Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 49210
Stand Up for Negative Exponents
This low-tech lesson will have students stand up holding different exponent cards to help students 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
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Suggested Technology: Basic Calculators
Instructional Time: 1 Hour(s) 30 Minute(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: negative exponent, exponent, negative power, zero exponent, integer exponent, negative power,
integer power, zero power
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Negative and Zero Exponent Worksheet.doc
Scavenger Hunt Multiplying and Dividing exponents.doc
Exponent Cards.docx
Negative and Zero Exponents Worksheet Independent Practice.doc
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to apply the properties of integer exponents to generate equivalent numerical expressions.
Students will be able to multiply and divide single, common-base exponential expressions with integer exponent results (not just whole-number exponents).
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should be able to expand exponent expressions. Example:
Students should be familiar with some of the properties of exponents.
Students should be able to multiply and divide single, common-base exponential expressions with whole-number exponent results.
Vocabulary: base, exponent, integer, power
Please note that it might be a good idea to talk about how the word base means something different in science and in geometry. In science it is a solution with a pH
level greater than 7, while in geometry it is the bottom of a three-dimensional shape. Tell students that power also has different meanings in different circumstances.
Guiding Questions: What are the guiding questions for this lesson?
What is the difference between 5 squared and 5 cubed? the exponent is one larger, the value is 5 times larger.
What operation do I need to apply to 4 squared to get to 4 cubed? multiply by 4
page 1 of 6 What operation do I need to apply to 7 cubed to get back to 7 squared? divide by 7
What is the value of a number divided by itself? 1
Teaching Phase: How will the teacher present the concept or skill to students?
Pick three students and have them stand in the front of the room with the cards for 2 cubed, 2 to the fourth and 2 to the fifth power with the "2" sides showing.
The teacher could use the students' names, for example Charlie has 2 cubed, Frank has 2 to the fourth and Fionna has 2 to the fifth.
What is the difference between Charlie's 2 cubed and Frank's 2 to the fourth? The value of 2 to the fourth is 2 times (twice) the value of 2 cubed. You multiplied by 2.
What about the difference between Frank's 2 to the fourth and Fionna's 2 to the fifth? Once again you multiplied by 2, since the base is 2, and the exponent increased
by 1, from 4 to 5.
What would the next number be? 2 to the sixth or 64
Have another student (Sean) come up and take the 2 to the sixth card.
Following this pattern, what would be the next number? 2 to the seventh or 128
Have another student (Heather) come up and hold the 2 to the seventh card.
What's being done to Sean's 2 to the sixth to get Heather's 2 to the seventh? Multiplying by 2 again
Have the students flip their cards to reveal the sides that have x instead of 2.
What is the difference between Charlie's x cubed and Frank's x to the fourth now? x to the fourth is 2 times x cubed. Remember how we multiplied by 2 before? Why
did we multiply by 2? Because it is the base value
So what is the difference between each of these students' expressions? The previous one is being multiplied by x, the base value, when the exponent increases by 1.
Say "Oops, I made a mistake. I forgot to include the lower exponent values. Johnny, can you come up here please?" Give him the x squared card.
How might I find the value of Charlie's x cubed after finding the value of Johnny's x squared. Multiply by 2. Charlie's x cubed is 2 times more than Johnny's x squared.
What if I were trying to determine the value of Johnny's x squared after finding the value of Charlie's x cubed? Divide by 2, the base.
So, what are we doing to find the next value, when the exponent is larger by 1 and the base is the same? Multiplying by the base again.
If the exponent is smaller by 1 and the base is the same, how do we find the next value? Divide by the base.
Have the students flip their cards back over to the sides with the 2s.
What would come before Johnny's 2 squared? 2
What would the exponent be on that? 1
Have a student (Olivia) come up and hold the power of 1 card with the 2 facing forward.
What value would come before Olivia's 2 to the 1? Zero will be a possible answer.Zero is the exponent, but not the value. Ask what operation we used to get from 2
squared to 2 to the first? Divide by 2. Is 2 divided by 2 equal to 0?
What if we had 3 instead of 2 as the base? 4? 5? They would all be divided by themselves and the answer would be 1.
So what can we conclude about anything to the power of zero? It will always equal 1.
Have a student come up and hold the zero power card with the 2 showing, then have everyone flip over to show the x-sides of the cards.
So, x to the zero power is equal to 1.
Then have them flip back to the 2. I want to go one more step lower. If I go one more step lower (have another student come up), what will the exponent be? What
will the value be? Remember, we divide by the base, 2, to get the next step down, so the previous value of 1 is going to be divided by 2 to get 2 to the negative 1.
Leave this as a fraction.
Have another student come up and hold the negative one exponent card. Write the fraction ½ on the board. I want to go another step lower. What do I do?
Show the divide by 2 on the board "Can you divide a fraction by a whole number?"Multiply by the inverse
Change the divide by 2 on the board to multiply by ½. "What does this equal?" ¼
Before I have another student come up and hold a card, let's look at this in terms of x.
We had 1/x for x to the negative 1 power. What do we need to do to get the 1/x to the next lower power?
(divide by x by multiplying by the reciprocal, 1/x)
When I multiply these fractions, what do I get? (
)
Have a student come up and get the negative 2 power card.
What is the next card going to look like?
The next one?
Why?
page 2 of 6 What can we conclude about negative exponents from this?
Have the students sit back down.
Write on the board,
Write on the board
Write on the board
. What does this equal in fraction form? 1/25
. What does this equal to? 1/27
. What does this equal? someone is bound to say
Point out how the base of the exponent will not include the coefficient, and that the
correct answer is
Write on the board
Write on the board
We know that
We get
. What does this equal to? 1
. What does this equal to?
, right? So, let's replace that in our expression's denominator.
This is the same as saying
.
What is an efficient way to divide by a fraction? What do you do? Multiply by the reciprocal.
So, this becomes
So, when we have a negative exponent in the denominator, how can we simplify the expression? Express the denominator (a power with the negative exponent) as a
fraction and multiply the numerator by the reciprocal of the denominator.
Let's look at this with a variable. What does
What about
?
?
Now let's take what we have just learned and apply it to some of the multiplying and dividing we have previously done. Before this our answers always had positive
exponents. This is no longer the case.
Use what you have already learned to multiply 5 squared times 5 to the negative third power?
=
= 1/5 Explain how you simplified the expression. Students
may subtract exponents or cancel the two 5/5s.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Students will be given individual whiteboards.
The teacher will post questions on the front board and give appropriate feedback to student responses. See the Formative Assessment Section for sample questions to
use.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Students will be given a worksheet Negative and Zero Exponent Worksheet Independent Practice.doc that the teacher will have them work on for independent class
work or homework.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
When we have an expression with a negative exponent, how do we write an equivalent expression, so that we have a positive exponent? Express it as a fraction, and
the denominator will have a positive exponent.
What is the value of any number to the zero power? 1
Now that we know how to handle negative and zero exponents, expect to see them as part of other questions.
Exit slip:
Why is any number to the zero power equal to 1?
because
or since
and
or because going down one in exponents is the same as dividing by the base, and the exponent before 0 is 1,
Give an example of a negative exponent term and it's fractional equivalent. Answers will vary
Summative Assessment
This activity / lesson should be assessed as part of the standard MAFS.8.EE.1.1 This only covers how to handle negative and zero exponents within the Standard.
page 3 of 6 Some samples of questions that might be added to an assessment to cover this particular part of the Standard are as follows:
Simplify the following expressions:
1.
Ans: 1
2.
Ans: 1
3.
Ans: -1
4.
1.
Ans:
2.
Ans:
3.
Ans:
Formative Assessment
Prior Knowledge:
Students will be given a bell ringer exercise containing questions involving single digit exponents with like bases being multiplied or divided with the answers always
having positive exponents.
Some Sample Questions
Simplify:
Write in exponent form:
Answers:
During or after lesson:
Individual whiteboard exercises or worksheet containing questions similar to the following:
Simplify the following expressions
answer:
answer:
answer:
answer:
answer:
answer:
answer:
answer:
answer:
answer:
answer:
answer:
answer:
page 4 of 6 answer: 1
answer: 1
answer: -1
answer: 1
Exit slip - details in the Closure section of the lesson plan
Feedback to Students
The teacher will provide feedback to the students throughout the exercise in developing the knowledge of how to handle negative and zero exponents.
If whiteboards are used the teacher can give instant feedback to the students through a simple thumbs up or thumbs down.;
The teacher will verbally question the class throughout the lesson. See teaching phase for details.
The teacher will provide answers to the Negative and Zero Exponent worksheet.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Students with vision problems: number the cards, have a two-sided page listing the contents of each card with the contents of the "2" side on one side of the page,
and contents of the x side on the other side.
Students who struggle or have special needs: provide extra time and support with guided direction. Pair students with a peer and encourage discussion of each
problem.
English Language Learners: provide dictionaries, translations, or examples of unfamiliar vocabulary.
Extensions:
Resource ID# 42392 - Extending the Definitions of Exponents Variation 1
You may want to give more advanced students sample problems that include more than one variable, or have negative combinations of multiple negative exponents
and zero exponents.
Suggested Technology: Basic Calculators
Special Materials Needed:
You will need to create double-sided cards or small posters. One side will have x to a power on top and and its expanded equivalent on the bottom. The other side will
have 2 to the same exponent with its simplified or fractional value.
You could also use the Exponent Card document to print out pages that you would fold in half to use.
Side 1:
Side 2:
You will need cards for the powers -2 through +7
Numbering the cards may be useful for certain accommodations.
Class set of Negative and Zero Exponent Independent Worksheets
Additional Information/Instructions
By Author/Submitter
This resource is likely to support student engagement in the following Mathematical Practices:
MAFS.K12.MP.8.1 Look for and express regularity in repeated reasoning
MAFS.K12.MP.7.1 Look for and make use of structure
page 5 of 6 SOURCE AND ACCESS INFORMATION
Contributed by: Matthew Funke
Name of Author/Source: Matthew Funke
District/Organization of Contributor(s): Brevard
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.8.EE.1.1:
Description
Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² ×
= =1/3³=1/27.
page 6 of 6