Abstract: An instructional system and method for representing and solving algebraic problems. A computer-implemented method of representing and solving mathematical problems comprises providing a graphical user interface for displaying an instructional system. The instructional system comprises a primary cog having a plurality of primary cog teeth and a secondary cog having a plurality of secondary cog teeth. The secondary cog is linked to the primary cog such that a rotation of the secondary cog causes a rotation of the primary cog. The method comprises representing a mathematical problem on the graphical user interface using the instructional system. The method includes receiving an input from a user and rotating the secondary cog based on the input to cause the primary cog to also rotate. The mathematical problem is solved when a particular tooth of the plurality of primary cog teeth reaches a predefined location.

Abstract: A tile instructional system for presenting a linear growth problem and for allowing the linear growth problem to be solved comprises an input tray having at least a first tile and a second tile. The first tile has a first marking and the second tile has a second marking. Each of the first tile and the second tile is configured to selectively expand in accordance with its respective marking. The system includes a tile bed comprising a plurality of tile slots. Each of the plurality of tile slots is configured to receive a tile. The linear growth problem is solvable by situating in the tile bed at least one of the first tile and the second tile and by causing each tile situated in the tile bed to expand in accordance with its respective marking to fill the tile bed.

Abstract: A liquid flow instructional system for presenting a proportions problem and for allowing the proportions problem to be solved comprises an input container and a plurality of output containers selectively and fluidly coupled to the input container. The system has an adjustable valve having a plurality of configurable regions. Each of the plurality of configurable regions corresponds to one of the plurality of output containers. An activable switch is provided for initiating flow of liquid from the input container to the plurality of output containers in accordance with a user configuration of the plurality of configurable regions. The proportions problem is solvable by filling each of the plurality of output containers to capacity without spillage. The correspondence between the plurality of configurable regions and the plurality of output containers is indicated by a visible indicator.

Abstract: A method for adaptively teaching content to a user comprises providing a plurality of teaching topics. The method includes sequentially associating a lead problem and a plurality of secondary problems with each of the plurality of teaching topics. Each of the lead problems and the plurality of secondary problems is an open-ended puzzle having a range of solutions and a difficulty rank. The range of solutions for each of the lead problems and the plurality of secondary problems has an optimal solution and a plurality of secondary solutions. A teaching topic is selected and a corresponding lead problem is presented to a user. User provides answer data via a physical input device having enmeshed gears. The user is provided another secondary problem in the selected teaching topic if the answer data matches one of the secondary solutions. The user is provided another lead problem if the answer data matches the optimal solution.

Abstract: A method for representing and solving algebraic equations that allows a user to view and solve algebraic equation with a virtual gear system. The virtual gear system includes a virtual primary cog and virtual secondary cogs. The virtual primary cog represents a range of outcomes for the virtual gear system and contains a number of teeth that is quantitatively greater than a numerical constant of the algebraic equation; amongst the teeth is a target tooth that represents the numerical constant. Each virtual secondary cogs represent a term of the algebraic equation and includes a coefficient and a variable. Each of the virtual secondary cogs contains a number of teeth equal to the coefficient. The equation is solved by rotating the virtual secondary cogs until the target tooth is aligned with a fixed pointer where rotation of the virtual secondary cog represents a value input for the variable of a term.

Abstract: A method for representing and solving algebraic equations that allows a user to view and solve algebraic equations through a novel user-interface. The user-interface includes a primary cog and a plurality of secondary cogs as parts of a gear system. The primary cog represents a range of outcomes for the gear system and contains a plurality of teeth that is quantitatively greater than a numerical constant of the algebraic equation; amongst the teeth is a target tooth that represents the numerical constant. Each of the secondary cogs represent a term of the algebraic equation and includes a coefficient and a variable. Each of the secondary cogs contains a number of teeth equal to the coefficient. The equation is solved by rotating the secondary cogs until the target tooth is aligned with a fixed pointer where rotation of the secondary cog represents a value input for the variable of a term.

Abstract: A method for representing and solving algebraic equations that allows a user to view and solve algebraic equations through a novel user-interface. The user-interface includes a primary cog and a plurality of secondary cogs as parts of a gear system. The primary cog represents a range of outcomes for the gear system and contains a plurality of teeth that is quantitatively greater than a numerical constant of the algebraic equation; amongst the teeth is a target tooth that represents the numerical constant. Each of the secondary cogs represent a term of the algebraic equation and includes a coefficient and a variable. Each of the secondary cogs contains a number of teeth equal to the coefficient. The equation is solved by rotating the secondary cogs until the target tooth is aligned with a fixed pointer where rotation of the secondary cog represents a value input for the variable of a term.

Abstract: A method for representing and solving algebraic equations that allows a user to view and solve algebraic equation with a virtual gear system. The virtual gear system includes a virtual primary cog and virtual secondary cogs. The virtual primary cog represents a range of outcomes for the virtual gear system and contains a number of teeth that is quantitatively greater than a numerical constant of the algebraic equation; amongst the teeth is a target tooth that represents the numerical constant. Each virtual secondary cogs represent a term of the algebraic equation and includes a coefficient and a variable. Each of the virtual secondary cogs contains a number of teeth equal to the coefficient. The equation is solved by rotating the virtual secondary cogs until the target tooth is aligned with a fixed pointer where rotation of the virtual secondary cog represents a value input for the variable of a term.