Monday, July 2, 2007

The Multiplication Principle - Developing SAT/Math Contest Skills

How many even 4-digit positive integers greater than 6000 are multiples of 5?

Students who have had many experiences with problems like this have a huge advantage on Math Contests and SATs. To level the playing field, you might want to consider giving your middle and secondary students a daily SAT/Math Contest Problem of the Day like the one above. Questions like this require:
(a) Careful reading skills (encourage underlining or circling of keywords
(b) Knowledge of the Fundamental Principle of Counting (most often termed the Multiplication Principle)
(c) Clear thinking
(d) Careful reasoning

The answer is 399 (pls correct this if you feel I erred!). The process is equally important. Students should be encouraged to list a few examples (preferably the first 2-3) of numbers satisfying the conditions. The challenge is to recognize HOW MANY conditions are subtly embedded in the dozen or so words in the question ! Some students will prefer to make an organized list and count by grouping, which is fine, but, as they develop, they should recognize that is the basis for the Multiplication Principle (and later on, permutations).


Mr. Lucchese said...

Maybe I am being stupid, which is entirely possible, but it seems like 399 is the number of 4 digit multiples of 10 between 6000 and 10,000 non-inclusive.

Jackie Ballarini said...


I think that is the answer to the question! Only multiples of 10 are allowed due to the even restriction, 6,000 not included due to the greater than restriction, 10,000 not included due to the 4-digit restriction. :)

Dave Marain said...

Tony, Jackie--
thanks for the comments...
I administered this question this morning to a group of about 36 students in an SAT prep class. These are strong students whose average SAT math scores are around 600. Most got this wrong because they missed a key word or words. Certainly the 'even' condition was the major stumbling block, but I believe students need to experience these kinds of problems. I know I have become a more careful reader as a result of writing problems of this type.
Many math educators, content specialists and assessment experts have reacted negatively to these kinds of problem. Typical comments:
Inappropriate for assessment purposes, unnecessarily tricky, more dependent on language skills, only good for math contests, the worst of what we want our students to be able to do...
Perhaps, but I didn't write this for an assessment. I wrote it because I believe that student's minds grow when challenged. You only learn to slow down and read more carefully when you misread or simply miss parts of the problem, the clues to the puzzle. I believe in challenging students to their limits, and not just honors students. Enough pontificating!

I teach this using the Multiplication Principle:
(4)(10)(10)(1) = 400 represents the number of possible 4-digit numbers we can form using 6,7,8,9 as the thousands' digit, any digit from 0-9 inclusive for the hundreds' and tens' digit and using 0 as the units' digit. Since 6000 is not allowed, we subtract 1 to obtain 399. I think that's right!

Of course when I gave this to the group this morning I accidentally changed the conditions which produced a different answer. My students were understanding of my carelessness. I guess I should practice what I preach!!

Do you believe that students will perform better on the harder standardized test questions if they experience these kinds of problems on a regular basis? I do! If you argue that we should not be teaching to a test, well, that's not the only benefit that accrues from doing these. Besides, test preparation is part of reality, isn't it? I will not apologize for that!

Mr. Lucchese said...

Even numbers! Eureka!!!

Jackie Ballarini said...

I love these types of problems. I too believe that all students should experience them (not just my math team kids).

As to the argument that it is not fair (too tricky, too language based...), I agree this is not a fair assessment - IF they have never seen this type of problem before. However students should be exposed to and grapple with these problems. Careful reading is a necessary skill, whether it be in a math text or otherwise.

As for teaching to the test, hmph. If a problem such as this creates the opportunity for meaningful dialog and learning (and "Eureka!!!" moments), then how is that a bad thing? In my own experience, those lessons are the ones I most remember (mostly telling myself, I'll never make that mistake again).

Tony--despite my "never again" claims, when I first skimmed the problem, I missed the 4-digit restriction. Needless to say, I quickly realized "infinty" was not the same as 399 and re-read the problem.

Dave Marain said...

tony, jackie--
Thanks again for your interest and support.
I often feel that no one sees things as I do and it's awfully lonely. Jackie, you've restored my faith! Now there's two of us!

It's strange. I work forever on developing extensive enrichment activities for this blog and get few responses, good or bad. I put up a challenge problem and I get reactions fairly quickly. Hmmm...

mathmom said...

For what it's worth, I didn't approach this using the multiplication principle, though I do use that often.

What I did was first combine the "even" and "multiple of 5" restrictions to see that the answers were going to be multiples of 10. Then I determined that there are 3999 numbers from 6001 to 9999 (inclusive). (If I'd counted wrong and got 3998 it wouldn't have made a difference in this case, but it's worth going over that with students, IMO. I see that error coming up all the time.)

1/10 of the numbers in the allowable range are multiples of 10. I notice that I am starting with one greater than a multiple of 10, so I will only get to a multiple of 10 at the "end" of each group of 10. Therefore I divide 3999 by 10 and "round" down, to get 399.

Dave Marain said...

oooh, nice! Now, our challenge is to find ways to develop student reasoning like this! You have confidence in your reasoning, you believe that your own ways of thinking can work. This doesn't happen by accident, nor does it happen by rigidly forcing methods down students' throats. However, students need to be given many problems of this type and, at some point, to be shown effective methods, procedures and strategies. In the beginning stages the student needs to learn how to make organized lists, look for patterns, count by grouping, etc. If all else fails they can always go back to this! They also need to believe it's ok to try their own ideas and discuss them to make sense of it all. This is clearly a reform approach, however, note that I balanced this by saying that effective methods need to be taught. Students cannot be expected to re-invent mathematics from the ground up! Unfortunately, this centrist position is entirely misunderstood and subverted by both sides of the Math Wars! Always has, always will be... Help me to fight the fight!!

Jackie Ballarini said...

re: "Help me fight the fight"

Why does it have to be a fight? Why can't we let students explore the problem (the way that mathmom did), compare solutions, discuss pros and cons of various solutions, then bring in our "expertise" to make suggestions. Although if a student doesn't fully grasp the Multiplication Principle, I'm ok with using whatever method works. To me, an "inefficient" student method the he/she can remember and understand is preferable to a student goofing up the same problem with a mangled attempt at the multiplication method.

Just for background, next year will be my first year teaching (although I'm teaching summer school now). I've been coaching the math team for 3 years, and working as a para for 10 years, mostly with LD and ED students. Next year I'll be teaching a "reform" curriculum.

So, I'm not so sure why it has to be a war. Why can't we take bits and pieces to find whatever works best for the students in our current classrooms?

Dave Marain said...

Ah, Jackie--
You properly dressed me down! You are absolutely right. It takes two to fight and if enough rational human beings such as yourself choose to do what is best for students and not engage in these infantile Math Wars, then eventually, the extremists will fade away.

I was using 'fight the fight' more in the metaphorical sense. You can tell from the problems I post and comments I make that I am ardently advocating for a middle ground, just as you articulated. Choose the most appropriate materials and instructional strategies that help students learn. Upgrade the curriculum and raise expectations. Unfortunately, many districts around the country have gone to one extreme or another and teachers are finding it difficult to make independent decisions in the best interests of their students.

I started this blog to provide illustrations of a middle ground position between the extremes of radical reform and rigid traditionalism. Hopefully, in some small way, my message is beginning to filter through...

Again, I will try to avoid such confrontational cliches - thanks for setting me straight!

Jackie Ballarini said...


Me, a rational human being??? Ah, wait until you get to know me a bit better, we'll see if you have the same opinion.

Didn't mean for my last post to be a "dressing down". And your message is filtering through. That is why I keep reading!

Have a great 4th!

Anonymous said...

Dave and Co.,

I read (and try to solve) just about all of your material. Stop questioning your efforts. They are a source of not only wonderful questions, but the decades of experience that is being taught to novice educators like myself, jackie and untold others...

Not to mention the mathmoms out there...

Even the experienced educators like JD have been using your material.

Keep doing this for as long as you can.

Dave Marain said...

jackie and anonymous--
thank you! I needed those words of support...

mathmom said...

I guess a use a combination of teaching methods and letting students explore on their own. Whenever we discuss problems the students have worked on, I encourage students to explain the methods they used. Usually there will be more than one method. Often there will be one I never would have thought of. Sometimes they don't come up with "my" method, so I will also show them how I did it. If I have "solutions" that might introduce yet another approach. I try to avoid saying any approach is the "right" or "best" one -- imo whichever makes the most sense to the student is the best. Though I will certain compliment a particularly "slick" solution that my kids come up with. By continuing to show them different ways to think about problems like this, they are more likely to hit upon something that works for them, that makes sense for them, and that can stick with them.

I find that for a lot of kids it takes more than 1 year of this for it to start to click. That is a great benefit I get from working with the same kids for many years in a row. I had two different girls (a 5th grader and an 8th grader) really "get it" this year. This would be, I think, the third year I was working with each of them. The older girl would have said that she was bad at math last year. :( I know she is headed for high school with much more confidence now! :)

I don't really think that problem solving is like long division, where there's an efficient method that it would be crazy not to teach. Problem solving is about creativity, about being exposed to a toolkit of methods and approaches, and finding one or a combination that will help you solve the problem. A lot of it comes down to experience -- remembering working on a similar problem.

One thing I always do is teach the logic behind the permutation and combination formulas without ever teaching the formulas (or at least, not until they've used them intuitively for quite a while). If I give them the formulas, they will forget them and have no clue how to do the problem next time they need them. If I teach them how to think about the problem, they remember that. (Maybe this is just a personal bias, since I usually have to re-derive the formulas each time I want to use them...)

In the case of this problem, I would expect beginning problem solvers to begin with a combination of a "make an organized list" approach, and then realize that there were going to be too many, maybe go to a a "start with a simpler problem" approach, and work to find a pattern. When we start out, and periodically thereafter, we talk about basic approaches like that, that they can think about when they are stuck getting started on a problem.

I am now starting to teach these basic ideas to even the youngest kids (starting with 5yos) so hopefully by the time they get to middle school it will be second nature. My newest crop of middle schoolers this coming year will all be experienced problem solvers and Math Olympiad participants. These skill definitely do develop over time and with practice, just like any other.

I agree with jackie and anonymous -- please keep doing what you are doing! I have multiple sources of problems that I use, but I always love the problems you post and the great discussions that ensue.

mathmom said...

You asked: "Do you believe that students will perform better on the harder standardized test questions if they experience these kinds of problems on a regular basis? I do! If you argue that we should not be teaching to a test, well, that's not the only benefit that accrues from doing these."

I believe students will perform better in all sorts of areas if they practice problem solving like this on a regular basis. The only caveat is that if you frustrate students with them, they may lose confidence in their abilities as math students. I always emphasize that they are working on hard problems and that I don't expect anyone to be able to answer them all. Much more than any gains on standardized tests, I think students will do better at higher levels of math instruction due to the mathematical problem solving skills (not particular techniques, thinking skills) that they will accrue from struggling with these problems.

Dave Marain said...

Your quote:
"Problem solving is about creativity, about being exposed to a toolkit of methods and approaches, and finding one or a combination that will help you solve the problem. A lot of it comes down to experience -- remembering working on a similar problem."

It's bizarre seeing my beliefs and thoughts coming from someone else! The critical part here is your confidence that all students can learn to think more critically if presented with these challenges. As educators, we are just providing opportunities. For some, the light will come on quickly, others just need more time and more guidance and patience on our part. What you're doing with middle-schoolers should be a model for others to follow...

mathmom said...

It's bizarre seeing my beliefs and thoughts coming from someone else!

That must be why I like your blog so much. ;-)