The Order of Operations (OoO for short) is used everywhere in mathematics because it encompasses many of the foundational rules that we’ve agreed to follow.
Alas, students have been given the cheap and dirty version of it for years. “Here, memorize this thing about your Dear Aunt Sally!” What the heck?!
There are subtleties in the Order of Operations that every person over the age of seven should know.
The series begins today.
The order of operations is a set of rules – like the drivers’ handbook for math. If everyone follows the rules, we’ll all be safe. But if someone makes a bad turn, we could be looking at a crash.
But the Order of Operations is only a set of rules for arithmetic! It isn’t even the best practice when it comes to expressions involving a variable like x. I’ll cover what I mean in this weekly series.
Here are the proposed articles:
Intro and mnemonics
Parenthesis
Exponents
Exponents, Multiplication and Addition
Multiplication and Division
Addition, Subtraction and Conclusions
Exponents of Negative Numbers
Another Reason to Ban PEMDAS (aka parenthesis aren’t an operation)
Mnemonics for PEMDAS
Well, there’s one: PEMDAS (pronounced just like it looks). That’s what the cool kids in high school always said. It was the same kids who said “soh-cah-toa” – which I thought sounded really goofy.
And then there’s “Please Excuse My Dear Aunt Sally.” And of course “Piranhas Eat Mostly Decayed Antelope Skin.”
Do you struggle explaining to your students what “simplify” means? Do you wonder why they won’t reduce fractions to their lowest terms?
Here’s how to explain and teach some of the top confusing terms in mathematics – without throwing you or your students into a tizzy!
Where
Really! “Where.”
As in “Where do the lines intersect?” or “Where is the highest point of the graph?”
“Where” in math almost always means “at what x-value?” Every now and then “where” can mean “at what point?” In the first case, “Where do the lines intersect?” can be just as easily satisfied by a point (x, y) as the x-value only.
When you get into calculus, the word “where” is used everywhere (pardon the pun). “Where is the slope of the tangent line equal to zero?” and “Where are there discontinuities?” among others.
Keep your eyes peeled in the textbooks for this tricky term.
Reduce
Typically this means to reduce to lowest terms. Which isn’t really helpful, since you still need the definition of “terms.”
Numerators and denominators (tops and bottoms of fractions) that have one or more common factors are not officially “reduced.” Once they have no common factors at all (except 1), they are said to be reduced or reduced to lowest terms.
A reduced fraction helps in many ways. You can “feel” the value more easily: 2/5 is a lot easier to picture than 14/35. Also, when you add fractions, starting out with all the numbers in their reduced form will often help to find common denominators.
Zeros, Real Zeros and Roots
Zeros, real zeros and roots all mean the same thing. It is where a graph passes through the x-axis. We can write “where are the zeros” meaning at what x-value (“where” from above) does the graph pass through the x-axis? We also will write “what are the zeros” and mean the same thing.
To find the zeros or roots, we set a function (or any equation solved for y that involves x) equal to zero. Then we solve for x. This will give the result that we want – the x-values where the graph will go through the x-axis.
We call them zeros because they are where y = 0. We call them real zeros because there can be complex or imaginary zeros that don’t exist on the graph.
Finding the zeros or roots of a function help when graphing it. If you know all the places it goes through the x-axis, you can then determine where it’s above or below the x-axis. Then you can roughly sketch the graph.
Simplify
Of all the terms in mathematics, this one is the most slippery. This meaning of this term is always in context to the book or situation. What it means is “manipulate the expression.” The end result may or may not be “simpler” than the starting expression.
Better textbooks use phrases that are clearer, like “multiply the polynomials” and “combine like terms.”
If “simplify” is the instruction to a problem set in a textbook, consult the examples to see what type of manipulation to do.
In later courses, students have to recognize when a situation requires a different form of the expression. Practicing various forms of manipulation (or “simplifying”) gives them tools to manipulate the expression to satisfy their needs.
Do you run into other terms in mathematics that are hard to explain to students? Hard to understand yourself? Share them in the comments!
Do you wonder if your children should be using a calculator “at their age”? Are you a fan of calculators, but have friends who aren’t? Are your friends “into” calculators while you oppose them?
I often hear people say that children 50 years ago understood math concepts more quickly. Although our parents weren’t taking classes called algebra in the 7th grade, they were doing algebra in the 7th grade.
Algebra is arithmetic.
There are two fundamental and rarely understood facts about algebra:
Algebra is arithmetic with one or more numbers in disguise.
Algebra has exactly the same rules as arithmetic.
Which means if you can do arithmetic you already know how to do algebra!
Our parents or grandparents, 50 or even 30 years ago, weren’t using calculators. They had to apply all the rules of arithmetic to get the job done. Which means that they had to apply all the rules of algebra.
Teaching them a class called “Algebra” was much easier because of this.
What are the rules?
The basic rules that non-calculator users must apply are the distributive property and the order of operations. The distributive property is the thing that calculator use eliminates.
Children could get practice mentally multiplying things like 3 x 86 and do 3(80+6) = 240+18=268. With this practice, they are ready for 4x(3y+2z) = 12xy+8xz.
If they never have to multiply 3 x 86 in their head, they never get the experience of the distributive property. Which means teaching them 4x(3y+2z) = 12xy+8xz will cause anxiety and frustration. They see it as “magic” or “something you made up just to confuse me.”
Give them the tools they need.
Refuse to let students have the calculator. Let them have the tool of the distributive property for algebra before you teach them “Algebra”. Give them the benefit our parents and grandparents had!
I mean: “What is three times a number (that number is four), plus two?”
The domain is all the possible questions:
What is three times a number (that number is five,) plus two?
What is three times a number (that number is six,) plus two?
What is three times a number (that number is seven,) plus two?
What is three times a number (that number is eight,) plus two?
<how long will I have to do this – Egad!>
Not only do the questions go on forever, but they also have all the fractions and decimals in between. And all the negative versions of those numbers (and zero).
So, what’s the point, you might ask. Looks like the questions go on forever and you can just pick any number.
The domain might not include all the numbers.
The two sticky points for the definition of “function” are bolded:
A function is a question with only one answer to a valid question.
The “valid question” part is where the domain comes in.
The numbers that make “valid” – meaning we actually can get some answer – are the numbers that aren’t negative.
Many functions have “all real numbers” as a domain. There are no limits on the things you can put in, other than numbers that aren’t imaginary or alligators.
For the most part, there are only two places where you have to be careful of limited domains. Those are
Numbers that cause a zero to turn up in the denominator
Numbers that cause negatives to turn up in square roots.
Here are two videos tackling each:
What do you think? Is this easier to teach when you consider “analyzing” the function rather than “solving” it? Share your thoughts and tips in the comments!
Other than being the most feared f-word in math teaching, a “function” is a question with only one answer.
Take the question: “How tall are you?”
We can change this to: “What’s the height of you?”
And if we wanted to compare your height to other people’s heights we can ask: “What’s the height of <insert person’s name here>?” This is the question template – the formula.
You can answer this question in inches, feet or cm, but the value of the answer remains unique, based on the person.
And that last little piece of the sentence is what makes the difference, based on the person.
The question changes with this little change. <cue music> This is the variable in the equation.
And we say, “Height is a function of the person.”
So where’s the fear come in?
As always, the notation is the kicker when it comes to teaching math.
Let’s change the question a little.
What’s the height of Enrique tomorrow if he grows three inches tonight?
Rather contrived, but work with me…
The question template is
What’s the height of <insert person’s name here> if he/she grows three inches tonight?
Which becomes
<height> = <height now> + 3
Or
H = N + 3
Egad!
And we haven’t even started with the f(x) stuff!
What’s this “domain” thing about?
I wrote the first sentence of this post a little too hastily. A function has only one answer if there’s a valid question.
If you ask, “How tall is love?” someone will laugh at you. Or think you’re from California.
Our question template included some specifics that you don’t normally get:
What’s the height of <insert person’s name here>?
If we instead ask, “What’s the height of x?” we would then have to ask: what kinds of things can we put in for x? Can we put concepts, like love? Or just objects? The kinds of things that you can put in for x is called the domain.
For our question, we would need to specify that x is a person.
What do you think? How does this feel when explaining it to your kids?
