Mathematicians are LazyLesson Plans > Mathematics
Mathematicians are Lazy
"...because mathematicians are lazy" is a phrase that frequently passes my lips when I'm teaching high school mathematics. I repeatedly underscore to my students that mathematicians are always looking for ways to avoid extra work. This amuses the students, and also helps them to remember some things that are often forgotten.
One is the Laziest Number
Despite the old song that says "one is the loneliest number, one is actually the laziest number. Let's see some of the ways that one is a lazy number.
I tell my students that every time they write a number or a variable, that number/variable has an exponent. If there's no exponent written, it's because mathematicians are lazy. If they see an x all by itself, really that's x1; we were just too lazy to write that exponent. (Eventually, some quick thinker will say, "So does that mean the exponent also has an exponent?" And I reply, "Yes, and that exponent's exponent also has an exponent. You can see why mathematicians prefer being lazy; we'd never get anything done if we had to write all those exponents!"
I also tell them that any term also has a coefficient. If you see xy2, you should think of that as 1xy2. In fact, when we're first learning about terms, if I have an expression like "5x + 3x + x" I'll always say, "And of course that x at the end has a lazy coefficient," and I'll write the one in. This repeated reminder of the lazy one helps the students who get confused about combining like terms when one of them has no coefficient.
This is also helpful when students face expressions like 5x + 3 - (2x + 1). I remind them that every group has a coefficient as well - we're just too lazy to write it. Thus, this is really 5x + 3 - 1(2x + 1), which allows them to do the distributive property, and they can see that distributing the -1 through the parentheses reverses the signs. If a student gets stuck on a problem like this, I'll just say, "Remember that mathematicians are lazy" and they'll automatically put in that lazy one.
"Did you know," I'll ask my students, "That every number is actually a fraction?" They'll reply something like, "No, six isn't a fraction." And I'll say, "Yes it is; we're just too lazy to write it as a fraction. There's actually a lazy one in the denominator." This is helpful to students who forget what to do when they see something like : 3 x 4/5. They remember that 3 is just a lazy fraction, and then they write it as 3/1 x 4/5, and can now use their multiplication rules.
Eventually, some smart aleck will realize that the expression x is really 11x1/11 - a lazy exponent, a lazy coefficient (with lazy exponent), and a lazy denominator (with lazy exponent). Students sometimes get a kick out of taking simple expressions and making them more complex by adding the lazy numbers in.
One is not the only lazy number: zero is also lazy. When you see the expression x4 + 3x3 + 2x + 1, the mathematician who wrote this was being lazy; he didn't feel like writing the x2 term, so he left it out. But really there should be a lazy zero in there: 0x2.
This is also useful when you're talking about lines on a coordinate plane. You can have a lazy y-intercept. If the equation of a line is y = 2x, it's actually y = 2x + 0.
Yes, two is also lazy. The most common place for lazy two to show up (or, technically not show up) is in radicals. When we are writing the cube root of something, we draw the funny radical symbol, and put a little 3 in the corner. We do the same thing for fourth roots, fifth roots, etc. But do we put a two in there if it's a square root? No, we don't. And there's only one reason we don't. We're lazy.
Why does this matter? Well, my students have learned that if you have something like the cube root of x12, you just divide the exponent by the root, to get x4. Similarly, the fourth root of x40 is x10, because 40/4 = 10 But after learning this, sometimes students will draw a blank when they go back to the easy square roots. But writing that lazy two there reminds them they can do the same process.
Yes, that's right - mathematicians are so lazy that we leave out x0. Why? Because x0 is 1, and 1 is lazy.
There are situations where it's helpful to remember that x4 + x3 + x2 + x + 1 is actually x4 + x3 + x2 + x1 + x0.
Once I started doing this "lazy" discussion with my students, I found that my students became much more adept at recognizing what to do next with a problem. It made me realize how much I've internalized the lazy shortcuts I take, and how much my students need to internalize them in the same way.
Of course, there are other situations where you can tell your students that mathematicians are lazy - here are just a few*:
- exponents are a lazy way of avoiding writing a lot of multiplications
- sigma notation is a lazy way of avoiding adding a lot of things together
- A fraction bar is a lazy way of avoiding writing both a division symbol AND parentheses to group the numerator and denominator.
- fractions are a lazy way of avoiding long division (seriously - if the answer is 7 divided by 3, then it's just 7/3!)
- function notation is a lazy way of avoiding writing the same rule over and over and over.
* I do realize that some of these are not "lazy" in the same way; 25 might be a lazy way of writing 2 x 2 x 2 x 2 x 2, but there is no analogous way of writing 2n, so exponents are a necessity, not just a convenience.