## The Calculator

When I entered college some years ago, the calculator was not in major use. The popular calculator then was the old TI-30. This calculator is still in my possession today. The value of the calculator has been worth it weight in gold. I continue to use this fine calculator even though it does not have the features of the new calculators. The best part is the joy of knowing I don’t have to charge or replace batteries.

Calculators are a necessity in today’s educational world. Long gone are the slide rules which I believe is better for learning. Now we have calculators to do the simple math I used to do it my head. This truly is not an advancement for humanity. These calculators are pushing us downward towards ignorance and stupidity. Look at our youth and see how many can do mathematics in their head. I still function faster than their attempts to use a calculator. Unlike my days in college where a calculator was primarily used to speed the work up and seldom if ever used during a math test. Today it is expected.

This brings me to a requirement for students going into the sciences or engineering fields. You need a good calculator but not one that is too good. Look at the SAT approved calculators for what a student should own. I have one child using a TI-83 and another using TI-84. Each of my sons has used this calculator since their high school years. Precalculus is one such class and AP statistics is another. You can get away without using one in precalculus but the textbooks and lesson plans are no longer geared for a calculatorless student. Even at the universities, there are classes where the calculator is a primary tool.

This is where I hoped parents emphasized what I call mental math during the students early math years. This would include algebra and early trigonometry. If your child can do calculations in their head then I would say the parent did well. Sadly, I do not see this with the public school students I interact with. They don’t have the ability to do the math in their head.

I highly recommend students using the TI-83 or TI-84 calculator if they are going into the fields of science or engineering. Use these calculator well before you need it. These are complicated computers that need understanding. It is painful to take a class and struggle using the calculator. Go to online sites where tutorials are given. My youngest struggled early on with his AP statistics class, because he was unfamiliar with the calculator.

Calculators are nice objects to have. I won’t deny that. Like any other tool, these items can cause unforeseen issues down the road. We become to reliant upon technology and cannot effectively integrate ourselves in a way to get the most out of ourselves. We lose the understanding and ability thus opening up the possibilities of failure whether it is upon us or what we design.

## Mathematical Mistakes

Though I am currently doing some history research, I thought it would be a nice idea to throw some mathematical myths out there for some people. There are plenty of myths or mistakes to go around, but a few of these should suffice. These reside in the algebra zone but apply elsewhere.

1. ax + b = a(x + b). An easy way to check this is to apply some values. a =2, x = 3 and b = 4. {2*3 +4 = 10} does not equal {2(3 + 4) = 14}.

2. a – (b + c) = a – b + c. Again, using a = 2, b = 3 and c = 4 you clearly see the mistake. {2-(3+4) = -10} does not equal {2 – 3 + 4 = 3}.

3.(a + b)^2 = a^2 + b^2. Apply the numbers and you see these do not equal.

4. (a – b)^2 = a^2 – b^2.

5. (a – b)/(c + b) = a/c

6. a(b + c)/(b + a) = (ac)/a = c

7. a/(b+c) = a/b + a/c I’ve seen my oldest make this mistake many times.

8. (ax + b)/(ac) = (x + b)/c

9. (a^x)(a^y) = a^(xy) Keep in mind ‘^’ means raised to the power.

10. a^(x+y) = a^x = a^y

These are but ten of the common mistakes students make. The easiest to double check yourself is to apply values to the variables and do the calculation.

## Real World Math

My sons have been involved in various STEM activities at various points in time. Few of these activities were realistic or even a sample of reality. The mathematics was limited. Nothing in the world of STEM informed my sons of the importance of mathematics. I don’t have a problem with this. These programs were an introduction to some part of science for my sons in order for the Educator and me to determine the level of interest.

There has been talk talk about using “real world” applications for math classes. This I find laughable. I have studied math and have a career where mathematics is used. Within my job are different mathematical requirements. Most of my use are your standard adding and subtracting. I thought about my work as well as those working around me. What math do we use?

The largest use of mathematics are the simple computations. These are addition, subtraction and multiplication with division a distant fourth. Seldom do I see calculators used. There is no time for a calculator check. Even when we go to the store, we do not use calculators to determine if we have the money. To me this is fundamental math. The real world of this usage is rather boring. Calculating tax on a purchase or determining the amount of money left over after a purchase. Even calculating how many miles I can go when I buy a few gallons of gas. These are the same calculations I do at work. Interestingly, I use estimations more than exact values.

I do use algebra for my work. Little of it is very complicated. My co-workers utilize algebra as much as I do. There are a few who use algebra extensively. Calculus is seldom used. I believe there is one or two people using calculus but not a daily basis. There are times I use calculus for my work. The real world application for us is no different than using the standard integration you see in college. Matrices are used by a few. Geometry is not used by me. Drafters may use it. Linear algebra is more common than most. This depends upon what the individual’s job is. Trigonometry is the big item in use. I use this to calculate volumes and other such requirements. I set the work up on excel and plug in the values. Quite a number of people use trig. We do not do research and therefor do not require what many would call higher levels of math.

This is a sample of the useage of mathematics. The reality is we are not tasked with paragraph lengthed mathematical problems. There is no hunt for what we want. What we do have is a basic understanding of math. We know how to add or do algebra. We know most of the trig functions and formulas. It is our basic knowledge that allows us to apply critical thinking to math.

When task with find the volume of a cylinder between two heights, I had to develop the method of finding the answer. My familiarity of trig and geometry allowed me to solve the problem. I understood the formulas and what they are.

I stick to a belief of knowing how to do the basics especially without calculators. If you understand the topic you can solve the problem.

## “C” is very important to you

Taking calculus in college was a transition for me. Sloppy math work was a detriment to me as it is to my oldest. I was forced to provide detailed work compared to what I did in high school. There was one part of calculus that I learned very quickly.

The letter “C” has no special value to people unless they are taking calculus. “C” is a constant that no one cares about. That is until you lose a point for forgetting about “C”. For those homeschooling families teaching calculus to their children, do not forget about the letter “C”. This is a constant you apply to the end of your answer when you integrate. If you integrate (2x)dx, your answer is x^2 + C. That “C” becomes a single point for most. If you’re lucky it is 1/2 point. My son learned this the hard way. After losing some points he basically etched into himself in order to remember. Thing was, he forgot the “C” in the next calculus class.

Never forget “C” when taking calculus.

## Rules of Algebra

What I see and hear in the colleges is a common observation and occasional complaint from math professors. Students do not know their algebra. Blame it on Common Core or Fuzzy Math; the problem continues to exist. Public schools are not correcting the problem, and private schools are not aware they have this issue to an extent. I am even certain homeschoolers have the same issue.

We do poorly with teaching algebra. I believe homeschooling parents become intimidated and allow their children too much free reign. No child is going to see the need to understand and learn the rules for algebra. It is a “set it and forget it” attitude. Mom and Dad see good scores but do not realize the missing pieces to the future math puzzle. This is why I may see more homeschooled student stray from the sciences. I hope not.

Schools have drifted away from the fundamentals to this critical thinking garbage. What good is critical thinking if you don’t know the rules? We can teach STEM and never see the results we want if the students do not know the fundamentals. Schools are more interested in science fairs or projects that truly teach little. The most it does is get a student interested in something they do not understand. Another problem with schools concerns those involved. We can lay blame on the teachers, students, parents and school district. They are looking for immediate results and not long term results. Sending teachers back to school does nothing to help education. Parents not monitoring their child’s schooling sends a poor message.

I can tell you that I was asked about school work every day growing up. My parents were involved in one way or another. They even got a tutor for me in math no less. They were involved and it was understood my schoolwork was first. A grade of B was tantamount to failure. The advantage I had over the school kids today is the lack of B.S. for me. There was none of this extra busy work given as we see in schools. There wasn’t a demand of community service. Community Service was separate from school and based on your family values.

There is a simple solution to the fundamentals. Practice, practice and practice. Quiz the student each day until they learn rules. I can still recall the rules from one of my math teachers in school where everyone was quizzed until they received a perfect score. This is what we need to do. I’ll go one step further. Have the child understand the proof. This takes more work but the reward far exceeds the effort. You can continue to quiz after algebra, too. I highly recommend this.

Science fairs and projects are not necessarily bad, but they should be done outside of the class during the weekend. You need to manage the available time for students. Evenings can be too busy. A few hours on Saturday is more appropriate. These events should not take up the entire time of the student. As a colonel of the army said to me recently, the academies are looking for youth who excel at what they like, play sports, music and do community work. This holistic approach is no different than what universities are starting to look for.

A final word to homeschoolers. Sign your child up for baseball, basketball, AYSO or some type of sport. Schools want a well-rounded student. Having great grades is not enough. Seek the organized sports for they are known. Do community service and not just through your church. There is a large bias against religion unless it is a religious affiliated school.

## Logarithm

My experience in transcribing my thoughts and knowledge is limited. Many times I do so in a hurried manner within a short period of time. Today I spent a little more time with this. I know if I take a few weeks this will come out more clearly, so I hope this give a general understanding of logarithms.

The definition of a logarithm is the “mathematical raising of a base number” to produce a given number. What does this mean?! We begin with looking at two graphs. The two graphs below are related. Can you see the relationship between the two?

The first graph represents 2^x where a = 2. The second graph is the log2(n) where n is the exponent of 2. If you rotate the second graph 90° to the left and then flip it left, you will duplicate the first graph. The logarithm function is the inverse of the exponent function.

Take 2^3, 2 raised to the power of 3. 2 is the “base” number and 3 is the exponent. The resultant of this is the value 8. The exponent is also called a logarithm. In our 2^3 = 8 example, 3 is the logarithm of 8 with a base of 2, Log2(8) = 3. Unfortunately, I am unable to properly write a log for the base should be a subscript. Keep in mind you write a logarithm as log#(#) where # represents a number.

The definition is:

**loga(n) = x means a^x = n**

Here are a few examples:

10^2 = 100 and log10(100) = 2

4^(-2) = 1/16 and log4(1/16) = -2

3^1 = 3 and log3(3) = 1

Take a look at this example: 7^0 = 1. Can you write the log for this? The answer is log7(1) = 0. Any log of one will be zero. You now have another defintion:

**loga(1) = 0**

What about this example: 5^1 = 5? Well, the log5(5) = 1, so you now know that any logarithm of the base itself is 1:

**loga(a) = 1**

Here is a tricky example to try. What is log2(2^3) ? The answer is simple when you realize a few things. You could take 2^3 and get 8, and now you have log2(8) which is equal to 3. Did you notice anything? Look at log3(3^5). What is the answer? It is 5. Hence we have another definition:

**loga(a^x) = x**

This is one of the areas I see students having difficulty. They do not understand the concept and go through school memorizing all of this without understanding. Math teachers or any mathematician likely does not recognize this deficiency. I struggled with this entering college. There are plenty of people that graduate with engineering degrees that do not understand this.

## Proof of log product rule

Youth 2 doesn’t know this, yet. He will be doing proofs for pre-calculus with me. They will be graded quizzes. He is going to hate it, but this will help him with understanding math. I believe we as teachers need to have our students understand how things are done whether it is in math or not.

One teacher I had in school, gave us the same quiz each day until everyone passed. If you passed the quiz, getting everything correct, you didn’t have to take it. I was one of the first. It would take many weeks before everyone passed with a 100%

Below is the proof for the log product rule. It is done in a pictorial format.

The product rule is: loga (xy) = loga (x) + loga (y)

You need to write the two logs in exponent form.

The next step is to multiply x and y together.

You next take the log of both sides. You are able to bring out the (m+n) as shown. Loga(a) = 1. If you substitute back m and n from the top, you get loga (xy) = loga (x) + loga (y).

Hopfully this rough proof helps you understand where the product rule comes from.