The Power of Weenie Numbers
In all my years of teaching middle school mathematics, I’ve been exposed to many an “impossible” problem which can be solved by studying the pattern involved. Some people have been endowed by our Creator with the ability to notice, recognize, and apply such patterns to solve given problems. Most people, however, must be taught these skills.
Part of the problem with mathematical achievement in this country stems from our commonly held belief that, in order to succeed in math, one has to be gifted, blessed, anointed. Not so; one has to be properly trained.
Let me illustrate this premise by stating a common middle-school math problem that any trained student should dismiss as “too easy.” Meanwhile, many students and adults may well reach with the old proverbial, “Say what?” when presented with this problem:
Problem: “What is the units digit in the expression 20082009?”
Yes, the base, 2008, is multiplied by itself a total of 2009 times:
2008 x 2008 x 2008 x 2008 x 2008 x ... x 2008… and so on.
How on earth can anyone be expected to multiply 2007 by 2007, just once, let alone 2008 times, and arrive at the answer in less than one minute? Impossible, isn’t it?
Not really! One problem-solving technique that I stress with my students is “Start small, then work your way up.” I work with a colleague who refers to “weenie numbers.” Let’s do just that: start small, with “weenie numbers!”
The expression “weenie numbers” is not my original, but rather the invention of my friend and colleague at The Barnesville School, Mr. M. Hitselberger. So, let’s examine how the use of Mr. H’s weenie numbers will help us solve this “impossible” problem.
20081 = 2008. So 8 is the units digit. Keep in mind we’re only interested in the units (last) digit. Nothing else matters. 20082 = ___4. So 4 is the units digit, since 8 x 8 = 64. 20083 = ___2. So 2 is the units digit, since 64 x 8 = 512. 20084 = ___6. So 6 is the units digit, since 512 x 8 = 4096.
20085 starts a new cycle, with the units digit going back to 8, then followed by 20086 ending in 4, 20087 ending in 2, and 20088 ending in 6, and so on. Every fourth power starts the new cycle. Dividing 2009 by 4, the remainder must be 1, so the units digit in 20082009 must be the same as 20081. The answer, therefore, is 8.
Welcome to the world of weenie numbers – a problem-solving technique that simplifies problems by reducing the frighteningly high numbers found in so many math problems, down to small, manageable ones.
Another example: “What is the value of 111,111,1112?” Reach for the calculator! Oops, the cheapo four-function calculator screen has room for only 8 digits, so how is one going to solve this problem?
So the next thing is to set up the problem and place 111,111,111 directly under 111,111,111 and multiply 1 x 1, etc.
Weenie numbers to the rescue:
1. What is 11 x 11? Easy, even some high school kids, for whom a calculator is more powerful than God, know that the answer is 121.
2. So, what is 111 x 111? Easy. That’s 12321. Easy, quick to do, even with pencil and paper.
3. See the pattern now? 1111 x 1111 is 1234321. It’s got to be. Trust yourself, no need to work it out.
4. That means that 111,111,111 x 111, 111, 111 must be 12345678987654321.
And it is. Weenie numbers – start small, work your way up.
Since time is of the essence in such mathematics lesson, the teacher should be aware of who “gets it” and who “doesn’t get it,” and intervene to ensure that the class proceeds in a fluidly smooth manner. There are more similarities than there are differences; it is the teacher's job to demonstrate these patterns to the students, so they can tackle “impossible” problems such as: “What is the units digit in abcdwxyz?” with the assurance that any such digit can be predicted, if only the problem is analyzed correctly and confidently.
These are the kinds of discovery lessons that any self-respecting mathematics teacher and MATHCOUNTS coach must use to guide students into such discovery – in an efficient, teacher-controlled environment. By doing so, students should be able to attack problems of a similar nature, the so-called “impossible” problems, with the assurance that these are doable and indeed quite possible.
The job of the mathematics teacher is to empower all our students to solve problems that depend on the study of patterns, which is basically what mathematics is all about.