Momentum and interest were gathering behind Algebra Suicide and crucially for Tomkiw’s identity as a poet, nearly all the critics recognized the fundamental poetic aspect of the project. Tomkiw was regularly credited as a burgeoning, successful poet and the nature of the songs as poems was central to critical appreciation of the band. In 1985, Algebra Suicide released their debut album, "Big Skin", on the label Cause and Effect. Its format represented the firmly dual nature of Algebra Suicide — a cassette of thirteen songs paired with a chapbook presenting thirteen poems in the same running order. Algebra Suicide began booking performances beyond the Chicago circuit, playing in Milwaukee, Lexington, Cincinnati, and Pittsburgh, with favorable reviews accumulating in alternative magazines like "Trouser Press" and "Option". Tomkiw continued to write prolifically, but chapbooks yielded to records and singles, and by 1986, Algebra Suicide had become the primary vehicle for publishing her poetry. In 1987, the band released "The Secret Like Crazy" on the Massachusetts-based label RRRecords and in Europe on the West German Dom Elchklang imprint, essentially becoming Algebra Suicide’s international debut. Shortly after, another German label, Pursuit of Market Share, released a live recording from a performance at Chicago’s Links Hall entitled "Real Numbers." Meanwhile, Tomkiw’s written poetry got a sudden boost when her palindromic poem “Six of Ox Is” was included in 1987’s "New American Writing", an annual anthology emphasizing contemporary American poetry. That led, a year later, to the poem’s inclusion in the first volume of The Best American Poetry, founded by poet and editor David Lehman who tapped John Ashbery as its inaugural editor. A year later, Tomkiw’s UK debut poetry collection, "The Dreadful Swimmers", was published by Wide Skirt Press, an independent imprint and literary magazine run by poet Geoff Hattersley, that had already published more than twenty of Tomkiw’s poems. In 1989, Algebra Suicide received a Chicago Artists Abroad Grant to perform overseas in Europe in the fall of 1990. Tomkiw and Hedeker booked a tour that took them through Munich, Cologne, Paris, Bordeaux, Louvain, and Zurich. The tour was a huge success. Preceded by strong reviews and advance buzz, the shows were packed, audiences rapt and appreciative. In May of 1991, Tomkiw got another Chicago Artists Abroad Grant, this time supporting her endeavors as a traditional poet.