For the past 6 years I’ve been explaining converse (of the inverse, converse, and contrapositive fame) by saying that you can switch your Converse® shoes from left to right (although potentially uncomfortable), but you certainly cannot negate your Converse® shoes.
Today a volunteer stepped up and she was willing to switch her left and right shoes:
Awesome.
Every two-digit number can be represented as AB, where B is the ones digit and A is the tens digit. Right? So for example the number 43, A is 4 and B is 3.
Imagine then that you took this two-digit number and you squared it, AB x AB, and when you did that the result was a three-digit number, CAB.
Here’s the question: What’s the value of C? So, for example if AB is 43, CAB might be 943. Of course this is a totally bogus answer, but you get the idea.
So again, what is the value of C, so that AB(squared)= CAB?
(spoiler – answer is here.)
This is from my fuzzy math series. It was sent in by Dave Atkins. On a recent Saturday afternoon, I saw a boy and his mother at the neighborhood diner where I often go for lunch. From my vantage point I could see they were working on some arithmetic problems. The problems seemed simple enough and the kid was getting all the correct answers. For example, the first one was 25 + 8 and he wrote down 33. And the next one was 12 + 5 and he wrote down 17. The next was 35 + 13 and he wrote 48. Then his mother posed the last two problems. 45 – 8. The boy said 47 but I thought the answer was 37. The next one was 42 + 15. The boy said 43. I thought it should have been 57. His mother accepted both of those answers. When I saw how the kid was dressed, I did too. What was going on?
(spoiler – answer is here.)
Great post: http://math-blog.com/2011/01/13/developing-math-intuition/
Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean?
Suppose we want to define a “cat”:
- Caveman definition: A furry animal with claws, teeth, a tail, 4 legs, that purrs when happy and hisses when angry. . .
- Evolutionary definition: Mammalian descendants of a certain species (F. catus), sharing certain characteristics. . .
- Modern definition: You call those definitions? Cats are animals sharing the following DNA: ACATACATACATACAT. . .
The modern definition is precise, sure. But is it the best? Is it what you’d teach a child learning the word? Does it give better insight into the “catness” of the animal? Not really. The modern definition is useful, but after getting an understanding of what a cat is. It shouldn’t be our starting point.
Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is.
I fully “buy into” this type of thinking. I used to give the students the formal definition and try and break it down. For the past couple of years I’ve taken a much more informal path with definitions. It seems like the students are better with building the formal on-top of the informal instead of vice-versa.
Here’s a great set of facts about the number 73 (a Chuck Norris Number, sidenote I might grow to like that show but I can’t stand the laugh track):
- The mirror of 73, the 21st prime number, 37, is the 12th prime number. The number 21 has factors 7 and 3.
- In binary, 73 is a palindrome – 1001001
- Of the 7 binary digits representing 73, there are 3 ones.
- Every positive integer is the sum of at most 73 sixth powers.
- In octal, 73 is a repdigit – 111
- Pi Day occurs on the 73rd day of the year (March 14) on non-leap years
Many more facts to be found at this great post: http://www.walkingrandomly.com/?p=3024
RAY: You’re given a hundred dollars and told to spend it all purchasing exactly a hundred animals at the pet store. Dogs cost $15. Cats cost a buck, and mice are 25 cents each.
TOM: Let me get this straight. You have to spend exactly a hundred bucks and you end up with exactly a hundred animals?
RAY: Right. The other only other criterion is that you have to purchase at least one of each animal.
The question is, how many of each animal do you have to purchase to equal a hundred animals purchased at exactly a hundred dollars?
(spoiler: answer is here.)
RAY: A family of four and their dog are trapped on an island, when rising floodwaters tear out the bridge that they had used just a few hours earlier. When they had just given up hope the son says, “I’ve got a small boat and oars.”
But their joy was short-lived. The manufacturer’s instructions printed on the boat stern said that the boat can carry only 180 pounds.
The dog is the only one who can swim.
The father weighs 170. The mother says she weighs 130 but it’s more like 155. The son is 90 pounds, the daughter is 80, and the dog is 15.
Here’s the question: What’s the fewest number of crossings they have to make to save everybody?
Update – 1/21/11
(spoiler – answer is here.)
This is a quick post describing one small reason (of many) of why I like SBG.
The setup for the gradebook is as follows: each student gets 2 shots (in class) to quiz on each topic, the grade is out of 4. They get one chance right after the end of the unit, and one a week later. If they improve from try 1 to try 2, then their most recent score is doubled (so the final grade is out of 8); otherwise add the two grades together. So here is a screenshot of my gradebook. The last row is the class averages. The “1” is the first attempt, “2” is the second, and “3” is any assessments taken afterschool.
Notice how on nearly every topic, the grade goes up from assessment 1 to assessment 2. In some cases it goes up significantly (concepts 16 and 17). Sometimes there is a concept where the grade goes down (concept 13) because it was tough, but the students stay after review and then reassess successfully: 2.8 -> 2.4 -> 3.4.
From Webmaths: Pretty cool website, a bit like Project Euler, but with more approachable difficulty level: http://www.artofproblemsolving.com/Alcumus/Introduction.php
Sample Problem:
The leveling up system they have could be an excellent way to get students addicted to solving some novel math problems. There are ways (I haven’t explored them) to set up an entire class of kids and monitor their progress through the levels.
After some of my favorite bloggers wrote posts talking about their favorite books of 2010, here and here, I’ve decided to do the same but with a twist. I don’t end up reading more than 5 books a year, but I do listen to numerous podcasts while on the daily walks with the dog and on the daily commute. This adds up to around two hours a day of free time, so I can get some audiobooks in from time to time as well (this year it was mostly Steig Larsson’s Millenium Trilogy and re-listens of JK Rowling).
Here’s the list (alphabetical order of favorites):
While I don’t enjoy David Gregory (or his whistle nose) as much as the incomparable Tim Russert, it’s an very good show that provides fair coverage to US politics. The show obsesses TOO much on stupid politics covering candidates 18 to 24 months before their election, but otherwise they do an excellent job of asking the hard questions.
I don’t listen to every minute of this podcast, as they get a bit too inside baseball with indie music, but they have provided me a first listen to some of my favorite music over the past 3 years. The end-of-year rankings are the best shows; where the commentators geek out in their debate over the songs of the year. Companion podcasts to this one include Tiny Desk Concerts, and Live in Concert.
My mom got me on to this show a long time ago, and it still holds up. Excellent car advise, hilarious commentary, and amazing puzzlers are all part of this classic.
Great interview show with Terry Gross. I don’t listen to every show, but some of the interviews are unmatched in their depth and un-NPR-ness, Sarah Silverman, Jon Stewart, Jay-Z, etc. Even the more NPR-er shows are great, topics include: economy, government, politics, sports, etc. Plus very good movie and TV show reviews.
Great material for economics and making it “real”. I would bet that the show producers and contributors would make excellent educators. They convert very dry topics to human interest stories, buy cialis online using paypal and they do an excellent job with dumbing down some very complicated economic concepts. Really. I’m also willing to bet that I have a better grasp of this current recession’s causes and impacts, than 90% of people who read the newspaper. This news source is really that good (and entertaining), and only in ~60 min per week.
This show is a bit like the media critique part of the Daily Show, only in more detail (and less funny). Really a great show, what I like most is the in-depth look at the topics. The most recent show was entirely focused on video games, how they have developed, and where they are going. This episode is important enough that I’m going to have my Computer Programming class take a listen.
from the howstuffworks.com people. Topics include: volcanos, migraines, dreadlocks, wicken, prohibition, bounty hunters, braille, ninjas, muppets, lobotomies, deja-vu…. Each week they put out 2 or so podcasts, each of which on a specific topic. Very hard to describe, but Chuck (Chuckers) and Josh are simply awesome. Hilarious, witty, and the banter is great. Can’t say enough about this podcast.
John Gruber and Dan Benjamin talk about Apple news. Yes its nerdy, but their insights into the world of Apple are unmatched.
The world’s experts talking about their most interesting material for 15 or so minutes. Can’t go wrong.
Hard to describe. A bit like Ira Glass (from This American Life), meets Stuff You Should Know, but with interviews, production, and no conclusions. Best show to date: Words, with companion video. Easily one of the best videos of the year.That’s it for now. I do listen to some others, but when these show up in iTunes, I get excited to get listening.
