MathMaine

Solving Math Problems At The Board

Many students perceive their teachers to know more than they really do (Math teachers in particular). If a Math student who just observed a teacher solve a problem at the board is asked “What do you think was going through the teacher’s mind as they solved that problem?”, I suspect the average answer would be something very specific, like “This is a second degree polynomial in standard form, so solving this problem will require precisely six steps, the first of which is…”

In reality, the jumble of thoughts in a math teacher’s mind probably go something more like: “What looks easiest to simplify first? Oh… I see two places, no make that three, that I could start… but which should I choose? Does this look like it will take a lot of room to solve? If so, I had better organize my work a bit more… OK – now that I have simplified things a bit, what options do I see from here?”

Teachers and mathematicians do not “see” the entire series of steps needed to solve a problem before they start work on it. Instead, they usually seek to take whatever step looks like it will simplify the problem the most, and have no clue (yet) what will follow that. Once that first step has been completed, the appearance of the result influences what is tried as a second step. It is an iterative process, one step at a time, with a re-evaluation of the situation being done after each step. There is much uncertainty in the process, even for a math teacher.

Keep Your Eye On The Variable

The following equations all have a similarity:

$y = |x - 8| + 5\\*~\\*y = 4(x - 6) - 7\\*~\\*(x - 3)(13x + 11) = 0\\*~\\*y = (x + 1)^2 - 9$

The similarity is that they all have expressions like (x- 6) or (13x + 11), which are often either translations or factors. Situations such as this occur with:
– Point-slope form of the equation of a line
– Vertex form of the equation of a quadratic
– Horizontal or vertical translation of any function
– Factors of a polynomial
– “Bounce point” of an absolute value function

Procedural vs Intuitive Approaches

Life is full of alternatives. Would like fries or coleslaw with your meal? Should you put on your right or your left shoe first? Should you attempt to solve a math problem using algebraic procedures, or your intuitive sense of the situation?

Life is also full of false choices: there are many occasions when you do not have to make a choice unless you wish to. You could have fries with a side order of coleslaw. If you wear loafers, you could slide your feet into both shoes at the same time. And many math problems can be solved quite successfully using a combination of intuitive reasoning and algebraic procedures.

Life Skills Learned In Math Class

One of the hardest questions for many math teachers to answer in a way that is relevant to students is: “why do I need to know this?” “For the next course you take”, the easiest answer in many cases, does not answer the question that was usually being asked. My answers to this question obviously depend on the topic being studied at moment, and I don’t have “good” answers for all topics… but here is my list of key life skills I learned directly or indirectly from math class, with some examples of situations where I find them indispensable.

Analyzing Linear Equations: a summary

Towards the end of the unit(s) on Linear Equations and their graphs, students can feel a bit overwhelmed. The following is an attempt to summarize and link the key concepts you need to be comfortable with.

Lines

What is the least amount of information you need to know in order to be able to identify a line exactly? Two pieces of information: a point that the line passes through, as well as either a second point on the line or the line’s slope.

Equivalence Deserves More Attention

Most students taking courses in Algebra or higher seem quite comfortable with the idea of “equivalent fractions”: improper or unsimplified fractions all of which evaluate to the same decimal value. An example would be

$\dfrac{2}{3}=\dfrac{4}{6}=\dfrac{12}{18}=\dfrac{60}{90}=0.\overline{666}$

To create such fractions, multiply whatever fraction you wish to start with by 1 (the multiplicative identity) in the form of a fraction whose numerator and denominator are the same:

$\dfrac{2}{3}\cdot \dfrac{2}{2}=\dfrac{4}{6}=0.\overline{666}$

$\dfrac{2}{3}\cdot \dfrac{6}{6}=\dfrac{12}{18}=0.\overline{666}$

The key concepts here are that
a) an infinite number of equivalent fractions can easily be created, and
b) while all these equivalent fractions sure look different, they all represent the same decimal value or simplified fraction.

Turning to algebra, the very similar concept of “equivalent equations” is helpful in explaining how to solve algebra problems. I like to define “algebra” as: a set of rules for changing the appearance of an expression without changing the quantitative relationship that it defines. This is exactly what was being done with the fractions above.

On the origin of algebra problems

As you are solving a math problem, have you ever wondered how textbook authors come up with algebra problems that have simple answers?

Just as you use inverse functions to solve an algebra problem, problem authors reverse the solution process to create a problem: they start with the answer. From there they use the same principles that are used to solve an algebra problem, except instead of simplifying the equation at each step, they seek to make it more complicated by substituting a series of equivalent expressions and/or performing a series of operations to both sides of the equation.

Suppose you want the answer to be

$x=3$

Pick something to do to both sides, say multiply by 2:

$2x=6$

Do something else to both sides, say add 4:

$2x+4=10$

I could add 4x to both sides… why 4x? Because I felt like it:

$4x+2x+4=4x+10$

I could rewrite the 4 on the left side as 2+2:

$4x+2x+2+2=4x+10$

Now to add another step to the solution process, I’ll factor part of the left side:

$4x+2(x+1)+2=4x+10$

And factor part of the right side too:

$4x+2(x+1)+2=2(2x+5)$

And voila! An algebra problem which is guaranteed to have x = 3 as a solution. Try creating several problems this way yourself. Now that you know how to create your own algebra problems, and once you get some practice creating them, you may find it easier to solve algebra problems.

Word Problems… !#$%@;*!!

Word problems can be… frustrating. Most of their reputation arises from their use of words to describe a quantitative problem. And if the problem’s author did not choose their words very carefully, you’ve got Trouble (with a capital T). So why are so many word problems assigned? Because they are more similar to the quantitative problems you might encounter in life than many of the practice problems in your textbook: you have to supply some insight and organization in order to arrive at a solution.

Just about every level of mathematics, not to mention chemistry and physics, seems to send periodic spasms of word problems your way. The SAT makes liberal use of them as well. Since you cannot avoid them, you had might as well learn how to solve them once and for all.

Problems fall into four categories

Math and science problems fall into four categories: Easy, Medium, Ugly, and Hard.

Easy Problems are ones you can solve with no difficulty in a short time. An example from Algebra I might be:

$3x+2=8$

The problems that come at the beginning of each group of problems in a textbook are usually Easy Problems. If you had a pulse during class when they were explained, you can probably do them without hesitation.

Lost points on a problem? What to do…

People don’t like losing points for errors in their solution of a problem. So, what can you do to insure you won’t lose points again the next time you are given a similar problem? Most folks seem to look through the corrections, then perhaps ask the teacher to solve the problem for the entire class, and perhaps even make a note to study that type of problem before the next test or quiz. Few do more than that… and the next time they are given a similar problem the probability they will lose points unfortunately remains greater than zero.

The solution I advocate to this dilemma involves a combination of repetition and self-awareness.

Every problem you lose points on, whether on homework, a quiz, or a test, should be copied neatly onto a new piece of paper, and put aside for a day or two before being solved again. When you sit down to solve it again, pay careful attention to how you feeling while you are working, to the pace of your work, and to your train of thought. Are you feeling frustrated or hesitating anywhere during the problem? Are you uncertain about which step to take next at any point? Do you have any doubt that your answer is correct before you compare it to the known correct answer? If you answered “yes” to any of these questions, then copy the problem onto a fresh piece of paper again, and put it aside for a day or two… then repeat.

Getting the most out of standardized test (SAT, ACT) practice books

1) For the SAT, I prefer the study guide and practice tests published by The College Board (it can be found at Amazon: The Official SAT Study Guide (2016 Edition)) as it contains good scoring guides and is published by the authors of the test. The College Board is now also providing on-line practice test resources, as well as an app for your phone which provides daily practice problems.

2) Most formulas needed to answer SAT questions will be given at the start of the test section. Overall, it is more important to know how to use a formula than it is to memorize it. In studying for the SAT, focus on understanding the mathematics instead of memorizing facts or formulas.

3) Always time yourself when taking a practice test. This will help you develop a sense of when it is time to move on to the next question without having to look at a clock.

Operations are taught in pairs

Many High School students I have worked with have not spent much time pondering the sequence in which math topics were taught to them. So, it can be interesting to step into the “waaay-back” machine and investigate this question a bit:

What was the very first arithmetic operation you were taught (probably in first grade)? What operation were you taught next? Why did your teacher choose this sequence?

If you followed the usual path, the first operation you learned was addition, and the second subtraction. Addition is the operation that describes things being joined or collected together: if I have three cookies, then two more are given to me, I add the two numbers to determine how many cookies I have. Addition is probably the most frequently used math concept in daily life.

Improve Your Math Grade: Use Scrap Paper

Never solve math homework problems on the piece of paper you intend to hand in (unless it is a problem that is very, very easy for you).

I used to do my work on the same piece of paper I intended to hand in, and when I encountered a problem I was uncertain about… I froze. I did not dare write anything incorrect on the page I was going to hand in. If I did, I might have to recopy all my work in order to end up with a neat-looking page to hand in. So, I would just skip the problem, telling myself I would ask the teacher about it in class next time. The result was that I greatly slowed my learning how to do that type of problem.

I now know that if I figure out how to do a problem all by myself, I will remember how to solve it for years. But, if I ask someone else how to do a problem I was struggling with, I will probably forget their solution within minutes. So, if my goal is to do well on a final exam a few months from now, I am better off trying to figure the problem out by myself.