Here's a warm-up you can give to your Algebra 2 (and beyond) students to welcome them back to math class after a summer of brain drain. NO CALCULATOR ALLOWED! This oughta' set the tone...
A total of 9991 M&M's were eaten by a group of Freshmen. Here are the facts:
(1) Each freshman ate the same number of M&M's.
(2) There was more than one freshman.
(3) Each freshman ate more than one M&M.
(4) The number of freshman was less than the number of M&M's each freshman ate.
How many M&M's did each freshman eat?
Work in your groups of 3-4. You have 3 minutes. First team to arrive at the correct number AND explain their method, gets to eat ______________.
Let me know how many groups solved it or your thoughts about the appropriateness of this question.
Thursday, August 30, 2007
9991 M&M's were eaten by a group of freshmen...
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14 comments:
They need to have seen this style factoring as a trick or gimmick in algebra I, which depends on the teacher from a few years back. I would consider it unfair to give a warm-up on which some students might be reasonably expected to have no chance of success.
You can disagree. I prefer things like this much later in the year (separated by months) from a time when I actually taught the topic and showed the gimmick.
You must have seen this comment last week. It's a topic that's always on my mind.
I wasn't able to solve it in 3 minutes. It was easy enough to set up the problem as "solve ab = 9991 for integer a,b". I used divisibility tests to rule out 1 through 10, and ruled out 11 and 13 with long division before I ran out of time. Then I used a one-line Perl script to find the answer:
perl -e 'for (2..100) { print "$_*", 9991/$_, "\n"}' | fgrep -v .
A Google search for [factoring 9991] revealed the trick: 9991 = 10000 - 9 = 100^2 - 3^2 = (100+3)(100-3) = 103*97. I knew that a^2-b^2 = (a-b)(a+b), but I didn't make the connection to 9991. Cool!
jonathan--
I probably should not have used the term 'warmup'! My intent was more to set a tone at the beginning of the year. Perhaps not a single student will come up with the factoring idea and that's fine. The teacher can use this as motivation for what they WILL be able to do if one values this skill (which I do). However, my experience is that working in groups, some will arrive at the result. If you know the technique, one can do this in less than a minute. If not, 3 minutes is not enough. Again a message was being sent.
My bigger concern is with phrases like 'number trick'. When introducing factoring a difference of squares, wouldn't it make sense to motivate the topic first with concrete numerical examples such as this one (easier at first!). Unfortunately this is not always the case from what I've seen in texts or observed in lessons.
I know you've run an extensive thread on factoring over at your site and I probably should have commented long ago, so this is coming late. However, all the pedagogical and curriculum issues surrounding factoring will not easily go away so I guess it can't hurt to re-open the discussion.
novemberfive--
think of 9991 as a difference of squares...
By the way, I toned this question down! The original one involved 999,919 M&M's!
Certainly, the difference of squares is a critical algebraic form that appears frequently throughout mathematics. Of course, what may push this problem over the top is that I didn't provide the form for the student. They had to think of it which is expecting a great deal. Oh well, let's see what others say. Moreover, why not try it and see what happens. It took me several years to start taking risks in the classroom. When I did, surprising things would happen. Hey, 'ya never know! What's the worst outcome? Frustrated students who think the course will be very challenging? They'll get over it!
Funny.. the first thing that occurred to me was to take the square root and then check primes nearby.. 97 was the first guess I made and there you go!
Dan--
I've seen students with strong quantitative reasoning skills do exactly that, although they would then test that more readily with the calculator. This is what happens when we allow students to try these kinds of challenges. There might be someone like you in that class! Based on this, would you allow the use of the calculator for this question, considering the time limit? Even though my intent was to have students appreciate the power of recognizing a number as difference of squares, I certainly would not want to limit their problem-solving approaches to just one method.
Also, Dan, how do you feel about using numerical examples such as these when learning factoring?
Here's another of my favorites:
Find factors of 1001. No difference of squares here but it is a sum of cubes! Of course, I can just picture the 'Dans' in the class thinking:
"Lets' see, 1000 = 10x10x10, so let's try primes near ----"
Thanks for the comment...
Additional hint that can be given: At the end of the exercise, most of the freshmen were sick.
TC
I tested 7, 11, 13 (about 15 seconds to test against 982) and then "saw" the difference.
The numerical examples are quite important. "x" is any number. If it's true for x, then we're saying it's always true...
We can use the "trick" ie the special factoring, to perform multiplication...
37^2 ?
(37+3)*(37-3) = 37^2 - 3^2, so
37^2 = 40*34 + 3^2 = 1360+9 = 1369
Calculators are faster, but less fun.
jonathan--
The 37^2 example is a wonderful 'reverse' use of the 'difference of squares'! Thanks for reminding me of that one.
tc--
We needed some comic relief here!
There used to be several of these numerical applications of factoring and distributive forms tested on the SATs BEFORE calculators were allowed. For example, they asked questions like
65^2 + 35^2 = 100^2 - k. What is k? Five choices were given such as (65)(35) or 2(65)(35). Now, of course, these questions are given in symbolic form.
I'm tempted to give a few more like these. Anyone have other favorites like jonathan's? In the old days, individuals would perform feats of mathematical magic on stage using these mental math stratagems (euphemisam for jonathan's 'tricks'!). Apparently some have even commercialized these methods from reading the website of one of our Carnival hosts! One thing is for sure. Middle school and even secondary students are always 'turned on' by our ability to mentally compute, which is fast becoming a lost art!
Whether or not I'd let students use a calculator depends on what I was trying to get at with the question. If I was looking at it as an exercise in problem solving and number theory, then I would let them use a calculator (and give them much longer than 3 minutes!). For my students, anyway, if I wanted to use this as motivation for factoring, I would definitely show them similar examples before expecting them to be able to produce something like that on their own. I completely agree that using numerical examples is helpful when teaching factoring (and the distributive property), both as motivation, and to help students see how the pieces all fit together. But once you show them how to do this, the problem in question turns more into routine practice than a problem with a lot of depth to it.
What's the worst outcome? Frustrated students who think the course will be very challenging? They'll get over it!
No, the worst thing is frustrated students who think they're no good at math. And once that happens, they give up and stop trying.
Unfortunately, I learned this the hard way. Fortunately, I get multiple years with the same kids, and was able to restore the confidence of some middle school girls I had inadvertently scared off (before they left for high school!)
If you just do "tricky" problems like this once in a while, you're unlikely to turn kids right off, but I do think it's something to be aware of. (For our middle schoolers, non-routine math problem solving gets a full 1/3 of their math class time, so I have to be a bit more careful about making sure a reasonable proportion of the problems I give are solvable with methods we've recently seen in class.)
I teach divisibility rules for 7, 11 and 13, so factoring 1001 is one of the neat problems I give in that lesson.
I didn't recognize this one as a difference of squares, and thus did not solve it myself in under 3 minutes. (It did not help that I read the 4th fact as saying the number of freshmen was one less than the number of m&m's they each ate, which I was quickly able to confirm had no solution...) The fact that you said it was for Algebra II and above should have alerted me to look for something like that. ;-)
I think actually a "one less" version of the problem, encouraging a solution like Dan's, is one I would give my middle schoolers.
but i already *know* 1001 = 7*11*13.
because i read about this silly trick
where you have each student
write down their favorite
3 digit number (314, say),
then copy it onto their calculator
twice in succession (314314, e.g.). . .
then, with a mystic air, you "predict"
that their number is divisible
by 13 (ooh, ahh!). so they
do the division (24178, not that
that actually tells us anything).
then (mystic air again) divide
by 7 (!), then by 11 (!!) ...
and the answer is: your original #!!
at this point, one spoils
the magic act by going into
teacher mode: "now ... what's going on?"
vlorbik
vlorbik--
Yup, that's one of my favorites too! That's all part of the math teacher 'bag of tricks' to hook them. Then we 'set the hook' and the real pain begins...
vlorbik -- I didn't know that one. I think I'll be able to use that. :)
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