How much do you have to modify a photo that is "All rights reserved" in order not to be infringing on their copyright?

First of all, the words “All Rights Reserved” are not part of the U.S. copyright law. They are part of an international convention, usually referred to as the U.C.C. That convention has largely been replaced by the Berne Convention. I will presume you mean that the works have a copyright notice on them. Please keep in mind that the copyright notice is not required. You should assume that any work you find on the Internet is protected by copyright law unless you know for certain otherwise.
You mention the “education clause” There is no sweeping exception from copyright infringement for educators. There are some favorable rules for educators, but they are generally pretty minor.
How much do you need to change a work to avoid infringement? There is no clear line, but basically you need to change it 100%. That is, you need to come up with your own works, rather than taking creative expression from others. Keep in mind that ideas are not protected by copyright law. Generally, you can use an idea you see in a work as long as you then provide your own creative expression. But see below, there are exceptions.
I tell my clients that as a rule of thumb you should ask someone to look at your work and at the original work. If that person thinks you based your work on their work, you probably committed copyright infringement.
There is a concept called fair use that will allow some copying. It is a complex area of copyright law. Any discussion of fair use goes way beyond what can be said on AVVO.
Copyright law is very complex. There are many rules, and there are many exceptions. As you will read over and over on AVVO, you really need to talk to an attorney in person about your specific situation in order to get an accurate answer.

You'd have to change it completely, to the point where people would think it is an original work and not a derivative work. The test for that is complicated in a court. But you don't want to wind up there. Make your own pictures, paintings, music, etc.

My peers are right, there is no magic amount of change that is going to make it "ok copyright infringement."
That said, you are a student at a major university. Your school has thought of this and has a policy or something that addresses exactly this type of issue. Ask your professor or search the school's website for some sort of copyright policy. It may take some digging, but it's there. If not, come over to the UW where they think of those things ahead of time.
Good luck!

You received some good responses. I understand what you mean by the "education clause" as there is a very clear exemption for use of copyright protected works by non-profit educational institutions within the face-to-face classroom learning environment (see 17 U.S. Code § 110).
But as you note, if you are asked to ignore that and play it like you would have to in the real world, you are probably greatly limited. I would first want to understand whether your use is falling under the realm of fair use in a very obvious way. This for example would be if you are commenting specially on the images you are using, or using them for social or political commentary. Or if you are transforming them to such a degree that you have obviated infringement.
There is simply no way to advise without knowing all the details. If this is of great importance, I suggest that you consult with a lawyer in private and discuss your objectives in more detail. You can start by calling around to several for a free phone consultation, get some insights then pick the best fit to work with.
Best regards,
Frank
Natoli-Lapin, LLC

There are very good responses here from the other lawyers. Let me just add some practical thoughts here because in the "real world" you won't always have the money or opportunity to consult with an attorney and you still need to make some decisions. The real problem here is that when you start by "copying" someone else's work and "modifying" it you are normally doing something that his still the exclusive right of the copyright owner. You are allowed to borrow "ideas" and unprotectable elements of a work, but anytime you start by copying the whole thing and modifying it, you are at risk. Yet artists all the time will use another work to maybe capture an outline of an object or something like that. It becomes hard to say for a lawyer, however, exactly where a line is between "ok" and "infringement" when you start by copying from something that is a protected work. So what's the best way to be safe? Start from a work that is already in the public domain or that at least is put out with a license that allows modification and reuse--consider things like WikiCommons or even use a Google Images search and use the advanced "Search Tools" to look for works that have "Usage rights" that allow commercial reuse with modification. If you start with something you know is legal to use, then you don't have to worry how far to go to make it unrecognizable compared to where you started. Hope this helps.

How much a photograph must be changed to not infringe its copyright can ONLY be determined on a case by case basis. There is no one rule. Sometimes the answer is none to very little, sometimes very significantly.
The applicable rule is that another's "fair use" of an existing photograph does not infringe its copyright. 17 U.S.C. § 107.
So what's a "fair use"? Depends on the particular use. You can read a few court decisions that address the question as applied to photographs via the links below. There are many, many more. Outside the academic environment you will need to discuss your potential "fair use" of another's work with your own intellectual property attorney.