This lesson helps students get a foothold on the intuitive nature of multiplying and dividing with scientific notation. It gives both practice and context, which I think are certainly necessary for mastery.
I start with a question around the Andromeda galaxy:
The Andromeda Galaxy is visible without a telescope. Amazingly, it is 2,200,000 light years from Earth. Here is andromeda galaxy. If you could fire a beam of light at the galaxy and know that the light would eventually reach the galaxy (without running into something cool like the Enterprise), in how many years would it reach the Andromeda Galaxy? How far would the light have traveled?
Space traveler's note: Light travels at about 5,900,000,000,000 miles per year
I leave the numbers in standard form because I want students to convert them into scientific notation and see how this helps us estimate the answer (I encourage them not to use calculators). I want them to recognize that the galaxy is about 2.2 x 10^6 light years away (which tells us how long light would take to get there) and that light travels at about 5.9 x 10^12 miles per year, so we can see that it is about 2 x 5.9 x 10^18 miles away. Here, I discuss why we round the 2.2 to 2 instead of rounding the 5.9 to 6. Perhaps the best way to help them understand this is to show them how different types of rounding would impact the answer. One could argue that rounding 5.9 to 6 would have a much larger impact on the result than rounding 2.2 to 2 (because of the place value).
As we discuss the answer, I stress how easy the scientific notation is to work. Of course, I point out why multiplication is the perfect operation to solve this problem (light travels a certain distance each year and we know how many years it travels. So we have a group of years each with a certain distance and multiply to get the total distance).
Here we have Multiplying and Dividing Scientific Notation for students to work with. I ask them to take a moment and read through the questions before they calculate. I ask them to write down if they think they will multiply or divide to solve each question. I write down their thoughts on the board, marking what they think and why. Students can debate ideas, but its important to give them a reference. I split this work time between individual and partner work. I give them about 10 minutes on their own and 15 with a partner. I believe it is important for them to have individual time with the mathematics of this lesson, because it is meant to give them a lot of practice and support around a wide variety of multiplication and division questions.
It is really important that students feel supported with these challenging problems. They need to see efficient approaches to each solution and recognize that both exponents and scientific notation are tools that can help them through almost all of these problems mentally.
For example, consider question 6:
(7 x 10^3)/50
Here students could write 50 as 5 x 10:
(7 x 10^3)/(5 x 10) = 7/5 x 10^2 = 1.4 x 10^2
Students may need help with 7/5 to understand that 5 goes into 7 once with a remainder of 2/5 or 0.4.
Furthermore, it is beneficial for them to realize that the Associative Property works nicely within the structure of the scientific number enabling flexible approaches to calculating a result.
Each question encourages pattern seeking and modeling with real world problems. If a question isn't referencing an interesting fact, the expressions are arranged to help students understand how to use patterns in their multiplication.
Consider question 1:
This question looks like endless drill, but it is really a string. This means that each expression exhibits a pattern in relation to at least one other element of the string. For example, parts 5 through 10 start with a simple problem (10)(9 x 104) and then continue with that idea by multiplying the first term to 100(9 x 104), 1000(9 x 104) and so forth. Here students can use their work with easier questions in a string to help them on the tougher problems in the string.