This lesson is meant to help students compare numbers in scientific notation and begin to understand why the first number in scientific notation is not larger than 10.
In the previous lessons we compared big and small numbers in a variety of ways, but I start this lesson with a quick review of scientific notation procedure.
I ask them to "convert the numbers 3000 and .0003 into scientific notation. Then write out your definition for scientific notation. When is a number in scientific notation?"
I circulate as they work for about 2 minutes and then share what I notice. I like to point out things that did work and things that didn't. The goal is to get group feedback that verifies or adjusts an algorithm without embarrassing any single student. Then I share my point of view:
3000 = 3 x 1000 = 3 x 10^3
.003 = 3/1000 = 3 x 1/1000 = 3 x 1/10^3 = 3 x 10^-3
As a class, we discuss that a number a x 10^b is in scientific notation when a has an absolute value greater than 1 but less than 10 and b is any integer.
"Although the convention is somewhat arbitrary, it is useful for the purpose of comparin numbers, which is our focus today."
I ask students to look at this Microsoft Info-graphic. I ask students to take two minutes to look at the graphic and "explain what happened to Microsoft between 2000 and 2013? Use the specific numbers to support your ideas." I might spend a moment explaining Market Value (total worth), Net Income (Profit from the current term) and Employees (all people working for the company).
As a class we walk through each section of the graphic:
Market Value: 555 Billion to 289 Billion
Net Income: 9,421 Million to 21,863 Million
Employees: 39,100 to 99,000
Students will comment on the basic trends, where they shrunk in Market Value, increased their Net Income and hired many more Employees. More importantly, I want them to explain "what made these comparisons so easy?" The idea is that phrases like 555 billion and 289 billion are easy to compare because they are both counting same number of units (in this case billions). The same is true for each category. But I want to push students here to think, "which number is the largest?"
This means they will have to compare numbers like 555 billion and 9,421 Million. I take suggestions like "write the number in standard form or scientific notation" and then give the students 2 minutes to order the numbers from least to greatest.
When we compare the lists, I would also encourage students to use their intuition and make statements like "9,421 Million is 9.421 Billion since every 1,000 Millions = 1 Billion." As we line the numbers up I would write them in different forms, row by row.
In the first row we would use the words from least to greatest:
(391 Hundreds < 990 Hundreds < 9,421 Million < 21, 863 Million <289 Billion < 555 Billion)
Then we would try it in standard form and finally scientific notation. Here the question is "does the scientific notation make the comparison process easier?" It may or may not for certain students, but the point is to start a conversation about how we can compare numbers in scientific notation (that is the activity of the day).
The table and questions in Scientific Notation Comparisons are designed to help students find an efficient and universal way to compare numbers in scientific notation.
To encourage them, I ask them to work in partners and work talk them through the table and questions. We talk about how they could "explain their reasoning." Here the basic expectation is that students will show how they compared numbers. I want to know if they are writing the numbers in standard form or if they are able to find a way to simply compare the exponents and factors of each number to determine which is larger. This process takes about 5 minutes.
I give them 20 minutes to work on the questions and think about how they can explain their reasoning to the class. As always, I circulate and help partners and groups support each other.
The goal of the summary is to review the work they did during the investigation and to develop an algorithm that they discovered in their work.
We start by reviewing the table. I pick a student's work to display with the projector and ask them to explain how they found each value. As always, I ask students to rephrase explanations and talk compare strategies. Students might have noticed that it was both tedious and sometimes difficult to write the numbers in standard form. I would comment on this by talking about the large number at the end of the table: " 100 Hundreds x 1000 Thousands x 1,000,000 Millions x 1,000,000,000 Billions." This number is meant to help students so how much easier it is to work in scientific notation for these types of numbers.
The following questions are relatively straight forward, but I would popcorn around and make sure that students constantly explain how they compared the numbers. This is meant to help the class formulate an algorithm. When students have a chance to put their ideas into words, they can find a way to remember and share a great idea and algorithm.
Getting to the algorithm is challenging, but I always read their ideas as I circulate during the investigation (the last question asks them to write their strategy and I read these to get a sense of their strategies). Since I know a little about their thinking, I can ask pointed questions. Some students might write something like, "a larger exponent means a larger number." However this is not true when we compare positive and negative numbers. So I ask questions like, "is this always true?"
So our approach is to start with a possible algorithm and then edit it as we go. It might start with an oversimplified statement, like:
"First, compare exponents. If the exponent is larger on one number, it is the largest. If not, compare the first factor in the same way. If the first factor is larger, then the number is larger."
After a small edit in regard to the comparison of positive and negatives:
"First identify the positive and negative numbers. Positives are always larger. Then compare exponents. If the exponent is larger on one number, it is the largest. If not, compare the first factor in the same way. If the first factor is larger, then the number is larger."
The tricky part is identifying a "larger" exponent. For example:
2 x 10^-3
3 x 10^-4
Here students need to remember that -3>-4 so 2 x 10^-3 is larger.
However, with negative numbers, things are a bit different:
-2 x 10^-3
-3 x 10^-4
Here -3 x 10^-4 is larger.
At this point I ask them to consider a separate algorithm for negatives. If we are comparing two negatives, we need to consider that "the larger exponent represents a smaller number."
We can keep going in the discussion by comparing other types of negative examples. Part of the conversation needs to be about the usefulness of an algorithm when confronted with so many variables. I ask students to consider each algorithm, but to always test their intuition by writing the numbers in standard form. However, when a number is too tedious to write out, I remind them to try a simpler example.
If I gave them this example:
-2 x 10^-98
-3 x 10^-99
They could simplify their thinking here by noticing that the first expression has a larger exponent and the second expression has a smaller exponent. In this way, the question will identical to our earlier example:
-2 x 10^-3
-3 x 10^-4
Here -3 x 10^-4 is larger.
This also implies that -3 x 10^-99 is larger (which it is).
This is a key habit of mind that students need to use as they solve complex mathematical problems. It isn't about memorizing a list of algorithms, but finding a way to reason through the solving problem process. Creating simpler examples is a very important step in this process.