In the last section we
saw both how to reduce Algebraic Fractions, which if you recall, are also
called Rational Expressions, but also how the math works. Because of the relationship between division
and multiplication, and multiplication’s commutative property, we can reduce
Algebraic Fractions like the one below:
When dealing with
Algebraic Fractions, your work is not done until you’ve reduced
completely. The example above is the
unfinished answer to the first problem we will do to introduce addition and
subtraction of Algebraic Fractions.
Since these have a
common denominator, we just add the like terms in the numerator. Note, in math when we say add, it means
combine with addition and subtraction.
Combining like terms, we end up with:
But since each term has a factor of 3, we can reduce
each term by 3:
Our answer has four total terms. While some share a common factor, not all
four terms share a common factor, so we are finished.
If the denominators are
the same, you just combine the numerators.
With subtraction it is
slightly trickier. Let’s simplify a
problem and see how this works.
5 4 + 3
5 (4 + 3)
The two expressions
above are not the same. The first equals
four, while the second is -2. The
parenthesis make a group of the four and three, which is being subtracted from
the five. The four and the three are
both being subtracted from the five.
But, great care is in order here.
It is easy to mess up these signs.
5 (4 3) = 4
Because 5 4 - -3 is 5 4 + 3.
Remember that fraction
bars also create groups in Algebra. So,
instead of parenthesis, you will see:
This is the same as:
An example with
variables would be:
Exponents are repeated multiplication, they do not change from
addition. Also, in order to be like
terms, the variables and exponents must be the same. x3
and x5 are not like terms.
Caution and care are in order when dealing with
subtraction and Algebraic Fractions!
Combining like terms we get
Since there is not a common factor between all
terms, we are done.
Adding and subtracting
Algebraic Fractions with unlike denominators involves finding the LCM (lowest common
multiple) of each denominator. We will
restrict our denominators to monomials for now, as to keep this appropriate for
beginning Algebra students.
Let’s begin with the
denominators a and b.
All we know is that a and b are numbers that cannot be zero, but
we don’t know their exact value. So, we
assume they are relatively prime, making their LCM their product, a×b.
We arrive at common
denominators through multiplication.
Don’t get confused here, we are multiplying by a number that equals one
if it were reduced, but we don’t want to reduce until we are finished. Also, note, that since e are multiplying by
one, the expression will look different but will have the same value. It is not unlike the difference between a
twenty dollar bill versus a ten and two five dollar bills.
Our denominator will be
To change a into ab, we multiply by b. We multiply b by a,
writing the variables in alphabetical order will help to recognize that they
are the same. (Sometimes students write ab and then ba. While they’re the same,
the order in which they’re written can confuse you.)
In many respects,
adding or subtracting Algebraic Fractions is easier because there is less
calculation taking place.
Let’s walk through an
uglier problem involving subtraction and negative signs.
We need to find the LCM
The LCM will be 5x2y.
So we need to multiply the second fraction by y over y.
Now we are multiplying the entire group of 4x 2 by y,
not just whatever term is written next to the y.
Normally I would not write the step above, but did
so to help make sure you understand why the fraction on the right is 4xy, not just 4x. We have to distribute the
y to the entire group.
Now with care for that
negative sign, let’s put it all together.
which will become:
Combining like terms, we get:
Before we can say we’re finished we need to check
for a common factor (GCF) that could be divided out. These don’t always exist but if one does, and
you had the answer correct up to that point, it would be a shame to mess up the
last little step, so check.
In review, you need a
common denominator which will be the LCM of the denominators. You must take care to both distribute
property in the numerator, and watch for sign errors, especially with
subtraction, when combining like terms.
The last thing is to check for a common factor between all terms when
you’ve finished combining like terms.
Instructions: Add or Subtract as
It is often the case,
for the mathematically-insecure, that the slightest point of confusion can
completely undermine their determination.
Consider a beginning Algebra student that is learning how to evaluate functions
A confident student is
likely to make the same error as the insecure student, but their reactions will
be totally different. Below would be a
typical incorrect answer that students will make:
The correct answer is
3, and the mistake is that -22 = -4, because it is really subtract
two-squared. And when students make this mistake it provides a great chance to
help them learn to read math, especially how exponents are written and what
Here’s what the
students actually read:
A confident student
will be receptive to this without much encouragement from you. However, the insecure student will completely
shut down, having found validation of their worst fears about their future in
There are times when
leaving traps for students is a great way to expose a misconception, and in
those cases, preemptively trying to prevent them from making the mistake would
actually, in the long run, be counter-productive. Students would likely be mimicking what’s
being taught, but would never uncover their misconception through correct
answer getting. Mistakes are a huge part
of learning and good math teaching is not about getting kids to avoid wrong
answers, but instead to learn from them.
But there are times
when explaining a common mistake, rooted in some prerequisite knowledge, is
worth uncovering ahead of time. This -22
squared is one of those things, in my opinion, that is appropriately explained
before the mistakes are made.
Be fair-warned…this might seem pedantic, and I agree, it is. But, with good cause. The point of this discussion is to cause students to pause and think about something they’ve always done in math without any understanding. It’s true, you can combine like terms, but why? To that point, and I do not say this to put anybody down, but I bet most math teachers have never wondered this on their own.
Why Can’t We Combine Unlike Terms?
I pose this question to Algebra 1 students every year, for over a decade now. The responses are always with an air of incredulity. Their looks say to me:
Come on Mr. Brown, you should be smarter than to ask such a dumb question!
When I press for an answer they’re suddenly tongue-tied. The most common answer, often spit at me with some venom is:
Because they’re not like terms.
Okay then, why can you add things that ARE like terms?
Are We in Violation of PEMDAS
The order of operations is the set of instructions by which math is to be performed. It’s kind of a big deal! Mess it up and you’re in a lot of trouble. And yet, how is it that we can combine like terms if there’s multiplication occurring first?
Consider 3x + 5x – 2a. Even the most average of 9th grade, Algebra 1 students will tell you that obviously equals 8x – 2. My question is this: How can you add 3x and 5x if there’s multiplication taking place. The three and the x are multiplying, right? And in the order of operations, doesn’t multiplication come before addition?
When I ask students why like terms can be combined even though there’s multiplication taking place and they add before multiplying, they just say, “cuz.” This is a classic example of something that’s been taught and accepted without understanding. It’s not a complicated issue, but promoting mathematical literacy is my crusade of late.
Multiplication is Repeated Addition
Consider the product of four and five. I remember being a young child and counting groups of five to find the product of five and a number. 4 x 5 = 5 + 5 + 5 + 5
Good to go on that point?
Then 3x = x + x + x
And 5x =x + x + x + x + x
Then 3x +5x = x + x + x + x + x + x + x + x = 8x
Because multiplication is repeated addition, and we’re adding the same thing repeatedly, we can combine the coefficients of x.
Why Can’t You Combine Unlike Terms?
Consider our original expression: 3x + 5x – 2a. Two times a is a + a. Three and five x makes 8x. But x and a are probably not the same thing (we don’t REALLY know because they’re unknown), so we cannot combine them with addition.
8x + 2a
x + x + x + x + x + x + x + x + a + a
This cannot be simplified further than 8x + 2a.
To combine unlike terms would be in violation of the order of operations. Since x and a are different, we would need to multiply 8 and 2 by them respectively, before adding them.
I hope this short discussion has caused you to pause and think about what you thought you knew about Algebra. Let me know what you think in the comments. Thank you, again, for reading.
Math was okay until they threw the alphabet in it.
I have heard, Math was okay until they threw the alphabet in it, so many times from so many adults, who I wish were trying to be witty, but are in fact completely serious. And these are reasonable, intelligent people. Why does the abstraction of a variable or an unknown become so confusing that it creates a huge disconnect?
I’d like to share with you an assignment that I believe will help students transition from the concrete application and properties of Real Numbers to the abstractions we deal with in Algebra and mathematics beyond.
The desired outcomes of this assignment are:
Improve mathematical literacy by encouraging students to read the mathematical meanings created by the spatial arrangement of numbers and symbols
Improve their understanding of the Order of Operations and to help student realize that the Order of Operations is not its own topic, compartmentalized, but rather an over-arching understanding of how math is performed.
Promote abstract thinking about numbers and their properties
Introduce some concepts that will come into play later in Algebra, like finding the x-intercepts of a quadratic once it is factored
How to Introduce the Assignment
Students must be aware that they will be dealing with abstract ideas and that there are sometimes more than one right answer. Also, a student can be right, but not completely right, they could also be wrong, but not always. By fostering a healthy discussion about these problems you can introduce the idea that in order for something to be mathematically true, it must always be true. If a single circumstance is untrue, then the statement is untrue.
Consider the problem: Given that a and b are real numbers, and the following is true, what do you know about the numbers a and b? a×b=0
A student might say, In this case a and b are both zero.
That is correct in one case, but there are many cases where that is not true. It is true that one of them must be zero.
The First Prompt
Given that a and b are real numbers, and the following statement is true, what can you conclude at the numbers a and b?
a – b = 0
At first have them think and write on their own. Make sure they’re all working, not avoiding this uncomfortable notion.
After a given amount of time (short, maybe one minute), instruct them to talk with two different people. Be clear that the expectation is that they take turns, one person shares, the other listens and responds.
After the time is up, have a whole-class discussion, but avoid being the authority until the discussion is winding down. Only be the authority on the subject to help summarize.
The Second Prompt
With the same properties of the numbers and the statement being true, provide them with the equation:
(a + 5)(b –7) = 0
Conduct conversations as you did with the first, maybe allowing more time for them to talk together as this is a more complicated situation.
The Third Act
Switching gears from properties of variables to applying some properties of real numbers will promote their understanding of when the associative and commutative properties as well as challenge their understanding of how math is written. We are really trying to promote their ability to read and write math and their fact that the spatial arrangement has meaning in math.
Have the students try and add parenthesis, as many as they like, to the following equation, so that it will be true. Have them try it on their own first, then provide a short amount of time for peer discussion.
3•2 – 72 + 5 = 80
When you conduct a whole-class discussion, make sure it’s student lead, your role is as a mediator, not a disseminator of facts.
Instruct students to create a similar problem by making a statement they know is true and removing the parenthesis. For example, they might make up:
8(5 – 3) + 11 = 27
But would only write
8•5 – 3 + 11 = 27
When they’re completed their problem, have them show you. Once everybody has a problem, hand out 3×5 cards. On the front of the card the student will write their problem, without the parenthesis. On the back, they’ll write their name. Have them pass the cards forward and you can distribute them at to another class the following day.
Last Thing for the Lesson:
The idea here is the same as with the previous activity, but we are accessing their knowledge from a different angle. They will be given an expression with parenthesis and be asked if the parenthesis can be removed without changing the value of the expression.
For example: (5 + 4) – 2, or 11 – (4 – 9).
Introduce these problems in the same fashion, with quiet thinking first, then small group discussions, then whole-class discussion.
The homework is critical here because it will challenge students to think and examine how the way in which math is written changes the meaning. It will also force them to think about numbers in a general fashion.
The last thing I’d like to mention is that this could easily be done over two separate days depending on the aptitude of the class you’re teaching.
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:
The Natural Numbers are infinite, meaning, they cannot be counted entirely. How do we know that?
The Integers are also infinite. How do we know that?
Is infinity a number?
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:
Is it true that 5 + 4 = 4 + 5 ?
If a and b are Real Numbers, would it always be true that a + b = a + b? (What if they were negative?)
Is it also true that a - b = b - a? How do we know that?
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.
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.
Click here to download a Power Point you can use in class.
Click here here download a PDF of the information covered in Real Numbers.
If you find these materials valuable, you could help me create more.
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.
expression would be 15x2. This is the product of 15, x, and x.
An algebraic fraction, or
rational expression of these two could be .
This expression can be
reduced and below we will see two ways to approach reducing algebraic
These are equal because when multiplying rational
exponents you multiply the numerators together and then multiply the
It is useful to separate 5x and 15x2 in
this fashion because a number divided by itself equals one.
So this would be:
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:
This method is less clear to see, but the math is the
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
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
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.
If we divide this GCF out of each term, we would be
Where this will come into
play is with reducing something like:
By dividing out the GCF of all three terms we are left
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
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
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.
The GCF of all the terms is five. This expression could be written as it is
And we could write that as follows:
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).
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:
You may object here
because in a previous section we showed how the order of division cannot be
changed without changing the value. For
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
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.
When I was in 1st grade I suffered punished from a wrongful accusation, well, kind of. And the punishment would land people in jail today. In front of the entire 1st grade class I was spanked, but not spontaneously. I was paraded to the front of the room and quite a spectacle was made of the ordeal. Then I was sent to the principal’s office where I suffered a similar fate. And at home, once again.
And while it is true, I did knock all of the lunch boxes off of the stand where they were stored, and they did spill open and they did make a big mess, it was an honest accident, no intent involved. And I was very willing to clean all of it up, all of it! The opportunity was not provided. Instead, I bent over, knees straight, palms placed flat on a chair at the front of the room while the teacher drew back her arm, equipped with a wooden paddle, and brought it forward squarely on my six-year-old hind end, with all of my peers in audience. Thrice over it happened.
It was spring and I had been spanked at school so frequently that my parents made a bargain for me, a bribe really. If I could manage to not get spanked for an entire week then I could go out for ice cream on Friday. And, being that it was spring, the local ice cream shop had my favorite ice cream in the whole world … peppermint ice cream!
Somehow I mustered the will-power to keep the tornado of energy that sprung forth from my body under wraps. I sat properly, as instructed by the teacher. And the teacher’s name, Mrs. Fortenberry, was pronounced clearly and accurately, all week. I raised my hand at appropriate times, passed papers forward neatly and returned from recess with punctuality. There was no cutting in line at lunch, no teasing other kids during class and the shenanigans that transpired when the teacher would turn around ceased to occur.
The staying power of six-year-olds is questionable, but I held this superb behavior through all of Monday, Tuesday, Wednesday, Thursday, and half of Friday. Upon returning from lunch on Friday I went to place my lunch box on the stand and it slipped. I had not been goofing off, running around, just an honest accident. But it was as if I threw a basketball at them, they toppled like dominos … to this day I marvel at the physics involved in knocking over all of those lunch boxes in such a fashion. Entropy is sometimes spontaneous in the presence of six year old boys!
My protests of innocence might as well have been mute.
That evening my family went out for ice cream. I’m sure there was plenty of the delicious peppermint ice cream eaten. I wouldn’t know, though, because I was left home by myself.
And while this may sound tragic, I realized something from this experience. I had my first epiphany. I realized that nobody believed me with just cause. They should not have believed me, my reputation was well earned.
From this unjust punishment I realized the power of reputation. I realized that if I wanted to be believed in uncertain circumstances, when light spilled over me in a questionable fashion, that I needed to have a reputation of being honest. The only way to establish such a reputation was through being honest.
My behavior did improve after this. Perhaps not getting the ice cream was the best thing that ever happened to me.
Sets of Numbers
Problem with Zero and Division
begin with the various types of numbers called Real Numbers. Together, these numbers can be ordered and
create a solid line, without gaps.
Numbers: These are counting numbers, the
smallest of which is 1. There is not a
largest Natural Number.
Numbers: All of the natural numbers and zero.
Zero is the only number that is a Whole Number but not a Natural Number.
ØIntegers: The integers are all of the Whole Numbers and
their opposites. For example, the
opposite of 11 is -11.
Numbers: A Rational Number is a ratio of
two integers. All of the integers, whole and natural numbers are rational.
that terminate or repeat (have patterns) are rational as they can be written as
a ratio of integers.
Numbers: A number that cannot be written as a ratio of two integers is
irrational. Famous examples are π, and
the square root of a prime number (which will be discussed next).
these make up the Real Numbers. The
name, Real, is a misnomer, leading people to conclude that the word real in this context has the same
definition as used in daily language.
That misconception is only strengthened when the Imaginary numbers are
introduced, as the word imaginary here
harkens back to a day when the nature of these numbers, and their practical
use, was unknown.
Is zero rational?
rational number is a number that is the ratio of two integers. Before we tackle the issues that arise from
zero, let’s reframe how we think about rational numbers (fractions) and develop
a different language for these to promote greater proficiency in Algebra and
allow for greater ease in understanding how zero causes real problems with
rational numbers. (If you understand the
nature of what follows you do not have to memorize or remember the tricks, you
the fraction . You
were likely taught to think of this fraction as division and would also likely
be taught to ask the question, “How many times does two go into eight?” That is sufficient for this level of
mathematics, but the Algebra ahead is seemingly more complicated, but by simply
rephrasing the language we use to talk about fractions, we can expose the
seemingly more complex as being the same level of difficulty.
of asking, “How many times does two go into eight,” the better question is,
“Two times what is eight?”
true that because
two times four is eight. Simply answer
the question “Two times what is eight,” and you’ve found the answer.
will come into play with Algebra when we begin reducing Algebraic Fractions
(also called Rational Expressions) like:
ask the question, “How many times does three x going into nine x squared,” you’ll likely be stuck,
especially when the expressions become more complicated.
asking, “three x times what is nine x squared,” is a little easier to
will be much more on reducing Algebraic Expressions later in this chapter. Let’s turn our attention to zero and how it
“behaves” in with rational numbers.
an integer, and again, a rational number is a ratio of two integers. Consider the
first expression asks, “Zero times what is five?”
second expressions asks, “Five times what is zero?” (Again, phrase the question in this fashion
to provide easier insight into the math.)
product of zero and any number is zero.
So, the answer to, “zero times what is five,” is … well, there is no
answer. There is no number times zero
that is five. There is not a number
times zero that equals anything except zero.
We say this is undefined, meaning, there is no definition for such a
second expression, “five times what is zero,” is zero. Five times zero is zero.
these two expressions is rational, the other is not a number at all. It does not just fail to fit within the Real Numbers,
it fails to fit in with any number.
Written as Fractions
the fraction This is a rational number because it is the
ratio of two integers, 1 and 3. Yet, the
decimal approximation of one-third is (the bar above the three means it is repeating
how to express a repeating decimal as a fraction. Let us begin with the number .
We don’t know
what number, as a fraction is , so we will write the unknown x.
repeating after the hundredths place, we will multiply both sides of the
equation by 100.
0.333333… we would multiply by 10, since the decimal repeats after the 10ths
place, but we would multiply 0.457457457457…by 1,000 since it repeats after
the thousandths place.)
The following step is done by a
procedure learned with solving systems of equations, which will be covered
later. (In fact, this procedure would be a great topic to review when systems
of equations is learned.)
first equation from the second.
Divide both sides
by 99 to solve for x.
Recall that x was originally defined as the
fractional equivalent of the repeating decimal.
1. Change the following into rational numbers:
is a the following called undefined: ?
all of the sets of numbers to which the following numbers belong:
There are many topics in math that are difficult for students to grasp because either the concept itself is elusive and maybe they don’t have the foundation required, or the procedures are complicated and riddled with traps. Algebraic Fractions or Rational Expressions, whichever name you prefer, is a great example of a topic that hits both of those.
Algebraic Fractions are intimidating from the get-go. Students see fractions with algebraic expressions and their heads practically explode! Where as exponents, are inviting and seemingly easy, at first anyway. But as students get into more complicated problems involving Algebraic Fractions, the path becomes clearer. To find a common denominator first find the LCM of the denominators, multiply each term, reduce and then combine like terms. To finish it up, reduce if possible by seeing if a GCF exists between all terms. If one exists, divide it out of all terms and done!
Once students get away from some of the basic properties and have to combine several ideas involving exponents together at the same time the students are hopelessly lost. An expression with exponents that needs to be simplified that might involve negative bases with negative exponents that are written in the denominator, then combining like-bases with the properties of exponents all while trying to traverse the muddy lines separating the order of operations (PEMDAS) when sometimes the first three (P, E, and M) are completely interchangable because it’s all multiplication, where other times you must take care of the group first and the multiplication last. It’s over-whelming and all comes back to my favorite harping point, the ability to read mathematics by understanding the meaning behind how it is written.
While I work very hard to build a strong conceptual foundation for my students, sometimes I start doing the framework of the house before the foundation has has time to set. This is often the case with Algebraic Fractions. But, if their procedural fluency is sufficient, they can often repair their weaknesses in the foundation, or never need to shore it up at all and suffer little, if any, from it.
There are other topics that can be taught with concept only, for example, graphing a linear equation by finding the intercepts. If students know what intercepts are, know how to find their coordinates, and know that linear equations are equations whose solutions form a line, they can put all of that together on their own.
Exponents are different. The basic properties of exponents can be taught without covering the “rules” at all. By simply knowing that exponents are repeated multiplication, and writing things out, students can get quite a long way. I’d suggest they can get through division like the problem below without using the “Negative Exponent Rule.”
Now this is not a practical way to go about simplifying an expression. The rules, or properties, prevent students from having to reinvest the wheel, as it were. However, if the rules are introduced too soon, before the foundation has set and hardened, then with exponents, there will be serious troubles ahead. And yet, visualizing this method displayed above, without using the properties to obtain the answer by some procedure, allows one to visualize that there are two more x’s in the denominator and four more y’s in the numerator.
But without a measurable level of fluency with applying the rules of mathematics, simplifying expressions involving exponents quickly becomes insurmountable. Consider a problem with multiple brackets, rational exponents, negative bases and negative exponents, like the one below. This would be overwhelming to approach with only a conceptual understanding, but no applicable short cuts in the proverbial tool box. With exponents it is imperative that a balance between the ability to approach any problem from a conceptual framework or to approach the same problem from a procedural framework with equal accuracy, must be achieved. Students with strong conceptual understanding but little practical procedural ability will struggle with a similar level of difficulty compared to students with little conceptual understanding but that are well versed in the “rules” of exponents.
Students and teachers are very accustomed to seeking and rewarding answer-getting techniques. This is what makes exponents a wonderful topic to spend a lot of time on. You can show students how focus on procedure alone is insufficient, and how multiple methods of finding solutions is a powerful tool because combined with a little experience clear paths to solutions unfold when there is no proper first step.
An example of a problem without a proper first step is below:
There are many, equally efficient, and mathematically correct ways to begin simplifying such an expression. Students that rely on procedures will struggle mightly with these types of problems because the first step depends on what you recognize first (conceptual understanding). How to respond to what is first recognized depends on the procedural fluency of the student.
It is a great exercise to take a problem like this and have one student perform some operation of their choosing, then another to perform a second operation and so on, until the problem is simplified completely.
By exposing students to problems where the initial step is highly variable and the method of arriving at correct answers vary upon what is noticed by students, they can develop a sense that math works this way.
Students should not consider math to work as follows: Well, this problem looks like the last problem we did and I used the quadratic formula last time, so let’s do that.
It is my intent that when teaching something like exponents (and radical expressions can be similar) I am trying to plant the seed of this idea that math can be approached from many points of view, and sometimes, the right answers are not so cut and dry.
I am currently working on a series covering exponents, square roots and rational expressions where both the needs of the student are addressed through tutorial/remediation videos, and instructional coaching, lessons and practice problems for teachers. The videos will be posted here on this website, but also on my YouTube channel. The lesson will serve to meet two callings. First, students have conceptual understanding of the topics and procedural fluency, but both done in a way that promotes mathematical literacy.
If you are interested in such a treatment of exponents, done in a way to encourage to view mathematics in a different light entirely, you can subscribe to my newsletter, or visit my YouTube channel and subscribe there. Either way, I’m finishing up the series on exponents soon and will post them when done.
Thank you for reading. And please, if you found this to be informative or useful, spread the word.
One of the biggest problems in public education, math in particular, is answer-getting. And this is endemic, not the fault of students, teachers or administrators, legislators or parents … alone. Students seek answer-getting methods and are rewarded for finding answers. Teachers are given massive amount of curriculum to be covered in a year while at the same time deeper understanding is demanded. It’s not any one person’s fault, nor is it one party’s fault.
Leadership is trying to fulfill expectations and avoid tripping over the strings attached to funding. For example, graduation rates are a coveted measure of success of a school. On the flip side, what constitutes the requirements for high school graduation are not closely monitored, or incentivized. There is a lot of pressure to promote students through the ranks of education, starting in Kindergarten, even when it is not always in the best interest of the student.
I won’t even get into how curriculum calendars and pacing guides became en vogue, but they are a massive burden on teachers. A teacher must decide to cover material on a predetermined schedule, and if they fall behind, they must hurry to catch up or face reprimand for not being on pace … This is complete insanity! Learning takes time and that time cannot be compressed. It cannot be compressed when students are cramming for tests (that aren’t strictly fact based), and it can’t be compressed during the school year either.
And yet, from the same mouths of the people who bestow the virtues of standardized testing comes the call for differentiation in education! What’s the saying about those who live in glass houses shouldn’t throw stones? I am working on a constructive solution to these types of things, but for now, I’ll do as my mother says and engage in silence on the topic as I have nothing nice to say.
All that said, all parties involved are well-intentioned. Yet, must traverse a jungle of pitfalls that land us in poor teaching.
There is one particular topic in Algebra that sticks around because it satisfies all of these pressures that, in the end, result in low-quality education. It is easily taught, gets quick answers and can be tested easily on a multiple choice (standardized) test: The Point Slope Formula. Now, I taught this formula for years. When I first heard its usefulness being called into question, when I first heard that it was a simple answer-getting technique, I bucked at the idea. I mean, after all, I am a good teacher who has high standards and plenty of rigor.
And that’s one of the issues with change. We are all part of the system, and while it is easy to break apart what everybody else is doing wrong, they’re part of the same system that produced you…look at yourself first. Back to the topic at hand…
A typical problem would say, Given the point (2, 3) and the slope of -1, find the equation of the line.
Students would use the formula below and they’d plug in 2 for x and 3 for y and -1 for m. Then they’d distribute and then solve for y. Dang, if they mess up distributing or the one step of inverse operations required, BOOM, there’s a chance for remediation or reteach or whatever the fancy phrase at the time is.
But none of that, the process or the reteach would ever touch the nature of the question at hand. None of it would connect the question to the concept of linear equations and how graphs are related to equations, which means the students could not build upon that concept in the future.
The concept at play here is the nature of the relationship between dependent (x) and independent (y), how every point on a line is a solution to the equation, and perhaps even how slope describes the relationship between the dependent and independent variables.
If a student knows the following, they can arrive at a solution in a way that addresses practical issues at hand and without having to be taught yet another thing!
The equation of a line can be written y = mx + b.
The values of x and y are an ordered pair, which means they’re a solution to the equation we need.
Consider the following two tables where I show the concepts (related to linear equations) at play using the two methods discussed. Then, I’ll show the simplicity of one method versus the other. As is often the case, the method that approaches a problem from the concept has a simpler solution.
I do believe there is benefit from teaching and learning the point-slope formula, especially if it is taught when learning order of operations or how to distribute. But as a method of answer getting when learning about linear equations, I believe, it is inappropriate.
I hope this post caused you to pause and reflect upon your own teaching. What other things do we teach kids that circumvent the concept at hand in order for a quick grab at an answer? Leave me a comment, let me know your thoughts.