Category: Math Mojo

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With so many things in bloom here in rural upstate New York, I figured it’s time for some new Ideas with Math Mojo.

One of them is the addition of a new segment, called, “Strange Powers of the Mind.” Look for a lot more weird stuff like this at Math Mojo. Sign up for the Math Mojo Monthly (”Comes out Quarterly, Mostly!”) Newsletter to keep up with the additions.
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To check multiplication of single digits by longer numbers with playing cards:

We’re going to use what I call “numbers crunching” to check. That is the same as using the nines-remainders. You do know how to get the nines-remainder of a number, don’t you? It’s very simple, but it takes a bit of explaining.

It also pays to know why checking with nines-remainders works. Both of those things are beyond the scope of this article, but I’m working on a booklet and a video about how to check your answers for all of the basic operations of math using “number crunching”. There are lots of tips and shortcuts that make this method absolutely simple and effective. Let me know if you’re interested by using the “Contact” box near the upper right hand corner of this page.

(This video will be re-edited and uploaded by the end of Wednesday, April 30)

If you know about crunching, you’ll be interested to know that practicing with cards like this is perfect for checking with crunching. It turns out that if you crunch all the digits from zero to nine, you get a crunch number of 0.

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Math Mojo has got some surprises for you. New lessons on how to improve your basic math skills, and videos! Professor Homunculus is getting his Video Mojo workin’ to bring you some great new stuff.

The first set of videos will be about how to practice multiplication using playing cards. So grab a deck of cards and let’s get going!

First, take out all the Spade cards from the deck - we’ll only be using those. Then, remove the court cards (the Jacks, Queens and Kings) from those cards. Consider the Ten to be a zero and the Ace to be a one.

Now you’ve got 10 cards, which represent the digits zero through nine.

Shuffle the cards. Now decide, in your mind, which digit you’d like to multiply by.

Deal the cards, face up, on the table so that you can see the faces of all the cards.

Get out a piece of paper and a pencil.

Depending on how advanced you are at multiplication, start at either the right (if you multiply the “school” way) or the left (if you know Math Mojo) of the spread deck, and start multiplying, writing only the answer (not the carries - never write the carries!)

In the video, we’ll be multiplying all the digits from 0 to 9, by 3. It’s simple to start with 3.

After you learn how to do it, try multiplying the cards by the other digits.

We’ll multiply by some higher digits in future videos.

You may have noticed that I don’t know my left from my right in this video. My bad!

Tomorrow we’ll practice checking, using this same example.

Something’s been on my mind for a long time. It’s the whole “public school atmosphere” thing.

I didn’t generally like school when I was a kid. I guess I went to pretty good schools, as far as schools go. I liked a lot of my teachers. I just didn’t like the “set-up.” I thought that whoever designed the whole process must have been a bunch of ignorant, arrogant jackasses.

Year after year, teachers and students complained about the same things. Some were reasonable, some weren’t.

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Surely most readers have had thoughts and observations about boys’ and girl’s differences concerning their typical learning strategies.

A reader (Susan G.) has been corresponding with Math Mojo for a week or so, and I’ve noticed that she’s made some great observations. She’s also written a e-book on word problems. I’m in the process of reading it, and it looks right-on-target. It will be a resource that teachers and home-schoolers will want to have.

I’ll be writing more about it in the future. In the meantime, if you want to find out more about it, please shoot me an e-mail (use the contact box on this page) asking about it, and I’ll put you in touch with her.

Susan had been kind enough to offer some of her expertise about education in general, and she’s been one of the readers who’s taken my request for proofreading to heart; she’s pointed out some typos in Math Mojo materials, which I’ll be correcting a.s.a.p. (Thanks for the heads-up on this stuff, Susan!)

She’s kindly given me her permission to publish some of the thoughts we’ve been sharing about math-ed.

Here’s and excerpt from a recent correspondence we’ve had:

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Today a concerned reader took issue with what he understands my methods to be. (See comment #4 at Augends, Addends and the Commutative Law of Addition.)
Fair enough, but I think he may have misunderstood my methods.

That could, of course, be due to the way I communicate them (or miscommunicate them). First let me say that none of the algorithms (ways of solving math problems) I teach are “mine.” “Math Mojo” is the name of my attitude, not the methods. The methods have been either gleaned from better sources than me (and most are hundreds, if not thousands, of years older), or I have “re-invented” them. That is typical for most people’s alternative methods.

Now to the issue; the reader stated:

After all these years (30) of struggling to teach children math, I finally realize why it is so difficult. A brief perusal of some of the mathematical girations you go through to multiply two numbers together explains a lot of why kids are poor at math. Commutative and associative properties are more easily understood when you have the basic tools to work with without adding zeros then subtracting the number from your cousins name on your mother’s side of the family. Teach the basics by rote then progress to the more abstract. Simple to complex seems to work.

Professor Homunculus’ reply:

I’m sorry you’ve come to that conclusion. If you’ve been teaching math for 30 years, you surely have some insights. But I can’t see see how you’d say, “simple to complex” seems to work. May I ask where it seems to work? And if it does, why is it a struggle for you, and why is it so difficult? Have you been teaching with the “girations” (sic) you say I use to make it so frustrating?

I’m not quite sure I understand the logic of your position.

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Tags: basic multiplication , learn multiplication , learning by rote , memorize multiplication , multiplication tables , rote memory , teach multiplication , times tables

Photography by Santarosa, Justin Wong and Brian. Edited by Brian

(This was meant to be posted on Monday. Sorry about the lateness).

Many of us who struggle to learn math (yes, I am one of them) suffer from assorted challenges, like ADD, procrastination, lack of focus, depression, and other things that are or aren’t nameable.

That’s no big, deal, unless we chose to make it one. Every challenge is just that, a call to step up and beat it. So we constantly seek methods, systems and other tools to help us. That’s partly what makes a challenge fun - finding new, cool things that other people never think about.

Recently I was speaking with a friend of mine. He seems to get a lot done, and I always admired that about him. I mentioned that to him, and he seems to think that he doesn’t really. At least not naturally, anyway.

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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.)

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Original photo by didbygraham

Q: When is a Compromise not a Compromise?
A: When it’s a Red Herring

We talked about “red herrings” in the previous posts about “The Case of the Missing Dollar.”

I occasionally do after-school presentations of MathMagic for the C.R.O.P. program in rural upstate New York. I’ve been participating in the program for years, as a local artist (magician). The program pays a very small honorarium, and a travel budget (Monday I drove 120 miles round-trip for peanuts) Most of the artists do it out of love of bringing their art to children who otherwise may never get exposed to it. It is a labor of love to all concerned, but it is a great mission.

The Creating Rural Opportunities Partnership (CROP) After School and Summer Program is a program which does just what it is named.

In it’s mission statement:

The goal of CROP is to provide intellectual development and opportunities for academic achievement for students grades K-8 with a strong focus on middle school.  In addition, CROP provides enrichment, health, wellness, life skills, recreational and cultural opportunities for 1260 middle school and primary grade students, community members and parents through a 21st Century Community Learning Center Partnership.

Over the last few years, funding for this excellent program has dwindled. Sound familiar? Art and after-school programs are typically among the first to get their throats cut when the government feels it needs to cut costs.
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Original Photo by Norsehorse Edited by Brian

Ah, I love it when readers beat me to the punch!

The comments to the original post pretty much sum up the paradox and it’s solution very well.

Khaled’s and Mark’s comments illustrate perfectly one of the things I wanted to point out about this puzzle. That point is:

Just because something is phrased a certain way is not reason to assume that that phrasing is the best way to represent the problem. And one way to critically examine the situation is to reframe it in a mathematical equation.

Khaled said, “Interesting how, once you assume that you can implicitly trust a given source, you can be led through any logic, or illogic, and have a lot of trouble pulling yourself back to a critical mindset.”

How true. Then Mark gave a good method to understand how to see where the paradox lies when he said, “I started to write an equation, because properly written equations can solve all counting problems, but then realized that this was pointless, because adding 2 dollars to the 27 dollars the guests paid did not reflect what happened.”

Exactly! The question was phrased to lead you to believe that because the facts were a certain way (which it accurately represented) you had to see it in a certain way (which was anything but accurate).

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