Category: multiplication

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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.

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I recently read a post on another blog concerning the two most important elements that children should master in math in order to succeed. The author suggests that basic skill with multiplication and basic mastery of fractions are the two essentials.

I am of the same opinion. The author also thinks that memorization and drills are the best way. On that, I’m not so sure. Yes and no.

Here’s my take:

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Have you ever heard of Hintikka’s Paradox? In a nutshell, it’s about, “Is it immoral to ask someone to do something that can’t be done?”

It’s so easy to say, “Just memorize it.” That’s the cry of people who want you to learn your “math facts.”

But do they ever teach a child how to memorize? How cruel to would it be to grade a kid on his reading ability if you never taught him how to read?

That’s the paradox of the memorization problem. Not every child even knows how to memorize.

There are lots of solutions, but I’ve never seen any good ones on math sites, or ever heard of teachers really using any. Let’s think about our part of helping out before we prescribe solutions we don’t provide.

Personally, I think rote memory is one rotten way to learn. Learning mnemonics is so much more fun, creative and effective. I don’t mean silly mnemonics that are taught in some of the baby-style trendy math methods that are out there. I mean a concerted effort to learn serious memory techniques. My favorite book to learn memory from is “How to Develop a Super Power Memory,” by Harry Lorayne. He’s written more since then, but despite its hokey title, it is the most straightforward of all the books I’ve read.

One solution for learning multiplication without rote memory is the method taught in my booklet “Numbers Juggling - Times Without the Tables.” the link for it is on the top of the right-hand column of this blog. I think this is the best solution for anyone who is dissappointed with the traditional “just shut up and memorize it” method.

Make sense?

Trust your brain - learn mental math and memory methods!

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A few years ago, a student wrote me the following question, which I think is very relevant for most children in the US, and that means for their parents and their teachers, too. Math inefficiency is a national epidemic, and it is a crying shame. Here’s what she wrote, and what I answered:

muliplycation

how can i rember my multiplycation? i don not know my multiplycation that well i need an answer. i am going to the sixth grade now i need to know my multiplycation. please answer my question.

I gave this response:

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This post is a continuation of the other posts about the video on YouTube entitled “An Inconvenient Truth” with M.J McDermott (not to be confused with Al Gore’s film) which concerns the dismal state of American basic math education in public schools. You can view it here.

M.J. had two good premises, but her conclusion does not jibe. “Their methods suck.” (True.) “My method is better.” (True.) “Therefore mine is the one everyone should use.” (Nahhhhh.)

Why don’t you experiment a lot and discover what works best for you, and keep refining it? It can be so much more fun and rewarding to do that. Respect your mind, not the opinions and emotional responses that were put there by others in the past. Try this stuff out, then decide.

It’s important to mention that people who think it’s OK not to learn the basic arithmetical operations because “you can do it with a calculator” are just plain damn dumb. That’s like saying, “Hey, this ‘walking’ stuff sucks. It takes effort! Why do we need to learn to walk? That takes years! Let’s just give everyone a wheelchair!’

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I’m tipping my mitt a little early, because the intermediate multiplication lessons are not finished yet. But I thought people should have easy access to the basic method for advanced multiplication. This is the tip of the iceberg, but everyone should at least be able to do this method. Anything less than that is either merely standard or substandard.

Here’s the “mystery algorithm” for 26 * 31, or any other set of two-digit numbers. Keep in mind that the description is much longer than the problem should take. After a little practice, it should take no longer than 10 seconds to do a problem like this in your head.

Step 1:
Multiply the digits in the ones columns of both numbers together. That would be the 1 ( of 31) and the 6 (of 26). That gives you 6. That’s the final digit of the product.

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(If you use a little imagination you can guess the title of this article.)

This article concerns M.J. McDermott’s youTube video about the sad state of basic math education in America. You can visit the video here, or you can simply scroll down to the next entry here in the Math Mojo Chronicles, where it is embedded.

It seems like M.J. McDermott has unleashed a firestorm that need to be unleashed. She’s gotten almost 60,000 hits in one week on youTube for a video about math! Imagine that! Good work, M.J.!

I’ve commented on that video several times, mentioning that there is at least one much better algorithm than what is called the “standard.”

Not one of the hundreds of other people who commented on the video seems to be aware of this, which is strange, because most of the comments to M.J.s video were posted by obviously thoughtful people. And one man who is obviously a lot better at mathematics than I am even made a video-reply to M.J.s video - but still stuck in standard mode.

Here is a synopsis of my part of the discussion so far:

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First a tip of the hat to thethinkingmother blog, an interesting homeschool blog, where I found out about the video that is the subject of this post.

The title of this post is a little strong, but I want to make an important point.

No, I’m not going to rag on Al Gore.

This post is about an interesting and provocative post on youTube, by M.J. McDermott. Ms. McDermott is a meteorologist on a Seattle news station, and she is concerned about the miserable state of math education in this country.

She has done a very nice exposé of some mindless math books and programs that have wormed their way into many school systems.
When you watch the video (I’ve linked to it here) you may be tempted to agree with just about everything, and become “incensed” at the state of affairs.

Me, too.

But it is all too easy to shake our little fists in the air and want to “get back to the good old ways” of doing things. The good old way in this case is what Ms. McDermott calls “The Standard Algorithm.” And there lies the rub.

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… a kiss is but a kiss. But on the other hand, a number can be many things. And there are many ways to remember numbers.

In lesson 3 of “Eating Math For Breakfast” you’ll learn a very basic but effective way to memorize a string of digits in order to do more math mentally.

Eventually Math Mojo will incorporate more mnemonic (memory) techniques into the website, but for now, this method will do. Mnemonics is a very powerful tool for magicians and is vastly underutilized in our education system.

In case you didn’t know, Math Mojo is part of Magic and Learning, a company that uses methods of magicians to teach thinking skills.

Today I have to start working on the “Math Mojo Monthly - Comes out quartly mostly” newsletter, to let subscribers know that the Chronicles have been reincarnated, and that these “Eating Math for Breakfast” lessons are available. .

I hope you have been trying out the lessons. They will become more fun to do in the near future, as I am adding a weirdness-factor to them as we go along.

(What the heck could that mean?)

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What makes us human? Opposable thumbs, memory? I think it’s the ability to determine to change, change, and be aware of the change.

Willingness to change is more important than the material, often.
It must be a big change.

I came up with the Idea of making an easily useable, understandable podcast that anyone from third-grade-level on should be able to understand, that will help them get better at basic math, so they can “eat it for breakfast.” There will be one lesson for each weekday.

These are very simple lessons, but it’s important to remember that simple does not mean easy. You will have to actively learn, and do the simple exercises in the lessons.

This first series of podcast lessons is meant to be the next step of learning how to turbo-charge your basic multiplication skills once you already know how to multiply one-digit numbers (like 6 x 7) using mental- and speed-math.

The first step is to make sure you know those one-digit multiplications “in your bones.” I feel the best way to lear them is with “Numbers Juggling - Times Without the Tables.”

These lessons are perfect for anyone who wants to learn how to multiply large numbers, like 7,693, times one-digit numbers, like 8, in their head, without pencil, paper, or heaven forbid, a calculator.

The lessons are free (for the time being) and you can find out more about them by clicking on the link on the left, entitled “Eating Math for Breakfast - Info., or you can click here.

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