It is tempting to start class by saying something like, "this is how scientific notation works." And this might work well for you and your students, but the problem with this approach is that in encourages students to often lose their intuition and number sense around scientific notation.
When we start with an algorithm, students see numbers like 1 x 10^3 and think something like, "the exponent is 3 so I move my decimal 3 places to the right and get 1000." This is fine, unless it is their only way of approaching this number. If it is there only way of dealing with scientific notation, then they will get trapped in the algorithm.
I want my students to both think about a quick algorithm and tap into their number sense as they work. They should be thinking, "1 x 10^3 is 1 group of 1000 since 10^3 is 10 x 10 x 10 and that is 1000." Having this flexible mindset will allow them to tackle all types of complex problems. I like to start with the intuition and develop the algorithms from their observations.
I start by giving them a simple string:
Write the value of each expression below in standard form.
For example, 10 is the standard form of 10^1
1 x 10^2
2 x 10^2
3 x 10^2
4 x 10^2
4.5 x 10^2
5 x 10^2
10 x 10^2
1 x 10^3
I let students take a moment to make sense of this, but we share strategies that help with each individual expression and help students see a pattern in the string as a whole.
Students use their knowledge of 10^2 as 100 to infer that 1 x 10^2 is another way of saying 1 group of 100 and so forth until they reach 4.5 x 10^2. This one is a bit tougher. Students can explain that this is like "4 and one half hundreds." Then they can find 5 x 10^2 and double it to find 10 x 10^2. We discuss why it makes sense that the last two expressions were equal.
"Does it make sense that 10 x 10^2 = 1 x 10^3?"
Some common responses:
"Yes, in both cases we are multiplying three groups of tens."
"Yes, if we divide the first term in 10 x 10^2 by 10 we make the expression 1 x 10^2. But this is not the same expression, we can balance it by multiplying 10^2 by 10 and then get 1 x 10^3.
"Yes, if we multiply he first term in 1 x 10^3 by 10 we get 10 x 10^3, but this is too big. So we divide 10^3 by 10 and get 10^2."
I also ask questions that get them thinking about possible algorithms.
"Is there a connection between the value of the exponents and the number of zeros?"
Students first notice that the exponent matches the number of zeros, but I make sure to point out that isn't true with 4.5 x 10^2 and ask them "whats happening there?"
Some students might answer this, but we return to this at the end of the class. It is one of the main take aways.
Since students are familiar with the concept of working with a string, The Strings of Scientific Notation, they can work individually or in partnerships. I circulate and encourage them to look for patterns in the strings and justify how they know their answer is right. I make sure to write down useful observations that lead us towards a viable algorithm. Any observations around the way the exponents impact the size of the number or the location of the decimal point will be useful.
We conclude class with a review of student observations about the patterns in each string, and, the answers students constructed along the way. We discuss how they know they are correct and what ideas they have for algorithms.
We focus on string expressions that they found challenging and I question them deeply around 1000 x 3 x 10^4. I want them to have many strategies in dealing with these types of expressions.
The main take aways are the following:
1. If your product is positive, increasing the exponent increasing the value of the number.
2. If your product is negative, increasing the exponent will decrease the value of the number.
3. Every time we multiply a number by one ten, the decimal in that number moves once to the right (since it is getting 10 times further from zero)
4. If x >0, multiplying a number by 10^x will move your decimal x places to the right.
These are in my words, but we would come up with our take aways in the words of the students. The list might be larger (depending on the class), but these are the basic observations I push for. I lead them towards these conclusions by asking leading questions like, "If I increase my exponent, will the product always get larger?"