This is Common Core State Standards Support Video in Mathematics. The standard is 8.EE.4. The standard reads: Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.
Now there’s a whole lot to this standard, and what adds a complication to it is that in order to do what the standard asks for, there’s a lot of prerequisite knowledge. First of all, in order to perform operations with numbers expressed in scientific notation, students have to understand the basic laws of exponents. That’s addressed in standard 8.EE.1, which states: Know and apply the properties of integer exponents to generate equivalent numerical expressions.
So just as a quick review, let’s take multiplication for example. We have 4 cubed times 4 squared. The students would expand it all out, and they see that they have 5 of them. So it’s 4 to the fifth power. If we look at the relationship, we have a 5 here and back over here, we had a 3 and a 2. So they’ll get the pattern that hey, we add the exponents. So when we’re multiplying and we have the same bases, if we are multiplying and we have two numbers with the same bases, we would end up adding the exponents.
Let’s try division. Again doing this manually, we would end up with several pairs of 4 over 4, which simplify to ones. So we get 4 times 4, which is four squared. Again noticing the pattern, we have a 2 here, and if we you go back and connect it to the original expression, we had a 5 and a 3, which means that hey, we had to have subtracted the exponents. That seems to be the pattern– that we would subtract the exponents when we’re dividing. Now let’s do a similar expression, but this time, let’s reverse it to where we have the larger exponent on the bottom. Again expanding this out and doing it manually, we have again several pairs of 4 over 4, which would simplify to ones. So we end up with 1 over 4 squared.
But we’re not finished. We need this expression to be something where it’s just a number raised to some power with no fractions involved. Now if we look over here, we see we have a -2. If we go back and look at the 3 and the 5 and the previous idea that well, we subtracted the exponents, it does turn out, because if you take 3 minus 5 we do get a -2. Now doing several of these examples, students are going to see this pattern right here, that the negative in the exponent simply indicates a reciprocal, which stated in general terms is this right here, that again, if we have something raised to a negative power, that is equivalent to that expression, but the reciprocal.
So now armed with the knowledge of the laws of exponents, students can take those properties and apply them to powers of 10 in scientific notation. It’s the same basic idea except now we have bases of 10. So here we have a multiplication problem. We’re multiplying, so we know we have to add the exponents, so we get 10 to the eighth power. This other example, again multiplication, so we have to add the exponents, except this time, it’s a 6 and a -2, which gives us a positive 4. So we get 10 to the fourth. Now for division, we know that we have to subtract the exponents. So here, we get 10 to the third. Another division example: again we have to subtract the exponents, and we get 10 to the minus 3 power. Now it’s a good idea to also have your students do some of these in fractional form like so. Here students will end up getting 1 over 10 to the fourth, which goes back and reinforces that idea that when I have a negative exponent like what I have here, 10 to the -4, it is simply the reciprocal.
If we closely examine the document with the Common Core State Standards, we will see that this standard 8.EE.4, is the only one of the Common Core Standards in mathematics that addresses scientific notation. Now when we look at the first part of the standard—Perform operations with numbers expressed in scientific notation including problems where both decimal and scientific notation are used—if we look at the first statement in the standard that deals with performing operations with numbers expressed in scientific notation, it should become apparent that students first, in order to do that have to know how to convert numbers in standard notation to scientific notation and vice versa, from scientific notation to regular standard notation before they can perform the operations with them.
So a little bit more review is in order. So again, let’s look at having to convert from scientific notation to standard notation and from standard notation to scientific notation. First let’s take this example which would be 8,700,000,000. In order to make this into scientific notation, the first important rule is that it has to be a number between 1 and 10. So we first have to convert this number to something between 1 and 10. This number’s called the significand. So now we have to make it times 10 to some power. The way that that is determined is look at what we have and then what do we have to do to get back to the original number.
So let’s look at this first example where we have to take this large number, which would be 8,700,000,000 and convert it to scientific notation. The first requirement is that in scientific notation this first part, which is called a significand needs to be a number between 1 and 10. So we’ve done that; we’ve taken what we have over here and made it a number between 1 and 10. Then the next part is we need that times 10 to some power. The way that is determined is to start where your decimal point is now where we made a number between 1 and 10, and where do I have to move to get back to the original value? So I’d have to move 9 places to the right. So then this has to be 10 to the ninth. Another example—again we have to make this a number between 1 and 10. So we have 6.4, and now with the decimal being here, to get back to my original number I would have to move 7 places to the left. So then this part has to be 10 to the -7.
Let’s look at the reversal where we have to take a number in scientific notation and put it back into standard notation. Well it’s just a matter usually of moving decimal points. So here I simply have to take the decimal and move it 9 places to the right. We’ll end up with this number 8,700,000,000 and in this case here, we have to take our decimal point and move 7 places to the left, which in this case will give us .00000064. If students know your basic properties and laws of exponents and you combine that with them knowing how to convert numbers in standard notation to scientific notation and vice versa, then now, that would be equivalent to students now being able to perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.
Of course this is the first statement, the first part of the standard. But before we go on it is important to make this point here. Any time you have numbers that are the result of measurement they’re subject to the rules of significant figures. The basic part of that rule says that solutions obtained from mathematical operations should never be more accurate or precise than the data that was used. Now significant figures do not appear in the Common Core State Standards in math. Significant figures do appear in most state high school science standards. But again, they’re not in the math standards at all. Now significant figures are part of and woven into scientific notation. When you have your significand, every digit there is a significant digit. Now how to handle this, well you’ll just have to consult with your science teachers and your science specialists in your school or your district to address how this topic should be handled at this grade level.
Now if we look at the last part of this first statement, including problems where both decimal and scientific notation are used, now students should not mix the two types of representation. Numbers involved in the computation have to be all of the same type. They either have to be all regular standard notation or all scientific notation. Since the standard calls for doing operations with numbers in scientific notation, that’s what we’ll do. Now there might be some instances where it might be better to do it in standard notation, but students already have a lot of experience and practice with that. So it’s important that they get a lot of experience and practice doing operations with numbers expressed in scientific notation.
So in a case where you have numbers expressed both ways, we need to convert in this case to all scientific notation. So we have to convert this number here. So we convert it to 5.712 times 10 to the sixth. What we should do now is use our commutative property. We need to switch the order here of the 10 to the sixth power and the 3.8. In this way we will have our significands together and our powers of 10 together. So we go ahead and perform the operation on both sides, and we get 21.7056 times 10 to the second power, where we combined the 6 and the -4 in the exponents and got 10 squared.
Now we have a problem. This is incomplete because this number here is not a number between 1 and 10. Now there’s a temptation to go ahead and move the decimal place and try to go directly to the answer. But there’s a good probability that students might get confused and move the wrong way or instead of adding 1 to the exponent, they’ll subtract 1, things like that. So here’s what they could do. Don’t even worry about the 10 to the second power. Pretend it’s not even there. Let’s just worry about converting this to scientific notation. Okay, so we make it a number between 1 and 10. To get back to the original number, we would have to move to the right one place. So then this has to be times 10 to the first power. Now once this is done, we can worry about combining our powers of 10. So then, that would be 10 to the third, so that would be our final solution.
Let’s try a division problem. Here it’s the divisor that is not in scientific notation, so we have to change that, so we have. Okay now wait a minute, what’s the deal with the parentheses? Well it’s very important with division, because if you don’t have the parentheses, the interpretation is that you’re only going to be dividing by the 7.2 and then multiplying by 10 to the negative 5 power. That will give you a totally different result, because what we have to do is divide by all of this, the product of 7.2 and 10 to the -5. So very important that this is done. What we can do to avoid that problem is just express this division problem using a fraction bar. Now what we can do is treat this as two different fractions where we have the significands over here and our powers of 10 over on the right. Perform the operations. Okay we’re not finished, because again we have that same problem. This is not a number between 1 and 10. So let’s convert that, and again, let’s not worry about the 10 to the eighth at all. Let’s just convert our .74 to a number between 1 and 10. Then to get back to the original number, we would have to move 1 place on the left. So that’s how come it’s times 10 to the minus 1 power. Now we can combine our powers. That would be -1 plus 8 is a 7, so that would be times 10 to the seventh power, our final solution.
Let’s try an addition problem. Now they’re already both in scientific notation; well that’s great. So we think that we’re ready to roll, but we’re not. What happens here is this; this would be very analogous to having something like 5y plus 7x. We cannot combine those because they’re not like items. So it’s the same thing here. Notice that over here I have 10 to the fifth. Over here I have 10 to the fourth. So again, I can’t just add them yet. They have to be like items. So for example, here I could only combine them if they were both Xs or both Ys. So let’s say we decide okay, let’s go with 10 to the fourth. Let’s make them both to the fourth power. So on the right-hand side, we’re done. But on the left-hand side, we have a little bit of work to do. Well the 10 to the fifth, we can break the 5 down to 1 plus 4. That’ll give us what we want. We’ll have 10 of the fourth power. So now, over here we need to go ahead and compute this. Just multiply 5.328 times 10 and we get 53.28.
Now what we need to do is use our distributive property and factor out the 10 to the fourth. So we factor that out, and we’re left with (53.28 + 7.2). So we can go ahead and do that addition, get 60.48 times 10 to the fourth. We have the same problem that we did before. We don’t have a number here between 1 and 10. So again, don’t worry about that 10 to the fourth. Pretend it’s not there. Change this to a number between 1 and 10. To get back to the original number, we’d have to move one place to the right, so that’s how come this is times 10 to the first. Now we can finish it out, combine the powers with the tens, and so we get 6.048 times 10 to the fifth.
The next part of the standard deals with using scientific notation and units of appropriate size for measurements of very large or very small quantities. So let’s take an example. Let’s say that your task was to spend $1,000,000,000,000.00 in 20 years. Seems like that would be a lot of fun, but what happens is a lot of times we really don’t have a true idea of the magnitude of these kinds of numbers like 1,000,000,000,000. So let’s express this using other units. Instead of in 20 years, about how much would I have to spend every day? So if we set this up to convert, we know that, well there are 365 days in a year. But since we are dealing with 20 years, we’re going to have some leap years thrown in. So let’s use the more precise measure of 365 1/4, 365.25 days in a year. Now when I do the computation, borrowing from science, the units of measure pretty much wipe each other out. So now we’re left with as our rate dollars per day. So now we have to take this number 1 times 10 to the twelfth, divide that by 365.25 times 20. So you’re dividing by that product actually and when we do that, this is what we get in scientific notation, which translates in regular notation to $136,892,539.40 a day.
That’s a lot of money. But what if we wanted a slightly different perspective. What about per hour? So let’s convert this. We know that there are 24 hours in a day. Again what happens here is that the day as far our units wipes itself out, and for our rate now, we have dollars per hour. Again doing the computation, we’ll take this number here and divide it by 24. This is our solution in scientific notation, which in regular notation would be $5,703,855.80. So rounded off, we’d be spending about 5.7 million dollars an hour. Let’s take it one step further. What about per minute? Well we know that there are 60 minutes in an hour. Again when we do the computation, the hours as far as units will get wiped out so that our rate now is dollars per minute. Now we do the computation and this is our solution in scientific notation. But in regular numbers this comes out to be a little over $95,000 per minute. So now we have a little bit better idea of the magnitude of what it would mean to spend 1 trillion dollars in 20 years. We would have to spend over $95,000 a minute of every day, 24 hours a day for 20 years.
Now let’s address the last statement in this standard—Interpret scientific notation that has been generated by technology. In all likelihood the technology that the students will be dealing with will be calculators. Now there are some aspects of calculators and scientific notation that might surprise students so they need to be aware of this. First of all, in many calculators, the result may appear in scientific notation, even though the numbers were input in just regular notation. Also scientific notation in calculators is represented differently than how it is typically written. And last but not least, the results in calculators can be programmed to appear in scientific notation even if the numbers are input in standard notation.
So let’s take a look at what this looks like. Let’s say we’re doing this multiplication problem; the calculator is programmed to just have regular standard notation, and we’ve input them in that manner, and then we perform the operation. Look what happens. What in the world is this? Well this is what happens; the calculators are programmed in such a way where if there are just too many digits, it’s going to automatically convert it over to scientific notation. So the first part of the number is a number between 1 and 10. But we have this E. Well, the way they calculator does it, basically the 14 is the exponent. So here what we have is 8.34015 times 10 to the fourteenth power. That’s what this notation is really indicating.
Now let’s say that we have this next problem. It’s just a plain old multiplication problem, and we multiply it out. We get the typical solution in standard notation. Now we can program the calculator to actually give us the answer in scientific notation even though we input the numbers in regular notation. Now it will depend on the calculator, but it should probably have some type of settings where you have the option like here to go from normal to scientific notation. So now from here on out whatever we input, the solution will be expressed in scientific notation. So now if we take that same problem and get our solution, notice that now it’s the same solution but it is expressed in scientific notation. Instead of 101,652 the answer now appears as 1.01652 times 10 to the fifth power. So this should make it a little bit more clear, some of the things that kids have to contend with when it comes to using calculators and using technology in scientific notation.
One last little note as far as using technology and scientific notation in particular; it’s very important that students input the numbers correctly. For example this problem here, notice that they are in parentheses, and that’s how the students should input them. If this problem was multiplication instead of division, the students could get away with not putting parentheses like here because it’s multiplication, so it’s all commutative. So they would get the same answer either way. In fact, if this were addition or subtraction they would luck out and get the right answer because the calculator is programmed with the order of operations. So it would do this multiplication before it would do the adding or the subtracting. But they wouldn’t be so lucky if it’s division, because again what’s going to happen is when it’s expressed this way without the parentheses, the calculator is going to divide by 7.2 and then take that result and multiply by 10 to the -5 instead of dividing the result of this here by all of this here, by that product. So again be careful. Make sure that the students input the scientific notation correctly, especially in the case of division; that divisor needs to be in parentheses.
As you’ve seen this standard is going to take some time because there’s not only several parts to the standard, but also there’s a lot of prerequisite knowledge that’s involved that teachers have to ensure the students know well.