Some thoughts and loose ends...
It's long overdue but better late than never - I've decided to feature two math websites/blogs for this month. I will try to do this on a monthly basis. It's important to me to acknowledge the outstanding work of other math bloggers particularly those who have been kind enough to support my efforts over the past two years. There are so many excellent sites to choose from and I don't want anyone to think I'm ranking these by this selection. I will acknowledge several others but I had to start somewhere.
FEATURED MATH BLOG/WEBSITES FOR DEC.
Maria Miller's HomeSchool Math blog.
In Maria's own words:
I love teaching and I love math. This blog is my way of reaching out and helping you to teach it too.
Maria has hundreds of posts covering many topics in math and math education as can be seen from the extensive Labels section in the right sidebar. Maria particularly enjoys demonstrating the bar model method from Singapore Math. Well worth reading...
Maria also has a comprehensive math resource site with links to free materials as well as other excellent products. In particular, I'd like to bring the following page to your attention: Problem Solving, Word problems and Math Projects - Free resources on line for K-8 students and beyond. Maria is very thorough and writes a short review of each. Make no mistake -- these resources are for public school parents, students and educators as well. Maria's enthusiasm, dedication, and knowledge are incomparable.
Finally, Maria has developed a series of highly regarded workbooks, her Math Mammoth series, for Grades 1-5. Affordable, easy to follow, a complete curriculum... Enjoy!
MathNotations was the featured website of the week on this excellent site. I would strongly recommend this site for all math educators. Jerry Johnson who maintains it somehow manages to update frequently with useful links to just about everything going on in math education. His weekly features include Problem of the Week, Quote of the Week, Statistic of the Week, Humor of the Week, Website of the Week (MathNotations made it!), Calendar Events and Famous Birthdays to name a few! This site receives thousands of visits each week. You will not be disappointed.
Talk about weird coincidences. Jerry found my blog and somehow remembered me from a presentation I gave back in the 90's at the Annual NCTM Convention in San Diego. Turns out that he was the program chair!
1. Anyone remember a MathNotations post from awhile ago regarding the classic boring of a hole through a sphere problem. This well-known Geometry/Calculus conundrum continues to tantalize and intrigue readers. Here's the idea of the puzzle if you forgot it:
Imagine you have a bowling ball with a diameter of 8.5 inches and you drill a hole exactly through the middle so that the remaining part of the ball is 6 inches high. Now imagine boring a large hole though a giant bowling with a diameter of 8.5 feet so that the part of the ball left behind is still only 6 inches high. Incredibly, the volumes remaining of these two drilled-out bowling balls are equal! Can you picture this? The scientific approach here might be to physically drill through two such objects of equal density and compare the volumes remaining by weighing them, but mathematicians usually try to avoid such 'hands-on' experiments, choosing to prove the result using methods from geometry or calculus. From the number of views this puzzle has received, I'm working on a video demonstrating the result using elementary calculus (volumes by cylindrical shells or disks). Because the writing on the board I'm using tends to be small, it will be difficult to read all the details but I'll do my best. I recognize that these videos are not of the quality you can find elsewhere on YouTube but I still think that working through such a fascinating problem makes it worthwhile and somewhat unusual. Considering that it reviews fundamental integral methods it might also be helpful for students who will soon be getting to this topic (or reviewing for exams). If you watched any of my other videos you know I explain the details thoroughly, trying to avoid skipping any key steps. That guarantees that the hole in the sphere video will be quite long (I'll probably break it up) and it will take some time to bring it to fruition. Despite the technical issues, if you want to see more of these, let me know. I'm also considering a few dedicated to the lower grades, e.g., how to develop algebra sense in middle schoolers (from numerical patterns to algebraic expressions). Any interest?
2. The problem from the other day regarding the angles of an isosceles triangle received some nice responses but the question would not have made a good contest or test problem. There's nothing worse than an apparently challenging question that students can answer correctly using incorrect reasoning! Here again is the problem:
I had indicated there were a couple of sticky points one had to be wary of (my readers caught on to the fact that the equal angles do not have to have integer measure and that the triangle is not equilateral), but, if the student overlooks both of these, the correct answer can still be obtained! Thus, if the student starts with a base angle of 51°, one obtains a vertex angle of 78°. Reasoning quickly, one arrives at 78, the correct answer, as the number of possible values (all integers from 1 to 78). Two errors which nullify each other and produce a correct result! That's why it's always preferable to have assessments in which students are required to explain their methods but it's simply impractical to have standardized tests with 45-50 such questions. I should have caught that but it does demonstrate the challenges of writing high-quality short-answer (objective-type) questions!
3. By now you've probably all forgotten about the MathAnagram for Oct-Nov-Dec.
Check the link if you're interested. Here it is again: