Student writers who are stuck on the five-paragraph formula often struggle in particular with introductions and conclusions. The formula for introductions and conclusions is most often a mechanical recapitulation of the topic sentences and the thesis. One way to break this pattern would be to use the kind of thesis described above, but, instead of cramming the entire x, y, z thesis (see above) into a giant granddaddy of a sentence, spread x, y, and z across the entire introduction.
Instead of mechanically recapping your topic sentences and the thesis itself in your conclusion, think of your conclusion this way: the opening of the conclusion can/should/might review your central point (thesis), but it should do so from the perspective of "we can now understand the thesis from a better vantage point than we did at first, because we have examined a lot of new information." In other words, an introduction projects the route you will travel (like a roadmap before driving cross-country), whereas a conclusion reviews the route traveled from a more experienced perspective (like a scrapbook from the roadtrip). Next, the conclusion might discuss what's now on the horizon: what you might write next on the topic (with more time and space), what new questions the current essay now raises, possible ways to improve upon the essay (like an error analysis in a lab report)-in other words, what might come next, if the writer were to continue building more knowledge in this direction. The key terms here are "next" and "more"; if your intro or conclusion are essentially the same, it means the essays hasn't gone anywhere. If you end up exactly where you start, then no ground has been gained. Things should appear differently at the end of your essay, and the purpose of the conclusion should be to highlight those differences (not to hammer the nail of your thesis once loud, last time).
Body paragraphs of essays can take a wide variety of forms beyond the five-paragraph formula. The most obvious first step would be to include more than three interior paragraphs (+ intro + conclusion = 5 paragraphs). "Variety" is the key term here. Paragraphs need not be cookie-cutter versions of each other. Some may be long; others short. They may arranged in terms of "point" versus "counter-point," or chronologically, or thematically, or in terms of specific-to-general (or vice versa), or by modes (description, exposition, etc.). The possibilities are limitless; so, you should explore them and avoid paragraph formulations that are mechanical, synthetic, clichéd, tired, formulaic. Like your sentences, if each paragraph looks, sounds, and reads almost exactly the same as the others, then you're being robotic-you're not really writing, composing, or communicating; you're filling in prefabricated boxes with words.