The process of building intuition around small numbers is quicker in this unit, since much of the same reasoning from previous lessons is now applied. The main obstacle we need to confront is the meaning and role of the negative exponent. Students need to recall that negative exponents are not negative numbers, but "negative" operations. Negative in the sense that they are the opposite operation of a positive exponent. They need to think about the expansion of a base to a negative exponent as starting with 1. For example, 10^-2 = 1/10/10, not just 10/10. This is crucial for work with scientific notation and is the starting point of this lesson.
Students enter the room and begin working on Prep for Small Numbers with me. We fill out the table together and compare the different forms of each number. As we work our way toward 10^0, we discuss how the pattern in our table supports the idea that 10^0 = 1 and that 10^-1 = 1/10. Students are encouraged to share any patterns they notice, but we give them about 5 minutes to finish the prep sheet alone. I circulate and look for any useful algorithms that deal with the connections between positive and negative exponents. For example, a student might say "10^-6 is the reciprocal for 10^6. So if I had to find 10^-6, I could think that 10^6 = 1000000 and then take the reciprocal and write 1/1000000." After they have had time to write their thoughts we have a brief class discussion around their ideas and algorithms.
If we know that 10^-2 = 1/100, how does this help us think about a number like 5 x 10^-2? I like to help students use the associative property to see that 5 x 10^-2 = 5 x (1/100) = 5/100. A key connection here is to recognize that multiplying by 1/100 is the same as dividing by 100. To help them work with this idea, we go through several examples.
"What is half of 50?"
"What is 50 divided by 2?"
"Is multiplying by 1/2 the same as dividing by 2?"
"Is multiplying by 1/10 the same as dividing by 10?"
Then we would look at an example to confirm that multiplying by 1/10 or 10^-1 is the same as dividing a number by 10.
50 x 10^-1
60 x 10^-1
70 x 10^-1
5 x 10^-1
6 x 10^-1
7 x 10^-1
These examples start with easy division and then finish with three expressions that are simply 10 times smaller than the first three. The goal is to derive the idea that 10^-1 means to divide the first number by 10 and furthermore that this division is easy (since we only have to move the decimal left for every division of 10.) We solidify this idea by trying another string:
5000 x 10^-3
4000 x 10^-3
3000 x 10^-3
5 x 10^-3
4 x 10^-3
3 x 10^-3
Here we start with numbers that are easily divisible by 1000 and then we move to 3 numbers that are easy to think about in contrast to the first three expressions. However, students are encouraged to start thinking that dividing by 1000 is the same as dividing by 10 three times and thus is the same as moving the decimal left 3 times. I encourage students to share this idea or simply compare each by first dividing by 1000 and then repeatedly dividing by 10 three times. They will certainly then notice that each process is equivalent. This is a nice moment for students, because they notice that 10^-3 has the exponent -3 which represents moving the decimal left 3 times (like -3 on the number line).
Students continue the investigation with two other strings and then have the opportunity to create their own string. I ask them to surprise us and make sure that all the expressions somehow build off of each other. The simple rule of constructing a string is to remember that each expression should be solvable on its own or in conjunction with a previous step. The idea of a string is that every expression is "tied" together in someway.
Here is the handout, the goal is to rewrite numbers in scientific notation. If the number is already in scientific notation, then write it in standard form. Strings for Small Numbers
The main take away in this lesson to track how the power of 10 moves the decimal. When the base is ten and exponent is positive, our decimal moves right. When the base is ten and the exponent is negative, the decimal moves left. This impacts the structure of the numbers and has implications for the number of zeros in standard form.
Zeros aren't fundamental, but they seem to really help students make sense of scientific notation. For example, they should recognize that 3 x 10^2 will have two zeros and 3 x 10^-2 will have 1 zero. I am careful with my wording here and have the students construct these "take aways." If we feel that something is too general, we rephrase it.
"How many zeros will 3 x 10^5 have? What about 3 x 10^-5?"
Here students might say that the negative exponent will have 1 less zero, since 3 x 10^-5 = .00003. Other students might confuse this with 0.00003 and conclude that there are 5 zeros. These are both equivalent, but can cause major confusion around scientific notation. Students panic and think "wait is there 5 or 4 zeros? Oh no I can't remember!" This is a good time to talk about the weakness of memorizing in mathematics. Whenever they simply memorize the way numbers work, they will always confuse themselves. However, if we step back and think about the fact that 10^-5 means to divide 3 by 10 five times, we recognize that the decimal moves 5 times to the left and thus hops past four place values, each of which are empty and thus must be zero.
Personally, I would avoid counting the number of zeros, but as I mentioned earlier, many of them like to incorporate this thinking into their algorithms and work.
Another major misconception is to forget the importance of the base 10 in these rules. Students often write that 2^-2 = .02, mixing up their thinking with 10^-2. In this summary I ask them about these types of contrasting expressions.
"How does 2^-2 compare to 10^-2?"
"Why is the base 10 so much easier to work with?"
"Whats harder to write in decimal form, 3^-5 or 10^-5? Why?"
After we discuss these landmarks, I review their work from the investigation, having students present on problems that highlight great strategies and help others who were struggling.