SWBAT evaluate negative integer powers of real numbers.

Students often struggle when negative exponents pop up. This lesson helps develop their meaning and overcome the difficulties.

10 minutes

Pair up students to begin this activity. Ask students to choose a partner they rarely work with. One of the two should have a scientific calculator. Pairing allows you to monitor more easily. As you do so, move around the class and listen in as the students speak with each other

Hand each pair of students one NEGATIVE EXPONENT ENTRANCE CARD.

Ask the pairs to follow each step in the card, discussing them with each other. Some students will be surprised to see with their calculator that the product in Question 2 is 1. By the time they get to question 5 they realize, and typically state that the products are always 1.

When students make this observation, write on the board:

**The product of b ^{n} and b^{−n} is always 1.**

Some students may quickly get it and state that the factors being used are reciprocals. Don't confirm this to the class as of yet and them linger a bit with this and begin the next next section.

15 minutes

1) Begin this section by writing the following pattern on the board. Tell the students to continue it until they reach 2^{-4}

**2 ^{4} =**

**2 ^{3 }=**

**2 ^{2} =**

**2 ^{1} =**

**.**

**.**

**.**

Students will probably use their calculator and write decimal answers for the powers with negative exponents. Tell them to write answers in fraction form as well. Students will quickly write 0.5 and 0.25 as fractions. Many will use their calculators to write the other decimal answers as fractions.

If you feel it is necessary, guide the class through the pattern a bit making them see how each exponent is one less than the one above it, so the value of each power is one-half that of the number above. Ask students to analyze the work carefully and find a general description of the pattern. Hopefully, they will come up with 2^{-n} = 1/2^{n}

Once the students get this, ask them to try it with another base. Suggest 10, since it is used in scientific notation, but also allow other bases. Ask if the pattern holds true. End by writing on the board for all to see: b^{-n }= 1/b^{n}

2) Since the previous lesson (Quotient of Powers) involved fractions involving powers with the same base, it is a good idea to now show the class that when the denominator contains the greater power, negative exponents can be used to simplify the expression.

Ask students to verify the Negative Exponent Property by using repeated multiplication (they usually understand quicker when you use the term expand), and simplifying.

See example EXAMPLE OF USE OF QUOTIENT OF POWERS

25 minutes

1. Hand each student in each group a NEGATIVE EXPONENT APPLICATION HANDOUT.

Tell students to put both their name and their partner’s name on their paper. The reason for this is that when you know someone is struggling, and you see on his paper that all answers and work are correct. If you see that his partner is a student that is doing well, you may want to keep this in mind and check this student out with discretion the following day. You don’t want students to just let their partner do the work and not learn the material.

2. It may be a good idea to hint to some students that seem to be struggling that each answer in problems 1 to 7 is a fraction. A common error with number 7 is that they bring the 5 down to the denominator along with the x. You may want to ask the student to separate the factors like 5 â x^{-2} â y ^{3}, then simplify and multiply.

3. Question 7 may give some students trouble, if they still don’t fully grasp b^{-n} is the reciprocal of b^{n}. The inductive reasoning in the pattern shown on the board in the previous section is quite convincing to students. They should go back and check it out.

4. In question 8, if students seem confused, ask them to change 1/25 into a power with negative exponents first. When they do this, they should be able to get n = -1 by using the power of products property.

5 minutes

End the lesson by asking each student to use the back of their handout to complete the 3 I's

Write them on the board and give students a few minutes to answer them. Collect the handout with the 3 I's answered in the back as students leave the room.

**I **really understood this idea…

**I** have a few questions about… before I can say I understand

**I** don’t even know where to start on …

See EXTENSION

Students know that the power -1 of a number b, is its reciprocal, 1/b

Therefore, they should figure that (a/b)^{-1} must be b/a

They should also be able to determine that (a/b)^{-2} is equal to (b/a)^{2}

Give them the general Negative Exponent property for fractions

(x/y)^{-n} = (y/x)^{n} and ask them to simplify the 3 extension problems.

Watch for the common mistake in #3, where students take the reciprocal of the fraction and change the negative exponent of m….(-5), and make it positive.

Remind them that taking the reciprocal of a fraction simply means inverting it and that the product must be 1.