The frontier orbital analysis is easier if we look at the ring closure direction. The electrons involved are those in the HOMO of the polyene, which will become the electrons of the s-bond.
Hence we simply draw a cartoon of the HOMO, and observe how the terminal orbitals will interact during a ring closure of a particular type:
We can see easily that for butadiene, a conrotatory motion connects terminal lobes of like sign, whereas for the hexatriene a disrotatory motion is required to accomplish this.
Therefore, the butadiene closure is allowed as shown, and the hexatriene closure is allowed with the opposite selectivity.
Although this analysis gives the right prediction, it is inherently less satisfying than the other methods because it does not account for all of the electrons involved.