The board of managers of such school shall consist of not less than five members and shall be composed of all the city, village and district superintendents of schools of the county in which it is located, in addition to such other members as may be necessary to make a total membership of such board of not less than five. Such additional members of the board shall be appointed by the board of supervisors or other governing elective body from the resident taxpayers of the county, who shall serve for terms of four years commencing on the first day of January succeeding their appointment. Such terms shall be so arranged that the terms of no two of the members so appointed shall expire in the same year, and for this purpose the terms of the members first appointed hereunder shall be as follows: In case one member shall be appointed, the term shall be four years, in case two members shall be appointed, the terms shall be four and two years, respectively, in case three members shall be appointed, the terms shall be four, three and two years, and in case four members shall be appointed, the terms shall be four, three, two and one year, respectively, which terms shall commence on the first day of January succeeding their appointment, and their successors shall be appointed for full terms of four years as above provided. Appointments to fill vacancies shall be for the unexpired portion of the terms. The members of the board shall serve without compensation. They shall receive their necessary expenses incurred in the performance of their duties. The amount of such expenses shall be charged against the county and shall be paid in the same manner as other county charges. The board shall organize by the election of one of its members as chairman and another as secretary.