### Examples

The expression is equivalent to which is equivalent to .

**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

### Connecting Benchmarks/Horizontal Alignment

### 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 $\frac{\text{64}}{\text{125}}$.

- For example, ($\frac{\text{5}}{\text{4}}$)

### 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, $\frac{\text{1}}{\text{2\xb3}}$.

- For example, 4² × 4
- 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 $\frac{\text{64}}{\text{125}}$.

- For example, ($\frac{\text{5}}{\text{4}}$)

### 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 4

^{5}. 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.

^{−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. 2

^{1}· 2

^{5}

d. 2

^{1}· 2

^{6}

e. 2

^{2}· 2

^{−8}

f. 2

^{2}· 2

^{3}

**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

## Formative Assessments

## Lesson Plans

## Original Student Tutorial

## Perspectives Video: Experts

## Problem-Solving Tasks

## Video/Audio/Animation

## MFAS Formative Assessments

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

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

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

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

## Student Resources

## Original Student Tutorial

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

Type: Original Student Tutorial

## Problem-Solving Tasks

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

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 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

## Problem-Solving Tasks

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

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