Exponents Tabled

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

Resource ID#: 59694 Type: Formative Assessment

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General Information

Subject(s): Mathematics, Mathematics (B.E.S.T. - Effective starting 2022-2023)

Grade Level(s): 8

Intended Audience: Educators

Keywords: MFAS, exponent, base, power, equivalent expression, zero power, integer exponents

Instructional Component Type(s): Formative Assessment

Resource Collection: MFAS Formative Assessments

Attachments

mfas_exponentstabled_worksheet.docx

Formative Assessment Task

Instructions for Implementing the Task

This task can be implemented individually, with small groups, or with the whole class.

The teacher asks the student to complete the problems on the Exponents Tabled worksheet.
The teacher asks follow-up questions, as needed.

TASK RUBRIC

Getting Started

Misconception/Error

The student does not understand that a nonzero number raised to the zero power is equal to one.

Examples of Student Work at this Level

The student states that:

$begin mathsize 11px style 3 to the power of 0 end style$ is 0 and $begin mathsize 11px style 4 to the power of 0 end style$ is 0.
$begin mathsize 11px style 3 to the power of 0 end style$ is 3 and $begin mathsize 11px style 4 to the power of 0 end style$ is 4.

Questions Eliciting Thinking

What does $begin mathsize 11px style 3 squared end style$ mean? Can you write this in another way?

Can you explain how $begin mathsize 12px style 3 to the power of 0 end style$ = 0 (or $begin mathsize 11px style 3 to the power of 0 end style$ = 3) would fit into the pattern shown on the table?

Instructional Implications

Review the meaning of positive integer exponents. Using bases with whole number exponents less than ten, model rewriting the expression in an expanded form (e.g., rewrite $begin mathsize 11px style 3 to the power of 4 end style$ as 3 x 3 x 3 x 3). Provide the student with sample problems of this type and ask the student to rewrite in expanded form and evaluate. Then provide examples in expanded form and ask the student to rewrite in exponential form.

Present the student with a table similar to the one used in this task that contains powers of two from $begin mathsize 11px style 2 to the power of 5 end style$ to $begin mathsize 11px style 2 to the power of 1 end style$ . Assist the student in identifying the patterns shown in the table as the powers decrease (e.g., for any two consecutive entries, as the exponent decreases by one, the value of the exponential expression decreases by a factor of two). Make explicit that any given entry (e.g., 16) can be divided by two (16 ÷ 2) to get the entry (8) for the next lower power of two $begin mathsize 11px style left parenthesis 2 cubed right parenthesis end style$ . Ask the student to divide the entry for $begin mathsize 11px style 2 to the power of 1 end style$ by two to determine the entry for $begin mathsize 11px style 2 to the power of 0 end style$ . Finally, ask the student to: 1) construct a table with a different base and 2) construct an explanation for why raising a (nonzero) number to the zero power should be one.

Moving Forward

Misconception/Error

The student is unable to write equivalent numerical expressions for numbers with negative exponents.

Examples of Student Work at this Level

The student understands that $begin mathsize 11px style 3 to the power of 0 end style$ and $begin mathsize 11px style 4 to the power of 0 end style$ are both equal to one. However, the student evaluates $begin mathsize 11px style 3 to the power of minus 1 end exponent end style$ and $begin mathsize 11px style 3 to the power of minus 2 end exponent end style$ incorrectly. The student evaluates:

$begin mathsize 11px style 3 to the power of minus 1 end exponent end style$ as 3 and $begin mathsize 11px style 3 to the power of minus 2 end exponent end style$ as 9.
$begin mathsize 11px style 3 to the power of minus 1 end exponent end style$ as $begin mathsize 11px style 3 over 1 to the power of 1 end style$ and $begin mathsize 11px style 3 to the power of minus 2 end exponent end style$ as $begin mathsize 11px style 3 squared over 1 end style$ .
$begin mathsize 11px style 3 to the power of minus 1 end exponent end style$ as -3 and $begin mathsize 11px style 3 to the power of minus 2 end exponent end style$ as -9.
$begin mathsize 11px style 3 to the power of minus 1 end exponent end style$ as -3 and $begin mathsize 11px style 3 to the power of minus 2 end exponent end style$ as -6.

Questions Eliciting Thinking

What pattern do you see in the second column of the table? How is 27 related to 9? How is 9 related to 3? Can you extend that pattern to determine what $begin mathsize 11px style 3 to the power of minus 1 end exponent end style$ should equal?

Instructional Implications

Present the student with a table similar to the one used in this task that contains powers of two from $begin mathsize 11px style 2 to the power of 5 end style$ to $begin mathsize 11px style 2 cubed end style$ . Ask the student to identify the patterns shown in the table as the powers decrease (e.g., for two consecutive entries, as the exponent decreases by one, the value of the exponential expression decreases by a factor of two). Make explicit that any given entry (e.g., 16) can be divided by two (16 ÷ 2) to get the entry (8) for the next lower power of two $begin mathsize 11px style left parenthesis 2 cubed right parenthesis end style$ . Ask the student to extend the pattern in the table to $begin mathsize 11px style 2 to the power of minus 1 end exponent end style$ by dividing the previous entry by two. Have the student continue in this manner to determine the values of $begin mathsize 11px style 2 to the power of minus 2 end exponent end style$ and $begin mathsize 11px style 2 to the power of minus 3 end exponent end style$ . Then ask the student to correct any errors made on the Exponents Tabled worksheet and create his or her own table using a different base.

Model the process of writing numbers with negative integer exponents in an equivalent form using only positive exponents, expanding, and then evaluating (e.g., $begin mathsize 11px style 3 to the power of minus 4 end exponent space equals space 1 over 3 to the power of 4 space equals space fraction numerator 1 over denominator 3 space cross times space 3 space cross times space 3 space cross times space 3 end fraction space equals space 1 over 81 end style$ ). Ask the student to evaluate numbers with negative integer exponents in a similar manner.

Almost There

Misconception/Error

The student’s explanation for why a number raised to the zero power should be one is incomplete.

Examples of Student Work at this Level

The student correctly evaluates $begin mathsize 11px style 3 to the power of 0 end style$ as one, $begin mathsize 11px style 3 to the power of minus 1 end exponent end style$ as $begin mathsize 11px style 1 third end style$ , $begin mathsize 11px style 3 to the power of minus 2 end exponent end style$ as $begin mathsize 11px style 1 over 9 end style$ , and $begin mathsize 11px style 4 to the power of 0 end style$ as one. However, when explaining why $begin mathsize 11px style 4 to the power of 0 space equals space 1 end style$ , the student:

States that anything to the power of zero is one or refers to the Zero Power Property.
States that four has nothing to multiply itself by.

Questions Eliciting Thinking

It is true that any number raised to the zero power is one, but why is it one rather than zero?

Can you identify any patterns in the table to suggest what a number with an exponent of zero should be?

Instructional Implications

Assist the student in observing patterns in the table to construct an explanation. In addition, provide the student with explanations based on the need to be consistent with the application of previously learned properties of exponents. For example, remind the student of the property $begin mathsize 11px style a to the power of n a to the power of m space equals space a to the power of left parenthesis n space plus space m right parenthesis end exponent end style$ (where a is a nonzero rational number and n and m are positive integers). Applying this property to $begin mathsize 11px style 4 to the power of 0 end style$ x $begin mathsize 11px style 4 to the power of 1 end style$ , yields $begin mathsize 11px style 4 to the power of 0 space cross times space 4 to the power of 1 space equals space 4 to the power of left parenthesis 0 space plus space 1 right parenthesis end exponent space equals space 4 to the power of 1 end style$ . So, $begin mathsize 11px style 4 to the power of 0 end style$ must be equal to one.

Got It

Misconception/Error

The student provides complete and correct responses to all components of the task.

Examples of Student Work at this Level

The student correctly evaluates $begin mathsize 11px style 3 to the power of 0 end style$ as one, $begin mathsize 11px style 3 to the power of minus 1 end exponent end style$ as $begin mathsize 11px style 1 third end style$ , $begin mathsize 11px style 3 to the power of minus 2 end exponent end style$ as $begin mathsize 11px style 1 over 9 end style$ , and $begin mathsize 11px style 4 to the power of 0 end style$ as one. To explain why $begin mathsize 11px style 4 to the power of 0 end style$ should equal one, the student refers to:

The pattern in the values of the powers of three in the table and applies it to a base of four.
The properties of multiplying and dividing integer exponents. For example, the student shows that $begin mathsize 11px style 4 to the power of 0 space cross times space 4 to the power of 1 space equals space 4 to the power of left parenthesis 0 space plus space 1 right parenthesis end exponent space equals space 4 to the power of 1 end style$ and concludes that $begin mathsize 11px style 4 to the power of 0 end style$ must be equal to one.

Questions Eliciting Thinking

Can you evaluate $begin mathsize 11px style left parenthesis minus 3 right parenthesis to the power of minus 3 end exponent end style$ ?

Can you evaluate $begin mathsize 11px style 1 over 3 to the power of minus 3 end exponent end style$ ?

Instructional Implications

Provide the student with more challenging problems involving negative integer exponents that include negative bases and fractional bases. Challenge the student to rewrite $begin mathsize 11px style open parentheses 2 over 3 close parentheses to the power of minus 4 end exponent end style$ in an equivalent form using positive exponents.

Accommodations & Recommendations

Special Materials Needed:
- Exponents Tabled worksheet

Source and Access Information

Contributed by: MFAS FCRSTEM

Name of Author/Source: MFAS FCRSTEM

District/Organization of Contributor(s): Okaloosa

Is this Resource freely Available? Yes

Access Privileges: Public

License: CPALMS License - no distribution - non commercial

* Please note that examples of resources are not intended as complete curriculum.

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Exponents Tabled

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General Information

Attachments

Formative Assessment Task

Instructions for Implementing the Task

TASK RUBRIC

Getting Started

Moving Forward

Almost There

Got It

Accommodations & Recommendations

Special Materials Needed:

Source and Access Information

Aligned Standards

Related Resources

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