Tag: division

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.

Related articles
July 4, 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
Two Reasons to Memorize Math Facts

I learned my math facts by “singing” them while looking at flashcards.

Having these facts ingrained with chanting or singing isn’t a bad idea. It might not “feel right” because we’re so into experiential learning these days. But if a kid can’t immediately access and use things like 8 x 7 = 56, he’s going to be slower than if he can.

And if he’s slower, he might get frustrated and start to think that he’s not good at math.

Also, knowing these cheap and dirty math facts helps with confidence. Even if a kid’s struggling with other things in math, knowing that he has this one thing (the “facts”) will help out.

I fight this battle often. Some people feel that math facts shouldn’t be memorized. But there’s so much value in it.

How about you? Which side of the fence are you on?

May 8, 2011
How to Teach Division in the Sandbox

Need to offer a better way to understand the concept of division and remainders? Try it in the sandbox!

The Discovery Toys are proportional, so they allow a really engaging way to see how division and remainders work.

This video shows how you can help kids put together the numbers 9, 4 and 1 to “see” division at work:

You can also do this in the bathtub or pool. And the are perfectly weighted and “massed” so they float.

I think this has something to do with “water displacement,” but I’m not sure. I’m a mathematician, not a physicist. 🙂

April 30, 2011
Using Toys as Curriculum Tools to Teach Arithmetic

Want to give your kiddos a jump start on multiplication and division? What to help the ones struggling with division to grasp it better?

The Discovery Toys Measure Up Cups can do just that. They are built as a curriculum tool, in the proper ratios, so that the #6 cup holds exactly twice as much as the #3 cup. This allows for engaging and beneficial play that gives kids a grasp on how numbers relate to reality.

For example, in this video, kids can compare the numbers 3, 6 and 9 to see how they relate:

You don’t have to say out loud, “three plus 6 is 9” or, “9 divided by 6 is one with three left over.” But these concepts are ingrained into the child’s brain as they see this work.

What do you think?

April 29, 2011