Science, Technology, Engineering, and Math (STEM) and Trade in Schools Act This bill amends the Higher Education Act of 1965 to specify that applicants who have expertise in career and technical education are eligible for Teacher Education Assistance for College and Higher Education (TEACH) grants. (TEACH grants support teachers who have expertise in high-need subjects or teach in schools serving students from low-income families.) In selecting teacher candidates for these grants, the Department of Education (ED) shall give priority to applicants who are members of a group that is underrepresented in teaching subjects related to science, technology, engineering, mathematics (STEM), or computer science. The bill amends the Elementary and Secondary Education Act of 1965 to: (1) provide a specified funding level for the STEM master teacher corps program, and (2) require a STEM master teacher corps to establish and consult with a private-sector advisory board that meets specified requirements.