# MA.8.NSO.1.3

Extend previous understanding of the Laws of Exponents to include integer exponents. Apply the Laws of Exponents to evaluate numerical expressions and generate equivalent numerical expressions, limited to integer exponents and rational number bases, with procedural fluency.

### Examples

The expression  is equivalent to which is equivalent to .

### Clarifications

Clarification 1: Refer to the K-12 Formulas (Appendix E) for the Laws of Exponents.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 8
Strand: Number Sense and Operations
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Exponent
• Expressions
• Integer
• Rational Numbers

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### 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 ($\frac{\text{1}}{\text{54}}$)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 ($\frac{\text{1}}{\text{54}}$)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.
(−$\frac{\text{2}}{\text{3}}$)−3 (0.8)²

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.

## Related Courses

This benchmark is part of these courses.
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.8.NSO.1.AP.3: Use the properties of integer exponents and product/quotient of powers with like bases to produce equivalent expressions.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Equivalent Powers Expressions:

Students are given numerical expressions and asked to use properties of integer exponents to find equivalent expressions.

Type: Formative Assessment

Exponents Tabled:

Students are asked to complete a table of powers of three and provide an explanation of zero powers.

Type: Formative Assessment

Multiplying and Dividing Integer Exponents:

Students are asked to apply the properties of integer exponents to generate equivalent numerical expressions.

Type: Formative Assessment

Negative Exponential Expressions:

Students are given expressions with negative exponents and are asked to identify those that are equivalent from given sets of expressions.

Type: Formative Assessment

## Lesson Plans

The Laws of Sine and Cosine:

In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate the measures of sides and angles of non-right triangles.

Type: Lesson Plan

Pythagorean Perspective:

This lesson serves as an introductory lesson on the Pythagorean Theorem and its converse. It has a hands-on discovery component. This lesson includes worksheets that are practical for individual or cooperative learning strategies. The worksheets contain prior knowledge exercises, practice exercises and a summative assignment.

Type: Lesson Plan

Triangles: To B or not to B?:

Students will explore triangle inequalities that exist between side lengths with physical models of segments. They will determine when a triangle can/cannot be created with given side lengths and a range of lengths that can create a triangle.

Type: Lesson Plan

What's the Point?:

Students will algebraically find the missing coordinates to create a specified quadrilateral using theorems that represent them, and then algebraically prove their coordinates are correct.

Note: This is not an introductory lesson for this standard.

Type: Lesson Plan

Multiplying terms that have the same base:

Students explore numerical examples involving multiplying exponential terms that have the same base. They generalize the property of exponents where, when multiplying terms with the same base, the base stays the same and the exponents are added together.

Type: Lesson Plan

Operating with Exponents!:

Students will participate in a gallery walk in which they observe patterns in algebraic expressions. Students will apply the properties of integer exponents to simplify expressions.

Type: Lesson Plan

Stand Up for Negative Exponents:

This low-tech lesson will have students stand up holding different exponent cards. This will help them write and justify an equivalent expression and see the pattern for expressions with the same base and descending exponents. What happens as you change from 2 to the fourth power to 2 to the third power; 2 to the second power; and so forth? This is an introductory lesson to two of the properties of exponents: and

Type: Lesson Plan

Scavenger Hunt for Multiplying and Dividing Powers:

Get your students up and moving and interested in simplifying expressions with whole integer powers. After getting your students to figure out what it takes to multiply and divide powers with whole number exponents, have your students scurry about the room to find the questions and answers for scavenger hunt exercise. The lesson includes questions and answers for the hunt, directions for the hunt, printable cards for the hunt, and step by step directions on how to get your students to figure out what they need to do when multiplying and dividing powers with whole number exponents.

Type: Lesson Plan

Math Is Exponentially Fun!:

The students will informally learn the rules for exponents: product of powers, powers of powers, zero and negative exponents. The activities provide the teacher with a progression of steps that help lead students to determine results without knowing the rules formally. The closing activity is hands-on to help reinforce all rules.

Type: Lesson Plan

Exponential Chips:

In this lesson students will learn the properties of integer exponents and how to apply them to multiplication and division. Students will have the opportunity to work with concrete manipulatives to create an understanding of these properties and then apply them abstractly. The students will also develop the understanding of the value of any integer with a zero exponent.

Type: Lesson Plan

## Original Student Tutorial

Applying the Pythagorean Theorem to Solve Mathematical and Real-World problems:

Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Experts

Fluency vs. Automaticity:

How are fluency and automaticity defined? Dr. Lawrence Gray explains fluency and automaticity in the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

B.E.S.T. Journey:

What roles do exploration, procedural reliability, automaticity, and procedural fluency play in a student's journey through the B.E.S.T. benchmarks? Dr. Lawrence Gray explains the path through the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

What is Fluency?:

What is fluency? What are the ingredients required to become procedurally fluent in mathematics? Dr. Lawrence Gray explores what it means for students to be fluent in mathematics in this Expert Perspectives video.

Type: Perspectives Video: Expert

Why Isn't Getting the "Right" Answer Good Enough?:

Why is it important to look beyond whether a student gets the right answer? Dr. Lawrence Gray explores the importance of understanding why we perform certain steps or what those steps mean, and the impact this understanding can have on our ability to solve more complex problems and address them in the context of real life in this Expert Perspectives video.

Type: Perspectives Video: Expert

## Problem-Solving Tasks

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Type: Problem-Solving Task

Ants versus humans:

This task requires students to work with very large and small values expressed both in scientific notation and in decimal notation (standard form). In addition, students need to convert units of mass. The solution below converts the mass of humans into grams; however, we could just as easily converted the mass of ants into kilograms. Students are unable to go directly to a calculator without taking into account all of the considerations mentioned above. Even after converting units and decimals to scientific notation, students should be encouraged to use the structure of scientific notation to regroup the products by extending the properties of operations and then use the properties of exponents to more fluently perform the calculations involved rather than rely heavily on a calculator.

Type: Problem-Solving Task

Extending the Definitions of Exponents, Variation 1:

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. It is good for students to learn the convention that negative time is simply any time before t=0.

Type: Problem-Solving Task

## Video/Audio/Animation

Exponentiation:

Exponentiation can be understood in terms of repeated multiplication much like multiplication can be understood in terms of repeated addition. Properties of multiplication and division of exponential expressions with the same base are derived.

Type: Video/Audio/Animation

## MFAS Formative Assessments

Equivalent Powers Expressions:

Students are given numerical expressions and asked to use properties of integer exponents to find equivalent expressions.

Exponents Tabled:

Students are asked to complete a table of powers of three and provide an explanation of zero powers.

Multiplying and Dividing Integer Exponents:

Students are asked to apply the properties of integer exponents to generate equivalent numerical expressions.

Negative Exponential Expressions:

Students are given expressions with negative exponents and are asked to identify those that are equivalent from given sets of expressions.

## Original Student Tutorials Mathematics - Grades 6-8

Applying the Pythagorean Theorem to Solve Mathematical and Real-World problems:

Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorial

Applying the Pythagorean Theorem to Solve Mathematical and Real-World problems:

Apply the Pythagorean Theorem to solve mathematical and real-rorld problems in this interactive tutorial.

Type: Original Student Tutorial

## Problem-Solving Tasks

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Type: Problem-Solving Task

Extending the Definitions of Exponents, Variation 1:

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. It is good for students to learn the convention that negative time is simply any time before t=0.

Type: Problem-Solving Task

## Video/Audio/Animation

Exponentiation:

Exponentiation can be understood in terms of repeated multiplication much like multiplication can be understood in terms of repeated addition. Properties of multiplication and division of exponential expressions with the same base are derived.

Type: Video/Audio/Animation

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

## Problem-Solving Tasks

How many cells are in the human body?:

This problem solving task challenges students to apply the concepts of mass, volume, and density in the real-world context to find how many cells are in the human body.

Type: Problem-Solving Task

Ants versus humans:

This task requires students to work with very large and small values expressed both in scientific notation and in decimal notation (standard form). In addition, students need to convert units of mass. The solution below converts the mass of humans into grams; however, we could just as easily converted the mass of ants into kilograms. Students are unable to go directly to a calculator without taking into account all of the considerations mentioned above. Even after converting units and decimals to scientific notation, students should be encouraged to use the structure of scientific notation to regroup the products by extending the properties of operations and then use the properties of exponents to more fluently perform the calculations involved rather than rely heavily on a calculator.

Type: Problem-Solving Task

Extending the Definitions of Exponents, Variation 1:

This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. It is good for students to learn the convention that negative time is simply any time before t=0.

Type: Problem-Solving Task