How To Teach Zero And Negative Exponents

Key concept of Zero and Negative Exponents

When students learn about negative and zero exponents in 6th grade, it is a big jump moving from multiplication to repeated multiplication. The next big jump in understanding exponents is with zero and negative exponents. The rules for zero exponents and negative exponents are not intuitive. It is helpful to start with an exploratory activity that helps students discover the rules. It drives deeper conceptual understanding and is less procedural.  

How do you teach negative and zero exponents?

Step 1: Review how to simplify positive exponents

Have students complete the following table independently.

When it looks like students are done, ask a student or multiple students to fill in the table on the board. 

Step 2: Find the patterns

Now, for a discussion. 

• Ask students to find the pattern as you move from the bottom of the table to the top of the table:
  2, 4, 8, 16, …
  • Allow students to talk or work together. They should notice that the pattern is multiplying by 2 as the exponent increases by 1.

• Now, ask students to find the pattern as you move from the top of the table to the bottom of the table:
  16, 8, 4, 2, …
  • Allow students to talk or work together. They should notice that the pattern is dividing by 2 as the exponent decreases by 1.

• As you discuss, remind students that the base is 2, so it makes sense that we would multiply by 2 as the exponent increases by 1, and divide by 2 as the exponent decreases by 1.

Step 3: Extend the pattern

Extend the table on the board to show decreasing exponents. 

Ask students to follow the pattern of dividing by 2. Encourage them to write their answers as fractions when they are less than 1. 

When it looks like students are done, ask a student or multiple students to fill in the table on the board. 

Step 4: Discussion

As a class, rewrite the fractions so that the denominators are written in exponent form, with a base of 2. 

Now for the final discussion. Discuss/ask the class about the following:

• “Let’s focus on when the exponent is 0. We got 1 for the answer. Using similar thinking, what would be 3⁰ and 4⁰?”
  • Allow students to talk or work together. Eventually, discuss that any base raised to the 0 power will always be 1. Add the rule a⁰ = 1 to the classroom’s growing list of exponent rules.
• “Now let’s look at the negative exponents. [Point to the middle column] Notice how negative exponents made the value of that number smaller and smaller as the exponent decreased.” 
• “We also found:
  2^-1 = 1/2¹

2^-2 = 1/2²

The digital co-teacher made with ❤️ by teachers

ByteLearn saves you time and ensures every student gets the support they need

Try for Free

2^-3 = 1/2³

2^-4 = 1/2⁴
Can we come up with a ‘rule’ for negative exponents?”

• Allow students to talk or work together. Depending on the students you may want to start them with “a^-n = ?” and let them fill in the rule. Eventually discuss that since the negative exponent makes the value of the number smaller, it becomes a fraction of that power. 
• Add the rule “a^-n = 1/aⁿ” to the classroom’s growing list of exponent rules.

Why teach zero and negative exponents this way?

Teaching negative and zero exponents this way allows students to discover the reasoning behind these exponent rules. This creates a deeper conceptual understanding.

Vocabulary for teaching negative and zero exponents

Exponent: An exponent (or power) is a small number placed to the upper right of a base. It shows how many times the base is multiplied by itself.

Power: The power of a number indicates how many times the base number will be multiplied.

Negative exponents: When a base has a negative exponent, bring the power to the denominator and make the exponent positive.
a^-m = 1/a^m

Zero exponents: Anything (except 0) raised to the power of 0 is equal to 1.
a⁰ = 1

Misconceptions and errors students are likely to have

• A negative exponent does not mean the value is negative!
  • Consistently remind students that negative exponents make the value small, not negative!
• Students sometimes will think something raised to the 0 power is 0.
  • Remind students of the rule: Anything raised to the power of 0 is equal to 1.

Resources

• Teacher slides to use this activity in the classroom
• Downloadable worksheet for practicing negative and zero exponents
• Quiz for assessing negative and zero exponents

Frequently Asked Questions On Zero And Negative Exponents