In this work, the fractional order thermoelasticity theory is used to investigate the thermoelastic problem of a thin slim strip considering the thermal conductivity is to be variable. The theory of thermal stresses based on the heat conduction equation with the Caputo time-fractional derivative of order α is used. The surface of the strip is subjected to a thermal shock and assumed to be traction free. By using the Laplace transform and numerical Laplace inversion, the governing equations are solved. The inverse of the Laplace transform is done numerically using a method based on Fourier expansion techniques. Numerical calculations for the considered variables are performed and the results obtained have been presented graphically. The effects of fractional order parameter and the variation of thermal conductivity on temperature, stress, and displacement are investigated.