On a recent chapter test covering probability, I gave an extra credit question of this type: 850!/848! . For those who are questioning what's so exciting about the numbers 850 and 848, the exclamation points indicate a mathematical operation known as "factorial". Factorial means to multiply all the whole numbers from 1 to whatever number you're looking at, so 850! means 850*849*848*...*3*2*1 and 848! means 848*847*846*...*3*2*1. And yes, I allowed the students to use calculators.
Math people can easily see why I chose these particular numbers. Unless someone has some ultra powerful calculator, the number 850! is too large for most calculators to handle.
This is one of those times where I get to see who thinks and who pushes buttons, who uses the calculator as a tool and who uses it as a crutch.
All the numbers in the fraction 850!/848! cancel except for 850*849 in the numerator, which gives an answer of 721,650.
One student listed the answer as "Overflow Error". There's an error, all right, but it isn't in the calculator's limited memory capacity!
Update: Interestingly enough, the calculator built into MS Windows handled the computation just fine, despite the answer's involving 10 to the 2,122nd power.
Most computers can only handle numbers up to 2^31-1, because numbers are stored in 4-byte chunks (one bit is reserved for the sign of the number). Windows calculator probably uses a special library that lets you have arbitrarily large integers.
ReplyDeleteIf you really want a powerful, versatile calculator try SpeedCrunch. It has a bunch of features such as calc-as-you-type, syntax highlighting, 50 decimals of precision, history, and smart correction. I used to use PowerToy Calculator for Windows XP but this is cross-platform and compatible with Windows 7. It handles this problem and problems dealing with much larger numbers plus it has a really nice list of constants.
ReplyDeleteGreat question.
ReplyDeleteOverflow error, you can't make it up!
quick question...do you think letting your students use a calculator hurts them when they go take the CST because they aren't allowed?
ReplyDeleteI go back and forth on this one. I take them away from my Algebra B & Algebra 2 classes, but with my Geometry I let them use them on certain chapters.
In fact I just posted about this same question a couple of days ago.
Thoughts?
I'm in a minority on this, I'm sure, but I'd prefer they use calculators with minimal capabilities, perhaps 4-function and square root. Honestly, I think students would get a much deeper understanding of math if they had to look up trig functions and logs in tables and just use the calculators for computation. At my school, though, we go to the opposite extreme, having TI-30's to issue to everyone in class who needs one, and (perhaps illegally) requiring graphing calculators in our calculus classes.
ReplyDeleteLove that they actually wrote "overflow error" as the answer. I can't tell you how many times students have come to me with calculator in hand saying "something's wrong with my calculator" when it's all operator error! I regularly tell my classes, the calculator is only as good as the person hitting the buttons. It's not magic. It doesn't give you the "right" answer.
ReplyDeleteagreed on the deeper understanding with the tables, but that won't happen anytime soon I think.
ReplyDeleteAs far as requiring graphing calculators, yeah it's illegal. The whole free education. In fact at my first school (a very wealthy area) a group of parents sued the school for making them buy one. So the school had to give them out.
You and I both know you are sitting on a lawsuit waiting to happen.
My son hits high school next year. First fee he hits, I'll be talking to a lawyer. I already warned the district.
ReplyDeleteGreat question Darren. Interesting, I remember getting a very similar question on a test in high school for extra credit. I was wondering why the teacher made it so easy. Most of the class got it because few of us even owned a calculator.
ReplyDeleteThe ones who had a calculator had the most trouble. Too bad that teacher didn't have the Internet to share the bitter laughs with others like you can.
I'm not laughing. I'm disappointed.
ReplyDeleteHonestly, I think students would get a much deeper understanding of math if they had to look up trig functions and logs in tables and just use the calculators for computation.
ReplyDeleteCould you explain this a bit more? How would understanding be deeper if we had to look up numbers on tables and then, say, multiply them on a calculator? That does not seem obvious to me.
I like good questions like that one.
ReplyDeleteToo many people use calculators not as a tool, but as a crutch. They *can* keep you from thinking--see the "overflow error" answer as Exhibit A.
Having to use a table reinforces what different parts of the logarithm are, what they mean, and how they are used. Assuming common logarithms, an integer to the left of the decimal point indicates a power of 10 (why?). The digits to the right of the decimal point indicate a number between 1 and 10 (why?). If you just crunch numbers with a calculator, you don't even have to think about the numbers--in other words, you don't think.
Having the table of logs and a 4-function calculator gives you only tools, it's up to you to use them. You begin to see, because you have to actually figure it out yourself, that logs turn multiplication into addition, and turn powers into multiplication.
They you'll be able to solve x^(2/3)=50 with only a table of logs, a pencil, and a piece of paper. When you can do that, and not just memorize the steps for doing so, *then* you understand logarithms.