Too much mathematical rigor teaches rigor mortis: the fear of making an unjustified leap even when it lands on a correct result. Instead of paralysis, have courage: Shoot first and ask questions later. Although unwise as public policy, it is a valuable problem-solving philosophy and the theme of this course: how to guess answers without a proof or an exact calculation, in order to develop insight. You will learn this skill by mastering six reasoning tools---dimensional analysis, easy cases, lumping, pictorial reasoning, taking out the big part, and analogy. The applications will include mental calculation, estimating population growth rates, understanding drag without differential equations, singing musical intervals to estimate logarithms, approximating integrals, summing infinite series, and turning differential equations into algebra. Your learning will be supported by regular readings that you discuss with other students, by short tablet videos, by quick problems to help you check your understanding, by weekly homework problems, review and and a final exam. You will work hard, and, by the end of the course, have learned a rough-and-ready approach to using mathematics to understand the world.