Category: speed and mental math

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Nice title, eh? Let me preface this with the admission that I know just about nothing about dyslexia. Clinically, I mean.

The reason for this post is that Angela (Mother Crone) left a very interesting comment on yesterday’s post concerning how mental math has helped her daughter, who is dyslexic.

How many screwbulbs does it take to light in a dyslexic?

(Yes, that was unbelievably cheap.) Although I have no insights into clinical dyslexia, I have fought my whole life against certain dyslexic-like symptoms. I also suspect that any person who is at least mildly aware of his or her thought-processes struggles with similar symptoms.

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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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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.

A few posts ago, I offered some tips about how to check large division problems without having to multiply huge divisors and quotients to get even huger dividends.

One of the drawbacks to using the “crunch” method, which I described, is that it is not 100% accurate.

Often, people who need to defend the status quo (you know who they are, they work in the principal’s office) insist that checking by crunching is not acceptable, because it is not foolproof.

Let me give an example:

You could divide 1206 by 18 and get 64

It looks like it works. But it doesn’t. The real answer is 67.

Sometimes you can transpose digits, or make a mistake, the crunch of which will work out to the crunch of the real answer. After all, there are only 10 digits which all integers can crunch to.

It is very seldom, though, that you will crunch mistaken digits and do the multiplication, and have the answer come out to a crunch number that still has the same crunch number as the real answer.

The reason that mistakes are so seldom, is that it is easy to add numbers like 1+8 and 6+4 and multiply the results.

Although mistakes still can be made, much less mistakes are made with this method than with the cumbersome method you probably learned in school. Consider this: What is easier to do,

(1+8)*(6+4)
=9*1
=0, (all of which you can do in your head, with no training, in seconds)
or
18*64? (Do you really want to multiply that mentally if you don’t have to?)

There are many small-minded people in education, who insist that methods other than theirs must be foolproof, when their own methods are even less foolproof.

The mission of MathMojo is not only to teach easier, more effective methods, but also to make math more meaningful to you. And one way to do that is to sharpen your critical thinking skills.

Here is a perfect opportunity to do just that. Can you see the flaw in the small-minded person’s argument? They set up conditions that new things must fulfill in order for them to consider using them. But they don’t put their own things under the same conditions.

That phenomenon is one of the most prevalent flaws of society. Catch it when you see it, and call them on it.

Hotcha!

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We’ve been talking about using factors to make long-division problems easier, sometimes being able to turn them into a manageable sequence of short-division problems, in which no paper and pencil (and certainly no calculators!) are needed.

Want to try another one? How about

962/52 ?

Well, they’re both even, so that’s going to be a piece of cake to start. Divide both by 2 and turn the problem into:

481/26

Can you factor them further? You can tell that 2 won’t be a factor. And a quick look at 26 tells you that 3,4,5,6,7,8,9 and 10 won’t factor into it. It’s only factor, other than itself and 1 is 13. When you factor 13 into 21, you get 2.

Now all you have to do is test if 481 is divisible by 13. If you know the trick to test for divisibility by 13, you could try that, but let’s just assume you don’t, and go ahead and divide it in our heads.

13 goes into 48 three times, with 9 left over. Carry the 9 to the front of the 1 in 481, and get 91. Divide that by 13, and whaddyaknow, it goes in exactly seven times. That gives us 37.

We have reduced the problem from

without much trouble, and no writing. I think you can handle 37/2 on your own from here.

Right, it’s 18, r. 1.

Remember, that’ the answer to 37/2. But if you want to check it as the answer to 962/52, you’re going to have to re-factor in the 13 and the 2 to the remainder. When you multiply the remainder (1) by the factors (2 and 13) you get 1 x 2 x 13, which is 26.

Checking 962/52 = 18, r. 26

962/52 = 18 r. 26

So the crunch to the problem is 0, remainder 26.

So the crunch to the answer is 0, remainder 26.

The crunch to the problem matches the crunch to the answer, so the answer is very probably right.

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In the last post we looked at the problem of 926/18, and we simplified it to 463/9, so we could make it a short division problem.

What if the problem had been 927/18?
Both numbers are not even this time, so it is not readily apparent if they have common factors.

If you know how to factor (if you don’t, you can get a lesson at The Pretty Good Guide to Prime Factorization at MathMojo.com.) then you factor both of these numbers by 9.

Here’s a hint: If a number can be crunched to 9 or 0, then nine is a factor of that number. If you want to know more about crunching, I refer you to “The See-Say-Write Method of Speed Addition“.

There are also many hints you can find about how to determine if numbers are divisible by other numbers. MathMojo will eventually cover this in depth, but I’m sure you can find info if you google “divisibility rules.”

Ok, so let’s factor 927/18.

Using short division by 9, we get 103/2. How easy is the problem now? Just cut 103 in half in your mind and get 51 remainder 1. But remember, like in the last post,
If you factor a division problem before you solve it, you must multiply the remainder by that factor after you are done.

So the answer to 927/18 is 51 remainder 9, (not 1).

Go ahead and check it. Remember how? If not, check out this post.

Check out the third and final post on this subject about Long Division Shortcuts. 

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(Is that title an oxymoron?)

Imagine you have to do this division:

926/18

How would you do it? Would you rewrite it with that funny division symbol (“division bracket,” or “right parenthesis followed by a vinculum over the dividend”)? Would you use a calculator? (Please say “no” to that!)

After you rewrote it, would you start by trying to figure out how many times 18 would go into 92? If you did, you would be doing it the way most people learned in school, and you would be wasting a lot of time and effort.
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They way we are usually taught to check division problems in school is unnecessarily complex. There is a better way. I always wondered why, after thousands of years of mathematics, schools generally haven’t figured that out. But I’d rather try solving the Riemann zeta-hypothesis than figure out why schools teach the way they do.

An astute reader in Iceland (yes, we get readers from the coolest places!) asked the following question:

I have a minor problem regarding crunching division problem. How would you crunch a problem like 275 divided by 11 = 25?
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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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