f you've read some of my earlier entries, it shouldn't be a mystery that I spend a considerable amount of time thinking about math education. It probably wouldn't surprise you to learn that I'm happy to talk with just about anyone about the subject. Unfortunately, it's a pretty emotional topic for a lot of people.
Earlier this week, I came home for lunch and was greeted with just such an opportunity. As it turns out, my daughter (who is in fifth grade) brought home some math worksheets to complete over spring break. She isn't as fond of math homework as I am, and so enlisted the aid of her mother and older sister. Two hours later, my wife had to walk away. Shortly after that, my older daughter (a high school freshman in "honors" math) also threw in the towel. "Dad, my homework doesn't make any sense ..."
The worksheet consisted of three riddles designed to challenge students on their understanding of rectangles. I'm fine with that, logic problems are a great way to keep sharp and relate very well to mathematics. So long as we don't have to do a lot of guessing and checking, I can keep pretty calm about these things.
a I am a rectangle.
b My dimensions are two consecutive even numbers.
c My perimeter is 20.
d My area is 24.
a I am a rectangle.
b My dimensions are two consecutive prime numbers.
c My perimeter is 1 less than the fifth square number.
d My area is 1 less than the sixth square number.
a I am a rectangle.
b My dimensions are consecutive counting numbers.
c The sum of my length and width is a prime number.
d My perimeter is 3 less than a prime number and 1 more than another prime number.
e My area is not a square number.
f My area is less than the 10th square number and greater than the 8th square number.
Riddles one and two were complete when I got there. In both cases, my daughter had chosen a strategy of listing the factor pairs of the area and then checking them against the other clues. It was a little messy where she wrote out her reasoning for each pair, but she got to the correct answer in both cases and her strategy was pretty direct. Good job! That third riddle on the other hand, had already used up most of the available space on the page as well as a page of scratch paper by the time I stepped in. Her original strategy wouldn't work, as she didn't have an area to start with. She did have a list of the first thousand prime numbers, but didn't know where to start. So, we carefully cleaned up her worksheet and got to work.
Clue A: Not helpful ... let's ignore that one.
Clue B: "Counting Numbers" is a little vague, but since we are talking about fifth grade geometry, it should be pretty safe to assume that they mean two "positive integers" where y=x+1.
Clue C: Now we know that our two unknown consecutive integers add up to an unknown sum, which is prime. Luckily, my daughter has that list of primes. This could be helpful, but won't be fun, as every prime larger than 2 can be shown as the sum of two consecutive integers.
Clue D: So, P-1=Prime and P+3=Prime. Okay, so all we need to do is narrow down our list of prime numbers to pairs that differ by four, and we know that our perimeter lies between one of those pairs. Not impossible, but not a piece of cake either.
Clue E: Are there any squares formed by multiplying consecutive integers? I didn't think so ... let's ignore that one too.
Clue F: Finally! A clue that we can use to narrow things down.
If the area lies between 64 and 100, and its dimensions are consecutive, any integer pair made of numbers smaller than 8 or larger than 10 are excluded as they would have a product too small or too large (respectively). So, our possible dimensions are either 8 & 9 or 9 & 10. Both 8+9 and 9+10 are prime numbers, which means Clue C isn't very helpful after all, so, we move on to Clue D and calculate our perimeters: 2(8)+2(9)=34 and 2(9)+2(10)=38. 34-1=33, which is not prime. Bummer. However, 38-1=37 and 38+3=41, both of which are prime. Happy news! We don't have to filter that stupid list after all!
Altogether, it's not the worst thing I've ever had to trudge through, but it's certainly not the best. The first and second riddles could have been solved pretty easily using algebra (not a fifth grade skill), but the third could not. That left us guessing and checking (and guessing and checking ...), which is a lousy way to teach kids to enjoy math (in my opinion). If I were less patient I might have gotten upset, refused to work through the problem, and sent her teacher a nasty note and some examples of questions that can reinforce elementary geometry without frustrating kids. Maybe I'd get some more traction on my website and an opportunity to interview with Glenn Beck. Yeah ... right.
About a week ago, a somewhat snarky response to a rather ridiculous math assignment was posted to facebook. In it, the student's father - a seemingly intelligent and articulate fellow - expressed his frustration with the common core curriculum as manifested in his child's homework. I don't blame him for his frustration; a lot of intelligent people get bent out of shape over homework. What I take exception with is that because his response labels the assignment a common core issue, those against the initiative have taken it up as evidence that the common core signals the impending apocalypse. They wave it around, scream about the government, and ask me how I can possibly defend such an asinine educational method. I'd go into a long response, but I'd rather pay attention to my wife. Besides, someone else has already covered it for me.