Standard #: MA.8.NSO.1.3

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.8.AR.1.1
• MA.8.AR.1.2

 

Terms from the K-12 Glossary

• Exponent
• Expressions
• Integer
• Rational Numbers

 

Vertical Alignment

Previous Benchmarks

• MA.7.NSO.1.1

Next Benchmarks

• MA.912.NSO.1.1

 

Purpose and Instructional Strategies

In grade 7, students were introduced to the Laws of Exponents with numerical expressions. They focused on generating equivalent numerical expressions with whole-number exponents and rational number bases. In grade 8, the learning extends to integer exponents. In Algebra 1, students will extend the Laws of Exponents to include rational exponents. 

• Instruction focuses on one law at a time to allow for conceptual understanding instead of just memorizing the rules. Students should be given the opportunity to derive the properties through experience and reasoning. During instruction, include examples that show the expansion of the bases using the exponents to show the equivalence. This strategy allows for moving beyond learning a rule or procedure.
• The expectation for this benchmark includes negative integer exponents but does not include fractional exponents.
• Students should develop and engage in understanding the rules of exponents from exploration. A strategy for developing meaning for integer exponents by making use of patterns is shown below:
  
  
  
  
• Students should develop fluency with and without the use of a calculator when evaluating numerical expressions involving the Laws of Exponents.
• Instruction includes cases where students must work backwards as well as cases where the value of a variable must be determined (MTR.3.1). Students should use relational thinking as well as algebraic thinking.

 

Common Misconceptions or Errors

• When working with negative exponents, students may not understand the connection to fractions and values in the denominator. To address this misconception, use expanded notation to show how to simplify to help support the understanding of exponents and values in the denominator of a fraction.
  • For example, ($\frac{\text{5}}{\text{4}}$)⁻³ can be rewritten as (($\frac{\text{5}}{\text{4}}$)⁻¹)⁻³ which can be rewritten as ()³ which can be rewritten as ($\frac{\text{4}}{\text{5}}$)³ which can be rewritten as ($\frac{\text{4}}{\text{5}}$) ($\frac{\text{4}}{\text{5}}$) ($\frac{\text{4}}{\text{5}}$) which is equivalent to $\frac{\text{64}}{\text{125}}$.

 

Strategies to Support Tiered Instruction

• Instruction includes teacher modeling the use expanded notation to show how to simplify. Have students practice the properties by generating equivalent expressions.
  • For example, 4² × 4⁻⁶ = $\frac{1}{4^4}$ which equals $\frac{\text{1}}{\text{256}}$ or 4 × 4 × $\frac{\text{1}}{\text{4}}$ × $\frac{\text{1}}{\text{4}}$ × $\frac{\text{1}}{\text{4}}$ × $\frac{\text{1}}{\text{4}}$ × $\frac{\text{1}}{\text{4}}$ × $\frac{\text{1}}{\text{4}}$ = $\frac{\text{1}}{\text{256}}$. Help students to discover how 2⁻³ becomes a positive exponent in the denominator of a fraction, $\frac{\text{1}}{\text{2³}}$.
• Instruction includes the use of a conceptual approach as opposed to memorizing the rules of the laws of integer exponents. Provide examples of the processes that lead to the rules for each law, such as the “Patterns in Exponents” table within Instructional Strategies.
• Instruction includes using expanded notation to show how to simplify to help support the understanding of exponents and values in the denominator of a fraction.
  • For example, ($\frac{\text{5}}{\text{4}}$)⁻³ can be rewritten as (($\frac{\text{5}}{\text{4}}$)⁻¹)³ which can be rewritten as ()³ which can be rewritten as ($\frac{\text{4}}{\text{5}}$)³ which can be rewritten as ($\frac{\text{4}}{\text{5}}$) ($\frac{\text{4}}{\text{5}}$) ($\frac{\text{4}}{\text{5}}$) which is equivalent to $\frac{\text{64}}{\text{125}}$.

 

Instructional Tasks

Instructional Task 1 (MTR.1.1)
Create an example that will show and explain the difference between −$b$ and $b$⁻¹.

Instructional Task 2 (MTR.5.1)
Create a pattern using the expanded form of the base, 4, between 4⁻⁵ and 4⁵. Explain why 4° is equal to 1.

 

Instructional Items

Instructional Item 1
What is the value of ($\frac{3^6}{3^{-4}}$)² ?

Instructional Item 2
What is the value of the expression given below.
Instructional Item 3
Which of the following expressions are equivalent to $\frac{1}{2^6}$? 
a. 2⁻⁵ · 2⁻¹
b. 2⁻² · 2⁻⁴
c. 2¹ · 2⁵
d. 2¹ · 2⁶ 
e. 2² · 2⁻⁸
f. 2² · 2³

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.