You wouldn’t believe how many seniors in high school haven’t mastered concepts from pre-algebra!

I don’t say this to be mean to anyone,

I say it to address a fact that many overlook.

At Alexander Mathematics and Physics Tutoring we start every lesson looking at two important skills:

• fractions
• and negative numbers.

I don’t care if you are 10 years old or 50 years old, this is where we always begin.

Why?

Because most people are not at all comfortable with these pre-algebra skills.

Even though they are first learned in the 4th or 5th grade!

I’ve seen it a million times, a student reaches a more advanced math class like pre-calculus and…

... the discomfort around fractions really starts to kick in.

The student will find themselves looking at a multi-step problem with many fractions scattered about and melt down.

That is why at Alexander Tutoring we ensure that every single one of our students is ultra confident in their pre-algebra skills.

This forms the base of the mathematical Jenga game.

If a block is missing from the base, eventually the structure will collapse!

Here’s a sentence we’ve all heard in math class:

“What you do to one side of the equation, you must do to the other side.”

Very true and important.

However we say it a different way:

“Whatever operation is currently written, you must perform the opposite operation to both sides.”

In other words if the equation reads:

x + 5 = 7

We first identify the operation being done to the variable.

In this case the operation is addition, the 5 is being added to the variable x.

x + 5 = 7

We then identify the opposite operation to addition, which of course is subtraction.

We then subtract 5 from both sides of the equation (where the “sides” are divided by the equals sign).

x + 5 - 5 = 7 - 5

x = 2

Quick Free Lesson

Want to see the most common mistake in the history of algebra?

See if you can tell me what went wrong here:

3x + 5 = 9

3x/3 + 5 = 9/3

x + 5 = 3

The above student correctly identified that the 3 is multiplying the variable x, and performed the opposite operation to multiplication which is division.

However did this student really do it to both sides of the equation?

The answer is no!

Only a portion of the left hand side was divided by 3.

Not the entire side!

In order to have a true statement we would have needed to divide both the 3x and the 5 by 3.

Some students have more algebraic brains

Some have more geometrical brains.

Geometry appeals to the visual learner.

While algebra appeals to the more analytical learner.

Whatever type of learner you are, we will make sure you are an expert at both algebra and geometry because they are both equally important.

One of the first things we teach at Alexander Mathematics and Physics Tutoring is the meaning of the number π.

Tragically ask any person on the street what π is and 100% of people will happily answer 3.14 and wait for you to hand them a medal.

If you then ask them why they memorized that number, as opposed to say a randomly generated number like 7.34 their stare will go blank. Nine out of ten students have no clue why they memorized the number 3.14.

A perfect example of what is wrong with how math is usually taught!

It is taught in a very "memorizational" fashion without any emphasis on the meaning behind the equations.

π is the number of times that the diameter of a circle would fit around the circumference of the circle. That is the meaning of π.

Here at Alexander Tutoring we use a compass to draw a circle and cut a piece of string to the length of the diameter.

We then ask the student to figure out how many times this piece of string will fit around the circumference.

That way the student can figure out for themselves what the meaning of π is.

After doing this exercise they will never forget the meaning of π.

It is also the origin of equations normally memorized.

For example if we present π as the number of times the diameter of a circle would fit around the outside, we can then write down the mathematical interpretation as:

π = C/D where C is the circumference and D is the diameter.

If we then multiply both sides by D we get:

(D)π = C/D(D)

C = πD

And given that the diameter is twice the radius (D = 2r) we arrive at

C = 2πr

The above equation is what is generally taught in school which the student is expected to memorize.

However we have shown that this equation originates from the formal definition of π which is much richer in content.

Once the student has a thorough understanding of π we will go on to derive the area of a circle.

This is another equation in geometry that the student is expected to memorize, but we show how the Greeks figured out the equation thousands of years ago.

This process brings in higher level concepts like the limit which gives a nice preview of calculus concepts.

For a detailed analysis please see the video in the “Our Method” section of the website. https://alexandertutoring.com/our-method/

This is where most students really get into trouble.

This is the largest jump in difficulty from the prior years math class.

It’s also the point in which all the skills you didn’t master from the last 4 years come back to haunt you.

Hate fractions and have avoided them for the past number of years?

Well now you have multi-step problems with fractions scattered about everywhere.

They are not the main point of the problems, but if you’re sweating bullets over each fraction you see then...

You may be vulnerable to a melt down.

Difficulties in pre-calculus are almost always directly correlated to a lack of algebra skills.

Now almost no students have mastered all of the skills from the prior years math class, so...

This is totally normal!

So we want to make sure that their algebra is in great shape.

That means:

• Two step equations
• Exponents
• Roots
• Graphing
• Sine
• Cosine
• Tangent
• And s.o.h.c.a.h.t.o.a

Once we make sure this foundation is in place, the student will be in a great position to learn the higher level pre-calculus skills.

We can then focus on solving:

• Trigonometric equations
• Logarithms
• The unit circle
• Polynomial long division
• And other high level pre-calculus skills

Calculus the first time around can be really daunting.

It introduces the concept of the limit.

This is pretty abstract.

It’s quite different than anything the student has seen in math class up to this point.

We make a point to introduce the concept of the limit to our students early in their career by showing a derivation of the area of a circle A = r2.

This simple derivation introduces the concept of the limit, you can see a detailed video of the procedure here:

https://alexandertutoring.com/our-method/

Many students will go the entire class without ever understanding what a limit is.

Therefore they will be lost the entire semester.

We make sure they understand how a limit works in extreme detail before moving forward

Also the student must understand that the derivative is the slope of the tangent line.

Many students don’t put this together even though it is a fundamental principle at the foundation of calculus.

From there we can learn the derivative rules:

• Product rule
• Quotient rule
• And the chain rule.

These methods for taking the derivative must be learned with extreme confidence.

Once you have these concepts down the rest of the class will run smoothly.