The Earth and Moon are orbiting each other - How Things Work - How Things Work Home Page
Since the earth is so much more massive than the moon, their combined center of mass is 81 times closer to the middle of the earth than it is to the middle of the moon. In fact, it's inside the earth, though not at the middle of the earth. As a result, the earth's orbital motion takes the form of a wobble rather than a more obvious looping path. Nonetheless, the earth is orbiting.
I hope that you can see that there is no reason why the earth should be fixed in space while the moon orbits about it. You've been sold a bill of goods. The mistaken notion that the moon orbits a fixed earth is a wonderful example of the "factoid science" that often passes for real science in our society.
Because thinking and understanding involve hard work, people are more comfortable when the thought and understanding have been distilled out of scientific issues and they've been turned into memorizable sound bites. Those sound bites are easy to teach and easy to test, but they're mostly mental junk food. A good teacher, like a good scientist, will urge you to question such factoids until you understand the science behind them and why they might or might not be true.
When my children were young, I often visited their schools to help teach science. In third grade, the required curriculum had them categorizing things into solutions or mixtures. Naturally, I showed them a variety of things that are neither solutions nor mixtures. It was a blast. Science is so much more interesting than a collection of 15-second sound bites.
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