Insights Into Algebra 1: Teaching for Learning
Exponential Functions Lesson Plan 2: Bigger and Smaller – Exponent Rules
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.
Two 50-minute class periods
Rules of exponents
Students will be able to:
- Understand, and develop a sensibility to, the magnitude of exponential growth.
- Develop laws of exponents.
- Define negative exponents.
Principles and Standards for School Mathematics, National Council of Teachers of Mathematics (NCTM), 2000.
NCTM Algebra Standard for Grades 6-8
NCTM Algebra Standard for Grades 9-12
Students will need the following:
- Scientific or graphing calculators
Teachers Activities and Assignment
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 … + 263. Finding the total on all 64 squares is not necessary to answer the question, since the running total surpasses 1 billion well before the 64th 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 31st square, since 230 = 1,073,741,824). Students may also use guess-and-check and say it would be on the 29.9thsquare. 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.
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 23) times her normal height.
- Find several combinations of cake and beverage that will cause Alice to be 32 (or 25) times her normal height.
- Find several combinations of cake and beverage that will cause Alice to be 4 (or 22) 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: 2m/2n = 2m – 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 27 � (1/2)4 = 23. Rewrite this as 27 � 2-4 = 23 and as 27/24 = 23.
5.During the discussion for question f, be sure to elicit the general formula 2c � (1/2)b = 2c – b. This formula will lead to the rule for negative exponents, 2-b = 1/2b, as well as to the general rule for divisibility, 2c/2b = 2c – b.
6.Assign practice problems for homework.
Related Standardized Test Questions
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
2x4y2 / x2y2, x 0, y 0?
Solution: 2x4y2 / x2y2 = 2(x4 – 2)(y2 – 2) = 2x2
- 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)
- 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 62 = 36 outcomes when two cubes are used, 63 = 216 outcomes for three cubes, 64 = 1296 outcomes for four cubes, and 65 = 7776 outcomes for five cubes. In general, there are 6n outcomes for n cubes.
- Taken from the Virginia Standards of Learning Assessment, Algebra I (Spring 2001):
Which is equivalent to:
C. b4 (Correct Answer)
Student Work: Factoring Assignment
Please click here to view the rubric Mike Melville used to assess this student’s work.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.