Looks good > On 12/30/15, jasmine wrote: >> Please help with finishing pattern, >> >> 2, 7, 26, 101, 400, ?
give a formula that tells you what the nth term in the series is, as a function of n. Right now you have the pattern in what we call recursive form, meaning that in order to find a given term you do things to prior terms. But what if you do not know the prior terms? What I want is a formula for the nth term where the only variable is n. This is sometimes referred to as the closed form of the pattern.
The recursive form would be written as follows, where a(n) means the nth term of the series (not multiplication):
a(n+1) = 4*[a(n)]-(n-1)
what we now want is a formula that looks something like this: a(n) = some function containing only n's and no a(n)'s.
Because if I were to ask what is the 11th term in the series, or the 157th term in the series is, you could not easily answer it without a TON of work, computing each term to find out what the desired term is. IOW to find the 157th term you would first have to find the first 156 terms. With the closed form you simply plug 157 in for n in your closed form.
Here is a simple example:
2,4,8,16,....
Pattern is previous term times 2.
Recursive form is: a(n+1) = 2*a(n)
But again, if I ask what the 136th term is you first have to compute the first 135 terns to use this form.
Closed form is: a(n) = 2^n
With this I can instantly see that the 136th term is 2^136, without having to compute any of the first 135 terms.
One final caution: while the solution given by the responder is correct, it is NOT a UNIQUE solution. There are literally (used in the correct sense) an infinite number of solutions, it is just that the one we came up with is the simplest.
Think of the five terms we were given as 5 points on a plane (the x coordinate is n, y coordinate is the nth term). So now you have 5 points on a coordinate plane. There are an infinite number of ways you can draw a line through those 5 points. Each one when extended would give different answers to what the 6th, 7th, etc., terms are.
Another simple example. Say the series is 1,2,4
Obviously the next term is 8, right?
Nope it is 7. Why? Because the rule is NOT double the prior term: a(n+1) = 2*a(n).
The actual rule I had in mind was first add 1, then add 2, then add 3, etc. Or: a(n+1) = a(n) + n
BUT, in the absence of any other information providing more specificity, both answers would clearly be correct.
Let...See MoreYikes, your answer was spot on (I too was a math major).
Which leads me to ask:
Why is it that a High School Chemistry teacher can comfortably teach General chemistry to students, a High School Bio teacher can teach AP Bio to students, but when it comes to helping a kid out in AP Calculus, many math teachers run away?
Let's face it: As math majors, we took two semesters of Calculus in our Freshman year in college, and Calculus III and Diffy-Qs (typically) in our second year. This is clearly analogous to what Bio and Chemistry teachers do during their earlycollege training.
So, Calculus is FIRST YEAR college material. So why the aversion ????
Although I am a pretty self-confident guy, I appreciate the affirmation. > Which leads me to ask: > > Why is it that a High School Chemistry teacher can > comfortably teach General chemistry to students, a High > School Bio teacher can teach AP Bio to students, but when it > comes to helping a kid out in AP Calculus, many math > teachers run away? > > Let's face it: As math majors, we took two semesters of > Calculus in our Freshman year in college, and Calculus III > and Diffy-Qs (typically) in our second year. This is clearly > analogous to what Bio and Chemistry teachers do during their > earlycollege training. > > So, Calculus is FIRST YEAR college material. So why the > aversion ????
weren't math majors? Heck by the time I was in college, having taken BC calculus in HS, I was taking Calc 3 and Linear Algebra in my first freshman semester. But then I come across math teachers who are seemingly clueless about math. Or even worse, administrators who are clueless about math who were math when they taught.
Like the former math teacher AP who didn't understand what a ring was when I told him that I used ring and field theory concepts to explain to my kids order of operations. It wasn't that he didn't see how those concepts could be used to teach O of O, he didn't know what a ring or a field were, or even seem to recognize the terms.
Personally I love calculus and love to help kids with it even now that I'm retired (from teaching 7th and 8th grade - so I wasn't exactly refreshing my calculus memory there). I used to go to a coffee shop after school every day and would sometimes help students who attended the local community college with their math when they asked. I even had a former student (I had her in 8th grade) come back and ask for help years later when she was in college taking calculus. And now I go to our local senior center and there are some college students who work there part-- time and I suspect I will be helping some of them, as well. I look at it as if I have retired from teaching but not educating.
Are you familiar at all with the work of Dr. Wu, at Berkeley? He is a math professor with a great deal of interest in math teacher preparation, and while his emphasis is on the early years (where I think we would both agree we are woefully weak) he has a very clear, correct- mathematics orientation and I got a lot of benefit reading through some of his stuff - even the elementary stuff, as he is so clear and focused.
If you're interested in checking him out, this is the address. math.berkeley.edu/~wu/
I found difficulty in explaining the answer of this question to the pupil write the smallest and largest numbers that will give the following: 8460000 when rounded to the nearest ten thousand would please help
> I found difficulty in explaining the answer of this > question to the pupil > write the smallest and largest numbers that will give the > following: > 8460000 when rounded to the nearest ten thousand > would please help
what a useless exercise. First of all it is horribly worded. I was a math major in college and have worked in a math field in business for 20 years and taught math for 15 years and I had to read it twice to see what they were after.
Secondly, it is mathematically incoherent, as there is no greatest number that rounds to that nor is there a smallest number that rounds to that There is a largest and smallest WHOLE number that rounds to that, but without making that restriction, which the problem doesn't, it is a problem without an answer. Except, you are going to frustrate the dickens out of the brightest kids who assume that if you ask for an answer there is one, but who are smart enough to see this problem does not have an answer.
Try a simpler version - what is the biggest number that, rounded to the nearest 10, yields 100. well, 105 rounds to 110, so that is too big. 104.9 would work - but 104.9 is bigger and still rounds to 100. But 104.99 is even bigger and still rounds to 100. In fact, any number you can give me that rounds to 100 can be made bigger, while still rounding to 100 - just add another decimal place and put a 9 in it.
It is like asking "what is the biggest number that is less than 5". There is no answer.
Now lets say that you add the restriction and ask for the largest and smallest WHOLE numbers (proper wording is why it is important that math amateurs not make up test). Then you have a problem that at least has an answer questions, but is still of very questionable mathematical or pedagogical value.
So if a kid says there isn't any answer and can come up with any kind of semblance of an explanation indicating that they knows why I would be giving them full credit, an attaboy/girl, and marking them as someone to watch and expect great thing from. Unfortunately, in most classrooms I think they get marked wrong because it isn't the answer in the book. Which is why we need teachers, not just books and computers.
BUT - only if the teachers really know what they are doing. Which isn't always the case in math.
harpazoOn 10/28/15, harpazo wrote: > On 10/26/15, HI wrote: >> can u show me all the steps for this bedmas > question: 2 raised >> to the power of 2[24 divided by 4(12 > divided by 3)] > > 2[24/4(12/3)] > > 2[24/4(4)] > > 2[24/16] > > [24/8] = 3 > > Answer: 3
promote that nonsense. It is a terrible way to "remember" the order of operations, because too many students fail to remember the left to right rule because it is not embodied within the acronym. A memory "aid" that leaves out a critical thing that must be remembered, i.e.,the left to right rule, is pretty useless. And in the case of the OP they are probably using a variant of words to describe the exact same thing as the one you cited.
A much better approach is to show the kids WHY the order of operations is the way it is. The reason that addition and subtraction have the same priority, e.g., is that subtraction is not a"real" operation, it is the inverse of the real operation, addition. without any left to right stuff Similarly with division being the inverse of the "real" operation. That is also why the commutative property worked for both addition and multiplication (the "real" operations) and does not work for subtraction and division.
But if they are too young or you feel you have to do SOMETHING to help them remember, there is a better way as well. Instead of pemdas, if you just change the md to dm and the as to sa, getting pedmsa you will find that this will actually work without any need for the left to right rule. And the reason it works is that resolving inverse operation before real operations can be proven to work perfectly.
Make up an example and try it.
When students ask questions here we should still retain our role as teachers and try to teach them , not just give them the answers. Yes, we are all good at math and can do the problems the students ask about. But the goal is to help them be able to do the problems, not show off the fact that we can do the problem.
Let b = the number of blue birds bought, g = the number of green birds bought, and y = the number of yellow birds bought. Now translate each of the two conditions that the collector a) spends 100 dollars and b) buys 100 birds into algebraic equations in terms of b, g, and y.
What are your two equations? A question about the problem: I assume you can't buy a fraction of a bird, but can you spend a fraction of a dollar? Can I buy 1 yellow bird for a nickel? An observation about my question about the problem: It doesn't matter. If you're gonna spend 100 bucks, no fractions of a dollar, and the blue and green birds won't create any fractions of a dollar, there's no way the spending on the remaining yellow birds could involve a fraction of a dollar.
harpazoOn 5/19/15, shawn wrote: > i have not been to 8th grade but i am going on to > highschool ive missed a year of education please help me > with math
On 12/30/15, jasmine wrote: > Please help with finishing pattern, > > 2, 7, 26, 101, 400, ?