# Teaching the meaning of negative exponents

One of the common misconceptions by students when they first encounter the concept of a 0 exponent is to think that a number to the power of 0 is equal to 0. For example, many students believe that 20 = 0. The correct result is actually 1, in other words 20 = 1.

This misconception stems from their initial understanding of what exponents are. Students are taught that 23 = 2 x 2 x 2, and so when they encounter the situation of 20, it is natural to believe that it is equivalent to the number 2 multiplied by itself 0 times, which should give a result of 0. What students often fail to understand is that each time an exponent increases by 1, the value doubles. Taken in reverse, the value of 20 should be half of the value of 21, which would give the correct conclusion that 20 = 1.

To teach the concept of the 0 exponent, I have decided to take the TGEM approach as discussed by Khan (2010) , using an activity I created using the Desmos platform. The activity can be viewed here:

https://teacher.desmos.com/activitybuilder/custom/597642998fb671717b38af33

This activity encourages students to brainstorm, and share their thoughts as to what 20 is equal to, before diving into an exploration that will eventually lead students to the value of 1 using a visual approach. The activity builds on the idea to discuss the concept of negative exponents and their meaning. I would utilize the above activity using the following steps:

1. Point students to the above link to allow them access to the activity. Ensure that a class code is created so that the class can join. Turn on teacher pacing for this activity to ensure students don’t work ahead, and to encourage discussion along the way.
2. On the first screen, pause and allow the students to read. Allow students to brainstorm what their initial thoughts are about the meaning behind the concept of a 0 exponent. Using the teacher dashboard, display the students input to look for commonalities in thinking.
3. On the second screen, ensure students understand that the numbers are doubling at each step. Students should be informed that they need to be precise, and that the “numbers are increasing” is will not adequately describe the pattern they see.
4.  On the third screen, ensure students understand that the numbers are halving at each step. Students should be informed that they need to be precise, and that the “numbers are decreasing” will not adequately describe the pattern they see.
5. On the forth screen, ensure students can now reach the conclusion as to what the value of 2^0 is. Spend some time explaining the idea that a power with an exponent of 0 is equal to 1, no matter what the base is.
6. On the fifth screen, ensure students continue the pattern to reach a conclusion as to the meaning of a negative exponent.

Khan, S. (2010). New pedagogies for teaching with computer simulationsJournal of Science Education and Technology, 20(3), 215-232. Available in Course Readings.

1. ChristopherRozitis says:

Hi Gary

I like the fact that you brought up this concept of a number to the power of 0. I do not remember learning this back in high school…I think I was just told the answer. I also, like the activity on the desmos activity. I have not used it before and I will see if I can incorporate it into my online classes.

I wonder if instead of submitting to the teacher, the student can get immediate feedback. It seems that way the desmos activity is set up that the teacher is going to have to do a lot of opening files. But I may be wrong.

A good next step might be to create a screen-capture video for what the students see and then what the teacher sees.

To keep the conversation going — make sure to respond to at least two other learners as well respond to all learners that respond to your own post. When responding to other learners, expand the discussion and please use references to support your ideas/thesis/concepts etc.

Christopher

1. Gary Ma says:

Hi Chris,

In some Desmos activities, the subsequent screens could be programmed in reaction to what the user has inputted in a previous screen. For example, if the user types in an equation in one screen (eg: y=2x+1), the graph of that line may show up in the next screen and the a question may be asked based on what you have entered. (for an example, see the activity here: https://teacher.desmos.com/activitybuilder/custom/56001cb3ccac42274a00be25 ) In that way, students receive some indirect feedback.

2. sarah winkler says:

Wow, I am pretty sure I learned this in my first year university math class, thanks for the refresher.

I like how you built in discussion and pacing to ensure all students are given time to think and interact with the presentation. I wonder how you could ensure you hear discussion from all students to ensure that they are mastering concept.

Desmos looks interesting, I hadn’t seen it before, I look forward to exploring some more. Sarah

1. Gary Ma says:

Hi Sarah,

Desmos quietly introduces discussion by allowing students to discuss via text as opposed to having to raise their hands in class. After students have entered theirs thoughts into the textbox, the teacher can choose to display the teacher dashboard in front of the class, which includes everyone’s inputs. The teacher can then simply point out what each student has entered in. I found that this way of eliciting participation is much more likely to get input from a greater variety of people, as they aren’t risking embarrassment in front of the class.

3. momoe says:

Hi Gary,

Cool desmos lesson. I also really like how you added in the pacing aspect to generate time for discussion. Do you think there’s some sort of way to demonstrate this information using simulation or animation? Do you think it would be helpful to include this type of visualization to your lesson?

1. Gary Ma says:

Unfortunately, the instructor cannot create animations in the activities they create on the activity builder. Having some animation would be useful in this case but I don’t believe is necessary.