Category: addition

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“In mathematics the art of proposing a question must be held of higher value than solving it.”

Georg Cantor

Here is an except from that comment:

“Frankly, I don’t care if an elementary school child can add long columns of numbers in their head - it is an almost worthless skill. I do care if they can think about mathematical concepts.

Better to teach them to come up with simple proofs (not memorized proofs) of basic facts in math.

Better that they should understand what a prime number is, and why we care about prime numbers.

Better that they should learn to enjoy slow, deep thought about puzzles and concepts.

That is where the gold standard in math education is.”

I wanted to revisit this thought, because Penny brought up some great points. I don’t disagree with any of them. But I must say that I, as well as a lot of the readers are coming from a different place. Penny is a brilliant research mathematician. A lot of us, on the other hand, basically have a history of thinking that we sucked at math (at least until we came upon Math Mojo, and learned that almost no one sucks at math, but some sometimes the way math is taught sucks.)

I wanted to address some of the points Penny made, because those points made me think a lot this month. Here’s

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original photo from Richard Masoner Edited by Brian

Not sure what got me thinking about this today, but I was musing about the commutative property, and how it applies to addition and multiplication. (Yeah, it was a pretty boring morning.)

Specifically, I was thinking about the word, “augend”. The augend of an addition problem is the first of the series of addends. It’s not a word that is usually taught, and I was wondering why not.

You should be aware that addition and multiplication have the commutative property, and subtraction and division don’t. That just means 4+6 is the same as 6+4, and 3*7 is the same as 7*3, but 8-2 and 2-8 are not the same, nor are 9/3 and 3/9.

So it doesn’t really matter which of the numbers is placed where in addition, so you don’t need to specify which is the augend, and which is not. You can call them all addends. Same with multiplication - you can call them all multiplicands. (Not so with minuends, subtrahends, dividends and divisors, though - they don’t commute.)

But I think we should teach about augends. There are several good reasons why, and you can read about them at a lesson I just put up at Names of the numbers in basic arithmetic operations. That lesson is all about why we give the different parts of arithmetic problems their names, (like dividend, divisor, and quotient) and why it makes sense to learn them.

Now that I feel like I’ve cleared this all up for you and me, I’ve got something that I’m not so clear on. Maybe some kind reader has some insight about it she or he’d like to share. It’s this:

Since you must differentiate the names of the parts of subtraction or division problems, what happens if a problem has more than two terms, like 8-3-2? Is there a name for the third term? What if there are fourth or fifth terms?

I assume that they are called, “the second (secondary?) subtrahend, the third (tertiary?) subtrahend, and so on, but I’m not sure.

Anybody got any insights?

You may want to check Names of the numbers in basic arithmetic operations first, though.

By the way, if anybody can write me and tell me why I chose the image that I used for this post, I’ll send them a free Math Mojo e-booklet. (Use the contact me near the top right navigation bar on this page.)

Update: You don’t have to write about that anymore - we have a winner! Mark (see below) got the booklet.

To clarify: The big dog in the picture is “Doggy Daddy,” and the little dog at the door of the train is “Auggie Doggy.” (They are Hannah-Barbera cartoon figures.) They are about to enter a commuter train. Get it? Augend/Auggie, commutative property/commuter train? (Groan.)

Make your own Cardboard Abax

Educators just love buzz words. One of the most frequently used buzzwords in math ed. is “manipulatives.” Of course, the greatest manipulatives there are, are your hands and fingers. (Ever wonder why they call them “digits?”)

In the last post, I talked about basic addition. The links lead to MathMojo pages where you could learn a better way to count and do simple addition on your fingers than the way you learned (or didn’t learn) in school. 

The next step in manipulatives is the Abax. An abax is the forerunner of the abacus. It was the ancient counting  board, that, in the West at least, was used deep into the 15th century, when we were still using Roman numerals. (Ever try to add or multiply with Roman numerals?)

Using the abax today, we use base 10 and Arabic numerals, so it is much easier. It’s even easier than using an abacus, because the abacus uses a modified base 10 system, using groups of fives as well. 

The use of an abax is about the most visual and tactile way you can teach basic counting and arithmetic. I’d never actually seen or heard of  one being used in a classroom, so I investigated. Now it turns out that the abax pages are the most visited pages on the entire MathMojo site. People have been writing for over a year for me to start selling them and the instruction booklets for them again.

I had stopped making abaxes because my router was on the fritz. I finally realized that was a lame excuse not to make such a great learning tool available, so I have created an online tutorial (pdf file) that you can download for free to make your own abax out of cardboard. It only takes a few minutes to make. You can watch a short video of how to do it here:

how_to_make_a_cardboard_abax.mov

You can also:
download the free pdf. instructions for how to make an abax here.

I’ve also made the booklet “Counting and Adding on an Abax” available for sale again. It is only $9.95 as a downloadable e-booklet (it’s also available as a physical booklet for shipping by mail) and is about the best first investment you could make in a child’s math education.

Order your own copy of “Counting and Adding on an Abax” here.

I’m also getting ready to send out the newest issue of “The MathMojo Monthly” (”Comes out Quarterly, Mostly”) newsletter. It’s been so long since I’ve published one, that this one is packed with math and information. If you haven’t signed up for it yet, you might like to head out to Mathmojo.com, where you can sign up for
it now.

After having spent some time trying to find the best way to deliver some MathMojo, I have arrived, so far, at podcasting. But not that old “audio-only” stuff.  I’ve made an “enhanced podcast” about the very basics of addition, which you can access here.

 What is an “enhanced podcast?” It’s one with visuals, like a PowerPoint presentation. Not every browser can see it, though, although most can. You may need a fast connection to hear and view it - at least a bit faster than dial-up, although it will work with dial-up if you have a lot of patience. You don’t need iTunes or an iPod to listen to or watch it, although if you want to subscribe to it, iTunes is the way to go. iTunes is free, and if you don’t have it, you can find out all about it and get it at apple.com.

What is “subscribing” and why should you do it? When you go to the above site, to view the podcast, there will be a button on it, from which you can “subscribe.” That means that every time a new episode is published, it will automatically be sent to your computer the next time you open your iTunes program. That way, you will always be up-to-date with new podcasts from MathMojo, without having to do anything further. 

A word about the podcasts. They were made on a mac. I love my mac. I never was a geek, but this thing is user-friendly. It’s user-promiscuous! Using Garageband, iWeb and a dotMac account, it is pretty simple to do podcasts. I hope to get more heavily into this technology, because it is a great way to communicate with the world.

I’ll also be putting up some videos on this blog, and on the main Mathmojo.com site, and on YouTube in the near future, so stay tuned.   By the way, if you are at all interested in the kind of magic I do, you can check out a very old video I made (in about ‘91 or so, when I was living in Germany), below. 

This post is about meaning and math

First we’ll learn a math concept - Lowest (or Least) Common Denominator (or LCD). Then we’ll talk about how it’s sometimes used in everyday life.

In layman’s (non-mathematician’s, in this case) terms, the LCD is the largest partition of something that will also go into another thing of the same kind.

What the heck does that mean? (more…)

I am thinking about an example from a GRE (graduate record exam) book
that was shown to me.
I think it was "Which is greater, x²+y² or (x+y)²?

Here is the poop on how to think about examples like that. When in doubt – substitute
(if you can) for whole numbers. (In the original post, I had written real numbers instead of whole numbers. See the comment below about this by astute reader Randall Jones for important information about the difference that makes in this equation.)

So, try, say,  "Which is greater, 5²+3² or
(5+3)²?"
In the first case, 5² = 25 and 3² = 9, so it would be
25+9, which equals 34.
In the second case, you would first do the 5+3 (because parenthesis come first
in the order of operations) and get 8. Then you would square that, and get
64, which is clearly greater than 34.
Therefore  (5+3)²  is greater than 5²+3².

For an easy substitution you can do in your head in seconds, substitute 1s for x and for y:
= x²+y² or (x+y)²
= 1+1 or 2 ²
= 2 or 4

What if the example had been a bit different, though? What if it had been:
"Which is greater, x²*y² or (x*y)² (using multiplication instead
of addition)?

This article is continued at Mathmojo.com.