As I await the blizzard of '09 here in the Northeast, some updates...
RSS FEED ISSUES FOR MATHNOTATIONS
Any problems getting my RSS feed via Google Reader or other aggregators? Blogger, which recently purchased Feedburner, required all users of Feedburner to transfer their feed. This may have caused a temporary disruption of the feed for MathNotations. I updated the feed address for redirecting my readers as of 2-28-09 and I inserted a new 'gadget' into the sidebar allowing for re-subscribing if needed. I noticed that the number of subscribers was cut in half when I transferred over so I'm hoping it will correct itself in a couple of days. Please email me at dmarain at gmail dot com to let me know if you're having any problems getting the feed. Either way, let me know. Apparently other Bloggers are having similar problems with their feed.
SAT Day is March 14th. How appropriate. May all of your students score at least 250π on the math section!
An SAT Ratio Problem
Here's a common SAT type of problem (above-average difficulty) which all students should know how to approach. I've chosen this because it demonstrates different mathematical approaches and test-taking strategies:
In Mr. Jonas' AP Stat class, the number of left-handed students is three times the number of right-handed students. If one-fourth of the lefties and one-third of the righties in the class play an instrument, what fractional part of the class plays an instrument?
(A) 7/12 (B) 5/16 (C) 11/36 (D) 5/18 (E) 13/48
Notes, Comments, Solutions, Strategies,...
(a) To make the wording more convoluted and difficult for students, I could have used the noxious "three times as many as" phrase. I chose to avoid this as it would weaken the reliability of this question IMO. Your thoughts?
(b) How would you categorize this problem? Prealgebra as it can be handled by ratio considerations? Algebra since it can be solved algebraically? Are questions like these typically included in middle school texts here in the US? Singapore texts in their primary materials?
(c) In your opinion, would most juniors in HS approach this algebraically or would they use the most common SAT strategy, 'Plug in'?? Is this question so obvious that it's rhetorical! More importantly, do most students really know how to use this method effectively? How many students know how to organize the information in the problem using a tree model? A Punnett Square model?
(d) How would teachers of Singapore materials explain this question? What model would their students be encouraged to use?
(e) What % of your students would have the level of ratio sense to do this:
(1/4)⋅(3/4) + (1/3)⋅(1/4) = 13/48.
As an aside, how many would attempt the arithmetic without a calculator!
(f) What is your opinion of the distractors? Could the intuitive student eliminate some answer choices quickly, narrowing the options to one or two and making an educated guess? Would this question make a better grid-in (student-constructed response)?
(g) If you felt that the proportion of left-handed students in this problem was highly abnormal, I can only reply that since I'm left-handed, I must be in my right mind!
I would like to share some other methods I've seen successful students use as well as delve further into the conceptual and skill foundations for this problem but I'll stop here for now. If you would like me to go further with this, let me know by commenting or emailing me directly.
Sunday, March 1, 2009
Updates: RSS Feed, Preparing for SATs on Pi Day, A Ratio Problem,...
Monday, February 23, 2009
Odds and 'Even' More Updates Week of 2-23-09
World Math Day '09
On March 4th, there will be the biggest online math contest on the planet! Registration and practice is now open for this event. Unite with students and schools from around the world to set a new world record! The Challenge - to correctly answer more than 182,445,169 questions in 48 hours.
Math Teachers at Play #1 - An Alternative Carnival of Math for PreK-12
Congratulations and Best Wishes to Denise at let's Play Math for this ambitious undertaking. Her inaugural edition is now up and running and I rate it a 10++++. She has links to nearly 20 math blogs and does justice to every one of them. Denise will need our support to keep this going.
Do any of you recall that I originally proposed a similar split in the 10th Edition of the Carnival of Math? It took Denise's courage and talent to make this a reality.
Dynamath - A New Student Math Blog
The author, currently a high school student and a former student of mine, goes by the moniker, Blabbermath. She introduced me to her blog a few days ago and I emailed her my comments and thoughts about her first few posts. She exemplifies everything I always looked for in that special student -- curiosity, insight, fearlessness in following her thought processes as far as they will take her and she is a darn good problem poser not to mention a talented writer. Try some of her challenges. I warn you - it can be addictive!
So who noticed the Amazing Decimal-Fraction Calculator in the sidebar? Surely you've always wondered about the pattern in the decimal representation of 1/9801? Now you can explore more decimal places than you ever thought possible:
1/9801 = 0.000102030405060708091011121314151617...
So when does this pattern start repeating? Go to the link in the sidebar to find out, then explore on your own...
1800-TEACHER.com Singapore's Education Network
Go to the Workbooks link and download samples of assessments for Primary 1 through 6.
Here's the invitation and terms from the authors of the site:
These worksheets are “school examination papers” contributed by parents and students over the years. The information contained herein is in the public domain. These worksheets are provided by 1800-TEACHER as a service to the community. You may download or print these worksheets for personal use only.
You need to register for free before you can download but it is worth it. For copyright reasons, I will not reproduce any questions here but I strongly urge you to look at the content and levels of difficulty of these assessments for the "primary" grades.
Pat's Blog
Pat Ballew has an excellent math blog and he and I often cross-reference each other. Pat expanded on the contest problem I posted the other day with a wonderful extension. This is a great question for your stronger 2nd year Algebra students. Building on each other's ideas is what Web 2.0 is all about! Thanks, Pat...
Friday, February 20, 2009
Updates and A Challenge Problem From Our First Contest
Updates...
- There are some 'new' math blogs (in some cases, new to me!) that I wish to bring to your attention. One of these was just started by one of my former students who is still in high school. She made excellent contributions to MathNotations last year. I'll post a link shortly. Look for creative thinking, challenging problems and an engaging writing style!
- Look for the Math Problem of the Day in the right sidebar. This is a 'gadget' made available by Blogger. Nice problems which are accessible to advanced middle schoolers and secondary students. They change every day so try them and check the solution link which is provided.
- There's a wonderful fraction calculator out there on the Web to which I will post a link and publish an article in a few days. If you haven't seen it, it will blow your mind! Imagine seeing the decimal expansion of any rational number to any desired number of places (within limits) instantly and that's not all it does!
- Yes, Pi Day is coming so I will post links to previous articles I've published and other excellent resources out there.
- Some of you know there's an excellent free resource of Singapore Math assessments for primary grades and more. You can download these and use them for your students. they make wonderful Problems of the Day and discussion points for your next department meeting! I'll post a link in a few days. Wait till you see the level of thinking and the content in the Grade 3 assessment, for example.
The following was question 5 from our contest. I'll leave it up for you to try. Feel free to comment or solve. This question proved to be of moderate difficulty for the teams. One has to be very careful about adhering to all the conditions regarding points P, Q and V. Have fun with it!
Problem 5 (2 pt question)
The graphs of y = 2x+3 and y = -x2 + bx + c intersect in 2 distinct points P and Q, where P is on the y-axis. Let V denote the vertex of the graph of the parabola.
(a) Determine all values of b for which the points Q and V coincide.
(b) Determine all values of b for which Q and V are distinct and the slope of line QV equals 3.
Tuesday, February 17, 2009
Results of First Math Notations Contest!!
"It was such a pleasure for me to see students so excited about math. Great job, Dave!"
Maureen Capuzzi, Montville HS
"I thought you might be interested to know that the moms appreciated your contest enough to ask for another one---for next week!"
Denise, Decatur Area Homeschoolers
"we all had a few 'a ha!' moments, which I think is the best part of math..."
ecv at fhs
And the envelopes please...
FIRST PLACE (11 out of 11 pts)
ARCADIA HS, Arcadia, CA
Advisor: Kerry King
SECOND PLACE
MONTVILLE Twp HS, Montville, NJ
Advisor: Maureen Capuzzi
THIRD PLACE (Tie)
FAYETTEVILLE MANLIUS HS, Manlius, NY
Advisor: Kate Nowak
FLORIDA MU ALPHA THETA TEAM, Tallahassee, Vero Beach, Fl
Advisors: Steve Friedlander, Brandi Williams
WYEDEAN HS, Chepstow, England
Advisor: Rhys Jeremiah
FOURTH PLACE (Tie)
DECATUR AREA HOMESCHOOLERS, Decatur, Il
Advisor: Denise Gaskins
WALLINGTON JR/SR HS, Wallington, NJ
Advisor: Stephanie Regetz
FIFTH PLACE (Tie)
FORT VANCOUVER HS, Vancouver, WA
Advisor: Nathan Shields
LAKE STEVENS HS, Lake Stevens, WA
Advisor: Kaleb Allinson
TAHOMA HS, Maple Valley, WA
Advisor: Dave Wright
Congratulations to our other participating teams, some of whom consisted of middle schoolers. This contest proved very ambitious for students below grade 11 but I hope it was a valuable learning experience for all. In the future, I plan to have separate contests for middle and secondary schools. THE NEXT FREE CONTEST IS SCHEDULED FOR APRIL OR MAY - STAY TUNED!
I also plan on posting some of the contest problems over the next few days. This has been an interesting experience for me and I learned much from the comments of the advisors and students. Next time I will limit the time to 30-45 minutes for ease of scheduling! If you think you might be interested in our next contest, you can always email me early and let me know. (dmarain at gmail dot com).
Thursday, February 12, 2009
Friday the 13th Parts I, II and III in 2009: Feb, Mar and Nov. Is this unusual?
Update: See detailed info on Fri 13th at bottom. Also, several revisions have been made.
Update on Contest:
All teams' Answer Forms have been submitted and scored. The highest score was a perfect 11 out of 11. There were several other outstanding scores as well. More details to follow...
If you suffer from triskaidekaphobia, uh, well, maybe you can be Rip Van Winkle starting tonight, Thu Mar 12th, and wake up on August 14th next year. Next year will be much less scary with only one Friday the 13th!
Note: Triskaidekaphobia only refers to fear of the number 13. See the Wikipedia article for the very long word denoting fear of Friday the 13th!
Can this be a learning experience for our students?
Here are some questions to ask:
Feb 13th 2009 happens to fall on a Friday.
Q: What day of the week will it be 28 days later? Explain.
Ans: Friday
Explanation: The days of the week repeat every 7 days and 28 is a multiple of 7.
Q: What day of the month will it be 28 days after Feb 13th? Explain.
Ans: It will be the 13th of the next month, in this case, March.
Explanation--
BASIC RULE:
If today is the nth day of a given month and there are k days in this month, then k days from today will always be the nth day of the following month!
Why? Well, we can write k = (k-n) + n, which can be interpreted to mean that k days from today will be the nth day of the following month.
Ok, an example would help here:
Let's say today is the 5th day of some month and there are 31 days in that month. We can write 31 = (31-5) + 5 = 26 + 5. the "26" brings us to the end of the month and then we add 5 more days which brings us to the 5th of the next month. Make sense?
Putting this all together, 28 days from Friday Feb 13th will be Friday March 13th. The key was that 28 is BOTH the number of days in the month AND a multiple of 7!
That wasn't hard, but now for the tougher question:
Q: Why is there going to be a third Friday the 13th in 2009, namely Nov 13th?
Explanation:
Mar-31 Cum. Total = 31 (not div by 7)
Apr-30 Cum. Total = 61 (not div by 7)
May-31 Cum. Total = 92 (not div by 7)
Jun-30 Cum. Total = 122 (not div by 7)
Jul-31 Cum. Total = 153 (not div by 7)
Aug-31 Cum. Total = 184 (not div by 7)
Sep-30 Cum. Total = 214 (not div by 7)
Oct-31 Cum. Total = 245 (Div by 7!!)
Thus, 245 days after Fri Mar 13th will not only be the 13th of Nov but it will also land on a Friday!
Why? Because, 245 is the cumulative total of the number of days in the 8 months starting in Mar but it is also a multiple of 7. No simple formula here, just grinding it out.
Suggested Additional Questions:
(1) Is 3 the maximum number of Friday the 13ths in any calendar year?
Ans: Yes!
(2) When is the next calendar year in which this 3-peat (three Friday 13ths in one calendar year) will occur?
Ans: See below!
(3) Is 1 the minimum number, i.e, could there be a calendar year in which there are no Fri the 13ths?
Ans: See below!
(4) In which months could a 3-peat occur in a non-leap year? A leap year?
Ans: See below!
(5) Related to (4): 31+29+31 = 91: What does this tell you about Friday 13ths in a leap year?
(6) Other questions: From the students!
The following is from the excellent Wikipedia article on Friday the 13th...
The following months have a Friday the 13th:
| The following years have Fridays the 13th in these months:
|
This sequence, here given for 2001–2028, repeats every 28 years from 1901 to 2099. The months with a Friday the 13th are determined by the Dominical letter (G, F, GF, etc.) of the year. Any month that begins on a Sunday will contain a Friday the 13th, and there is at least one Friday the 13th in every calendar year.
The longest period that can occur without a Friday the 13th is fourteen months, either from July to September the following year (e.g. in 2001/2002 and 2012/13), or from August to October in a leap year (e.g. in 2027/28).
| Patterns for non leap-years:
| Patterns for leap years:
|
Thursday, February 5, 2009
Odds and Evens 2-6-09: Contest Updates, ADP Algebra EOC Tests, "Numb and Numberer", a MathNotations Podcast?
MathNotation's First Math Contest
The Feb 3rd contest is 'virtually' over! We had a dozen schools participate including high schools, middle schools, a HomeSchooling team, a math fraternity team and a team from the UK! Our British team has had to delay the contest until they dig out of the deep snow that's been blanketing and paralyzing the UK for several days. I sure hope I am not in some way responsible for that!
The contest has been well-received although much of it proved too advanced for our middle schoolers. I will post results and selected problems in a few days after I sort it all out. I am planning on one more free contest in the spring. This one will probably be restricted to secondary teams. If interested, let me know early on. I am also planning 4 contests for the 2009-10 school year. Details about dates, registration and fee will be forthcoming.
"Is trillion the new billion?"
What are the odds that a CNN feature story appeared a few days after the Super Bowl post I published - Was the First Super Bowl More or Less Than a BILLION Seconds Ago?
Heres' an excerpt from this wonderful piece on CNN.com:
"To put a trillion dollars in context, if you spend a million dollars every day since Jesus was born, you still wouldn't have spent a trillion," McConnell said.
CNN checked McConnell's numbers with noted Temple University math professor and author John Allen Paulos.
"A million dollars a day for 2,000 years is only three-quarters of a trillion dollars. It's a big number no matter how you slice it," Paulos said.Here's another way to look at it.
"A million seconds is about 11½ days. A billion seconds is about 32 years, and a trillion seconds is 32,000 years," Paulos said. "People tend to lump them together, perhaps because they rhyme, but if you think of it in terms of a jail sentence, do you want to go to jail for 11½ days or 32 years or maybe 32,000 years? So, they're vastly different, and people generally don't really have a real visceral grasp of the differences among them."
Everyone is tossing around the words million, billion and trillion. With the national debt now topping $10 trillion, following a $700 billion bank rescue and proposed $800 billion-plus stimulus package, have we become numb to the numbers?Prof. Paulos is making an excellent case here for our middle schoolers developing better sense about large numbers. This is another view of the Number Sense standards which have been promoted for over two decades. Ah, the Power of Ten! Gee, that would be a cool name for a game show...
Update on ADP Algebra I, II EOC Tests
Do you frequently check the Achieve website to get the latest info on these tests, revised benchmarks and latest developments. For a current overview on the Algebra I test, look here.
I don't yet see released items or a set of sample questions but there is some very helpful information there, in particular the Algebra I Expected Knowledge document you can download. This can serve as the basis for curriculum development and provides sorely needed consistency. It will help to address questions like: Should we still teach those mixture problems? How much emphasis should we place on factoring quadratic trinomials?
MathNotations Podcast
Ok, what's this nonsense about podcasting? Another venture that I will try and not follow through on? Perhaps, but I'm never afraid to challenge bold frontiers that people my age are not supposed to consider! I would like to offer MathNotations visitors another way to connect with issues in math education. Once I get this set up, I'll let you know. There are a few technical obstacles I need to overcome but I'm getting close.
A sample of what I'm considering -- A 10-15 minute 'broadcast' you can subscribe to and download to your iPod or computer.
Here's the first topic I'm preparing:
Using current technologies to provide supports for our students outside the classroom.
What does that mean? Well, we all know that many students, who actually want to complete their math homework, become frustrated and give up when they can't get started or hit a wall. OR it's the night before a quiz, test, midterm, final, SAT and there are a couple of problems they can't do. To whom do they turn? Parents? Friends?
What are the Web 2.0 options?
Email, chats, forums, free help sites, mathcasts, videos on YouTube? Parents and students want to know what is available and I'd like to open up a dialog here and make some recommendations of those avenues that are cost-effective (aka 'free'!) and are effective for learners (i.e., high quality). Not every parent can afford high-cost tutoring, so where to turn? I can post recommended links but a podcast allows more informal discussion. I may also be able to have an audio dialog with some of the more knowledgeable people in these areas.
Our next Mathanagram? Prof. Rademacher to Infinity and Beyond?
Well maybe....
It's getting harder to keep this going so, for now, enjoy his picture!
Of course, you could visit MathNEXUS, and the Math Person of the Week challenge in the sidebar. Prof. Johnson, of Western Washington University, who maintains this exceptional site, provides new weekly goodies for all math educators. A truly great resource...
Saturday, January 31, 2009
Was the First Super Bowl More or Less Than a BILLION Seconds Ago?
Hey, another kind of "over-under"!!
Just some food for thought to put the number one billion and history in perspective...
A billion seconds ago it was about 1976.
A billion minutes ago Jesus was alive.
A billion hours ago our ancestors were living in the Stone Age.
A billion days ago no creature walked the earth on two feet.
And a billion dollars lasts 8 hours and 20 minutes at the rate our
Government spends it.
There are many references for this on the web and I'm sure you've seen it before. You can check the accuracy (perhaps the last one is a little off!). I still like sharing this with students as it not only puts the concept of a billion in perspective but it does offer a wonderful application of 1-significant digit estimates, scientific notation and orders of magnitude. Can you imagine asking students to estimate that a billion seconds is roughly 32 years without a calculator!!
Where are these kinds of estimates currently in our math curriculum? More likely occurring in a science class? Do they belong somewhere in our classes or are they just amusing curiosities? You can guess where my thoughts lie!
Wednesday, January 28, 2009
Algebra 2 as a Graduation Requirement? Related Issues...
The deadline for registering for the First MathNotations Math Contest on Tue Feb 3rd is drawing to a close but you still have an opportunity to register! Look here for details and email me if you want a team of your students to participate!
As I've reported for some time (look here), here in NJ the Commissioner of Education has been promoting higher standards and more ambitious graduation requirements, choosing Algebra 2 as the cornerstone. I've had misgivings about this as a requirement for all students for several reasons although I'm a strong supporter of the American Diploma Project's Algebra 2 benchmarks and the End of Course Test for all students who choose to take the course because of their educational goals.
A recent article (1-27-09) on the website pressofAtlanticCity.com gives an excellent account of the debate raging over this topic at the State Legislative level. I will reprint a good portion of the article and then reprint the comment I posted on the site. I strongly encourage my readers to read the entire article and all of the comments posted thus far. It is a microcosm of much of the current debate in math education. Several of the commenters provided a commonsense view of these issues and gave me food for thought.
Education Commissioner Lucille Davy and a panel of education and business professionals appeared Monday before the Assembly Edu-cation Committee to discuss the Department of Education's High School Reform project.
The requirement that all students take algebra II has been controversial, and on Monday it dominated a discussion that attempted to identify just what students need to know to succeed and compete in the 21st century.
Davy insisted the algebra II requirement would not be so rigorous that it would lead to high rates of failure or students dropping out.
She said it would be a continuation of algebra I, but schools could offer more rigorous honors courses to those who would need them.
But Rutgers math professor Joseph Rosenstein, of the New Jersey Math and Science Coalition, wondered if the proposed courses might then get so watered down they would no longer really be algebra II.
"Most of our students don't need algebra II," said Rosenstein, who supports requiring more practical applied math courses.
Rosenstein said if courses were tailored just to meet state requirements, students who should take a true algebra II course might not get the higher level of work they need.
The algebra II issue has also frustrated vocational high school officials, who worry that too many requirements will make it impossible for students to complete programs in high school.
"These are students who benefit from applied learning," said Thomas Bistocchi, superintendent of the Union County Vocational School, adding that their goal is to have students graduate as industry-credentialed professionals. "We just want students who want to become plumbers have the time to do it," he said.
Davy said there will be flexibility in how the coursework is offered, so that it could be integrated into vocational coursework, but opponents wonder if that could be done and still teach what would be tested.
Stan Karp, of the Education Law Center, said reform is needed, but the state needs better education, not just more requirements. He said teachers and students will need better preparation to meet the new requirements.
"Less than half of the high schools now require those courses," he said. "What is it going to take to get there?"
Asked about the cost of reforms, Davy said the state already spends the most of any state and should not need more money, just a better reallocation of existing funds.
Business representatives said they just need students who can perform modern jobs.
Dennis Bone, president of Verizon, said students need the foundation of skills to be able to adapt to new and changing technology.
"We are being revolutionized by technology," he said. "Billboards now are electronic, run by someone sitting at a computer, not climbing a ladder."
"So what does algebra II have to do with that?" Education Committee Chairman Joseph Cryan, D-Union, asked.
Dana Egresky, of the New Jersey Chamber of Commerce, said that if taking algebra II can help a carpenter solve more problems on the job, then that is the carpenter who would get the job.
Assemblyman Joseph Malone, R-Ocean, Monmouth, Burlington, suggested asking professionals ranging from carpenters to doctors how they actually use algebra skills.
"We need to do a better job at finding out what people actually need to know, not what we think they should know," he said.
Here was my comment:
I thoroughly agree with Prof. Rosenstein that not all students will need the skills/concepts of a more advanced algebra class. While I admire Commissioner Davy's desire to significantly raise the bar for NJ students there are some underlying issues that must be addressed first. How many of you believe that the majority of NJ students have demonstrated proficiency in the foundational arithmetic and prealgebra skills needed to be successful in a legitimate Algebra 1 course, never mind Algebra 2? As a retired math supervisor, believe me, that question was rhetorical!
However, we must clearly distinguish between the issue of a graduation requirement for all and the need for consistent, clearly stated and rigorous standards for a 2nd year Algebra course. Despite opinions to the contrary, I believe the latter is necessary for most college-intending students. The American Diploma Project (NJ is a member of this consortium) has developed precisely those kinds of world-class standards and the result is the new End of Course Test in Algebra 2. This test, which many NJ students have already taken, requires a deeper conceptual understanding of topics such as mathematical modeling which separates the Algebra 2 of the 21st century from the Algebra 2 course many of us remember. And, yes, there are still some mechanical skills which students need to master away from the calculator!
I strongly advocate that NJ adopt these higher standards for those students who will go on to take more advanced math courses. Clearly, it isn't for everyone and therefore we should reexamine it as a grad requirement for all.
Dave Marain
I felt it was important to make a clear distinction between Algebra 2 as a high school graduation requirement and the need for a high-quality curriculum which should be uniform for all students who need to take the course. Many commenter ranted about the evils of testing, the "who really needs algebra anyway" argument, allowing politicians to make educational decisions they know little about (imagine acknowledging that it should be left to math education professionals!) , the skills needed for the 21st century, etc. Fascinating stuff...
Is this same discussion happening in your district or state? Your thoughts are important to me. Do you take strong exception to my comments? Do you agree with the NJ Commissioner of Education or has she gone too far? What do you say to the many adults who argue that, in their occupation, they haven't ever used any of the 'stuff 'they learned in Algebra 2?
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