11 Nov 2005 - 03:53

drilling the properties

I mentioned before in 'Six Weeks On' that I'm still supplementing Ben's math education with a handmade-by-Mom worksheet every night. The topics are a mishmosh of things that I'm currently feeling he needs more work on.

The other day I realized that he cannot name the fundamental properties of arithmetic, and he cannot identify them when they come up in equations. There are probably two reasons for this. The first is that he recently jumped from doing supplementation with Saxon 6/5 last year, to doing Saxon 8/7 full time this year; in Saxon 8/7, they're reviewing the properties, so we missed Saxon's introduction to them. The other reason is that they weren't taught in Everyday Math, as far as I can tell.

Probably the Everyday Math people thought they were dry and stuffy, but they aren't; they are critical to being able to do algebra, and if you can label them with language (stuffy as that may seem), then you will also know how to identify and use the proper strategies for solving equations.

Here's a list of all of them:

• Identity properties.
  • of multiplication: 1*a=a.
  • of addition: a+0=a.

• Associative properties.
  • of multiplication: (ab)c=a(bc).
  • of addition: (a+b)+c = a+(b+c).

• Commutative properties.
  • of multiplication: ab=ba.
  • of addition: a+b=b+a.

• Distributive property: a(b+c) = ab+ac.

(Saxon spends a lot of time in the early grades doing exercises that show that subtraction and division do not have the commutative and associative properties. It's pretty cool.)

So why are the identity properties important?

Because they are the ones you use every time (for example) you solve a simple linear equation like 2x-5=9. To isolate the variable (i.e., get it on one side by itself), you first use the additive identity property to get rid of the 5, then use the multiplicative identity to get rid of the 2. The identity properties tell you how to undo the operations that have been done to the x.

Why are the associative and commutative properties important?

They're important to know for algebra, because they allow you to do lots of convenient regroupings and rearrangements of parts of an expression without having to worry about the fine details.

For example, imagine if these properties weren't true. Then it would be much harder to solve 2x - 5= 9, because if you tried to add 5 to both sides, then you would have to wonder whether (2x - 5) + 5 were the same as 2x + (5 - 5); it might not be.

Drilling this stuff

I've been giving Ben drills on identifying the properties of arithmetic in action for two nights running now, and so far, it's not clicking. That's okay; we'll keep at it.

I'm trying to give him the properties as he'll actually need to use them in algebra.

Tonight's drills:

• 136 x 12 = (100 x 12) + (30 x 12) + (6 x 12)
• 14 + 0 =14
• (3/3) x 8 = 8
• 16 + (6-6) = 16
• p(r+s) = pr + ps
• (a+b)+c = a+(b+c)

He's stinking at identifying these, so we'll keep on at this level for a while.

Any other suggestions on how to Drill The Properties?

Back to main page.

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Comments

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I don't have any clever idea, but sometimes it's more fun to find bugs. So, perhaps you can construct examples that show violations of the properties:

9 - 6 = 6 - 9

3 + ( 12 / 4 ) = 3 + 12 / 3 + 4

Okay. Maybe not. :(

-- DanK - 11 Nov 2005

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I'm going to be fussy (and hopefully not derail your point too far), but I don't like your examples of the identity property.

In your first example, you're really using the fact that additive inverses exist in Z. (N has the identity property, too, and you can't always solve by subtracting in N.) And in your second example, if you're working in Z, it's because Z is a UFD. (Otherwise it's the existence of multiplicative inverses in Q.)

A much nicer example of the multiplicative identity property would be the unit conversions. Or anything where you manipulate the expression by multiplying by a fraction equal to one. (Do you know how many times a semester I hear myself saying, "Look here [points at board], you will see that I've done something to one side of the equation but not to the other. Why is this OK? Do you see why this works? This expression [points at fraction whose numerator equals its denominator] is just equal to one. Do you see why this is one? Do you remember what happens when you multiply something by one? That's right, when you multiply by one, you have the same thing as what you started with. So I haven't changed anything; the value is unchanged.")

How to get your kid to recognize these? Not a clue. I've had students in abstract algebra struggle with the properties. (At least until a few years ago, the state of NY was all hardcore on these at the high school level. There's one question on NYS Regents Course III from the late 1990s that asks, "Which field property does the set of integers fail to satisfy?")

-- RudbeckiaHirta - 11 Nov 2005

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Carolyn,

Will you be posting any of the sheets as math lessons for us? I would love to use anything you've got.

-- SusanS - 11 Nov 2005

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"...you will see that I've done something to one side of the equation but not to the other. Why is this OK? Do you see why this works? This expression [points at fraction whose numerator equals its denominator] is just equal to one. Do you see why this is one?"

"How to get your kid to recognize these? Not a clue."

I agree. It's easy to say that anything times (or divided by) one remains the same. In my past (college) teaching of math, nobody had a problem with this. However, use this in context of an equation or expression, and you get blank stares. I know that at some point in my math ecucation, I finally realized how important these identities were. (especially the idea that anything is a fraction if you divide it by one) It took me about three years in high school to feel that I could do anything with algebra. Part of it, perhaps, had to do with finally understanding the power and use of these identities.

As for teaching, I just did what RH did. Over and over. Practice, practice, practice. And some say there is no linkage between mastery and understanding. They say that conceptual understanding is all you need. Conceptual understanding gives you the identities. True understanding gives you algegra.

-- SteveH - 11 Nov 2005

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I think we're feeling the effects of skipping 7/6, too. (I'm doing 8/7 myself, and that's the book I use for supplementing...)

I'm thinking I should go through 7/6 and see what things I need to pull out.

-- CatherineJohnson - 11 Nov 2005

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golly, yes

I second Steve.

Post the sheets!

btw, I'm probably going to put up a page JUST FOR EVERYONE'S WORKSHEETS.

-- CatherineJohnson - 11 Nov 2005

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The Russian Math lesson on moving variables & numbers from one side of an equation to the other is incredibly cool.

-- CatherineJohnson - 11 Nov 2005

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Changed my life.

-- CatherineJohnson - 11 Nov 2005

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Seriously.

-- CatherineJohnson - 11 Nov 2005

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Hey Dan!

I love that idea!

I'm going to write some sheets with bugs!

-- CatherineJohnson - 11 Nov 2005

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Look here [points at board], you will see that I've done something to one side of the equation but not to the other. Why is this OK? Do you see why this works? This expression [points at fraction whose numerator equals its denominator] is just equal to one. Do you see why this is one? Do you remember what happens when you multiply something by one? That's right, when you multiply by one, you have the same thing as what you started with. So I haven't changed anything; the value is unchanged.

Saxon does a beautiful thing that I'm afraid I'll have to scan rather than describe....

OK, why don't I do that.

-- CatherineJohnson - 11 Nov 2005

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I saw this for the first time in Saxon 6/5, and I found it enchanting.

Magical, really.

And not 'mathemagical,' as J.D. says.

-- CatherineJohnson - 11 Nov 2005

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OK, it's not supposed to be upside down.

-- CatherineJohnson - 11 Nov 2005

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-- CatherineJohnson - 11 Nov 2005

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Practice, practice, practice. And some say there is no linkage between mastery and understanding.

I've emailed with Willingham about this a little.

To me, it seems as if no one really knows how conceptual understanding emerges&mdashand it does emerge; I would call it an emergent property.

Willingham seems to imply that at some point understanding emerges from domain knowledge & automaticity.

-- CatherineJohnson - 11 Nov 2005

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Willingham seems to imply that at some point understanding emerges from domain knowledge & automaticity.

That is my experience with math.

-- KDeRosa - 11 Nov 2005

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"domain knowledge & automaticity"

Isn't this called experience? Isn't experience more than just being able to do your job faster?

This seems so obvious to me that I wonder if I am missing something. There are different levels of understanding so it's a vague point to begin with. Some concepts are easy to understand and some are difficult. I guess I don't understand why some seem to think that the opposite is true; that understanding can exist without domain knowledge and automaticity. Of course, they talk about "conceptual" understanding, but all of this is so vague that it could mean anything, like trying to whack a phantom mole.

-- SteveH - 11 Nov 2005

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That is my experience with math.

Interesting....

I wonder if this is a Core Principle??

I'm also thinking this may be yet another Lost in Translation moment.

Constructivists may have seen that understanding isn't something a teacher or a book can simply hand to you. Hence all the talk about 'passive' versus 'active' and 'chalk and talk.'

But from there, they decided that knowledge can't emerge from anything associated with traditional teaching, including practice.

maybe...

-- CatherineJohnson - 11 Nov 2005

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"domain knowledge & automaticity"

Isn't this called experience? Isn't experience more than just being able to do your job faster?

I think it's experience, yes.

I think this is what people are talking about when they use the term 'experience.'

-- CatherineJohnson - 11 Nov 2005

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There are different levels of understanding so it's a vague point to begin with. Some concepts are easy to understand and some are difficult.

Definitely.

Some understanding you can get from reading it in a book, or having a teacher just tell you.

I'm sure that happens when you've already got a lot of 'hooks' in your brain; you're ready to understand the concept; it's not brand new or foreign.

-- CatherineJohnson - 11 Nov 2005

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Conceptual understanding gives you the identities. True understanding gives you algegra.

Great!

Wit & wisdom!

-- CatherineJohnson - 11 Nov 2005

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"I'm going to be fussy (and hopefully not derail your point too far), but I don't like your examples of the identity property.

In your first example, you're really using the fact that additive inverses exist in Z. (N has the identity property, too, and you can't always solve by subtracting in N.) And in your second example, if you're working in Z, it's because Z is a UFD. (Otherwise it's the existence of multiplicative inverses in Q.)"

I agree with you that this is fussy.

I invite you to create your own user page with some other examples of how to drill the properties, as I will no doubt need them soon!

-- CarolynJohnston - 11 Nov 2005

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lol!

-- CatherineJohnson - 12 Nov 2005

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meanwhile i'm lazy to decide if something IS or IS NOT fussy

-- CatherineJohnson - 12 Nov 2005