### Examples

The expression is equivalent to .**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**8

**Strand:**Algebraic Reasoning

**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:
- 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.

- For example, determine the numerical value of 6³.

### Instructional Tasks

*Instructional Task 1*

**(MTR.2.1)**Two students were working on generating equivalent expressions for (15$x$$y$²)³, and showed their solutions below.

*Instructional Task 2*

**(MTR.5.1)**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

## Related Access Points

## Related Resources

## Lesson Plans

## Problem-Solving Tasks

## Tutorial

## Student Resources

## Problem-Solving Task

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

## Tutorial

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

## Problem-Solving Tasks

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

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