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**Overview:**

This lesson will teach students about several of the rules regarding exponents. It uses a situation from *Alice in Wonderland* in which Alice’s height is doubled or reduced by half depending on what she consumes to introduce negative exponents and the rules for dividing powers.

**Time Allotment:**

Two 50-minute class periods

**Subject Matter:**

Rules of exponents

**Learning Objectives:**

Students will be able to:

- Understand, and develop a sensibility to, the magnitude of exponential growth.
- Develop laws of exponents.
- Define negative exponents.

**Standards:**

*Principles and Standards for School Mathematics*, National Council of Teachers of Mathematics (NCTM), 2000.

NCTM Algebra Standard for Grades 6-8

http://standards.nctm.org/document/chapter6/alg.htm

NCTM Algebra Standard for Grades 9-12

http://standards.nctm.org/document/chapter7/alg.htm

**Supplies:
**

Students will need the following:

- Scientific or graphing calculators

**Steps**

**Introductory Activity:**

**1.**Have students consider the following problem individually before the lesson begins:

Rallods in Rednow Land

Which is more money?

- One billion rallods
- The amount obtained by putting 1 rallod on one square of a chessboard, 2 rallods on the next square, 4 on the next, and so on, until all 64 squares are filled.
(NOTE: To find the total number of rallods on the chessboard, students must add 1 + 2 + 4 + 8 … + 2

^{63}. Finding the total on all 64 squares is not necessary to answer the question, since the running total surpasses 1 billion well before the 64^{th}square.)

**2.**Have students discuss their intuition regarding this situation. Without calculating, which scenario do they think would yield more rallods, a or b?

**3.**Allow student groups a few minutes to calculate and discuss the result in choice b.

**4.**Have students consider the following problem:

On what square of the chessboard would the total number of rallods first exceed 1 billion?

**5.**Give student groups a few minutes to calculate an answer to this question. Then, have the groups share their results and discuss their methods for obtaining the answer (which is the 31^{st} square, since 2^{30} = 1,073,741,824). Students may also use guess-and-check and say it would be on the 29.9^{th}square. Although this answer doesn’t make sense in the context of the problem, it will allow for a discussion as to whether or not it is okay to have decimal exponents.

**Learning Activities:**

**1.** Have students discuss the effect of cake and beverages on Alice’s height in *Alice in Wonderland*. Have a student from each group describe his or her group’s discussion to the class. Students should understand that when Alice eats an ounce of cake, her height doubles, and when Alice drinks an ounce of beverage, her height is halved.

**2.**Give students time to discuss the problems below in their groups:

- What happens when Alice eats several ounces of cake and drinks the same number of ounces of beverage?
- Find several combinations of cake and beverage that will cause Alice to be 8 (or 2
^{3}) times her normal height. - Find several combinations of cake and beverage that will cause Alice to be 32 (or 2
^{5}) times her normal height. - Find several combinations of cake and beverage that will cause Alice to be 4 (or 2
^{2}) times her normal height. - What happens if Alice consumes more ounces of beverage than ounces of cake?
- If Alice eats c ounces of cake and drinks b ounces of beverage, what is her height? Describe her height using a mathematical expression.

**3.**Have a volunteer from each group present the group’s solutions for each of the above questions.

**4.**Use the students’ solutions to develop the rules for negative exponents and for divisibility of exponents: 2^{m}/2^{n} = 2^{m – n}. For instance, in question b, students may have shown that eating 7 ounces of cake and drinking 4 ounces of beverage would cause Alice to be eight times as tall, or 2^{7} � (1/2)^{4} = 2^{3}. Rewrite this as 2^{7} � 2^{-4} = 2^{3} and as 2^{7}/2^{4} = 2^{3}.

**5.**During the discussion for question f, be sure to elicit the general formula 2^{c} � (1/2)^{b} = 2^{c – b}. This formula will lead to the rule for negative exponents, 2^{-b} = 1/2^{b}, as well as to the general rule for divisibility, 2^{c}/2^{b} = 2^{c – b}.

**6.**Assign practice problems for homework.

The questions below dealing with concepts related to exponents have been selected from various state and national assessments. Although the lesson above may not fully equip students to answer all such test questions successfully, students who participate in active lessons like this one will eventually develop the conceptual understanding needed to succeed on these and other state assessment questions.

- Taken from the Massachusetts Comprehensive Assessment, Grade 10 (Spring 2002):

What is the simplest form of the expression

2x^{4}y^{2}/ x^{2}y^{2}, x 0, y 0?Solution: 2x

^{4}y^{2}/ x^{2}y^{2}= 2(x^{4 – 2})(y^{2 – 2}) = 2x^{2} - Taken from the Massachusetts Comprehensive Assessment, Grade 10 (Spring 2002):

On January 1, 2000, a car had a value of $15,000. Each year after that, the car’s value will decrease by 20 percent of the previous year’s value. Which expression represents the car’s value on January 1, 2003?

**A. 15,000(0.8)**^{3}(Correct Answer)

B. 15,000(0.8)^{4}

C. 15,000(0.2)^{3}

D. 15,000(0.2)^{4} - Taken from the Colorado State Assessment Program, Grade 10 (2002):

Number cubes are the basis for many games. Each face of a number cube is identified by a number from 1 to 6. Some games use one number cube and some games use multiple number cubes.

Part C – Complete the table below to show the number of possible outcomes when 2, 3, 4, and 5 number cubes are used.

**Solution:**There are 6^{2}= 36 outcomes when two cubes are used, 6^{3}= 216 outcomes for three cubes, 6^{4}= 1296 outcomes for four cubes, and 6^{5}= 7776 outcomes for five cubes. In general, there are 6^{n}outcomes for n cubes.- Taken from the Virginia Standards of Learning Assessment, Algebra I (Spring 2001):

Which is equivalent to:

A. 1/b

^{3}B. b

^{3}**C. b**^{4}(Correct Answer)D. b

^{8}

**
Teacher Commentary:**

Please click here to view the rubric Mike Melville used to assess this student’s work.