                                                  direction of the force.                   
Power              kg*m2/sec3 J/sec W             Watt: the amount of power which     1     
                                                  gives rise to the production of           
                                                  energy at a rate of one Joule             
                                                  per second.                               
Mass Density       kg/m3                          kilogram per cubic meter            1     
Quantity of        A*sec  C                       Coulomb: the amount of charge       2     
Electricity                                       transported in one second by an           
(Charge)                                          observation point in a                    
                                                  conductor carrying an electric            
                                                  current of one ampere.                    
Potential          kg*m2/sec3*A W/A V   J/C       Volt: the electric potential        1     
Difference                                        difference between two points             
(Electromotive                                    of a substance while carrying a           
force)                                            constant current of one ampere            
                                                  and dissipating power between             
                                                  the two points at a rate of one           
                                                  Watt.                                     
Electrical         kg*m2/sec3*A2 V/A [[Omega]]    Ohm: the amount of electrical       1     
Resistance                                        resistance of a substance which           
                                                  produces a one volt drop in               
                                                  electromotive force when a                
                                                  current of one ampere passes              
                                                  through the substance.                    
Capacitance        sec4*A2/kg*m2 A*sec/V F        Farad: the amount of                1     
                                                  capacitance produced between              
                                                  two parallel plates with an               
                                                  electromotive force of one volt           
                                                  between the plates while                  
                                                  holding a charge of one Coulomb.          
Magnetic Flux      kg*m2/sec2*A V*sec Wb          Weber: the amount of magnetic       1     
                                                  flux produced by a single-loop            
                                                  circuit wherein an                        
                                                  electromotive force of one volt           
                                                  is maintained while the current           
                                                  in the circuit is reduced at a            
                                                  uniform rate to zero in one               
                                                  second.                                   
Inductance         kg*m2/sec2*A2 V*sec/A H        Henry: the amount of magnetic       1     
                                                  inductance produced by a                  
                                                  closed-loop circuit wherein an            
                                                  electromotive force of one volt           
                                                  is maintained by a current in             
                                                  the circuit which varies at a             
                                                  uniform rate of one ampere per            
                                                  second.                                   
Magnetic Flux      kg/sec2*A V*sec/m2 T           Tesla: the amount of magnetic       1     
Density            N*sec/C*m   N/A*m              flux density which produces a             
                                                  force of one Newton on a                  
                                                  straight conductor which is               
                                                  carrying a current of one                 
                                                  ampere when the conductor is              
                                                  perpendicular to the magnetic             
                                                  field.