The Balance Applet is a game. There is a balance and 3 different shapes. The weight of one shape is given. The player is asked to find the other shapes' weights using the balance. (We added a TRY button that gives the truth value of an attempt.)

Basics:

In all the activities of this applet students are working on The Illinois State Goal 6 : Demonstrate and apply knowledge and sense of numbers, operations, ratios and proportions. Especially compare and order whole numbers [6.a.2] and solve 1 and 2 steps problems involving whole numbers [6.b.2]

Working with the applet will also address ILS Goal 8. 8b: Interpret and describe numerical relationships using tables, graphs and symbols, and Goal 8d: Use algebraic concepts and procedures to represent and solve problems. Basic reading of a game frame involves translating a situation to numerical sentence and manipulating small numbers. (4 operations and sense of order) That is what is needed to play a Trial and Error Strategy.

From Game to Strategy:

From Game to Strategy Brings different Math to the game:
The idea of a Strategy, or Algorithm, introduces Reflecting and Planning to the activity and with it more of Documentation and Presentation. On the basic level we deal with State Goal 8.b: Interpret and describe numerical relationships using Tables, Graphs and Symbols. But it gives opportunities to wider use, more in line with the definition of solving problems (IL. Learning Standards pg. 16): Recognize and investigate problems; formulate and propose solution; support by reason and evidence.

When we ask that a strategy will be a written document we add a great communication component. This one is best achieved with teams work. We found that different ways of looking at things really pays off here in finding a strategy and the sharing clears ideas. The two are very important components for a complicated task like this one. [Bonus: those who don't get the ideas right away are great for "debagging" a strategy because they follow instructions, not intentions.]

1. Forget the balance. Put numbers systematically into the solution's slots, try each combination with the TRY button until you hit a winner! (Systematically means that you have a way to go over all the possible combinations)
2. Working with 2 shapes, one with known weight and one with unknown weight. Playing so that on each tray you have only pieces of the same shape. Load until you reach equilibrium and then calculate the unknown weight by equating the total weights on each tray.
3. Elimination: For each frame of the game, we write what we could conclude about the weights; usually from inequalities we could eliminate impossible values and so limit our search, until we reach one possible value for each shape. This search is best when done on the frames of strategy 2).
4. Using the three shapes we create 2 sets of "independent" equilibriums. Translate each into a linear 2 variables equation and solve the system using algebra. This strategy requires knowledge that is usually unavailable to 5-6 graders.

We found that most kids use combinations of the first 3 as their way to find the weights

Additional topics for: From Game to Strategy.

1. Play the Guess game (One player chooses a number from 1-100. The other(s) have to find the number by Yes or No questions.). An easy game to play by random guessing. Looking for a minimal steps strategy adds interest and mathematical content. (The winning strategy is a Binary Search.)
2. Introducing structured Search version II as a refinement of version I. We add reasoning to each step of version I. This change reduces number of steps, in most cases, but adds brain's work.
3. Compare 2 strategies using statistics: Average time it takes per game; Difficulty level; Fun level and so on.
4. Presentations:
   1. Present strategy 1 (The TRY button) as a set of instructions and/or a Flow Chart.
   2. Present strategy 2 (Play toward equilibrium) as a Flow Chart and/or set of instructions.
   3. Present the Binary Search for Guess Game as a Flow Chart and/or set of instructions.
   4. Present strategy 3 (Structured search version II ) as an elimination processon a Number Line.
5. What happens if we change the rules:
   1. What strategies, if at all, are working if we allow big numbers as weights?
   2. What strategies, if at all, are working if we allow fractions (Rational numbers) as weights? Try half unit as a start.

Math properties of strategies:

On the strategies as a base we could build a rich mathematical research with reasoning and proofs as outlined by NCTM Pg.56: Make and investigate mathematical conjectures. Develop and evaluate mathematical arguments and proofs. Select and use various types of reasoning and methods of proofs. This supports also working on Ill's Goal 8: Using algebraic and analytical methods to identify and describe patterns and relationships in data, solve problems and predict results.

1. How many steps it takes to find both unknown weights at most if we play strategy 2 (Structured Search version II), and we limit the weights to whole numbers under 10 units.
2. We could ask to prove that the strategy will always work, as long as we play with whole numbers. (Even with very big ones.)
3. Design a way to find out if a given 3 numbers are the solution for a given game without the TRY button. (Especially if you didn't work with strategy 2)

Or we could deal with the same subjects by dividing the scope for the students to sub questions like:
A1) When we look for one unknown weight where does the search ends?
A2) What will be the end of search point for some given examples? The answers could be found either by simulating the process or by playing the balance version that allows us to use our own input?
A3) What mathematical expressions could describe the relations among the 2 weights and the weight on each tray at equilibrium? We could make and investigate conjectures.
Check them against the previous examples or by using the balance version that allows us to use our own input. Or we could look for patterns on results of A2 examples. ( Putting them on a table might help)
A4) What is the connection between the number of steps and the point of equilibrium? (The number of steps is the sum of pieces used because we play by adding one piece at a time.) The answer could be found by looking at patterns or by reasoning.
A5) Having answers to the previous questions make us ready for the last step: the number of steps for a full game is twice the number of steps of one search. For the at most, (largest possibility) we check cases starting with the biggest possible number, 9. (The result is 17 for half search or 34 for full one.) With some tricks we might lower it a little.
A6) We could ask to show , by counter example, that a number (E.G. 11) could not be an upper limit for steps of one weight search. This experience could be a trigger for a meaningful search on these subjects with different tools.
B1) The proof that the Strategy always works could be divided to 2 parts. a): There is always, for any 2 whole numbers, a candidate for equilibrium.(In fact there are always many such candidates) And b): Our way of playing will take us to one of the equilibriums. (In fact it will take us to the lowest one.)