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10 Apr 2006 - 04:14

## introducing the absolute value

Ben is encountering absolute value now, in Saxon 8/7, for the first time (he's in 6th grade, which I think is an appropriate place to see it for the first time -- and of course to master it at the appropriate level).

Absolute value should be on the all-time list of Things That Kids Get Stuck On (now where did we put that? Oh, yeah.... here). Simply encountering an absolute value can be a major trauma for an older kid in algebra or calculus, and really understanding them in all the guises they appear in is a major challenge (remember problems such as "graph the region |x-3|>5 on the number line"? Well, that region has two pieces, but the region satisfying |x-3|<5 has one. Things like this really throw kids off).

So you want to make sure, at every stage, that the absolute value work that you do is Learned To Mastery before you move on. You want to do this anyway, of course, but with absolute values it's even more imperative.

So Ben had his first introduction to absolute value in the week before he came down with rotavirus (at least we're guessing it's rotavirus), and did not retain it, and he was showing signs of the kind of winging-it behavior you see with kids doing absolute value (my guess is that the winging-it thing comes from the fact that, unlike other symbols in math, if you completely ignore the absolute value signs, you sometimes get the problem right anyway).

The problems were similar to these:

|10| - |-10| = ?

|10-3|-|10+3| = ?

Here's how I taught it. Let us know any other tricks you have for teaching absolute value; we're all working on getting our Three Different Explanations.

1. I started our session by capitalizing on the 'function boxes' that Saxon does. Those problems look like this:

.. and the kid is, at first, asked to figure out what the value of the question mark is. Later he'll be asked to state the rule of the function (i.e., 'the rule is: multiply by 3'). A Saxon kid sees a million of these, starting in second and third grade.

So, when I got ready to discuss absolute value, I showed him this function box and asked him what the rule is:

He couldn't put it into words, and he didn't connect it up with the absolute value symbols; but it's easy to see what this function does; it strips the negative sign off of anything you put into it. "That's the absolute value function," I told him. "That's what absolute value does to things".

Then we did a few examples together where we took the absolute value of numbers, so he could confirm for himself that all the absolute value symbols do is to strip the negative signs off of numbers.

2. Then the problems got a bit more complex, such as the ones above, with expressions inside the absolute values that had to be evaluated before the absolute value is taken. Ben has lately been doing order-of-operations problems with parentheses, such as these:

10 + 6(4+3) -16 = ?

So I capitalized on his familiarity with those; I told him that absolute values are like parentheses that take away negative signs. They are like parentheses in that, in a complex expression, you go inside them and do that step first; you calculate the value of what's inside them; and then if the value is negative, you get rid of the negative sign. Then you get rid of the absolute value symbols.

So for a problem such as this:

|10-3|-|10+3| = ?

.. you begin by calculating what's inside the absolute value signs, as though they were parentheses:

|7| - |13|

and then strip the negatives (if any) off what's inside the absolute value symbols, to get:

7 - 13 = -6.

This problem is a good one to illustrate the dangers of a common misperception about absolute values: that the answers to problems involving absolute values always have to be positive. It's not so, as this problem shows.

-- CarolynJohnston - 10 Apr 2006

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We've had a brutal year on absolute value, entirely due to spiralling instruction.

Very, very difficult.

I think I posted some 'out loud' problems for absolute value (or maybe for computations involving absolute value)....let me go look.

-- CatherineJohnson - 10 Apr 2006

-- CatherineJohnson - 10 Apr 2006

I'm going to re-post this procedural technique here, because it's been a lifesaver for Christopher:

Here is how Christopher does this problem:

-1 - ( - 2 )

He pencils in a vertical line across both of the minus signs in the middle, turning them into plus signs:

- 1 + ( + 2 ) =

That works for him every time, no matter what the numbers, and he isn't thrown off by the same problem written with an absolute value:

-1 - | - 2 | =

This reminds me of Carolyn's belief that you need to get math into a child's hand.

-- CatherineJohnson - 10 Apr 2006

wow!

that's a very interesting idea!

I'm going to check Prentice Hall to see how they covered it

-- CatherineJohnson - 10 Apr 2006

they probably used triangles....

-- CatherineJohnson - 10 Apr 2006

oh hey!

Just opened the book to page 40 - there's a picture of a kid mowing the lawn wearing protective goggles!

-- CatherineJohnson - 10 Apr 2006

fyi

-- CatherineJohnson - 10 Apr 2006

nope, no triangles

they've just got the standard number line with absolute values defined as opposites

for me personally that works pretty well, but it wasn't great for Christopher because he needed to spend time subtracting positive & negative integers on the number line. (Haven't we put up illustrations of that before?)

Instead, Prentice Hall goes next to integer tiles; then has a couple of problems on a number line; then straight on to the various rules.

Virtually no practice and no further explanation.

-- CatherineJohnson - 10 Apr 2006

Can't find exactly what I'm looking for, but this isn't bad (unfortunately, no answers less than 0, but you could rewrite these problems as 'algebraic addition')...I'll keep trying):

-- CatherineJohnson - 10 Apr 2006

I found one!

2 + (-3) = -1

source: addition and subtraction of real numbers

-- CatherineJohnson - 10 Apr 2006

absolute values are like parentheses that take away negative signs

I love that!

I'm going to use both of these ideas with Christopher when he finally gets out of school and I can reteach pre-algebra.

-- CatherineJohnson - 10 Apr 2006

This site is good, too.

-- CatherineJohnson - 10 Apr 2006

-- CatherineJohnson - 10 Apr 2006

Have we talked about James Brennan before?

He has a complete book on algebra posted online.

-- CatherineJohnson - 10 Apr 2006

...when he finally gets out of school and I can reteach pre-algebra...

Sigh. That says it all, doesn't it? I'm actually very happy with my kids' school, but I, too, am looking forward to getting it out of the way so my kids can try to learn some Spanish.

-- DanK - 10 Apr 2006

That says it all, doesn't it? I'm actually very happy with my kids' school, but I, too, am looking forward to getting it out of the way so my kids can try to learn some Spanish.

LOL!

Do your kids get Spanish at school??

That reminds me, I'm going to have to get some decent Spanish teaching for my home-charter-schooling venture....

-- CatherineJohnson - 10 Apr 2006

We've reached a point at which our school is actually an obstacle to Christopher's education.

You should see our latest 'Irvington Insight' newsletter.

It's shocking!

8 pages of glossy paper (the most expensive you can use) and not one word about academics. In eight pages!

It's all character education, wellness, and Jason Project.

-- CatherineJohnson - 10 Apr 2006

That reminds me, I'm going to have to get some decent Spanish teaching for my home-charter-schooling venture....

If you lived in Texas or Colorado, you could have a Maria instead of a Martine, and she could do the Spanish thing (instead of French -- I think you said Martine is French, right?).

-- GoogleMaster - 10 Apr 2006

The curriculum wasn't especially coherent, but what they did learn they learned pretty well - there's a lot more teaching-to-mastery at the grade school level than you'd expect given practices around the country, and the two schools were nice, warm places with good relationships all 'round.

AND we weren't having constant, constant projects and meaningless tests.

If you wanted to teach more content at home, you had time to do it.

Now we're jumping through one hoop after another, and constantly having to advocate and butt heads with the administration; it's like having a 3rd kid with special needs almost.

WAY too much going on, at every level.

-- CatherineJohnson - 10 Apr 2006

all the absolute value symbols do is to strip the negative signs off of numbers.

Not quite. What an absolute value does is gives you the distance of the number from the origin ( The origin is 0, or [0,0], or [0,0,0], depending on whether you are in 1-d or 2-d or 3-d space. Calculating the origin for 4-d spaces and higher is left as an exercise for the reader).

Ben probably doesn't need to worry about this until he gets to complex numbers, which can be represented in 2-d space. But I'd emphasis in your explanations that the absolute value gives you the distance from zero on the number line. This will prepare him for understanding the absolute value as a distance on a plane.

Here's a link with an explanation of absolute values and complex numbers.

-- TracyW - 10 Apr 2006

yup, Martine is French

-- CatherineJohnson - 10 Apr 2006

I think Andrew, who is nonverbal, understands French

sure looks that way

and I THINK he understands it way better than Christopher does

-- CatherineJohnson - 10 Apr 2006

I don't know anything about complex numbers (I don't think), but 'distance from the origin' is how I've seen absolute value defined in all of my books

but I really like the function approach - I especially like just seeing it & sort of having it drill down into my brain - definitely gonna show Christopher

-- CatherineJohnson - 10 Apr 2006

Remember this is an introduction. There are a number of different ways to introduce absolute value -- I suppose you'd call my way 'functional' and the distance-from-the-origin way 'conceptual'.

My approach here goes along with my general philosophy that you teach the kid procedural knowledge first, and then back it up with conceptual knowledge. If you've introduced the notion of absolute value as distance from the origin, how exactly have you helped the kid to approach solving a problem like

|10-3| + |6-19| = ?

As I've said before -- procedural knowledge is only the beginning of my demands -- but it is also literally the beginning of my demands.

FWIW, I've also taught a lot of college-aged students who can't do problems like the one above.

-- CarolynJohnston - 11 Apr 2006

If you've introduced the notion of absolute value as distance from the origin, how exactly have you helped the kid to approach solving a problem like |10-3| + |6-19| = ?

I'm aiming at a slightly more basic level than Ben's problem. My thinking was directed at the question of what |-3| and |4| are, with an eye to the future of "what is |-2+4i|?".

When I tutored this (for students whose ages were a bit older than Ben), I would get my students to draw number lines and measure the distance between zero and the number. Then write down the relationship in a list, a bit like your function boxes.

Any more doubts about what an absolute value was -" back to the number line. Find the number 6. Subtract 19 from it. Plot it on the line. Oh, we're at -13. What's the distance between -13 and 0? [out comes the ruler] Yes, you're right, 13. So the absolute value of |6-19| is?"

It may have been a bit pedantic of me (I took up tutoring when I had just learnt a lot about technical drawing so was stuck thinking in quite a visual way). But it seemed to work.

I'm getting inclined to do some volunteer tutoring again, now I've learnt so much from this weblog about tutoring maths. For a start, I should have been digging back into my student's history more to find out where the maths went wrong.

-- TracyW - 11 Apr 2006

Any more doubts about what an absolute value was - back to the number line.

This is great for promoting understanding -- not so great for promoting speed, though. A kid needs both.

But in the end -- it's only 2 out of 3 different ways to explain the concept! So we still need a third... ;-)

-- CarolynJohnston - 11 Apr 2006

"So we still need a third... ;-)"

How about difference? (Referring specifically to problems of the form, |6-13|.)

The absolute value of both 13-6 and 6-13 is the difference between 6 and 13. This feeds back on the "difference explanation" of subtraction.

-- DougSundseth - 11 Apr 2006

Well when I do absolute values now in 1-d space I just drop the negative sign without thinking about the number line consciously at all (any more dimensions and the mental picture of the graph forms in my head quickly). Speed will come.

Plus I had appointed myself an objective of teaching students how to check their work when they weren't sure by reducing it to something they were sure of. E.g. check a simplified algebra equation by plugging in some numbers and seeing if the equality still holds. Drawing number lines was part of it.

-- TracyW - 11 Apr 2006

WebLogForm
Title: introducing the absolute value
TopicType: WebLog
SubjectArea: MiddleSchoolMath
LogDate: 200604100013