How does thermal energy flow as you hold a glass of ice water?

By conduction from your hand to the glass. By convection from your hand to the glass. By convection from the ice to your hand . By radiation from the ice to your hand.

The thermal energy is transferred from your hand, to the glass, to the water. It starts out high as molecules are vibrating fast, and gets slower until it reaches the ice water and starts heating it up very slowly.

Because a model would make the investigation easier to understand

Solubility would be your answer.