always expressed as an addition to or subtraction from an integer. The integer was written on one line and the fraction in its two parts on the next line. If the fraction was marked by a small circle or cross , it is subtracted from the integer; if no such sign appears, it is understood to be added. For example, Bhaskara I writes: ६ १ २ १ १ १० ४ ५ ९ which is the equivalent of 6 1 2 1 1 −1 4 5 9 and would be written in modern notation as 6, 1, and 2 − (i.e., 1). The horizontal fraction bar is first attested in the work of Al-Hassār (), a Muslim mathematician from Fez, Morocco, who specialized in Islamic inheritance jurisprudence. In his discussion he writes: "for example, if you are told to write three-fifths and a third of a fifth, write thus, The same fractional notation—with the fraction given before the integer—appears soon after in the work of Leonardo Fibonacci in the 13th century. In discussing the origins of decimal fractions, Dirk Jan Struik states: The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet De Thiende, published at Leyden in 1585, together with a French translation, La Disme, by the Flemish mathematician Simon Stevin (1548–1620), then settled in the Northern Netherlands. It is true that decimal fractions were used by the Chinese many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century). While the Persian mathematician Jamshīd al-Kāshī claimed to have discovered decimal fractions himself in the 15th century, J. Lennart Berggren notes that he was mistaken, as decimal fractions were first used five centuries before him by the Baghdadi mathematician Abu'l-Hasan al-Uqlidisi as early as the 10th century. In formal education Pedagogical tools In primary schools, fractions have been demonstrated through Cuisenaire rods, Fraction Bars, fraction strips, fraction circles, paper (for folding or cutting), pattern blocks, pie-shaped pieces, plastic rectangles, grid paper, dot paper, geoboards, counters and computer software. Documents for teachers Several states in the United States have adopted learning trajectories from the Common Core State Standards Initiative's guidelines for mathematics education. Aside from sequencing the learning of fractions and operations with fractions, the document provides the following definition of a fraction: "A number expressible