In this activity, students extend their understanding of exponents to include negative exponents and explain patterns involving place value when a value is multiplied by 10 or . Students use repeated reasoning to recognize that negative powers of 10 represent repeated multiplication by and generalize to the rule (MP8).

A table is used to show different representations of decimals, fractions, and exponents. The table is horizontal to mimic the structure of decimals and to help students connect this work with place value. Monitor for different strategies used to complete the table.

The goal of this discussion is to reinforce the idea that negative exponents can be thought of as repeated multiplication by , whereas positive exponents can be thought of as repeated multiplication by . 

Invite students to share their strategies for completing the table. Some students may describe multiplying by as multiplying by the reciprocal of 10. In grade 8 the focus is on negative exponents with whole number bases, and use of the word reciprocal is not necessary at this time. Record their reasoning for all to see. Here are some strategies students may use:

• Notice that the exponent decreases by 1 as the table moves to the right.
• Notice that for values with a positive exponent — the corresponding decimal and fraction have 1 less zero as the table moves to the right, and the values are greater than or equal to 1.
• Notice that for values with a negative exponent — the corresponding decimal and fraction have 1 more zero as the table moves to the right, and the values are all less than 1.

Students may also mention the following strategies:

• They “moved the decimal to the right or left.” 
• They used the value of the exponent to know how many zeros to write.

Validate student thinking as these observations make sense based on the information given in the table. But emphasize the idea that multiplying by 10 or increases or decreases the value by a factor of 10, and therefore increases or decreases the place value of the digit 1. This will make it appear that the decimal place is moving, or that the exponent is equal to the number of zeros.  

Then introduce and explain the visual display prepared earlier. This display should be kept visible to students throughout the remainder of the unit.

Continue to reinforce student understanding of this idea by writing out an expanded form of each expression when discussing the following questions:

• “What is written with a positive exponent?” ( )
• “What is written with a positive exponent?” ( )
• “What is written with a negative exponent?” ()

In this activity students make sense of negative powers of 10 as repeated multiplication by and use this structure in order to distinguish between equivalent exponential expressions (MP7).

The goal of this discussion is for students to understand that the exponent rules work even with negative exponents by making a clear connection between the exponent rules and the process of multiplying repeated factors that are 10 and . 

Display the expressions and from the first problem for all to see. Ask students what is the same and different about these two expressions. (They are both equivalent to . Both expressions contain one positive exponent and one negative exponent.) 

Reinforce students' understanding of the exponent rules by writing out an expanded form of each expression when discussing the following questions:

• “What do the 3 and -2 in  mean in terms of repeated multiplication?” (The 3 means that there are 3 factors that are each , and the -2 means that there are 2 factors that are .)
  So
• “What do the 2 and -3 in  mean in terms of repeated multiplication?” (The 2 means that there are 2 factors that are each 10, and the -3 means that there are 3 factors that are each .)
  So