Tag: multiplication

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
How to Teach a Weird Math Rule: Negative Times a positive Is a Negative
Thanks to Molly for this question she left in the comments, “Why is a negative times a positive, a negative?”

“Sign – One Way” by Andrew Schmidt

It’s common to hear a child say, “They created this rule just to make me mad.”

And on the heels of that usually comes a frustrated teacher’s comment of, “No, this has a purpose.”

When then faced with, “Okay… so what’s the purpose?” a teacher or parent might become more frustrated. It’s not always clear the purpose of things like the rules for multiplying differently signed numbers.

It may also be that the grown-up secretly believes that this was all created just to make us mad. After all, when the last time you had to multiply a negative number times a positive number in the real world?

I, personally, can’t ever remember a time when that case has come up outside of a math class or tutoring session.

Then why create the rules at all?

The truth is that many math rules are created because we need a rule to continue working – something to agree on among peers working on the same project. We need everyone on the same page.

This agreement somehow gets filtered down to the kiddos as “this is the absolute way it is – do it because we said.”

So it very much looks like we created rules to make them mad.

Good deduction, kids.

How do they pick the rules?

“Street Chess” by Petr Kratochvil

Some math rules are created arbitrarily just so we have something (anything) to agree to. Like the rules of a game. The order of operations is one such situation.

There is no natural reason why we’ve chosen to do multiplication and division before we do addition and subtraction. No more so than the people in the US have decided to drive on one side of the road whereas the people in England, Australia and the Bahamas drive on the other.

We’ve agreed to it.

It keeps us from crashing.

Sometimes, we agree to a rule because it is “less bad” than any of the other possible rules we can think of. The result of any number raised to the zero power is one of these. 3⁰ feels like it should result in either zero or 3. But if you mess with all the other rules (in particular the one about subtracting exponents here $\frac{3^7}{3^7}$ ) you’ll see that 3⁰ really needs to be 1. If it were anything else, it would mess up the rule of $\frac{\text{anything}}{\text{anything}}=1$ .

How do we teach crazy math rules?

Why is a negative times a positive equal to a negative? Is it because it messes other things up or is it an arbitrary rule? Instead of explaining the reasoning to a child, allow them discover it.

Have a “King for a Day” day where the students get to rewrite a particular math rule. Allow them to sit as a class or family and decide on the rules for multiplying signs numbers.

Give them plenty of time to experiment and see if the new rule messes anything up. (If your administrators or math curriculum specialists fret about you allowing students this time, note to them that the students are behaving like real mathematicians. They are inventing a concept or rule and testing it out to see what happens.)

If the students discover that the new rule has dire consequences when applied to the rest of our number system, they then see the reason for the rule. And they’ve learned through experience and solidified the rule in their minds!

If the students discover that there are no consequences of the new rule, they can then determine that the rule was created arbitrarily and agreed upon for convenience. It is a rule for rules’ sake.

You can discuss with them how things would be different if they kept the new rule and everyone else in the world followed the old rule. Again experience and discussion will solidify the “real” rule.

So the next time you’re faced with “why?” regarding a math rule, give it a shot. Let the children be the rulers of their own math world!

Will it work? Did it work? Share your thoughts in the comments!

Related articles
June 29, 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 Adding Evens and Odds is like Multiplying Positives and Negatives
I was playing cribbage with Husband when I found my hand to be 8, 4, 2, 2 and the cut card was also a 2. I focused hard to see if I could come up with 15. (If any group of cards add to 15, you get two points.)

My resolve was strong. Then I remembered that for any group of numbers to add up to an odd number, one of them must be odd.

While shuffling for the next hand, I started thinking about adding evens and odds:
- Even + Even = Even
- Odd + Odd = Even
- Odd + Even = Odd
This looked familiar. Then I remembered the rules of multiplying positives and negatives:
- Pos x Pos = Pos
- Neg x Neg = Pos
- Neg x Pos = Neg
I was so excited to notice this. Alas, I’m pretty sure I’ve noticed it before. I love rediscovering patterns as much as discovering them!

Look for patterns everywhere.

Always keep your eyes peeled for similarities. When you see them, if your children are old enough, use leading questions to get them to observe the same thing. If they aren’t old enough, keep a journal to share things with them later. Encourage your kids to look for patterns in everything. Use toys and manipulatives to improve this, but also use nature and everyday activities. What patterns or nifty things in math have you noticed recently?

Related articles
June 7, 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
Is Zero Positive or Negative?

When we teach math at home, we start with the counting numbers. Then we move to either the homeschool or classroom setting and build on those with negatives. And zero is sometimes left behind.

Zero seems to be an understood concept because it’s so much a part of our lives, as grownups. But zero is a very unusual bug.

Is Zero Positive?

We think of the numbers as either positive or negative.

But actually, all real numbers can be divided into three categories: positive, negative and zero.

This is one of the reasons zero can be forgotten in all this mix. With so many other numbers to be positive and negative, zero’s truly in a class by himself.

Notice I wrote “All real numbers can be divided into three categories.” Imaginary and complex numbers can’t be classified this way. You can order imaginary and complex numbers, but you have to do it without labeling them as positive or negative.

Why can’t we just say zero is positive?

I guess we could. But you might not want to when you see this…

In math, sometimes, we do things because if we didn’t, they’d mess everything else up.

This is one of those situations.

So let’s say 0 is positive, just for the minute.

Take the rule, “When multiplying, if the signs are different, the result is negative.” So then (positive) zero times negative 3 is (negative) zero. Like this:

0 x -3 = +0 x -3 = -0

But we determined that zero was positive!

So this messes things up. Therefore, zero needs to be in a class by himself.

But zero feels positive and negative, sometimes!

Indeed if a kid doesn’t owe his sister anything, that’s a positive feeling.

If you owe someone $3, then you can say you have negative 3 dollars. So if you owe them nothing, don’t you have a negative zero dollars?

This is more of a direction than a positive or negative sign. The “vector” (arrow in a direction) that has length zero can technically be pointing in any direction. It doesn’t matter. And that’s what’s going on when you think of zero as positive or negative. You think of owing (money going in a direction) nothing.

But remember, that’s only thinking of zero as positive or negative. He really isn’t either.

He’s just zero.

Check out the classic Schoolhouse Rock Multiplication Rock video and let me know what you think of all this zero talk.

May 25, 2011
Multiplication Tricks Roundup
Should kids know their multiplication facts?

When teaching math, whether in the home or in the classroom, having the multiplication facts memorized speeds things up. If you allow a calculator, you’re still reducing the speed of doing other math calculations or thinking.

But is there some way to memorize these and make them fun?

I scrounged the web and found some resources for multiplication tricks.

Poems as Multiplication Tricks

Inspired by @Leighbra at Hearth to Heart, I found a list of math poems that can help kiddos (and grownups) remember some of the facts. My favorite: I ate and ate and got sick on the floor. Eight times eight is 64.

Seven

This is an impressive list of various ways to multiply by seven. Tricks to Multiply by Seven

Nine

This one shows a way to multiply 9s without using your fingers. Which means you don’t have to waste time putting down and picking back up your pencil! Tricks to Multiply by 9

Using Fingers

This article gives three tricks to multiply using your fingers. Not my favorite, only because I hate to put down my pencil. But if you are doing purely mental multiplication, this article’s a great resource! Three finger tricks for multiplying

Multiplication Tricks for Other Numbers

Here’s a couple more that give multiplication tricks for many numbers:
- Multiplication Tips and Tricks by Math Is Fun (with a handy chart!)
- Multiplication tricks by Home-Ed
The one I made up

I created this while in class one day. I have a hard time remembering the difference between 7 x 8 and 9 x 6. So I thought about it and saw that 56 = 7 x 8, which is 5, 6, 7, 8 in order!

Do you have one? Please share with us!
May 24, 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