We are concerned with the following nonlinear second-order three-point boundary value problem on time scales − x Δ Δ ( t ) = f ( t , x ( t ) ) , t ∈ [ a , b ] 𝕋 , x ( a ) = 0 , x ( σ 2 ( b ) ) = δ x ( η ) , where a , b ∈ 𝕋 with a < b ,    η ∈ ( a , b ) 𝕋 and 0 < δ < ( σ 2 ( b ) − a ) / ( η − a ) . A new representation of Green's function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution for the above nonlinear boundary value problem are established by using the iterative method.