Lesson objective: Extend understanding of how whole number exponents work to include integer exponents.
Students bring prior knowledge of exponents from 6.EE.A.1. This prior knowledge is extended to integer exponents as students use patterns to formalize that \(a^{n}\) = \(\frac1{a^n}\). A conceptual challenge students may encounter is understanding that a negative exponent doesn’t give a negative product.
The concept is developed through work with the illustration/diagram in the prior knowledge and the table, which make patterns in exponents (with the same base) more visible.
This work helps students deepen their understanding of equivalence, because when students develop an understanding of integer exponents, they are able to write equivalent expressions, such as \(5^{2}\) = \(\frac1{5^{2}}\) = \(\frac1{25}\).
Students engage in Mathematical Practice 8 (look for and express regularity in repeated reasoning) as they use patterns to derive rules about integer exponents. Using multiplicative patterns is the foundation of this conceptual lesson. In the prior knowledge, the diagram is a representation of how whole number exponents work. As integer exponents are explored, tables and expressions are the representations used.
Key vocabulary:

base

exponent

expression

factor

integer

multiplicative inverse

power
 reciprocal