Math 8 Standard 7 08-30
9/5/02
11:32 AM
Page 199
TIMSS
NAEP
Standard Indicators
8.7.1-8.7.12
Don’t Break the Law (of Exponents)
Purpose
Students will solve problems by choosing strategies, explaining their
reasoning, making calculations, and checking results.
Materials
For the teacher: chalk, chalkboard
For each student: paper, pencil, copies of Black Line Masters (BLMs)
Discovering Laws of Exponents and Understanding the Law, calculator
(optional)
Activity
A. Introducing the Problem
1. Tell students that today they will be discovering patterns that can
be used to simplify expressions containing exponents.
2. Explain that, in math, we frequently use laws found by other
mathematicians to shorten the time it takes to solve problems.
Give examples such as the Pythagorean Theorem or the Quadratic
Formula.
16 × c ÷ (b )
3. Write the following expression on the chalkboard:
cb
4. Call on volunteers to help you solve the problem at the
chalkboard. Solve the problem by writing in expanded form.
16 × 16 × c × c × c × c × c × b × b × b × b
[
]
c×c×b×b×b
5. Explain to students that they will discover laws of exponents that
will help simplify expressions like this one.
2
5
extending
THE
ACTIVITY
Create a matching
game for groups to
play. On one set of
cards, write exponential
expressions. On the
second set of cards,
write the simplified
versions of those
expressions. Ask groups
to use the laws of
exponents to match
equivalent expressions.
−2 2
2 3
(continued)
Standard 7 / Activity 8
Indiana Mathematics Grade 8 Curriculum Framework, October 2002
Standards Links
8.1.4, 8.1.5
page 199
Standard 7
B. Solving the Problem
1. Divide the class into groups of three students. Distribute one copy
of the BLM Discovering Laws of Exponents to each student.
2. Tell students to look at the first column of each row. Explain that
students may use their scratch paper and/or their calculators to
substitute values into the expressions for the terms a and b.
3. Ask students to look for patterns and write algorithms based on
those patterns.
4. Tell students to test their algorithms for several different values.
Once groups are convinced the algorithm is correct, write it in the
second column on the BLM.
5. Allow ample time for groups to complete their charts.
Math 8 Standard 7 08-30
9/5/02
11:32 AM
Page 200
Activity (continued)
6. Discuss the results with the class. Write algorithms on the
chalkboard. Make sure students correct any mistakes on their
chart before going on to section C. (Correct laws are found in
the answer key on the back side of the BLM.)
C. Understanding the Laws
1. Tell students that now they have discovered the Law of
Exponents, it is time to show they understand what the laws
mean.
2. Distribute one copy of the BLM Understanding the Law to each
student. Allow students to remain in groups of three to complete
the BLM.
3. Tell students to copy the correct algorithm for each law onto the
BLM. Ask the students to then verbalize the law. Refer students
to the example for Law One.
4. Allow students time to work on the BLM and then ask students
to share the results with the class.
Classroom Assessment
Basic Concepts and Processes
While groups work on solving the problem, ask students the following
questions to gauge their understanding of the Standard Indicator:
What patterns did you find for law [insert law number here]?
How did you go about finding the algorithm for law [insert
law number here]?
Standard 7
Which values did you use to test the law [insert law
number here]?
page 200
Standard 7/ Activity 8
Indiana Mathematics Grade 8 Curriculum Framework, October 2002
Math 8 Standard 7 08-30
9/5/02
11:32 AM
Page 201
Names:
Discovering Laws
of Exponents
Work with your group to find a rule to simplify each exponential
expression. Write the algorithm for each rule in the second column.
bm × bn
bm ÷ bn
(bm)n
(a × b)m
(a ÷ b)m
b−m
a −n
b
( )
Standard 7 / Activity 8
Indiana Mathematics Grade 8 Curriculum Framework, October 2002
Black Line Master 1
page 201
Math 8 Standard 7 08-30
9/5/02
11:32 AM
Page 202
Discovering Laws
of Exponents
Teacher Directions
Distribute one copy of the BLM Discovering Laws of Exponents to each student. Have students
work in groups of three. Ask groups to substitute values for the variables and look for patterns in
the solutions. Tell students to write an algorithm for the pattern and test the algorithm using
several different values. Have students write the algorithm in the second column on the BLM.
Discuss the results with the class. Make sure the students have the correct algorithms, as shown
in the answer key, before continuing on to the next section of the activity.
Answer Key
bm × bn
bm+n
bm ÷ bn
bm-n
(bm)n
bmn
(a × b)m
am × bm
(a ÷ b)m
am ÷ bm
b−m
a −n
b
( )
Black Line Master 1
page 202
1
bm
b n
a
( )
Standard 7 / Activity 8
Indiana Mathematics Grade 8 Curriculum Framework, October 2002
Math 8 Standard 7 08-30
9/11/02
11:12 AM
Page 203
Names:
Understanding the Law
Copy the rule you found for each law. Underneath each rule,
write a verbal description of the law. Use Law One as an example.
Law One: Product Law
bm × bn = bm + n
The product of two exponents of the same base is equal
to the base raised to the sum of the exponents.
Law Two: Quotient Law
bm ÷ bn =
Law Three: Power of a Power
(bm)n =
Law Four: Power of a Product
(a × b)m =
Law Five: Power of a Quotient
(a ÷ b)m =
Law Six: Negative Power
b−m =
Law Seven: Inverse Powers
−n
( ba ) =
Standard 7 / Activity 8
Indiana Mathematics Grade 8 Curriculum Framework, October 2002
Black Line Master 2
page 203
Math 8 Standard 7 08-30
9/5/02
11:32 AM
Page 204
Understanding the Law
Teacher Directions
After groups complete the BLM Discovering the Laws of Exponents, review the laws with
the class. Distribute one copy of the BLM Understanding the Law to each student. Allow them
to work in their groups to copy the correct law from their earlier work and write a verbal
expression for each law. Read the example given for Law One in class. Regroup and discuss
the results with the class.
Answer Key
Law One: Product Law
bm × bn = bm+n
The product of two exponents of the same base is equal to the base raised to the sum
of the exponents.
Law Two: Quotient Law
bm ÷ bn = bm−n
The quotient of two exponents of the same base is equal to the base raised to the difference
of the exponents.
Law Three: Power of a Power
(bm)n = bmn
The power of a power is equal to the product of the powers.
Law Four: Power of a Product
(a × b)m = am × bm
The power of a product is equal to the product of the powers of the individual terms.
Law Five: Power of a Quotient
(a ÷ b)m = am ÷ bm
The power of a quotient is equal to the quotient of the powers of the individual terms.
Law Six: Negative Power
b−m = 1bm
A base number raised to a negative power is equal to the reciprocal of the base number raised
to the positive power.
Law Seven: Inverse Powers
−n
( ba )
n
( ab )
=
The negative power of a fractional expression is equal to the reciprocal of the fraction raised
to the positive power.
Black Line Master 2
page 204
Standard 7 / Activity 8
Indiana Mathematics Grade 8 Curriculum Framework, October 2002