Is this what your model is?
If so, I have an easy experiment for you:
On the equinox, at sunrise where you live, draw a straight line about 55' long on a reasonably flat surface, and plant a straight 6' long pole about a foot into the ground, pointing at the sun on the horizon. Then mark on the model the location on Earth at which the sun is directly overhead. One hour later, plant another pole 6" farther down the line, pointing at the sun. Then mark the location at which the sun is directly overhead. Repeat this every hour until sunset.
Now draw a line on the model, showing the path of the sun, then look from the end of the first pole to the end of the last pole and ask yourself why the ends of the poles are all in a straight line, and not a half circle, like they will be on the model.