Lesson Plan 2: Bigger and Smaller - Exponent Rules
Students will need the following:
- Scientific or graphing calculators
1. Have students consider the following problem individually before the lesson begins:
Rallods in Rednow Land
2. Have students discuss their intuition regarding this situation. Without calculating, which scenario do they think would yield more rallods, a or b?
Which is more money?
(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.)
- 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.
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 29.9th 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.
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:
3. Have a volunteer from each group present the group's solutions for each of the above questions.
- 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.
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.