Education Technology

Expressions and Equations / Extending Exponents

Grade Level 8
Activity 11 of 18
In this lesson students investigate how patterns in the computations for $$a^n, (ab)^n, (a^m)^n$$ that held for positive integer exponents extend naturally to integer exponents.

Planning and Resources

Objectives
Students should be able to extend properties of postive integer exponents to all integer exponents, apply properties of exponents to express numbers in scientific notation, and interpret and explain numerical expressions involving exponents that are 0, negative, or $$\frac{1}{2}$$ or $$\frac{1}{3}$$.

Vocabulary
exponent
base
factor
cube
square
power

Standard:

Lesson Snapshot

Understanding

This lesson allows students to investigate how patterns they developed for postive exponents extend to integer exponents.

What to look for

Common misconceptions include not understanding which element in the expression is the base of the exponent, multiplying the exponent and the base, reversing the role of the base and the exponent, and assuming a negative exponent changes the sign of the base.

Sample Assessment

Which of the following expressions are equivalent to $$\frac{3^{-8}}{3^{-4}}$$?

a. $$3^{-12}$$       b. $$3^4$$       c. $$3^2$$

d. $$\frac{1}{3^2}$$           e. $$\frac{1}{3^4}$$       f. $$\frac{1}{3^{12}}$$

Answer: b and e

The Big Idea

Patterns in the computations for $$a^n, (ab)^n, (a^m)^n$$ that held for positive integer exponents extend naturally to integer exponents.

What are the students doing?

Students recognize patterns that emerge from generating equivalent expressions involving integer exponents and generalize rules for working with exponents.

What is the teacher doing?

Encourage students to explore equivalent ways of writing numerical expressions involving integer exponents.