We get students excited to work with large numbers by starting with a fundamental arithmetic "trick." I set up pattern strings on the board and ask students to solve the individual expressions but also describe the characteristics that "tie" each string together.
2 x 1
20 x 1
20 x 10
200 x 10
200 x 100
2000 x 1000
2000000 x 10000
12 x 10
12 x 100
120 x 100
12 x 11
12 x 110
1200 x 1100
12000 x 1000 x 10000 x 100000
String 1 is meant to help students recognize the simple pattern of multiplying positive digits with following zero digits. When they see 2000000 x 10000, they should be able to think, "2 x 1 = 1" and this is similar but on a much larger scale, so we can just count the number of zero digits and include them in our answer." They can finish by thinking, "2000000 x 10000 = 20000000000." The nice thing about the string is that each expression prepares them for this insight.
String 2 is tougher but even more valuable. Students start with a simple expression. 12 x 10 = 120 and thus 12 x 100 = 1200. After they complete that part of the string they confront 12 x 11. They might know this product or use a stacking algorithm, but I encourage them to use 12 x 10 as their guide. I want them to recognize that 12 x 11 has one more group os 12 than 12 x 10. Since 12 x 10 is 120, 12 x 11 = 132. They continue and can reason that 12 x 110 = 1320 and 1200 x 1100 = 1320000. Finally they confront the expression most useful to this lesson. Here they can use the associative property to view the expression as (12000 x 1000) x (10000 x 100000) = 12000000 x 1000000000 = 12000000000000000.
A critical and necessary part of this discussion is to ask, "why can we just add in the zero digits?" I would give them a minute or two to turn and talk with a partner to come up with an example of their own. For the share, we would use their example.
4 x 2 = 8
40 x 2 = 80, since 40 = 4 x 10, 40 x 2 = 10 x (4 x 2) = 10 x 8
The use of the associative property and the concept of balancing an equation is a major component here, but it will give them incredible flexibility in their work with large numbers and scientific notation.
To really get a grip on the patterns emerging from some of our larger numbers, I give out a partially completed table and ask students to fill in the gaps.
How high can you count table is meant to to bring out patterns and help them make sense of the names, size and relative size of numbers.
A few notes on How high can you count table:
The first column lists out the numbers in standard numeral form (I didn't write "standard form" for the header since that is a reference to scientific notation in Britain). I want students to notice that the way in which the numbers grow. I started the sequence on a thousand to help them see that all the number names are based on powers of 1000. I also want them to consider omitting writing all the numbers out. In fact, it would be great if students recognize how irritating it would be to write out these numbers. This is part of the reason it is natural to use scientific notation. It is an efficient way of dealing with our number system.
The third and fourth column are labeled "thousand notation" and "ten notation." I let students know that I made up these notations, because they will help us understand the key role of 1000 and the way in which we can convert to powers of 10. I typed out the first group of cells to help students get started, but might leave those blank and review them with the class. There is certainly some power behind the process of writing while thinking (as opposed to just listening).
The final column is for scientific notation and should be a natural extension of the other columns in How high can you count table. My students are already familiar with basic properties of exponents and should be able to manipulate "ten notation."
I review aspects of the table with each partnership and group as I circulate. The goal is to get students ready for the use of scientific notation. I want them to realize that it is an efficient and logical way to represent large numbers. I want them thinking that if they were facing these large numbers before the "invention" of scientific notation, that they too would use the system of scientific notation to describe these numbers. I believe that this makes a big difference. It is the difference between learning a system imposed on them (just another math formula to memorize) and learning a system that helps them make sense of the world of numbers.
The table review is a fairly quick process because I have reviewed most of the table with them during the investigation. The real challenge is in the extension summary, where students make sense of the scale between these large numbers.
How do you sum up all of the large numbers ever used in mathematics? You don't. Instead you give them something they can use to understand any large number they might encounter. With this lesson, that one thing is to recognize the scale between large numbers. When we jump from a million to a billion, I want students to understand we are increasing the size of our number by a factor of 1000. With this understanding, they can always estimate an answer and think about any problem in a general way before they analyze the specifics. If they are dividing 4.34 billion by 2.1 million, I don't want them to think "let me grab a calculator." I want them to think, that is a little over 2 thousand. Then they can check the exact value if they need to. Having this mental estimate will allow them to comprehend and calculate.
All of my lessons in this unit are designed to evoke that type of thinking. I like using a visual here to give students of what it means to make something "1000 times larger." Here is how I do it:
I draw a tiny dot on the board and say, "suppose this dot represents a billion. How long should a line be in order to be a thousand times larger?"
If students ask for an exact measurement of the dot, I will measure it on the board and also give them the lengths of the whiteboard I am using and the perimeter of the room. This will help them answer with both a number value, "I think the line will be x cm" and a concrete value, "I think it will go halfway around the perimeter of the room."
After we discuss and agree upon a value (the size is astonishing), I will offer the same exercise and use a slightly larger line. This time I will ciculate and see how they approach the answer. If I give them the length of this line, will the recognize that the new expansion can be found by using a scale factor from the smaller line? Will they simply multiply the lenght of the line by 1000? Can they use both approaches?
After comparing strategies again, I like to bring this discussion back to the original investigation.
"What number are we talking about here? What is 1000 times larger than a billion?"
We finish by talking about a trillion and other large numbers. A trillion is more than just the next number name after a billion, it is a thousand thousand times larger than a million.