By the time they reach middle school, all students have been taught to add fractions. However, not all have "learned" to add fractions. The common mistake in adding fractions is to report that a/b + c/d is equal to (a + c)/(b + d). It is certainly necessary to correct this mistake when a student makes it. However, this occasion also presents a valuable opportunity to enhance the student's mathematical confidence while also strengthening the student's number sense. Rather than placing all the emphasis on why the formula produces wrong answers, teachers should also acknowledge that the student has "invented" an operation that is both useful and interesting. The student will be proud and intrigued to learn that the operation has utility in geometry, statistics, calculus, and other areas. This article offers examples and activities that explore properties of the mediant and can be used to strengthen weaker students' basic numerical skills while honing the problem-solving abilities of the best and brightest.