Zero As an Exponent

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SWBAT to apply rules of exponents for adding and subtracting exponents and then derive the rule for zero power.

Big Idea

After playing a matching game that provides some practice with recently learned rules for multiplying and dividing exponents, students will discover that a zero exponent simplifies to one.

Warm Up

8 minutes

For the Warm Up, students will simplify several exponential expressions with like terms that will require them to apply the previously learned rules. As students work, I move about the classroom watching for student errors or misconceptions. The most likely one to elicit incorrect answers is the the 4th problem, m^6/m^2. Most students answer m^3 because 6 is divisible by 2. When I see this, I ask students to recount the rule for dividing exponents and ask them if they need to revise their answers.

Let's Play!

20 minutes

To reinforce the two rules learned previously, I introduced the Exponents Match Up game in the Let's Play! lesson section. This mult & div exponent match game  is made up of three pages of cards, each with five pair.  I printed, copy, cut and organized the cards into three sets: A, B, and C (I used envelopes and copied the cards onto three colors of paper). Each pair of students worked to match the exponent expression with its simplified form.  They then record their matches in their journals and then trade for another set until all three are completed.

While students work to make matches, I move about the room to redirect and ask guiding questions as needed to keep students on task. When the timer sounds, I direct students to locate their exponent foldable in their journal.

Discovering the Zero Exponent Rule

7 minutes

Once students locate their foldables, I explain that we have another rule to discover, the rule for zero exponents.  I then show students five equations, all with zero exponents, that are equivalent to 1. Students quickly see the pattern, so I ask them to write the rule in the zero tab of their exponent foldable (see Organizing Rules of Exponents lesson for the foldable PDF).

For me, it is critical that students not only know the rules, but can explain the "why" behind each one, so I then show two examples that demonstrate why any base raised to the zero power is one.

The first example is a string of exponents, beginning with 2^4 followed by 2^3 and 2^2 all the way down to 2^0.  I ask the students what 2^4 simplified would be. We then simplify 2^3 and 2^2. I stop and ask students if they notice a pattern in the answers and typically, a student volunteers that the resulting numbers are decreasing by half each time. When the other students confirm this pattern, I then ask, "What will 2^0 simplify to?"  Almost always, a student or two respond with "Zero!", so I ask, "Is half of 2, 0?" to which most students self-correct, "1".

I then show the next example: n^4/n^4. I remind students that n^4 means n x n x n x n. I then ask them what they know about n/n.  If I get no response, I ask what is 3/3 and I get the response of one. I remind students that this is true because the identify property of division says that any number divided by itself is 1.  If we simplify our expanded expression, we get 1 x 1 x 1 x 1 which equal 1.

Ticket Out the Door

5 minutes

For closure, I want students to be able to show their understanding of the zero exponent, so I provide a prompt for the Ticket Out the Door. I distribute 3 x 5 cards for them to record their responses.

As students leave class, they turn in their card to me. I quickly read the response and allow students who have written an explanation to leave. For those who attempt to turn in blank or incomplete cards, I send back to their table to finish. This is a great opportunity to hold students to expectations. Most give up on the idea of resisting work when they realize I will check to make sure it is done.