# MA.8.AR.1.1

Apply the Laws of Exponents to generate equivalent algebraic expressions, limited to integer exponents and monomial bases.

### Examples

The expression 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.)
Strand: Algebraic Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Base
• Expression
• Integers
• Monomial

### Vertical Alignment

Previous Benchmarks

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### Purpose and Instructional Strategies

In Grade 7, students applied the Laws of Exponents to evaluate and generate numerical expressions, limited to whole-number exponents and rational number bases. In Grade 8, students extend their knowledge of the Laws of Exponents to generate equivalent algebraic expressions with integer exponents and monomial bases. In Algebra 1, students will use their knowledge of the Laws of Exponents to generate equivalent algebraic expressions with rational and variable exponents.
• At the onset of learning about exponents, students learn that it is a way to write expanded multiplication in a more condensed form. The understanding that the number which is referred to as the base is multiplied times itself based on the value of the exponent is foundational.
• This benchmark can be paired with MA.8.NSO.1.3 which helps students work within numerical expressions with integer exponents and rational bases. Students should move from numerical expressions to algebraic expressions to best enhance their conceptual understanding of the Laws of Exponents.
• A strategy for developing meaning for integer exponents is to make use of patterns as shown below:

• Teach one law at a time to allow for conceptual understanding instead of memorizing the rules. Students should not be told the properties but rather should derive them through experience and reason. During instruction, include examples that show the expansion of the bases with the use of the exponents to show equivalence.
• For mastery of this benchmark, monomials can be defined in the following way: a base may be a product of a coefficient and one or more variables with integer exponents. This limitation should not prevent students from understanding that a negative exponent can be represented equivalently as a positive exponent with the reciprocal base (changing numerator to denominator or denominator to numerator).

### Common Misconceptions or Errors

• When working with negative exponents, students may not understand the connection to fractions and values in the denominator.
• Students incorrectly multiply the exponent with the base number.
• Students may incorrectly apply the Laws of Exponents.

### Strategies to Support Tiered Instruction

• Teachers should review exponents as condensed multiplication and write out expanded form, and provide opportunities to notice patterns as discussed in MA.8.NSO.1.3. Teachers can use the “Patterns in Exponents” chart shown in the Purpose and Instructional Strategies section with the right-side blank so that students can begin to complete and understand the patterns of exponents.
• Teachers should re-emphasize the structure of exponents, and how they are used by multiplying the base by itself the number of times as notated by the exponent.
• Teacher provides a review of the relationship between the base and the exponent by modeling an example of operations using a base and exponent.
• For example, determine the numerical value of 6³.

6³ which is equivalent to 6⋅6⋅6 which is equivalent to 216.

Two students were working on generating equivalent expressions for (15$x$$y$²)³, and showed their solutions below.

The teacher said Rachel and Justina both have the correct answer. Do you agree with the teacher? Explain your reasoning.

Create a pattern using the expanded form of the base, $x$, between $x$−5 and $x$5. Explain why $x$0 is equal to 1.

### Instructional Items

Instructional Item 1
Write $x$5$x$8 with the variable $x$ used only one time.

Instructional Item 2
An expression is given.

Write an equivalent expression with only two exponents and no negative exponents.

Instructional Item 3
Write $y$−3$z$−4 with only positive exponents.

*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.AR.1.AP.1: Use the properties of integer exponents and product/quotient of powers with like bases to produce equivalent algebraic expressions limited to positive exponents and monomial bases.

## Related Resources

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

## Lesson Plans

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

For a given circle m and point A in a coordinate plane, students will be able to show whether this given point A lies on the circumference of the given circle m using the Pythagorean Theorem. Subsequently, this can be used to prove that an airplane lies within or outside the radar's range with a given radius of detection.

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

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

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

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.

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.

## Tutorial

Exponents with Negative Bases:

In this tutorial, you will apply what you know about multiplying negative numbers to determine how negative bases with exponents are affected and what patterns develop.

Type: Tutorial

## Student Resources

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

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.

## Tutorial

Exponents with Negative Bases:

In this tutorial, you will apply what you know about multiplying negative numbers to determine how negative bases with exponents are affected and what patterns develop.

Type: Tutorial

## Parent Resources

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