Multiplication as Proportional Co-Variation Shows Sign Rules

① This GeoGebra example dynamically demonstrates how multiplication can be understood as proportional co-variation, using geometric line segments to visually reveal the sign rules for multiplying positive and negative numbers. ② Core mathematical concepts include proportion, multiplication, sign rules (same signs yield positive, different signs yield negative), and the geometric construction of ratios as in Euclid's Elements, Book VI, Proposition 12, historically referenced by Campanus of Novara in the 13th century. ③ Construction details: The applet employs GeoGebra commands such as Intersect, Point, Line, Segment, Angle, and Textfield, with objects like a_1, C_1, multiplier, unit, c, B, multiplicand, a, E_1, product, I, and more. Users adjust the multiplier and multiplicand via sliders or input boxes, observing how the product segment changes and reading text labels that state the sign rules. ④ This resource is ideal for middle school (grades 7–9) mathematics classes learning rational number multiplication, ratio, and geometric visualization, as well as for teacher demonstrations connecting arithmetic and geometry in undergraduate mathematics education.