MathNotations

Virtual Studio has all the required live media pro...

2009-11-07T02:21:32.216-05:00

Virtual Studio has all the required live media processing facilities to ensure successful completion of projects on time and within budgets. Its facilities includes: Talk shows hosting, using virtual sets; live & on-demand web-cast streaming studio; recording by voiceover artists; production facility for audio video presentations; option for live call outs or live phone-ins; radio shows; panel discussions; training-educational videos with software demos audio-video processing, optimization & encoding facility. All this with 100% power backup; multiple connectivity for live web-casts; online project management system.

it's a really nice blog thanks for add my comment....

2009-11-06T15:48:11.136-05:00

it's a really nice blog thanks for add my comment...

Las Vegas Card Game is the only place where you can get card games, online card games, betting games, online card games news, Here you can play black jack 21, 3 card poker, baccarat, seven card stud, slots and many popular games, So if you want to play with fun then come with us and let’s play card games with lasvegascardgame.com.Card Games News

Thank you so much for doing this. You helped my s...

2009-11-06T02:48:42.128-05:00

Thank you so much for doing this. You helped my sister and I not only finish, but understand the problem. Thank you for devoting your time to this project!

I understand. Examinations measure proficiency and...

2009-11-05T11:11:24.776-05:00

I understand. Examinations measure proficiency and understanding. I simply think that examination questions should be related to mathematical history and research. There are short steps from "What squares are sums of two squares?" to "What numbers are sums of two squares?" to "What numbers are sums of three squares?" to "What numbers are sums of four squares?" to Waring's problem.

You might want to discuss with students the distinction between G(n) and g(n) in Waring's problem; similarly, discuss Vinogradov's result on sums of three primes.

Eric-- Nice to be talking about math again, number...

2009-11-05T08:26:21.157-05:00

Eric--
Nice to be talking about math again, number theory in particular.

I've always believed that an understanding of the proof that odd primes are uniquely representable as a sum of squares iff they are of the form 4n+1 is a rite of passage for young mathematicians. Now I'm really curious about Zagier's one-sentence proof of this theorem! Must be a very complex sentence!

Back to the contest...
I specifically chose a fairly straightforward derivation for the test to give all students a chance to at least engage the question. The fact that so many struggled confirms my belief that we should spend a more time early on getting our students to formulate explanations of their reasoning and putting pencil to paper. Writing one's ideas in a coherent manner takes years of practice, never mind the deeper understanding of big mathematical ideas.

Let's go further with this. Fermat made an observ...

2009-11-04T21:57:57.390-05:00

Let's go further with this. Fermat made an observation about positive integers that are the sums of two squares. The proofs are a bit complicated; you can look them up on Wikipedia or Mathworld.

What numbers are sums of two squares (or 1 square--0 counts).

0 = 0²
1 = 1²
2 = 1² + 1²
3: no
4 = 2²
5 = 1² + 2²
6: no
7: no
8 = 2² + 2²
9 = 3²
10 = 1² + 3²
11: no
12: no
13 = 2² + 3²
16 = 4²
17 = 1² + 4²
18 = 3² + 3²
19: no
20 = 2² + 4²

Now, look at the primes in the list. 2, 5, 13, and 17 are sums, while 3, 7, 11, and 19 are not. Do your students notice a distinction?

Now look at the composites. 10 is in the list; its factors 2 and 5 are in also. In fact, if m and n are such sums then their product is. Why? It helps to be familiar with complex numbers.

If m = a² + b², then m = (a+bi) (a-bi).

If n = c² + d², then n = (c+di) (c-di).

mn = (a+bi) (a-bi) (c+di) (c-di)

= (a+bi) (c+di) (a-bi) (c-di)

If (a+bi)(c+di) = (e+fi), then (a-bi)(c-di) = (e-fi), and

mn = (e+fi) (e-fi) = e² + f².

So, we can guess that 2, and odd primes of the form 4n + 1 appear in the list; if that's true then all their products do. Can 7 or 11 appear as a factor of a number in the list? Of course. 7² = 49 appears. And that leads to Fermat's observation:

A positive integer is the sum of two squares if every prime factor with the form 4k+3 appears to an even power.

школьницы ебутся http://free-3x.com/ порно малоле...

2009-11-04T10:01:10.315-05:00

школьницы ебутся http://free-3x.com/ порно малолетки молодые молоденькие free-3x.com/ порно молодых мальчиков [url=http://free-3x.com/]free-3x.com[/url]

it's a really nice blog thanks for add my comment....

2009-11-02T19:22:04.580-05:00

it's a really nice blog thanks for add my comment...

Welcome to Thebettingonline. We are here for to be the most online betting. For read our online casino gambling and betting guide click on www.thebettingonline.com. We also suggest you types of betting action Opening bets, Calling, Rising, Checking, it will help you to win the Bet. The Betting Online

Mathematics of Linear Algebra

2009-10-10T05:09:46.328-04:00

Mathematics of Linear Algebra

I find that students with more experience with thi...

2009-10-05T15:04:56.380-04:00

I find that students with more experience with this type of problem tend to better handle the "organized" part of "make an organized list". Since I don't think there are really any high school level skills that enter into this problem, and the hard part (IMO) is making the organized list of matched pairs, I think experience will win out over school level here.

It's always comforting when we agree on an answer!...

2009-10-05T13:04:54.749-04:00

It's always comforting when we agree on an answer! I felt pretty sure of the result but I believe in hedging my bets, particularly since i was doing this mentally around 6:30 AM!

It would be interesting to "kid-watch" on this one, although the younger set may need clarification or one example. Do you think high schoolers' approaches would be more sophisticated than middle schoolers or about the same?

I started by considering all pairs in which "1" is the least digit. "1" could not be paired with any primes so that left "14" (no good), "16" (that paired with "23"), "18" (paired with "24") and "19" (no good).

Each pairing like "16" with "23" leads to 2x2x2 = 8 outcomes.

Then I considered pairs in which "2" is the least possible digit, i.e., no "1's." This produced "23" (already covered above), "24" (already covered), "26" (paired with "34") and "29" (pair with "36").

This leads to 2 more sets of eight results.

I'll stop the process here to allow students to check these and complete them.

Since we think in similar ways, everything you described in your comment applies to me as well!

I came up with 40 also. I ended up just enumerati...

2009-10-05T09:35:00.500-04:00

I came up with 40 also. I ended up just enumerating a list of the pairs of digits whose product equals the product of another pair of digits. I didn't feel like I had a great way of thinking about that (though some rough ideas about primes and squares and cubes emerged) so it felt kinda brute-force-ish to me. Of course the list was reasonably short in the end, so that was ok. But like you, I wasn't terribly confident of my answer in the end. (At least I suspect you weren't since you invited folks to disagree.)

Mathmom-- Let me know if you try it with your sons...

2009-10-03T06:45:31.040-04:00

Mathmom--
Let me know if you try it with your sons or other middle schoolers.

I believe that some high schoolers will struggle with the "tricky bits" or, as I would like to say, "the devil is in the details." There is a lot to process here. The student taking physics will definitely be more comfortable with the "rate" approach since they do many motion problems and the units issue should be trivial for them.

Going back to your proportion method, I believe that the units used, "miles over seconds" or "mi/sec" is a powerful way to introduce rate. I believe math teachers need to stress the importance of units as much as science teachers do. Formulas such as R = D/T emerge naturally from such discussions. Once the student buys into (mi/hr) x hr = mi, for example, formulas like D = RT express themselves!

The mathematical basis for dimensional analysis however is noot always fully explained in science classes. If RATE is constant, then the middle schooler needs to see that D = RT express the idea that Distance is directly proportional to Time while Rate is the constant of proportionality. The 1st year algebra student needs to make the connection that D = RT is just a special case of y = mx, a direct variation function!

Let me extend this to "varies directly as the square of". Science and math students later on will learn that y = ax^2 expresses this relationship and they will graph it and solve applied applied problems. However, how many students look at a formula like A = πr^2 and realize that this is a special case of direct square variation where π is the constant of proportionality! Sometimes we have to model "making connections"...

My older middle schoolers (~7th - 8th grade level,...

2009-10-02T23:32:49.057-04:00

My older middle schoolers (~7th - 8th grade level, either in a second year of pre-algebra, or starting Algebra 1) are pretty good about recognizing ratio problems. They also know the D=RT relationship. But I'm guessing that there are enough "tricky bits" (figuring out how far each train has to travel, which is stated in an indirect way, translating between hours and seconds) that I'd only give them a 50% chance of solving this one correctly on the first try, without any hand-holding. Maybe I'll try it out with them sometime -- if I get to it, I'll let you know.

I wouldn't give this to my younger middle schoolers (5th-6th grade level, just starting pre-algebra) yet. Sometimes I do a lesson on distance/rate/time problems and if I do, I'll throw this one in to see what happens.

Three of clubs-- Thank you for your insightful and...

2009-10-02T10:50:40.357-04:00

Three of clubs--
Thank you for your insightful and supportive comments. Considering that I wrote this problem for average middle schoolers and that algebra and formulas are not required, it's sad to think that some high schoolers would give up on this.

Mathmom--
I absolutely agree that this this should not be considered a difficult problem, never mind a contest question! Your proportion model is exceptionally important for middle schoolers. Do you believe that most middle schoolers would be able to solve it this way or solve it at all?

Is this a "contest problem"? No, IMO, this is a s...

2009-10-01T21:19:18.121-04:00

Is this a "contest problem"? No, IMO, this is a straightforward problem. I think our pre-algebra kids who have worked with ratios and can solve a simple equation with cross-multiplying could solve this problem, though I think the conversion from hours to seconds would mess up a lot of them. I would hope they would do something like:

100 miles          9/20 mile
---------      =      ----------
3600 seconds      x seconds

And solve for x. This doesn't require remembering that D=RT or any other version of the formula, and I always prefer methods that don't require kids to remember a formula

caught my mistake...thanks I hadn't really focuse...

2009-10-01T08:35:26.853-04:00

caught my mistake...thanks

I hadn't really focused on the question of appropriateness, being too caught up in trying to teach each student at his level and making sure there was something constructive for each member of the class to be doing.

It is, though, the most seminal question. You've certainly thought through more of the "should and ought" than I have. Some of your problem depends on formal teaching, but the part I find most interesting is informal understanding ... the stuff which we expect people to get, except, they don't.

At what age do we expect people to be able to know that you need to add the length of the tunnel to the length of the train? Or, better still, reverse their thinking, and have them going in the same direction?

When confronted with a class what should you expect and what should you assume? When I first started teaching high school math, I assumed my students knew the material from the previous term. That expectation was despite finding college juniors and seniors in advanced math classes still trying to turn (x+1)^2 into x^2+1. My expectations are running kind of low right now.

With regard to your problem, incidentally, I'm taking a year off from teaching, or I would field test it for you. My own guess though, fewer high school seniors could solve it than identify the Bill of Rights. So, appropriate? or, desirable?

three... I calculated A's time to be 25.6 seconds,...

2009-09-30T15:07:52.810-04:00

three...
I calculated A's time to be 25.6 seconds, not 19.2. Did you remember to divide A's distance by 75 mi/hr rather than 100 mi/hr?

I'm also interested in your thoughts about the appropriateness of this not very difficult problem for 6th or 7th graders(or 4th & 5th in Singapore!!)??

I tried to write an "RT = D" problem which did not require algebraic methods other than manipulating the basic formula to find time.

Am I wrong in calculating that train B has to trav...

2009-09-30T12:44:02.788-04:00

Am I wrong in calculating that train B has to travel 9/20 of a mile which it does in 9/2000 of an hour or 16.2 seconds?

Train A travels 8/15 of a mile in 19.2 seconds. So isn't the difference just 3 seconds?

You got it, Emily!

2009-09-30T00:28:26.610-04:00

You got it, Emily!

Oops! For the challenge 1, I thought it was 10^100...

2009-09-29T23:34:21.946-04:00

Oops!
For the challenge 1, I thought it was 10^1000 - 1...

Okay, so I think the correct answer to challenge 1 is,
3000 9's.

Thanks.

Emily, I agree with your 2nd and 3rd answers but c...

2009-09-29T05:49:43.334-04:00

Emily,
I agree with your 2nd and 3rd answers but consider the following pattern:

1000^1 -1 = 999 (three 9's)
1000^2 - 1 = 999999 (six 9's)
etc....

There are many other ways to look at #1 using laws of exponents but the best strategy for these types is usually to "Make the problem simpler and look for a pattern!"

Keep commenting!

Following are my answers to the questions. Challe...

2009-09-28T23:47:04.150-04:00

Following are my answers to the questions.

Challenge 1: 1000 digits
Challenge 2: 15
challenge 3: one-third mile

About problem 2: You should also remind your stud...

2009-09-24T23:35:47.731-04:00

About problem 2:

You should also remind your students of a famous formula in which s=(a + b + c)/2 appears. Why would that formula tend to give evidence for the solution of problem 2? You should also ask your students how practical that formula is.

As a point of style, I always try to use capital letters only for the vertices of triangles and for their angles. I reserve small letters for their sides.

Jmoser-- I admire your attempt to show them that a...

2009-09-24T15:39:58.375-04:00

Jmoser--
I admire your attempt to show them that all three forms are equivalent. I've done the same with some success but we always returned to the pie chart or some other visual model.