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!