This is Common Core State Standards Support Video for Grades 9 through 12 in math. This particular standard is going to be F.BF.1a. Actually, there are two parts. The first one is the main standard: write a function that describes a relationship between two quantities. But the more specific one that we’ll talk about today is determine an explicit expression, a recursive process, or steps for calculation from a context.
Now the key here is that last phrase, from a context. So, based on the context, you determine the steps for the calculation, a recursive process, and then an explicit expression for it. Now it is a sequential process for determining the calculations that lead to a recursive process, which then, in turn, establishes the patterns that the students would need to establish some type of function or some type of formula. Now notice that the standard that we’re covering today is not the same as another standard that’s in that same cluster. That second one in that same cluster states that you should write arithmetic and geometric sequences, both recursively and with an explicit formula, and use them to model situations and translate between the two forms. But the standard that we’re covering today, it does establish a foundation for the standard that deals with arithmetic and geometric sequences. But again, there is a difference.
Now based on the idea “from a context,” let’s go ahead and start with a first example, doubling your money. I’m sure a lot of you all have seen this particular problem. You start off, in this case, the first day of September, and you’re given two choices. Which one of these do you want and why? Would you rather have a million dollars, or would you rather start with 1 penny today, on the first of September, and you’ll get the amount of money that would result from doubling the amount every day until the last day of the month. So you would get the money that would result on September 30. So again, which would you prefer?
So that’s the problem for the students. Now our first option would be to do a recursive process. The steps for the calculation here would actually be fairly simple because you’re starting off with a penny, and then you’re going to double that every day. So I start off with a penny. Then the first day, I double it. I’m going to have 2 cents, and then I double again. I’m going to have 4 cents and so forth. And that’s going to get pretty tedious because they’re going have to do that all the way down to September 30. So that’s going to be a lot of work, but that’s what’s involved with a recursive process. You have to figure out the next one by doing something to the one before.
Now the silver lining is that doing the calculations gives the student some kind of pattern that can be used. We do have the technology now to make the calculations a lot easier and where it’s not a whole lot of drudgery for the kids. Now there are some calculators that once you perform a calculation, in this case, we’re going to take a penny and double it, and there’s our answer of 2 pennies. And then I want to take that answer and double it again. And now that’s 4 cents. But what happens with this calculator is that now it knows that I want to keep doing that same calculation over and over. So I could actually do this very quickly using my calculator. Well, let’s see. Let me go back and make sure that I am at the right place. I start off with a penny, so that would be 1. The second day I’d have 2 cents. The third day I’ve got 4, the fourth day I’ve got 8. So that would give me, I’m starting with 4, so now we go 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30. Viola! That’s $5,368,709.12.
So we’ve done this recursively, but we let the calculator do a lot of the work, and we did it very quickly. We already know our answer and everything, but what if we didn’t have a calculator that did all that work for us? So we need to develop some type of explicit function, something that will get us there directly. Let’s see, what did we have to do to get the amounts? Let’s see, to get the amount, we started off, and we would take a penny, and we doubled it. So I know the computation is going to involve doubling all the time, and if you’re doubling over and over, that means you’re multiplying by powers of 2. So that’s going to be our unknown, and we started off with a penny.
Now let’s try to see if it works. If we substitute a 1 for the x, that would be 2 to the 1. Lets see, that’s two times. Well, wait a minute. It doesn’t quite match up because for the first day, I get 2 cents. So I’m going to have to back it up a step. Let’s see, how would I do that? Okay, I just need to back it up here. Now let’s see if it works. If I plug in a 1, that’s 0. Anything to the 0 power is 1, so I would start with a penny. If I substitute a 2, that would be 2 to the first. So that’s 2 times a penny, that’s 2 pennies, so yes, it does check out. So this would be my explicit formula, and if you were to take your calculator and take a penny, and then multiply that times 2 to the 30 minus 1 power, because it’s 30 days in September, and you get 2 to the 29th times a penny, and you would get the 5 million plus answer. So this takes care of the first example. Again, recursively, using some technology to help us out with that recursive process, and then using the pattern to find a function for it.
Okay, let’s take a second example, earning interest. You deposit $5,000 in a bank account. You’re going to earn 6% on your money, and that’s APR, which stands for annual percentage rate. Now it’s compounded quarterly, so the task is to find out how much money you’re going to have at the end of 5 years. So with no other assistance, I’d have to use a recursive process. So I’m going to start with $5,000, and I have to multiply that by, let me see. Now here’s the problem. You’re earning 6%, but that’s for the year. It’s compounded quarterly, so you’re not going to earn 6% every quarter. You’re going to earn 1/4 of that. So I have to take that and divide by 4. So that’s basically 1 1/2%, which as a decimal would be .015.
Now there’s a faster way to do this. Instead of multiplying by .015, that will only give us the actual interest that you’ll earn; but if you multiply by 1.015, then that will give you the answer directly. It’ll give you the $5,000, plus the 1 1/2% interest that you would have earned. So if I were to multiply the 5,000 times 1.015, I get $5,075. Now the problem is, I’m going to have to then take the $5,075 and multiply it. Let’s see, this was after 3 months, quarterly. So after 6 months, a second time, now I’m going to have to do the same thing again. Let’s see, compounded quarterly for 5 years, that’s 20 times that it’s going to be compounded, so I would have to do this 20 times. But I have to take whatever the current value is and multiply it by 1.015 to get the next value.
So this is going to be pretty tedious. But I think I can use my calculator again. Now I’ve done it the first time, but let me repeat the process. I start off with $5,000, and I’m going to multiply that by 1.015 and get my solution. Then, I want to do it again, so I’m going to multiply by 1.015. Now my calculator knows that that’s what I want to keep doing over and over. So after 6 months, I’ve got $5,151.13 if you round it up. I’m going to compound this 20 times. Let’s see, that’s already been twice, so that’s 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. So, at the end of 5 years, 20 compoundings, compounded four times a year, we will end up with $6,734.28 rounded.
Now that was a lot of work. So it sure would help if we had a formula for this. Now your students may not necessarily have to develop the formula. This would be kind of a tough one to develop, although they could, given the correct information and enough time and information with the pattern. But they could also do some research, and there is a compound interest formula, which states that the amount that I would currently have would be whatever your original amount, the principal was, times 1 plus the interest rate as a decimal, divided by n, which is the number of times that it’s compounded every year. And then, you’ll want to know, well, how many times is that going to happen? Well, you would take the number of times it happens per year, and multiply it times t, which would be the time in years. So now, if we wanted to get this directly for our problem, we would substitute $5,000 for the principal, 1 plus, the interest rate was 6%. So we put .06. It’s compounded four times a year, and then it’s compounded four times a year for 5 years. So basically, we would have 5,000 times parenthesis 1 plus .015, raised to the 20th power. And that would give us our solution, and we would get the same answer.
Let’s try a third example. Let’s not neglect geometry. We seem to go primarily with the arithmetic stuff and algebraic stuff. So let’s say the students had this problem: find the sum of the interior angles of a polygon with 11 sides. Okay, well, let’s start just drawing some pictures, the interior angles, all right. Initially, we would have a triangle with three sides, and we know the interior angles of a triangle add up to be 180. Now if we had a quadrilateral of some type, let’s say the students, they just don’t know, but you can use this fact right here, and work from there. So if I wanted to draw a diagonal, note that a quadrilateral splits up into two triangles. So I know that these three angles would all add up to be 180. These three would add up to be 180. So that’s a total of 360. Let’s say we had a five-sided figure, and let’s try doing something like we did before. If we draw a diagonal from one vertex, a pentagon, a five-sided figure, is going to have three triangles. So again, those angles would add up to be 180, 180, and 180.
Now notice that I would have all the angles for the bigger pentagon because, here’s 1 angle, 1 angle, 1 angle for the first one. And then, for the second triangle, I’ve got this, which completes this bigger angle here. And then I’ve got this angle here, this angle there, this angle here, this one here, and then this other one for that second triangle, which completes this bigger angle there. So yes, it checks out. And, let’s see 180, that’s 3 times, that’s 540. So a five-sided figure has 540 degrees total. So we’re going to have to keep doing this until we hit one with 11 sides, which would be a lot of work. So we started off with a 180 for a triangle. And then we added another 180 to get us 360 for the quadrilateral. And then we added 180 again for the pentagon. So that’s 540, so that’s five sides, 6, 7, 8, 9, 10, 11. So an 11-sided polygon has 1,620 degrees total for the sum of the interior angles.
So we used technology to help us, and that sure was a big help as far as all the calculations. So what’s left is, and this is a big part of what you want the kids to do, is to use those patterns and all the information to do some deeper thinking and come up with some type of explicit function for this process. If we go back to what we did before, we started off with 180, and then we had 360. Okay, so that means that we added 180 to get to 360. And then we added another 180 to the 360 to get the 540, and so forth. Now the pattern here is, well, it was all based on triangles. And if you connect back to the polygon, we’re also talking about the number of sides.
So how do we combine that information together? Let’s see. The triangle had three sides, and there was one triangle. For the pentagon, sorry, for the quadrilateral, it had four sides and there were two triangles formed. For the pentagon, it had five sides, and we had three triangles formed. Now look at the relationship; look at what’s happening. It looks like when we draw the diagonal, the diagonals from one of the vertices, it looks like we’re going to end up with the number triangles being two less than the number of sides. Okay, so to get the sum of the interior angles, it looks like we take 180 and multiply but not times the number of sides. If n was the number of sides, it would actually be two less than the number of sides. That would give us the number of triangles. So if we were to do this directly, if we had an 11-sided figure, that would be nine. So we would have to multiply nine times 180, which is the 1,620 degrees that we got earlier from the recursive process.
So this wraps up this particular standard. But again, the key is you’re always working from a context. You do a recursive process. If there’s a whole lot of calculation involved, you might want to incorporate some technology to help yourself out with that. And then, based on whatever patterns that you end up with, then comes the deeper thinking to establish some kind of explicit formula or function that will help out. The nice thing is that this is what will happen after doing all that work with the recursive process, is that your kids will realize the utility of a function, that it does enable you to find a specific value without having to find all of the previous values.