Square Root Day occurs only nine times per century, and one of them is today: 3/3/09. You might have been busy celebrating New Year’s Day on the first Square Root Day of the century: 1/1/01, and you might have been distracted by Groundhog Day on 2/2/04, but make sure you don’t miss this one, or you’ll have to wait another 7 years until the next one on 4/4/16. 🙂
Since we won’t be in school for Pi Day (3/14), which falls on a Saturday this year, you might want to mention this one to your classes, if you read about it in time.
Happy Square Root Day!
Last night, around midnight, needing a lesson plan for my middle school group, I strolled over to MathNotations and clicked on his Number Theory category for ideas. The Four Factors Problem filled the bill. In this investigation, students are asked to find all the numbers less than 100 with exactly four factors, and describe the two categories of numbers that they find.
I decided that first we should find all the prime numbers less than 100 using a Sieve of Erastothenes. Having the prime numbers in front of them is useful when trying to figure out how many factors a number has, since they might notice that 74 is 2 x 37 but then not immediately know if 37 can be factored further or not. So, we spent the first 20 or so minutes of class crossing out multiples of all the primes up to and including 7 on a 100 chart. We also figured out why we only had to go up to 7.
We counted 25 primes less than 100. The kids noted that none of those numbers have exactly four factors, since primes have exactly two factors. (Yay!) One girl also pointed out that no perfect squares could have exactly four factors, because squares always have an odd number of factors. (Double Yay!!) I asked the kids to each guess how many numbers we would find under 100 with exactly four factors. Their guesses ranged from 14 to 25. I also asked them to guess whether they would find more numbers less than 50 or greater than 50 with exactly four factors. Everyone (myself included) thought we would find more such numbers below 50. (Perhaps you would like to make your own guesses before reading further — I will post the actual number and distribution of the numbers at the end of this post. )
I had to clarify Dave’s “higher order” question: “These numbers fall into 2 categories. Describe these categories.” The first “guess” I got as to the categories were “odd and even”. Well, yes, some of them are indeed odd, and some of them are even. 😉 So I clarified that I was looking for descriptions of two categories of numbers, all of which have exactly four factors, and that al numbers with exactly four factors must fal into one of the two categories. I hinted that the categories have something to do with the prime factorizations of the numbers.
Most of the kids got most of the way through finding the numbers during class. I assigned finding the rest of the numbers (after comparing my results with a few fast finishers, we told everyone how many numbers they were looking for) and coming up with the two categories.
We have in the past discussed how to find the number of factors a number has, based on its prime factorization. (I can’t find a good page to link for that, so perhaps I’ll write it up myself at some point.) I wonder if any of them will think about it in those terms. I’ll report back to let you know how they did with that part of the challenge.
Answers: There are 32 numbers less than 100 with exactly four factors. And more than half of them are greater than 50!
While I was off being distracted by life this summer, Kara Hazen of Hazen Happenings and Mathman at When Will I Use This? had nice things to say about my Blog. Kara listed me among three “non-classmate” blogs she added to her RSS reader. Mathman even gave me my very first blog award, which is only slightly diminished by the fact that he gave it to everyone on his blogroll. 😉
Thanks for the kind words, folks!
I’ll have more content posted soon, I promise! 🙂
“The best part about math is that, if you have the right answer and someone disagrees with you, it really is because they’re stupid.”
I found the above in an amusing list of quotes from an honors linear algebra class that was linked from Epsilonica, a blog I recently discovered, via his great post In Praise of Proving that Zero equals One.
Danielle posted a question on my About page that I thought I’d answer in a separate post. She wrote:
I have a question… my daughter gets number triangles for math warm-up. E.g. Use each number only one time (1-9) and the sum of the numbers along each side of the triangle must equal 20.
She’s been doing them by trial and error, but that takes quite a long time the more complicated they get. Is there a formula or method that she’s missing?
Thank you for any help you can give her!
There isn’t really a formula that can be used to solve generic puzzles like this, but the key to solving them is thinking about which numbers get used more than once. In a triangle puzzle, this is the numbers in the corners. Each of these numbers gets included in 2 of the sums, while the rest of the numbers get included only once.
In the example you gave above, your daughter should first think about what the sum of all the available numbers is if they were each used only once: 1+2+…+8+9 = 45. Since we need the total of the 3 sides to be 60 (3×20) we know that the numbers that are used twice must add up to 15 (the difference between 60 and the 45 we’d have if each number were used just once). So, the corners of the triangle should add up to 15. Unfortunately, there are a lot of ways of making 15 out of 3 of the numbers from 1 to 9. But it does cut down the trial and error a fair bit to start. I tried a few combinations and was able to find answers pretty quickly using that constraint. I hope this insight makes these puzzles a little more fun and less frustrating for your daughter, Danielle!
Here’s another type of fill-in puzzle. You don’t use sums to solve it, but the key to it rests in realizing which spots are “special”. I’ll leave it at that for now so as not to give it away for those who’d like to try to solve it.
In case anyone was wondering, I’m still here. Real life intervened over the summer and distracted me from blog reading and writing. But now I’m preparing to go back to my volunteer role in teaching math problem solving at my kids’ school, and I’m excited about that. Hopefully I’ll find time to write about it as I go.
I found this forum due to the fact that someone there linked to my last blog post. (The link is on page 10 of the discussion if you want to check out the context.) There’s a great discussion there about some non-orthodox math education options (mainly, I believe, intended for homeschoolers or parents of gifted kids wanting to supplement at home).
The thread was started by a parent who had just heard a presentation by a local mathematician who advocated that kids shouldn’t learn math using a textbook at all — it should all emerge naturally from various scientific/engineering explorations. Although I don’t particularly agree with that idea (not least because “natural” situations tend to involve ugly numbers, which make hand computation impractical, and argue for the use of calculators — I’d be in favor of calculator use in such cases, but not if it’s the only exposure to computation the student is going to get!), the thread contains a lot of good ideas about math education, many relevant to both gifted and non-gifted students, and many as relevant to classroom instruction as to homeschoolers.
On page 11 of this discussion, there is a link to a great-looking free online math course intended for “adult learners and high school teachers”. NOT a course in computation or algebra or trig, but a course in cool math topics, such as prime numbers, combinatorics and game theory. Although they are aiming it at HS teachers, I think it could be a great resource for math-phobic elementary teachers, not because they would necessarily pass on that particular content to the kids they teach, but hopefully to help improve their attitude toward math, so that they could pass on some excitement about it (and also give age-appropriate introductions to many of the topics, even at the elementary level).
I haven’t had a chance to actually go beyond the overview of the course — perhaps I will have a chance to review it more completely at a later date. But it definitely seems worth a look.