Many students find certain mathematical concepts extremely difficult. Fractions, percents and decimals, in particular, are hard to conceptualize, especially if they are used in computations such as addition, subtraction, multiplication and division. Some of this difficulty in understanding may be attributed to the way in which fractions, percents and decimals are taught. If they are merely explained orally and/or in writing on a chalkboard, students may not grasp the abstract concepts or calculations as well as they could if they had actual objects representing fractions, percents and decimals to see and touch. In addition, complex ideas such as fractions exceeding one ("improper fractions") or converting from fractions to mixed numbers (whole numbers summed with fractions, for e.g. 13/4) are difficult to model even in writing.
Currently, there are various teaching aids which attempt to assist students in visualizing, describing and understanding some of these fraction concepts, all of which suffer some significant shortcomings or limitations. For example, Fraction Bars.RTM. by Scott Resources (P.O. Box 2121, Ft. Collins, Colo. 80522) is a program which uses activities, games, worksheets and tests to teach fraction concepts and operations. Fractions Bars.RTM. includes a set of color-coded plastic strips which contain fractional increments and shaded portions to represent parts of a whole. In addition, Rainbow Fraction Tiles.TM. by Learning Resources (675 Heathrow Drive, Lincolnshire, Ill. 60069) is designed to aid students in manipulative activities essential for understanding fractions. Rainbow Fraction Tiles are color-coded plastic tiles of different sizes with fractional designations printed on each one. The length of each tile corresponds to its fractional designation. For example, the largest tile (which represents one whole) is red and has the number "1" printed on it, the next two tiles (which together equal the length of the one whole tile) are pink and have the fraction "1/2" printed on each, etc. Unfortunately, these two teaching devices are rather limited in their application. Although the Fraction Bars and Rainbow Fraction Tiles offer an effective means by which to compare fractions and learn basic concepts, performing a range of mathematical calculations, such as addition, subtraction, multiplication and division (particularly where the result exceeds one), with these devices is very difficult because the fractional components cannot be easily combined into a new unitary whole. Consequently, comparison and exploration of fraction concepts is limited.
Fraction Stax.RTM. by Ideal School Supply Company (Oak Lawn, Ill. 60453) overcomes some of these limitations by making it possible for students to compare, order, add and subtract fractions. Fraction Stax.RTM. is a three-dimensional fraction model that includes sleeve-like color-coded fraction pieces and a nine-peg base on which the pieces are mounted. The length of each color-coded piece corresponds to the fractional designation which is printed on it. The piece with the number "1" designation is "the whole" and thus the longest piece. The other pieces are halves, thirds, fourths, fifths, sixths, eighths, tenths and twelfths of the whole. The pegs on the base are the same length as the whole. Accordingly, this teaching aid is limited because the fraction pieces cannot be mounted higher than the peg without the risk that they will be knocked over by any slight movement, the fraction pieces are very difficult to use without the base and the number of fractions displayed is limited by the number of pegs on the base. Consequently, improper fractions and mixed numbers cannot be displayed or taught and more complex calculations involving operations of improper fractions or mixed numbers cannot be performed. Further, teachers cannot show students the transition or interchangeability between mixed numbers and improper fractions. This is a major impediment to teaching because a significant portion of math curriculum devoted to fractions concentrates on "conversion," that is from improper fractions to mixed numbers and vice versa.
Therefore, it would be highly desirable to develop mathematical pieces which can be combined independent of a peg, stand or tray. It would also be highly desirable to develop mathematical models which can readily display improper fractions, mixed numbers and other rational numbers. Furthermore, it would be highly desirable to develop mathematical models that can be used to teach complex mathematical concepts and calculations and illustrate the conversion of improper fractions to mixed numbers and vice versa. Finally, it would be highly desirable to develop mathematical models of fractions which can be easily manipulated by a small child.