Standard #: MA.8.AR.1.1


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


Examples


The expression begin mathsize 12px style open parentheses 3 x cubed y to the power of negative 2 end exponent close parentheses cubed end style is equivalent to begin mathsize 12px style 27 x to the power of 9 y to the power of negative 6 end exponent end style.

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: Algebraic Reasoning
Standard: Generate equivalent algebraic expressions.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Base
  • Expression
  • Integers
  • Monomial

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

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:
    Patterns in Exponents
    Patterns in Exponents
  • 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.
      Base -> 6^3 <- Exponent
      6³ which is equivalent to 6⋅6⋅6 which is equivalent to 216.

 

Instructional Tasks

Instructional Task 1 (MTR.2.1)
Two students were working on generating equivalent expressions for (15xy²)³, and showed their solutions below.
Table
The teacher said Rachel and Justina both have the correct answer. Do you agree with the teacher? Explain your reasoning.

Instructional Task 2 (MTR.5.1)
Create a pattern using the expanded form of the base, x, between x−5 and x5. Explain why x0 is equal to 1.

 

Instructional Items

Instructional Item 1
Write x5x8 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−3z−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

Course Number1111 Course Title222
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

Access Point Number Access Point Title
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

Lesson Plans

Name Description
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.

Airplanes in Radar's Range

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.

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.

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.

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:begin mathsize 10px style a to the power of 0 equals 1 end style and begin mathsize 10px style a to the power of negative n end exponent equals 1 over a to the power of n end style

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.

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.

Problem-Solving Tasks

Name Description
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

Name Description
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.

Student Resources

Problem-Solving Task

Name Description
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

Name Description
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.



Parent Resources

Problem-Solving Tasks

Name Description
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.



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