Why is x to the Zero Power Equal to 1?

This is my third year teaching, and my third try at getting kids to open up and talk to each other. One of the ways I try to accomplish this is by encouraging students to ask questions. However, it seems like too often I hear, “I didn’t ask during class because I didn’t want to sound dumb.” Many students keep their questions to themselves out of fear that other students will think their question is obvious. I try to tell my classes that all questions are good enough to be asked and that there are many students with the same questions; we just need someone to be brave enough. 

My research group and I decided that to get students to ask more questions, we first had to create a safe environment where questions are valued so students can ask anything without fear. I tell my students all the time that it is important to ask questions and it doesn’t mean they aren’t smart, it means they are learning – and likely they have helped another student who had the same question. One way that I have helped students open up to the idea of asking questions is making it part of the assignment. I tell them they need to think of one question to ask their teammates. I also try to let them ask in their small groups before I have them share with the whole class.

Earlier this semester, after my classes began to feel like a little math family, we started the lessons about simplifying exponential expressions. I always teach these lessons using direct instruction and then use the “red light / green light” study team teaching strategy. We started with expanding and simplifying expressions using the number 1, then the students and I made a list of “rules” that we were finding as they worked. On the second day, the class was comfortable with adding exponents when multiplying, subtracting exponents when dividing, and multiplying exponents when raising a power to a power. Then, we got to the part of the lesson where students try to make sense of x to the zero power. The lesson has students use the idea of the “giant one” and the new rule of subtracting exponents when dividing to make the connection between x to the zero power and one. I had a lot of students ask clarifying questions throughout this lesson, trying to understand how exponent properties work. But once we got to the zero power some students were really struggling to understand how and why x to the zero power was one, “Why wasn’t it x? Or 0?” they kept asking. This turned into an amazing class discussion. One student in particular, AE, was so confused and I tried to break it down again for him. He just wasn’t getting it. I ended up having the entire class trying to help AE to understand the meaning of x to the zero power, and I even had a few students go over to AE’s table to try to help him! Student TM got up and went over to AE’s table saying that even though x to the zero power didn’t make sense mathematically, it made sense because it was a pattern based on previous data – he was writing on AE’s notes to try and help but looking back at these notes I don’t know how helpful it was. Then another student, MC, said x to the zero power isn’t zero because it’s not multiplying by zero. Slowly, AE started to make sense of this and finally said that he got it! It was definitely a group effort and totally worth the additional time out of the lesson.

Even though teachers often feel crunched for time, sometimes it’s important to spend extra time on content, letting the students rephrase material and ask each other questions. I feel it is in these moments where students truly learn. Proof of this came to me the other day, almost 2 months after we had the class discussion about the power of zero. A student was working on her final review and she asked me about x to the zero power, then she said, “Wait, isn’t this the thing we all talked about in class that one day when we were trying to help AE get it? So, it’s one, right?” I thought it was so powerful that our conversation had stuck with her! 

As we move forward in the year with our study one thing I plan to do is use exit tickets, tests, quizzes, or other assignments to ask the students to write down anything they wonder about or any questions they have about class, current or old material, or anything they didn’t want to ask out loud and could privately ask on paper. I am hoping this will help them to feel more comfortable in asking questions out loud and I will continue encouraging students to ask questions and try to make sure I hear from all students.

These are AE’s notes that TM wrote on to try and help explain why x to the zero power is one. I think that TM made more sense in his verbal explanation than this circled note above (2 to the negative 2 is negative 0.02? ). Not sure where he was going with this, but we talked about negative exponents after our conversation about a zero exponent.  
Example from another student’s notes from this day. I chose these notes (and the ones below) to show how the students made sense of – or at least copied down – that by using a “giant one” and the law of exponents, they could prove that x to the zero power is equal to 1. 
Example from another student’s notes from this day. 
Example from another student’s notes from this day. 

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