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.

Confusion type: unexpected answer

When the answer to a problem shows up in an unexpected or unfamiliar format, the student very well may fail to realize that result is the answer to the problem.  

Confusion:
Are we there yet?
Where is the answer?
This cannot possibly be the answer, can it?
This type of confusion can occur at any level.
It most commonly shows up in Algebra, the first time students are asked to find equations of lines, circles and so on. Before that point, chances are they had only been asked to find numerical solutions to equations, or systems of equations, or to simplify algebraic expressions.
So, up to that point, for them an equation is understood as the problem, while the answer needs to be a number, or a simplified expression. Therefore, in their minds, an equation cannot be the answer to any problem.
This can happen again and again, in many forms, even in Differential Equation courses. For example, when the answer to some problem takes the form F(x, y) = c, meaning the variable ‘y’ is interpreted to be an implicitly defined function of ‘x.’ The same kind of confusion as mentioned above can take place if the student is strongly expecting the answer to be an explicitly defined function, in the traditional form y = f(x). At that point the solution given by the equation F(x, y) = c may seem like nothing, like no answer at all, just one more step in the process of finding the “real” answer. However, this impression is of course mistaken because in those particular cases F(x, y) = c is the real answer, since an explicit definition of the form y = f(x) for ‘y’ is nowhere to be found.

Ratio Word Problems in Standardized Tests

Look for all the numbers the problem does not show you

Standardized tests like GMAT, GRE, CBEST, and ASVAB include ratio word problems. These may be, for example, problems about mixing water with alcohol, or about the ratio of girls to boys in a classroom, or any other type of situation where it makes sense to talk about ratios. There is a consistent pattern that shows in nearly every ratio word problem found in standardized tests. They give you the basic proportion between two parts, and then they ask you a question about the total. Or they give you the ratio between the total and one of the parts, and then they ask you a question about the other part. To give a simple example, let’s consider this problem:
In a certain school, the ratio of girls to boys is 5 to 7. How many students are there in a classroom with 15 boys, if the same girls to boys ratio applies to that classroom?
Notice how the given ratio is that of girls to boys but then the question is about the total number of students in the classroom. That is typical of these ratio word problems, and it tends to confuse some students, especially at the beginning of their preparation period. If you are preparing for a standardized test, when you see this type of problem make sure you keep track of all the quantities involved, all the different parts as well as the total, not only the parts that come with the numbers in the given ratio. Look for the numbers the problem is not giving you. More often than not, the key to the solution is in those numbers that pertain to the situation but are not shown in the phrasing of the problem.

Look for Solutions with Less Math and More Logic.

One instance where often “less is more.”

The following question is an excellent guideline for solving math word problems:
“How can I solve this problem by doing the least possible amount of math?”

Oftentimes there are several pathways from the setting of a problem to its final solution. Some routes are safer, while some are riskier, more error-prone. Some routes are faster, while some are time-consuming. Some routes are clearer, while some may be confusing.
Usually, the routes with more elementary operations (especially long division), and bigger numbers, tend to be lengthier, longer, and riskier, because adding, multiplying, and dividing big numbers or expressions requires a laser-focus attention. There are plenty of opportunities for doing silly mistakes during these calculations. Besides, it is easy to lose sight of the big picture when worrying about the accuracy of the calculations.
Factoring whole numbers and algebraic expressions is a good habit because it allows you to simplify some expressions before diving into the calculations, so you can operate with smaller numbers, gaining time, and accuracy.
Using logic is a very good habit, too. Many problems lend themselves to solutions that involve more reasoning, and less calculation. This is usually a good thing because these solutions tend to be clearer, and shorter.
Organizing all the information about a problem in a way that makes sense to you, is an excellent habit because this way you keep track of where you are and what you are doing all the time through the problem, and having all these references available makes it easy to retrace your steps, and identify any possible mistakes.
Go visual at any opportunity. Pictures, drawings, charts, graphs, and tables often are a huge help in writing down the right equations, or even in avoiding equations altogether sometimes.
There are many problems you can solve with a drawing and a little logic. Just because the problem is a math problem, that does not mean you need to write down an equation to solve it.
Focus on your possibilities, on what you can do. Organize the information in a logical way, using a drawing, or a table. Above all, try to spend the least possible amount of time and energy doing long, detailed, time-consuming calculations. Instead, simplify the expressions, and ask yourself logical questions about the problem.

Missing Pieces of Information

Some search for doors, sometimes some do not want to see them

Last week I showed a student how to solve two linear equations in two unknowns. He knew perfectly well how to solve one equation with one variable but did not know how to combine two separate equations into one.
Also last week another student made the remark: “I do not know how to start solving this problem. What does ‘isoceles’ mean?” As soon as I gave him the definition of an isoceles triangle he successfully proceeded to solve the problem.
Earlier today another student asked me: “What is a frequency histogram?” When I explained the concept to him, he found it very clear. He said: “Just that? Documenting the numbers in a graph? That is pretty simple!”
Most times students take the initiative, and they spontaneously ask the meaning of terms they are not familiar with. Sometimes however, some students are near some sort of saturation point, and they do not want to even think about the remote possibility that maybe there is a concept they do not know, or a technique they have not seen, and they need this new information to solve the problem at hand. In these rare occasions they keep trying to solve the problem with only the insufficient tools they already have in their problem-solving toolkit.
Writer Kenneth Grahame said “The strongest human instinct is to impart information, the second strongest is to resist it.” So, I choose my words carefully when telling them there is something extra they absolutely need to know first before having any chance of solving the problem. Many times I let them finish their attempts, and check the solution in the back of the book so they realize their approach was wrong without me telling them so before hand, because that could increase their resistance.
There are several problem-solving techniques or approaches that seem indeed artificial, weird, or mystifying the first time around. Once you see how they work, and you use them a couple times, they become perfectly natural, and then you wonder why you never thought of that before.
A perfect example of this I saw also last week with another student.
It was a probability problem involving three coins. For me it is quite amazing to watch time and again how students keep trying to solve these problems by reasoning only about the three separate coins, as if the relevant probability space had only three points. The strong insistence in this naive approach is only matched in its consistency by the strong surprise students show the first time you show them the full eight-point probability space by branching out the development of the experiment at each successive flip, and recording the eight different combination triples. It is really interesting. Somehow these once missing pieces of information act like doors to a whole new realm of math knowledge when they are presented and opened. Many times the student’s reaction reminds me of that feeling of “Wow! I never thought that was a door!” I get when watching some sci-fi movies.

Finding Problems That Motivate Students

Time units may help with multiplication practice

According to my experience each child is a unique learner. Generally speaking, students are more motivated to solve a particular problem when the problem relates to something they find real, meaningful, or important. However, different children usually assign different degrees of importance to the same thing. So it is always helpful to find a topic that holds a student’s interest, and that can be easily connected to math.
For example, let’s say you want to help a student who is struggling with multiplication tables. You can start a session by asking how many hours are there in a day, how many minutes in an hour, days in a week, and so on and so forth. Most third graders will know the answers off the top of their head. Then you proceed to ask how many minutes are there in a day, how many hours in a week, and so on and so forth. Children will realize these are multiplication problems but each child will react with a different degree of enthusiasm –or apathy– to find the answer.
Students who somehow have a strong connection to time units will diligently work out these multiplication problems all the way down to the number of seconds in a week. With a little bit of help they will continue working until they get all the numbers right, even if they struggle with the multiplications tables all along. They do this because they want to know the answers. In their mind, these problems are real, not just an empty drill.
Some other kids do not care at all how many seconds are there in a day, so for them this particular set of problems will not be very motivating. You will have to find a different set of problems for them.
Even when students understand the concept of multiplication, and the basic rules to multiply numbers, they may feel that some questions do not justify doing all the work necessary to find the answer. It depends on how real or important the questions seem to them, because that is what makes them want to find the answers.