## Why Teaching Properties of Real Numbers is Important

If you are going to do a fraction review, the lesson here might be of some help.  I believe things are best reviewed in context, but this is a decent set of information that also introduces the real numbers and some other basics of math.

The PDF icon to the left has a lesson outline you can feel free to use with the PowerPoints of in any way you see fit.

The structure is all there in the lessons, but they're not over scripted.  Remember, I believe the majority of a lesson should be spontaneous.  It should be anticipated and prepared for, but how the lesson really unfolds depends on the audience.

Below you will find an overview of how and why I teach real numbers as well as two PowerPoint icons you can download and use as your own.  I only ask that you share where you found them.

Anything you purchase from Amazon.com through the banner below goes to producing more materials, and at no cost to you.

### What Good Is It?

The Real Number Line has always been one of the dullest lessons I have to teach.

Natural Numbers are the set of numbers you can count on your fingers, beginning with one.  The Whole Numbers are the Natural Numbers and Zero...Integers are ...

Blah Blah Blah

I have to teach it because it's in the curriculum.  And I always wonder, what use is it if a student knows the difference between a whole number and a natural number?

It is hypocritical of me to complain in such a fashion because I laud the virtues of education being greater than a set of skills or a body of knowledge.  Education is about learning to think, uncovering something previously unknown that ignites excitement and interest.  Education should change how you see yourself, how you think about the world.  It should enrich our lives.

Teaching the Real Number Line can be a huge first step in that direction, if done properly.

### Math is About Ideas, Not Just Computation

There are some rich, yet entirely approachable, mathematical ideas that can be introduced with the Real Number Line (RNL).  For example, a series of questions to be posed to students could be:

1.  The Natural Numbers are infinite, meaning, they cannot be counted entirely.  How do we know that?
2. The Integers are also infinite.  How do we know that?
3.  Is infinity a number?
4. Which are there more of, Natural Numbers of Integers?  How can you know, if they're both infinite?

The idea of an axiom can be introduced.  Most likely, students assume math is true, or entirely made up, but correct or incorrect, because it is written in a book and claimed to be such by a teacher.  The idea of how we know what we know and if math is an invention or a discovery can be introduced by talking about axioms.  For example:

1. Is it true that 5 + 4 = 4 + 5 ?
2. If a and are Real Numbers, would it always be true that b = b?  (What if they were negative?)
3. Is it also true that b = b - a?  How do we know that?
4. Is the following also true:  If a = b, and b = c, then a = c?  How do we know?

The idea here is not to teach students the difference between the Associative Property and the Commutative Property, but to use these properties to introduce students to math as a topic that can be discussed, and that it is not about answer getting, but instead about ideas.

For more on this topic and a few other related items, visit this page.

### Why Are Some Rational Numbers Non-Terminating Decimals?

If you had a particularly smart group of students, you could pose this question.  I mean, after all, 1/3 = 0.3333333333333333...  And yet, we are told rational numbers include decimals that can be written as a fraction (the ratio of two integers).

How it works is sometimes very clear and clean.  For example, 0.7 is said, "Seven tenths." And "Seven tenths," can also be written as the ratio of seven and ten.  And the number seven tenths is of course equal to itself, regardless of how it is written.  The number 0.27 is said, "twenty seven hundredths," which can easily be written as the ratio of twenty seven, for the numerator, and one hundred, as the denominator.  And this can continue so long as the decimal terminates.  But try the same thing with the a repeating decimal and you do not end up with things that are equal.

The algorithm to convert a repeating, but non-terminator decimal into a fraction is pretty straight forward.

But that does not address why a rational number would be a non-terminating decimal.

Click the PPT Icon to the left to download a lesson on converting repeating decimals into fractions for honors students.  It includes a proof of why the square root of two is irrational.

The simple reason that some rational numbers cannot be expressed as terminating decimals has to do with our numbering system.  We use base 10 numbers.  Our decimal system provides us an easy way to write fractions with denominators that are powers of 10.

That means that we have ten numbers, including zero, that fill up one column, like a car’s odometer.  When you travel 9 miles the odometer will read 000009.  When you travel the tenth mile the odometer will read 000010.

Not all of our number systems are base ten. While the metric system is base 10, or at least translates into powers of ten, the Imperial system is base 12 from inches to feet, but 5,280 feet to the next unit of miles, and so on.

Time is another great example of bases other than ten.  Seconds and minutes are base sixty.  You need sixty seconds before you have an hour, not ten.  But hours are base 24 because 24 hours are needed to make one of the next category, which is days.

In time, 25 minutes of an hour is the ratio:

But in base ten this is 0.4166666666666666... Our decimal system does math in base ten, not base sixty.  This is not 41 minutes!  A typical mistake would be two say 25 minutes is 0.25 of an hour.

Back to our original example of 1/3.  Not all numbers can be cleanly divided into groups of ten, like 3.  If we had a base 3 numbering system, where after the third number we moved, then 1/3 would just be 0.1.  But in our numbering system, 0.1 is one tenth.

Other numbers, like four, translate into ten more easily.  Consider the following:

The only issue remaining is that 2.5/10 is not a rational number because 2.5 is not an integer and rational numbers are ratios of two integers.  This can be resolved as follows:

Let's try the same process with 1/3.

As you can see, we will keep getting ten divided by three, forever.

This is a great example of how exploring a question can uncover many topics within the scope of the course being taught.

I hope this has caused you to pause and think of how exploring questions, relationships and properties in mathematics can lead to greater understanding than just teaching process and answer getting.

The video below is a fun way to explore some of the attributes of prime numbers in a way that provides insight into the nature of infinity.   All of the math involved is approachable to your average HS math student.

Here is a link to the blog post that goes into a little more detail than offered in the video:  Click Here.

If you find these materials valuable, you could help me create more.

## Algebraic Fractions Part 1

Algebraic Fractions Pt 1

Algebraic Fractions $–$ Rational Expressions

In this section we will learn how Algebraic Fractions can be multiplied, reduced and added or subtracted. This particular entry will cover reducing and how reducing uses the greatest common factor of all terms.

It is often the case that students that once struggled with fractions gain insight and confidence with the rational numbers.

An algebraic fraction, or rational expression, is just a ratio of two algebraic expressions. The difference between an algebraic expression and a number is the variable, or unknown value. (Note that they’re called expressions and not equations because they’re not equal to anything.)

For example, 5x, is the product of five and x. Since we do not know what x equals, we cannot carry out the multiplication. So, we just leave it written 5x.

Another algebraic expression would be 15x2. This is the product of 15, x, and x.

An algebraic fraction, or rational expression of these two could be $\frac{5x}{15{x}^{2}}$ .

This expression can be reduced and below we will see two ways to approach reducing algebraic fractions.

$\frac{5x}{15{x}^{2}}$ = $\frac{5x}{5x}\cdot \frac{1}{3x}$

These are equal because when multiplying rational exponents you multiply the numerators together and then multiply the denominators together.

It is useful to separate 5x and 15x2 in this fashion because a number divided by itself equals one.

$\frac{5x}{15{x}^{2}}$ = $\frac{5x}{5x}\cdot \frac{1}{3x}$

$\frac{5x}{15{x}^{2}}$ = $1\cdot \frac{1}{3x}$

So this would be: $\frac{5x}{15{x}^{2}}$ = $\frac{1}{3x}$.

To reduce you find what factors the numerator and denominator share and recognize that those shared factors are being divided by themselves, resulting in the number one.

The greatest common factor is what gets divided out of both the numerator and denominator. Another way to see this is below:

$\frac{5x÷5x}{15{x}^{2}÷5x}=\frac{\frac{5x}{5x}}{\frac{15{x}^{2}}{5x}}=\frac{1}{3x}$

This method is less clear to see, but the math is the same.

Regardless of the method, the key piece of information required to reduce is the greatest common factor. The greatest common factor of two expressions is the largest expression that divides into the expressions in question.

For example: has a greatest common factor of 3xy, because 3xy is the largest thing that divides into all three terms.

Let’s look at another example and use a table for factoring.

15a4b3, 20a7b, 30a7b2

 15a4b3 20a7b 30a7b2 3$•$5$•$a$•$a$•$a$•$a$•$b$•$b$•$b 2$•$2$•$5$•$a$•$a$•$a$•$a$•$a$•$a$•$a$•$b 2$•$3$•$5$•$a$•$a$•$a$•$a$•$a$•$a$•$a$•$b$•$b

These are all of the factors of each of these expressions. To find the GCF we can make a list of the repeated factors, factors that are in common between all three expressions.

5$•$ a$•$a$•$a$•$a$•$b

If we divide this GCF out of each term, we would be left with:

 15a4b3 3$•$5$•$a$•$a$•$a$•$a$•$b$•$b$•$b = 3b2

 20a7b 2$•$2$•$5$•$a$•$a$•$a$•$a$•$a$•$a$•$a$•$b = 4a3

 30a7b2 2$•$3$•$5$•$a$•$a$•$a$•$a$•$a$•$a$•$a$•$b$•$b = 6a3b

Where this will come into play is with reducing something like:

$\frac{15{a}^{4}{b}^{3}-20{a}^{7}b}{30{a}^{7}{b}^{2}}$

By dividing out the GCF of all three terms we are left with:

$\frac{15{a}^{4}{b}^{3}-20{a}^{7}b}{30{a}^{7}{b}^{2}}=\frac{3{b}^{2}-4{a}^{3}}{6{a}^{3}b}$

Reducing:

To reduce an algebraic fraction all terms must have a common factor. Terms can be separated by the fraction bar or by addition or subtraction. The expression below has three terms, two in the numerator and one in the denominator. In order to reduce, all terms must share a common factor. What students will often do when reducing this expression is reduce the a’s, just leaving the expressions of $–$b. Sometimes, they will realize that a divides into itself one time, so they will write
1
$–$ b.

If we assign some relatively prime numbers for a and b, and evaluate each of these figures we will see that they are not all equal. If the reducing was correct, each expression would be equal.

Let a = 5, b = 3

Only Figure 1 is correct. The others are incorrect because in order to reduce all terms must have a common factor. Here is why.

The order of operations governs the process in which we perform mathematical calculations. The fraction bar groups together the terms in the numerator, even though there are not any parenthesis. Operations grouped together must be carried out before any other operations. Division is reducing, which takes places after the group’s operations are completed.

So why can we divide (reduce) before carrying out the group’s operations? Consider the following example for an idea of why this is before reading the why this works.

$\frac{15x+5}{10x}$

The GCF of all the terms is five. This expression could be written as it is below.

$\frac{15x+5}{10x}=\frac{5\left(3x+1\right)}{5\cdot 2x}$

And we could write that as follows:

$\frac{5\left(3x+1\right)}{5\cdot 2x}=\frac{5}{5}\cdot \frac{3x+1}{2x}$

And five divided by five is one. The product of one and anything is, well, that anything. Multiplying by one does not change the value (that is why one is called Identity).

$\frac{15x+5}{10x}=\frac{5\left(3x+1\right)}{5\cdot 2x}=\frac{3x+1}{2x}$

The reason we can reduce before completing the operations in the group (numerator in this case), is because of the nature of multiplication and division being interchangeable. For example:

$5\cdot 3÷5=5÷5\cdot 3$

You may object here because in a previous section we showed how the order of division cannot be changed without changing the value. For example:

$\begin{array}{l}8÷4=2\\ 4÷8=\frac{1}{2}\end{array}$

$\frac{15x+5}{10x}=\frac{5}{5}\cdot \frac{\left(3x+1\right)}{2x}=\frac{3x+1}{2x}$

Without going into too much detail, division is multiplication by the reciprocal, and there is multiplication by the same factor taking place in both numerator and denominator. So when reducing, you are simply dividing out that common factor before multiplying it, which is mathematically sound.

Regardless, it must be understood that to reduce an algebraic expression each term must contain a common factor. In the expression remaining from the example above, two of the terms contain a factor of x, but not the third. To reduce (divide), before adding the numerator together, would be in violation of the order of operations.

Practice Problems:

Reduce the following:

1.     $\frac{32{x}^{2}{y}^{4}z}{14{x}^{5}y}$

2.     $\frac{3a}{9{a}^{2}}$

3.     $\frac{5{x}^{m}{y}^{3}}{15{x}^{m}y}$

4.     $\frac{27{a}^{2}b+3{a}^{2}b}{99{a}^{5}{b}^{3}}$

5.     $\frac{7xy-2}{4{y}^{2}z}$