Understanding the idea of deconstructing a number and changing its form without changing its value is key moment for students developing their number sense. The switch between standard and scientific notation is one of many ways to reflect this process. Teaching this concept is a great opportunity to help your students learn about the flexibility of numbers and the strategic choices we constantly make in picking a number format.
I start this lesson with a string that plays on the prior knowledge. They know how to deconstruct a number out of scientific notation and they know quite a bit about the world of exponents. So I ask students to write out the string and tell me what they notice.
2 x 10^3
2 x 1000
1 x 10^6
1 x 1000000
3.14 x 10^5
3.14 x 100000
I give students two minutes of individual time to find that we have three sets of equal expressions. If they finish early, I ask them to "write another piece that would fit onto this string." When they are ready, I share some "pieces" that students have created. I ask them, "does this piece fit?" And "how do you know?"
For example, a student writes:
1.5 x 10^2
1.5 x 100
We would reflect that these three expressions are equal. I would ask them to explain, something along the lines of "that first expression says 1.5 one hundreds and that is deconstructed in the second expression and the third expression is just the number 150, which is the product of the previous expression and literally equal to one and a half hundreds."
I would then popcorn around the room and ask other students to explain the other three pieces of the string. When they are finished, I would introduce a piece that doesn't fit (one from a student or one that I create on the spot):
15 x 10^2
15 x 100
"Why doesn't this fit? What is different about the first expression?" Here we would talk about the value of the number 15. And ask pointed questions to help students deduce that all the numbers in the string are below 10. Eventually we would agree that there has to be a definition of scientific notation.
With the class, I would generate something like this:
A number a x 10^b is in scientific notation when a is a number at least as large as 1 but less than 10 and b is an integer.
Naturally, they will want to know why this is. I save that conversation for the summary.
Deconstructing Standard Form. I did not mention any standard algorithms in the intro because I want students sharing and creating their algorithms during this investigation. I am careful to write down their interpretation of the conversion algorithms so that I can lead a conversation in the summary around their ideas.
One of the most common questions I ask as I circulate is "which number is larger?" For this question I am referring to any pair of equal numbers in both standard and scientific notation. It is a critical check for me. I need to make sure they know that the number value is not changing. I want to make sure they know that they are simply changing a number's appearance.
Again, here are the strings: Deconstructing Standard Form
Strings 1 and 2 in Deconstructing Standard Form are essentially identical. They are constructed so that one expression leads to an obvious answer in the following expression. For example,
1 x 10^3
In this case, 1 x 10^3 helps them recognize that 3000 = 3 x 1000 = 3 x 10^3.
In the second string, the process is identifcal, except that we are dealing with negative values. This leads me to ask, "what was wrong about our original definition for scientific notation?" Here students realize that we should have said that "scientific notation is of the form a x 10^b, where a has an absolute value that is at least as large as 1 but less than 10." I like to show the inequality along with the verbal definition.
The third string is much more difficult, but it is designed to push their thinking toward an efficient and universal algorithm in converting to scientific notation.
The string starts by helping them develop an entry level algorithm:
Here students deduce that we can "count the number of zeros and that is our exponent."
However, this approach fails in the next group:
Here students deduce that the "exponent equals the distance that the decimal has to travel in order for the expressions to be equivalent." For example, 12 = 1.2 x 10^1. The exponent is 1 because the decimal has to move one space from 1.2 to 12.0, the original value. Students need to make the connection that number in scientific notation is equivalent to its original expression.
The last group will push their thinking around conversions.
12.3 x 103
123 x 102
1230 x 101
12300 x 10
This group is meant to help them recognize intuitive approaches for converting. Students might evaluate and then convert. For example, they might write that 12.3 x 10^3 = 12300 = 1.23 x 10^4. But they also might reason that 12.3 = 1.23 x 10. So 12.3 x 10^3 = 1.23 x 10 x 10^3 = 1.23 x 10^4 (a much more advanced and often more efficient approach).
Now that students have had a chance to start dabbling in the conversions to and from scientific notation, I need to know where they are at. The exit ticket is meant to be a quick and easy evaluation for me.
question 1: convert 243000 to scientific notation
question 2: convert 1.4 x 10^5 to standard form
question 3: convert 356 x 10^8 to scientific notation
I use the results to guide the beginning conversation of my next lesson.