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.
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.