This paper is written for mathematics educators and researchers engaged at the elementary and middle school levels and interested in exploring ideas and representations for introducing students to ratio and proportion and for making a smooth transition from multiplication and division by whole numbers to their counterparts with fractions. Book V of Euclid’s Elements offers a scenario for deciding whether two ratios of magnitudes, embodied as a pair of line segments, are equal based on whether the ratios of magnitudes, when multiplied by the same whole numbers, *n* and *m, *each yield common products. This test of proportion can be performed using an educational software application where students are presented with a *target ratio* of commensurable magnitudes, *A*:*B,* and challenged to produce a *selected ratio*, *C*:*D*, that behaves like the target ratio under the critical conditions. The selected ratio is automatically constructed such that *C*:*D* = *m*:*n*, on the basis of a lattice point (*n*, *m*) chosen by the student. By adding partitive and Euclidean division to Euclid’s model, five new scenarios with similar goals are proposed. Representations in the Euclidean plane, on a number line, and in the Cartesian plane provide feedback that students may use to help identify a ratio of whole numbers corresponding to the targe ratio of magnitudes. The representations serve to highlight fractions as members of equivalence classes. The model remains to be investigated with teachers and students.

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Posted 15 Mar, 2021

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Posted 15 Mar, 2021

###### No community comments so far

This paper is written for mathematics educators and researchers engaged at the elementary and middle school levels and interested in exploring ideas and representations for introducing students to ratio and proportion and for making a smooth transition from multiplication and division by whole numbers to their counterparts with fractions. Book V of Euclid’s Elements offers a scenario for deciding whether two ratios of magnitudes, embodied as a pair of line segments, are equal based on whether the ratios of magnitudes, when multiplied by the same whole numbers, *n* and *m, *each yield common products. This test of proportion can be performed using an educational software application where students are presented with a *target ratio* of commensurable magnitudes, *A*:*B,* and challenged to produce a *selected ratio*, *C*:*D*, that behaves like the target ratio under the critical conditions. The selected ratio is automatically constructed such that *C*:*D* = *m*:*n*, on the basis of a lattice point (*n*, *m*) chosen by the student. By adding partitive and Euclidean division to Euclid’s model, five new scenarios with similar goals are proposed. Representations in the Euclidean plane, on a number line, and in the Cartesian plane provide feedback that students may use to help identify a ratio of whole numbers corresponding to the targe ratio of magnitudes. The representations serve to highlight fractions as members of equivalence classes. The model remains to be investigated with teachers and students.

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure 11

Figure 12

Figure 13

Figure 14

Figure 15

Figure 16

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Figure 18

Figure 19

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