Category: Algebra

The Order of Operations Explained: Multiplication and Division
This is the 5th in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

Last week’s article on the order of operations included a quick mention that division is the same as multiplication – but different. Now’s the time to explain that a bit.

The only thing mathematicians like to do more than create, is destroy. That’s how we get imaginary numbers, dividing by zero and raising things to infinity.

Once we create with multiplication, we want to know what happens when we destroy with the opposite of multiplication.

Enter Division, stage right.

Multiplication is created from the need to quickly add a bunch of numbers that are all the same. They each must be equal to make this work:

6 + 6 + 6 + 6 + 6 is shortcutted to 5 X 6.

Division is the breaking up into pieces that are all equal.

Technically we can break 30 up into these 5 pieces: 4, 8, 7, 6 and 5. But “division” requires (or implies) that we are dividing equally. So 30 would have to be broken up into 5 equal pieces of 6 each.

Of course this is in a purely mathematical world. When you get into a toddler world, things will be different.

How division is the same as multiplication.

We represents multiplication with little x‘s or dots or stars. Like this:

3 X 5 or 3 • 5 or 3 * 5

We represent division with this cute little symbol: $\div$

But we can write it in many more ways. Notice the cute symbol $\div$ looks like a fraction with dots on the top and bottom. That’s not a coincidence. A fraction means division.

$7 \div 3$ is the same as $7\times\frac{1}{3}$ .

Remember the poem:

When dividing fractions
Don’t bat an eye
Just flip the last
And multiply!

Well, you can change this to:

When dividing numbers
Don’t bat an eye
Just flip the last
And multiply!

And “flipping” the last just means taking the “assumed 1” that’s underneath it and putting it on the top.

Now go forth and divide…

So when you’re teaching the MD in PEMDAS, the order of operations, remember that D is the same as M. And if things get a little confusing, demand some parenthesis before doing the problem.

Share your thoughts in the comments!

Related articles
July 10, 2011
The Order of Operations Explained: Exponents, Multiplication and Addition
This is the 4th in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

The Order of Operations can be boiled down into three “real” operations.

Parenthesis are merely a way to group things – they aren’t a real operation. So they doesn’t count as a real operation.

Since division is really just multiplication turned upside down, we don’t need to include it separately, either. Likewise, subtraction is addition on its ear. So we throw him out, too.

Now we have only three: exponents, multiplication and addition.

Exponents are the shortcut for multiplication.

In a previous article about remembering exponent rules, I recorded this video about exponents:

You see that 3 x 3 x 3 x 3 is 3⁴. The exponent is the shortcut for multiplication.

Multiplication is the shortcut for addition.

Likewise, when we get overwhelmed with adding up the same numbers over and over, like in the video, 4 + 4 + 4 + 4 + 4, we can use multiplication to shortcut it: 5 x 4.

Follow the shortcut evolution.

The shortcut evolution is like this
1. Addition came first.
2. Then we created multiplication to make addition easier.
3. Then we created exponents to make multiplication easier.
So when you do arithmetic, we should do the “recent” shortcuts first (exponents) and then the “older” shortcuts (multiplication) and then the “regular” arithmetic (addition). Remember that subtraction is addition and division is multiplication.

And keep in mind that we need to watch the grouping or anything isolated with parenthesis, absolute value bars or in a fraction. The grouping/isolation tools always trump the other operation rules.

What if your answer isn’t the same as the back of the book?

Some textbooks actually have multiplication done before division. This means that the problem 9 ÷ 3 x 2 will show a different answer (3/2) in that textbook’s solution pages than what you would get following the OoO I’m describing here (6). These texts are rare, but I’ve seen them. So keep your eyes peeled.

The Order of Operations is a set of rules that we’ve agreed on. Which means that as long as a textbook clearly defines their order that they will follow, they can do things like this.

If you get confused as to which to do first, demand parenthesis. (Yes, you can do that.) Or put parenthesis into your child’s textbook to help them out.

The problem sets should be there to enforce, not confuse.

What do you think? Does this help or hinder the way you’ve always viewed the Order of Operations? Share your thoughts in the comments.

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July 4, 2011
How to Teach Math Concepts at the Dinner Table
Would you like to teach math everywhere you go? Well, here’s one from the table!

Daughter enjoys playing with our salt-and-pepper shaker holder at dinner. She takes out the salt, then takes out the pepper, then replaces the salt, then replaces the pepper.

The order in which she does these four operations vary. Including switching the salt and pepper.

She’s slowly putting together the pieces that will one day become the commutative property.

She’s also practicing substitution…

She’s learning that the salt and pepper can be switched (commutative). And she’s learning that one can be interchanged for the other (substitution).

…and the associative property!

She attempted to put her small milk cup into the holder. It fit, but only with pushing. She then removed the milk cup and attempted to put it in the other side. (At her age the things grown-ups understand are not obvious to her.)

Although non-equality isn’t part of the associative property (which is if a=b, b=c then a=c), the comparison of three things is.

Here are the things she’s learning from this dinner session:

This fact she discovers from interchanging them in the holder.

By putting them in the holder in a different order, she learns that the equality is commutative.

Since the milk cup won’t fit into the spot the salt was just in, she learns this.

And trying to shove the milk cup in the other side yields this fact.

So pull out the stops – give the children everything. And let them explore. If they have the gift of language, you can hint at some of these properties, but be careful not to go into a full “lesson” at dinner. Teaching math at the dinner table should be fun.

Where have you seen math properties in your world? Share your stories in the comments – or link back to your story on your blog!

Related articles
June 30, 2011
The Order of Operations Explained: Exponents
This is the 3rd in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

Exponents are the second in the list for the Order of Operations (OoO).

When we want to find the result of 3² x (2 + 7), we have no problem. We know to do parenthesis and then exponents, then multiplication.

When you teach algebra, you’ll have to teach some distributing of exponents. But that’s still okay. And the rules of exponents are pretty straight up.

So why a whole article on exponents?

In the order of operations, the “Exponents” rule represents a bunch more than just superscripts or tiny numbers flying up and to the right of things.

Roots are exponents, too!

Tree Roots by Linda Allardice

Not the ones from trees, but things like square roots and cube roots. Consider $\sqrt9 + 2$ . You do the square root first because it qualified as an “exponent.”

But if you had $\sqrt{9+2}$ , the 9 + 2 is under the radical sign (the square root sign) so it’s bound together in the “Parenthesis” rule.

This one isn’t that hard with arithmetic, but when you come to algebra and start “undoing” these things – it’s important to remember that roots fall into this category.

Fractional exponents are exponents.

This one seems pretty “duh” so it’s easy to see how they fall into the “E” of the order of operations. But what are fractional exponents really?

$9^\frac{1}{2} \text{ means }\sqrt9$

So fractional exponents are the same as roots.

Note that some fractional exponents are roots and “plain” exponents all mixed up. Like this one:

$27^\frac{2}{3} \text{ means } \sqrt[3]{27^2} \text{ as well as } (\sqrt[3]{27})^2.$

This is a big fat full concept that needs a little more explaining. I’ll write more on these in another article.

Logs fall under the E.

Axe In Stump by caroline steinhauer

As my algebra and computer math teacher in high school, Mrs. Kelley, used to tell us – logarithms are exponents. It took me a long time to figure out what the heck she meant. But when I did, I thought it was brilliant.

This is a true statement: $\log_3 9 = 2$ . Let’s analyze it.

Based on the definition of logarithms, this means that 3² = 9. Which we know is true.

Notice who the exponent is in this: 3² = 9: 2 is the exponent. And 2 is the same as $\log_3 9$ because the equals sign in $\log_3 9 = 2$ means “is the same as.” So the logarithm $\log_3 9$ is the exponent 2.

Still with me? Either way, it’s okay. It’s a weird concept that I can go into detail in a video soon.

The thing to remember here is that logarithms fall into the “Exponents” rule of the order of operations.

So if you have $\log_3 9 + 7$ , you have to do the $\log_3 9$ first and then add the 7 after.

Want more on exponents?

In the meantime, you can check out more than everything you always wanted to know about exponents on the Wikipedia Exponents page. Rebecca Zook created a great video on logarithms. And check out this explanation and problems to work on fractional exponents.

And let me know what you think. Did I miss something?

Related articles
- How Calculators Inhibit Learning the Distributive Property in Algebra
- What’s the Domain, Why You Need It and How You Get It
June 26, 2011
The Order of Operations Explained: Parenthesis
This is the 2nd in the series The Order of Operations Explained. For the other articles in this series, click here to visit the introduction.

I mentioned in the introductory article for this series that “the order of operations isn’t best practice for expressions involving variables.”

When you involve a variable, you can’t just “do” the arithmetic. Like in the case of 2(3x + 4)=11. You can’t add 3x and 4 to get a result before moving on. You have to use the distributive property.

And even the distributive property won’t work sometimes – as in the case of absolute values shown below.

Parenthesis mean isolation.

When we say “parenthesis” in the Order of Operations (OoO for short), we mean anything that’s grouped together and isolated. This could mean with actual parenthesis. This could mean [square brackets] or {curly brackets}.

Although grownups seem to know that square and curly brackets are the equivalent of parenthesis, children don’t. This has to be said out loud.

“Parenthesis” in the OoO can also mean |absolute value bars|. This one’s not so clear.

When you try to solve the equation 4|2x+3|=20, you have to start with dividing by 4 to isolate the absolute value chunk. There’s not a “rule” for distributing the 4.

(Although, it would be worth it, and fun, to see if your children can come up with some rules for distributing within absolute value bars. This would be some real mathematics at work for them – experimenting and discovering.)

Tops and bottoms of fractions are implied parenthesis.

It’s also the case that the numerator (top) and denominator (bottom) of a fraction are isolated places. These fall under the OoO as parenthesis.

Check out this older video I did. It shows how this works with fractions:

Parenthesis are for deviation from the other rules.

David Chandler of Math without Borders commented this in the previous article of this series:

The rule is to do higher level operations first. Use parentheses whenever your intention is to deviate from this rule.

If you can focus on this instead of a mnemonic device, you can get students to internalize what’s going on with the OoO. It’s important, however, to make sure they remember about other bracket shapes as well as isolation.

Let us know your tips and thoughts on the P in PEMDAS!

Related articles
June 19, 2011
The Order of Operations Explained: Intro and Mnemonics
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:
1. Intro and mnemonics
2. Parenthesis
3. Exponents
4. Exponents, Multiplication and Addition
5. Multiplication and Division
6. Addition, Subtraction and Conclusions
7. Exponents of Negative Numbers
8. 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.”

What’s your way to remember it?
June 12, 2011
How Calculators Inhibit Learning the Distributive Property in Algebra
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:
1. Algebra is arithmetic with one or more numbers in disguise.
2. 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!

Related articles
June 1, 2011
What’s the Range of a Function?
You’ve taught what a function is. And the kids are starting to understand what the domain is all about.

But then they ask, “What’s the point in the range?”

As I wrote in a previous post, a function is a question with only one answer to a valid question. The domain is the set of all valid questions. The range of a function is the set of all answers you can get.

Simple? Sounds like it – but kids the world over still struggle with the question, what’s the point of the range?

To be or not to be a function.

Why is it important to know all the answers of an equation? It has to do with the equation being or not being a function.

If you have an equation like

$y=\sqrt{x}$

you have more than one answer per question.

Here are some valid questions associated with this equation:
1. What is the square root of a number, specifically the number 1?
2. What is the square root of a number, specifically the number 1.69?
3. What is the square root of a number, specifically the number 4?
4. What is the square root of a number, specifically the number 9?
The answers to these questions are:
1. 1 or -1
2. 1.3 or -1.3
3. 2 or -2
4. 3 or -3
Notice that there is not “only one” answer to each question. So this equation isn’t a function!

But that’s no fun at all!

You can force an equation to be a function by limiting the answers.

By limiting the answers (AKA limiting the range of a function) you can force an equation to be a function. So if we write

$y= + \sqrt{x}$

We have just limited the range of answers to be only the positive square roots of numbers.

The practical application for kids is the graphing.

In this image above it’s the blue curve:

You can see that we get only the “upper half” of the curve. If you look at “squishing” a function (like the garbage compactor in the movie Star Wars) you can see the range of a function (all y-values) becomes the vertical line:

The line starts at zero and goes up forever. (In the video it stops, but that’s only because I have a hard time displaying forever on a computer screen.)

The handy thing about knowing the range of a function before you graph is that you know how much space on the paper you need – or how small to make your units!

Does this help? Share your range of experiences with this in the comments! (And pardon the very bad pun.)
May 27, 2011
What’s the Domain, Why You Need It and How You Get It
As you teach domain and range, do you get the question, “Why are we doing this?”

No doubt the question, “When am I ever going to use this?” comes up too, right?

I’ve asked myself that question my whole teaching and tutoring life. And now’s the time for an answer.

A function is really a question.

As I wrote in a previous post, a function is a question with only one answer to a valid question.

When I write:

y = 3x + 2 where x = 4

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!
May 20, 2011
What’s a Function?

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?

May 19, 2011