So you say division is repeated subtraction? I'd like to see you prove it!

The truth is, subtraction is challenging. I was glad to hear Dr. Moldavan confirm as much in class. I have the procedural knowledge down pat (if I didn't at my age, I'd have bigger problems), but now that I'm thinking in terms of understanding instead of performing, I'm seeing that I don't have the comfort level with subtraction that I'd like if I'm going to  teach this fundamental concept.

I don't think I'd find it useful to practice a bunch of subtraction problems. But I would like to explore in more depth what it means to say that "division is repeated subtraction."

Maybe I should see what other resources are out there besides Khan Academy. But on the other hand . . . why fix what isn't broken, am I right? 

What can I say? I'm a loyalist.

So let's start at the beginning--with their video The idea of division:

Well, that WAS interesting. This video explains that you can think of 24 ÷ 3 as either 3 equal groups within 24, or as groups of 3 within 24.(This concept reminds me of Bonnie Jeanne's post on arrays and how you can flip them on their sides).

So on to repeated subtraction.

Wait . . . 

The Khan Academy search bar doesn't bring up anything for repeated subtraction?!

You betrayed my loyalty, Khan Academy!

After taking a moment to pick up the shattered pieces of my math life, I'm ready to move on. . . .

One of the things I've learned in this course is that unlike most other subjects, I learn math best when someone explains it to me. Videos are much more useful to me than textbooks are in this context. So I'll go straight to the source for All Things Video: YouTube.

Whoa--here's a link to a video combining two of the core concepts I'm exploring on my PLP: Division as Repeated Subtraction Using a Number Line!

Oooh. I really like her description of division using the example of 6 ÷ 2:

"This question is asking you, 'How many times does this number [2] fit into this number [6]?' "

I might reword her question as:

 

"How many groups of 2 can you divide 6 into?" Because that wording includes the word divide--and we're teaching division.

 

Okay, so I see in the video that the presenter is subtracting 2s to get to 0 . But I still don't get why division is considered repeated subtraction; after all, couldn't I just as easily start at 0 and see how many 2s I have to add to get to 6?

 

Let's see what else we can find. . . . Check out this video--Interpret division using repeated subtraction:

 

This video is only three minutes long. But it has two key phrases that break this concept open for me.

Phrase 1 is at 2:15: "We had ten objects. We wanted to divide by two. Meaning we wanted to make groups of two, or give two to each group." 

•  When I give two to each group, I am subtracting 2 each time. I am giving those twos away. Giving something away and subtracting it are the same to me conceptually. And now I understand why we would say "division is repeated subtraction."
  

Phrase 2 is at 2:49: "The quotient is how many times we subtracted the divisor."

• This is one of those simple phrases that one person might gloss right over, but that could shift someone else's understanding (namely mine). Basically, the answer to any division problem (quotient) is how many times the second number (divisor) fits into the first number (dividend).

 
But perhaps most importantly, now I understand something else: "multiplication is repeated addition" is very different from "division is repeated subtraction":

 

• Multiplication actually is repeated addition: 3 x 3 is simply a shorter way of saying 3 + 3 + 3. But

• Division is not necessarily repeated subtraction: For example, you can also start from 0 and add your divisor until you hit your dividend to get to the quotient--which does not use subtraction at all. 

 

In other words: "multiplication is repeated addition" is a mathematical fact. "Division is repeated subtraction" is a mathematical process.

 

How cool is that?!

 

Adding and subtracting negative numbers

It's kind of subtle, but if you've been following along with this blog at all, you may have noticed . . . negative numbers are not in my comfort zone. So the more I learn about them, the better. Today I'm focusing on adding and subtracting negative numbers--which is only fitting, since I started this exploration after getting that problem in our problem set.

Again, this is gonna be a shocker, but . . . we're going to the Khan Academy!

Not surprisingly, they have content that looks like it could be useful. Here's a video on adding and subtracting negative numbers. Let's take a look:

Okay, that wasn't particularly helpful. It's basically the exact same thing I did on my first blog entry about negative numbers. It does make me feel a little better, though, that I couldn't understand in that first entry how to use the number line to show -3 - (-5): The answer is, You can't--not without converting it to -3 + 5, anyway. Which is what he does at 2:40.

Maybe it will help if I can explain why subtracting a negative number is the same as adding a positive number? I was able to get there in my first post on this subject by using examples and logical argument, but truth be told? I still don't understand why subtracting a negative is the same as adding a positive. Maybe this video will help? After all, I'm not a betting man, but it is called  subtracting a negative = adding a positive:

That does help a little. The concept is,

• if Steve has a net worth of negative $3 (so -3, basically),

• and his uncle wants to help him out by taking away that negative net worth (so -3 - [-3], basically),

• then the easiest way to do that is by giving Steve $3 (so -3 + 3, basically).

I'd probably avoid using the term "net worth" with young kids, though. Speaking of which: When are students supposed to learn this? Khan Academy usually has the Common Core standard listed at the top of a lesson, but that's not the case here. Let me do a little digging. . . .

Oh.

That's much later in the curriculum than I thought. Seventh graders should be a lot easier to teach negative numbers to than second- or third graders. I feel better already! I may even feel good enough to move on to another topic. But let me do some practice first--which, Ill admit, I mostly want to do because I want to get to that sweet, sweet Level 2 of Mastery! 

So close!

What can I say?

 I'm feeling a little cocky now, so let me try a problem set with negative numbers but in a way I haven't studied them yet--negative symbol as opposite:

Ouch.

 I'll keep trying!

More on conceptualizing negative numbers

In my last post I explored how to add and subtract negative numbers on the number line. I got to a place where I think I can teach that to my students, but it still feels cumbersome--maybe because I can see on a number line that subtracting a negative is the same as adding it, but I still don't understand why.

As I was scrolling through the lessons with my good friends over at the Khan Academy, I found the Negative Numbers and Coordinate Plane section. Sounds promising. . . .

Oh my gosh they have a whole practice section on section on negative numbers on the number line! Let me click through that right quick . . .

Question 1

Question 2

Etc etc etc. . . . These are not what I need, although I do like how they do the number lines both horizontally and vertically which may give students the chance to conceptualize numerical order slightly differently.

Let's see what else they have. . . . A-HA! Here's an introductory video on negative numbers. This might be just my speed:

The most helpful explanation of the concept of negative numbers starts for me around 2:10: the higher the negative number, the smaller it is. He says it this way:

"Negative 100 is a lot less than negative 1."

He adds that we can think about this idea as a negative being a lack of something. So if I lose $100 from my bank account, I have way less money than if I lose $1 from my bank account. I might say it this way:

"How not-much of something do I have? If I have 100 not-muches of something, and you have 10 not-muches of something, who has less?" (Now that I see that all typed out, it could be even more confusing. But who knows? Maybe it unlocks something for someone. . . .)

I can also connect this concept to my last post about the number line: When I subtract, I go to the left--and when I go to the left, the numbers get smaller. So anytime a student is confused by this concept, she can simply draw a number line--all she has to do is memorize what the number line looks like, i.e., that the numbers immediately to the left of the 0 start at -1. So which number is smaller, -10 or -30? And she will see that the answer is -30, because she has to go to the left of -10 to get to it:

Gotcha.

I'm really starting to see how grasping the concept of negative numbers--especially adding and subtracting them!--can be very confusing for students. After all, I'm a student, and I'm confused myself!

Next time let's look in more depth at adding and subtracting negative numbers. Thanks for stopping by!

Why is subtracting a negative number the same thing as adding it?

I've always been a fan of the number line. So easy, right? To add, you go to the right. To subtract, you go to the left. 

But then we got this problem in our problem set:

 Explain, using both a chip model and a number line model, how it is possible to have a positive difference when subtracting two negative numbers—e.g., -2 – (-5).

  

If I solve -2 - (-5) using the number line rule above--i.e., subtraction moves to the left--I get the wrong answer:

Oh, dear.

It's only because I secretly know that subtracting a negative is the same as adding a positive that I was able to get this problem right: -2 - (-5) is the same as -2 + 5. And then the number line works:

That's better.

But why? Why doesn't it work to subtract -5 just by moving to the left on the number line? I guess the rules are different for negative numbers? 

As usual, I started with the good people at the Khan Academy. There's no obvious section for "why can't I just move to the left when I subtract -7?", so I tooled around a little until I found this in the Arithmetic Properties section: Apparently there's something called the Inverse Property of Addition? Sounds promising.

Hmmmmm. . . . 

So the inverse property of addition just says if you add the negative version of a number to itself, you get 0:

5 + (-5) = 0

17,462 + (-17,642) = 0

And as the video above shows, to get to 0 from 5 I have to move 5 spaces to the left:

Got it.

Plus I already know from my number line rules that to subtract I go to the left and to add I go to the right. I moved 5 spaces to the left to get to zero--which means I subtracted 5. 

So if

5 + (-5) = 0

and

5 - 5 = 0,

then

5 + (-5) = 0 is the same as 5 - 5 = 0. And that means adding a negative number is the same thing as subtracting it!

So then. . . . Might the inverse also be true? That is, is subtracting a negative number the same thing as adding it? 

"It's a long shot, but it just might work."

 Let's see how this might play out:

I secretly already know that -2 - (-5) = 3, and that on the number line it looks like this:

Oh, right, right.

 And what did I do to get to 3 from -2? Look at that--I moved 5 spaces to the right . . . which means (say it with me) I added 5!

So if

-2 - (-5) = 3

and

-2 + 5 = 3,

then

-2 - (-5) = 3 is the same as -2 + 5 = 3. And that means subtracting a negative number is the same thing as adding it!

BAM!

So it appears that the sacred rules of the number line,

add to the right

subtract to the left,

Still work with negative numbers, as long as we remember the following:

Subtracting a negative number is the same as adding it.

Adding a negative number is the same as subtracting it.

As long as we keep those rules in mind, we're good! 

Still seems cumbersome to teach a second grader, though. But I'll work on it. . . .