Gotta say it's really sad that I'm Junior's lifeline when it comes to getting physics homework done. When I left the gym early this evening, I turned on my phone and there were four messages from Junior and his little brother asking when I was getting home and could I help them with their homework.
"I have physics, and I don't understand it," Junior told me. When I called him back, it was pretty late and I asked if he had gotten any of it done. "Yeah, some," he said. "There's just two parts I don't understand."
One of them was scientific notation and how to convert back and forth between scientific and standard. There were some problems where you had to multiply and divide numbers in scientific notation and I vaguely remembered you could just add and subtract the powers of 10 (the integers), but I wasn't really sure.
"Call Picasso," suggested Junior. "He's smart."
"Yeah, you're right," I said, and we called him. He's done it at Curie, too, and he thought yes, we could do that. So we did. (If we were wrong, you can call us out in the comments.)
Then we went back and checked something he had done earlier. Junior was supposed to explain and make a sample graph of a direct relationship between two things. He explained it in general and made a general graph, but he hadn't given a specific example. When we looked back, he came up with a good one: "The more kids you have, the more you have to do laundry."
"You see that from having your little brother around, right?" I said. They laughed. (There's nothing like a new baby in the house for laundry.) So he put labels on his graph--I think loads of laundry was the x-axis and number of kids was the y-axis.
4 comments:
Hey Marshfield -- the best way I know to make dividing and multiplying numbers in scientific notation stick is to realize why the shortcuts you mentioned work. You're right about the adding and subtracting exponents, but it's helpful to know why that works, just in case the problem gets mixed up, or to check your work.
Here's an example:
(2.1 x 10^3) x (4 x 10^4) = ?
If you were to write those out in standard form you'd get this problem:
(2100) x (40000) because the exponents on the power of ten move the decimal of the integer over to the right that number of spaces. ((Side note: Lots of people get confused and think of the exponent as adding on a certain number of zeros, but that's not really the case as you see in the example with 2.1))
Multiplying out the problem in standard notation gives you 84000000. To convert this back to scientific notation you have to remember that the integer must have only one digit to the left of the decimal place so you use 8.4, (NOT 84) as your digit, giving you 8.4 x 10^? Now, ask yourself, what power of ten would move the decimal over the appropriate number of spaces?
10^0 = 8.4
10^1 = 84
10^2 = 840
10^3 = 8400
10^4 = 84000
10^5 = 840000
10^6 = 8400000
10^7 = 84000000 BINGO!
The solution is 8.4 x 10^7
Look back at the original problem:
(2.1 x 10^3) x (4 x 10^4) = ?
Instead of doing it the long way like we just did above, you can short cut it by multiplying the integers together then multiplying the powers of ten together:
(2.1 x 4) x (10^3 x 10^4) =
(8.4) x (10^7) =
8.4 x 10^7 just as we solved!
The same idea goes for division. The important thing for division, though, is to be sure to keep the order of the problem the consistent. For example:
(6 x 10^5) / (2 x 10^3) =
(6/2) x (10^5)/(10^3) =
3 x 10^2 because the exponents are subtracted (5 - 3 = 2)
I still recommend getting the hang of switching between standard and scientific notation because it helps to check your work. The above problem, then, becomes:
600000 / 2000 =
300 =
3 x 10^2
If the problem were reversed, it would be quite different:
(6 x 10^3) / (2 x 10^5) =
(6/2) x (10^3)/(10^5) =
3 x 10^(-2)
It's okay to have negative exponents. That just means that you're dealing with a fraction instead of a number greater than one. 3 x 10^(-2) = 0.03 or 3/100
Hope that helps some! This can be confusing sometimes but I'm sure it'll get easier with practice!
Thanks, Erin! I was trying to remember this and explain it to Junior at the same time, and that was quite a struggle. Having your explanation as a handy reference will help the next time we work on this topic.
a little note on the graph... the number of kids should be the x-axis (along the bottom) becuase it's the independent variable. The laundry should be the y-axis (up the side) because it's the dependent variable.... the amount of laundry depends on the number of kids, not the reverse.
hope that helps in the future!
Junior may well have gotten it right, I couldn't remember by the time I wrote that post. But it shows you why I shouldn't be helping Junior with physics homework, since I can't remember this stuff!
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