Inspiration

Negative exponents can be a confusing topic for many students.  Due to time constraints, I’ve focused more on the rules in lieu of introducing negative exponents conceptually.  As a result, students become confused on how the negative exponent effects the base.  I wanted to change this outcome.

One night, while my family and I were watching, Creed, I began tinkering with Desmos.   (Side note:  Sylvester Stallone should have won the Oscar for Best Supporting Actor).  I created a variety of slides.  The next day, when I sat down to organize my work, the glaring need to incorporate a zeroth power activity was apparent.

The following activity was therefore created:  The Zeroth Power

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Part 1 – Desmos

Slide 1

Slide 1 is packed with information and needs to be broken down.  Students will need to understand

• The meaning of the slider
• The meaning of the x/y chart
• How the slider effects the graph
• How the slider effects the x/y chart
• How the x/y chart and the graph are connected.

Slides 2 & 3

Slide 2 is designed to kick off the conversation surrounding slide 1

Slides 4 & 5

1. Now that students are aware of the connections in slide 1, it’s time to set up the discussion about the power of zero.
2. Before moving onto slide 6, call on students to share out their coordinates. Record coordinates on the board
3. Reinforce the meaning of the x and y columns.  The x-column represents the exponent.  The y-column represents the value of the base raised to the given exponent.

Slide 6 & 7

Slide 6 requires students to describe, in their own words, the meaning of each coordinate.

Slides 8 & 9

Give the students 10 seconds to choose their base value and move their slider.  Then go to the overlay. Ask students to discuss their observations, with a partner.  Then submit their answer in slide 9

At this point, students should’ve arrived at the conclusion that the graph for each base has one point in common, (0, 1).  With that conclusion, the discussion medium changes from Desmos to a vertical non permanent surface (aka over-sized white board)

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Part 2 – Vertical Non Permanent Surfaces (#VNPS)

I love VNPS and the concept of a vertical classroom.  In my ideal school, all classrooms would have white boards as walls.  As a coach, I feel it’s my duty to expose teachers to the magical powers of a vertical classroom.  How?  I bought three 4ft x 8ft shower boards at Lowe’s.  The cost – roughly $13 per board.

Lowe’s will cut the boards for free!!  All 3 boards were cut into 2ft x 2ft squares.  Since I work at 2 schools, I’ve stored half at each school.

Back to the lesson

At this point, the purpose of the activity is to understand why the exponential graph of each base crosses at the (o,1).  Why is the value of any base raised to the power of zero is equal to 1.

1. Ask each group to pick a base value
2. Using their base value, the group’s are to generate an x/y chart similar to the one in slide 1.  All work displayed on their white board (#VNPS)
3. As a group, they are to find a pattern that explains why (their base)^0 =1
4. Group members Prep their Rep, by requiring each member practice explaining the pattern.
5. The teacher circulates through the room, listening to group discussions and asking questions.
6. The teacher picks a few groups to share their work and pattern.  Due to the size of the white board, groups can easily share their work with the class.

Conclusion

The zeroth power is the gateway concept to understanding negative exponents. It’s necessary for students to visually see that any base raised to the zeroth power equals 1, then discover the pattern and explain their reasoning.

The 2 mediums, Desmos & VNPS, support a student’s development of the concept. Desmos allows students to interact and explore this concept which nicely segues into a conversation about negative exponents and eventually asymptotes.  The VNPS provides a big space to process and clarify thinking.

*My original goal was to create working with negative exponents.  Here it is…

Negative Exponent Exploration

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My Desmos Page:  Desmos Activities