Powers of 2

From my early days as an undergrad math student, I knew that powers of 2, the numbers of the form 2ⁿ, have many properties, and are very important in many areas of mathematics.
However, it was only after I became a math tutor that I realized how common questions involving powers of 2 are in standardized tests.

Before becoming a math tutor I had held positions at colleges and high schools as an associate professor, adjunct professor, math teacher, and teaching assistant. Up to that point I had memorized only the first six natural powers of 2, from 2¹ to 2⁶, these are:
2, 4, 8, 16, 32, and 64, the powers of 2 that are two-digit numbers.

Later, when I became a math tutor, and I realized how often standardized tests present students with questions involving powers of 2, then I memorized larger powers of 2, up to 2¹², or:
128, 256, 512, 1024, 2048, and 4096.

It is really a big advantage at test time, for a student to be able to recognize a power of 2 when they see such a number in a problem, because then they can write that number in its exponential format, and use the algebraic rules of exponents to work out the problem’s math easier and faster than dealing with the number in its decimal form, using only arithmetic operations, because the longer route is more time-consuming, and more attention-intensive, therefore riskier and less efficient.

So, if you are preparing for a standardized test, from the CBEST or the ASVAB to the GMAT or the CSET, know your powers of 2. Memorize them well from 2¹=2 to at least 2¹⁰=1024, and you’ll increase your chances of scoring a few extra points in the exam.

Writing upside-down, and sideways

An indirect measure of tutoring experience

Last week, a student made this comment to me:
“Wow! You write not only upside-down but also sideways!”

Then I realized I have gradually acquired this ability over the years as a direct result of my continued math tutoring practice.

As a tutor, many times you have to correct a result, or an equation the student has just written. You are sitting across the table from them, and the correction may be a minor one. Reaching for their notebook across the table, grabbing it, turning it, putting it in front of you, writing what you want to write, and giving the notebook back to the student may not be extremely time consuming; however, the couple seconds it takes to move the notebook back and forth across the table may sometimes add up to something of a hassle if you find yourself having to make a lot of corrections and/or suggestions to keep the session moving forward.
In such cases, especially when the corrections are minor, once your hand is on the notebook at the other end of the table, it is much easier just to write whatever you have to write, right then and there, without ferrying the notebook to your side of the table.
For me the process started inadvertently, just by changing minus signs to pluses. From there it went on to changing y’s into x’s, inserting parenthesis, adding missing zeroes at the end of a number, and things like that. Still easy stuff but increasing in difficulty a little bit at a time.
The easiest digits to write upside-down are 0, 1, and 8. Before long, you can write all the digits upside-down. One day, all of a sudden you find yourself writing whole formulas upside-down. By this time, most likely you have started writing a few things also sideways, because often the student sits next to you but at a 90 degree angle.
I never had any independent practice writing upside-down or sideways on purpose. The only times I write sideways or upside-down are during my tutoring sessions. So I can say this ability, in my case, is a direct result, and therefore an indirect measure, of my tutoring experience.