First of all I want to mention that Henry at Why Homeschool is looking for homeschooling carnival submissions for an anniversary edition next week. Check here for instructions.
And then, about fractions. I have now updated and improved my two fractions workbooks.
In these books, I've strived to teach all fraction operations visually, and explain why the various rules for fraction operations work.
I know, most math books have pie charts for fractions as well, but here's the difference: in my books, the kids do a lot of exercises using those pies, and not just see them once in the beginning of the lesson.
And these are not manipulative exercises but simply picture exercises in the book.
I've always felt you need to first get the kid to understand what fractions are and only then start dealing with rules.
Those rules so often just get confused with each other. (Have you ever had a child try add fractions by adding the numerators and the denominators...?)
I've strived to create lessons that gently guide the student in discovering the "rule" for multiplying or dividing fractions, while doing all those picture exercises.
We need to strive that our students first remember some simple pictures in their minds before they start applying some half-forgotten rule and just hoping it's right.
So that's the goal of these two fraction workbooks. They are sold as PDF files that you download and then print; the first book is $4, and the other is $3.50.
Check also the free sample pages!
Wednesday, December 27, 2006
Learning fractions visually
First of all I want to mention that Henry at Why Homeschool is looking for homeschooling carnival submissions for an anniversary edition next week. Check here for instructions.
And then, about fractions. I have now updated and improved my two fractions workbooks.
In these books, I've strived to teach all fraction operations visually, and explain why the various rules for fraction operations work.
I know, most math books have pie charts for fractions as well, but here's the difference: in my books, the kids do a lot of exercises using those pies, and not just see them once in the beginning of the lesson.
And these are not manipulative exercises but simply picture exercises in the book.
I've always felt you need to first get the kid to understand what fractions are and only then start dealing with rules.
Those rules so often just get confused with each other. (Have you ever had a child try add fractions by adding the numerators and the denominators...?)
I've strived to create lessons that gently guide the student in discovering the "rule" for multiplying or dividing fractions, while doing all those picture exercises.
We need to strive that our students first remember some simple pictures in their minds before they start applying some half-forgotten rule and just hoping it's right.
So that's the goal of these two fraction workbooks. They are sold as PDF files that you download and then print; the first book is $4, and the other is $3.50.
Check also the free sample pages!
And then, about fractions. I have now updated and improved my two fractions workbooks.
In these books, I've strived to teach all fraction operations visually, and explain why the various rules for fraction operations work.
I know, most math books have pie charts for fractions as well, but here's the difference: in my books, the kids do a lot of exercises using those pies, and not just see them once in the beginning of the lesson.
And these are not manipulative exercises but simply picture exercises in the book.
I've always felt you need to first get the kid to understand what fractions are and only then start dealing with rules.
Those rules so often just get confused with each other. (Have you ever had a child try add fractions by adding the numerators and the denominators...?)
I've strived to create lessons that gently guide the student in discovering the "rule" for multiplying or dividing fractions, while doing all those picture exercises.
We need to strive that our students first remember some simple pictures in their minds before they start applying some half-forgotten rule and just hoping it's right.
So that's the goal of these two fraction workbooks. They are sold as PDF files that you download and then print; the first book is $4, and the other is $3.50.
Check also the free sample pages!
Monday, December 18, 2006
Square root of 11
Is square root of 11 an irrational number? How do you know from using a calculator? Thank you.
Well, I happen to know that if the square root of a natural number is NOT a whole number, then it is an irrational number. There are no other possibilities.
Of course you can't tell by the calculator. The calculator will show you 8 or 10 decimals, but you won't know from that if it's going to continue or not, or if it is periodical or not.
But pure mathematics and established, proven theorems will tell you that! : )
A proof that the square root of 2 is irrational
Lots of proofs of the same... plus one proving that any root is irrational if it's not a whole number
Square root of 11
Is square root of 11 an irrational number? How do you know from using a calculator? Thank you.
Well, I happen to know that if the square root of a natural number is NOT a whole number, then it is an irrational number. There are no other possibilities.
Of course you can't tell by the calculator. The calculator will show you 8 or 10 decimals, but you won't know from that if it's going to continue or not, or if it is periodical or not.
But pure mathematics and established, proven theorems will tell you that! : )
A proof that the square root of 2 is irrational
Lots of proofs of the same... plus one proving that any root is irrational if it's not a whole number
Friday, December 15, 2006
Cute circle terminology song
This was sent to me... I thought it was a very cute, melodic song about learning circle-related terminology such as radius, diameter, and circumference, and the two formulas with Pi (area formula and circumference formula):
Here's a direct link in case you want to spread it around:
http://www.youtube.com/watch?v=-z4SUypJZxo
This is done by Dave Mitchell who also has a website Arithmecode.com.
Here's a direct link in case you want to spread it around:
http://www.youtube.com/watch?v=-z4SUypJZxo
This is done by Dave Mitchell who also has a website Arithmecode.com.
Cute circle terminology song
This was sent to me... I thought it was a very cute, melodic song about learning circle-related terminology such as radius, diameter, and circumference, and the two formulas with Pi (area formula and circumference formula):
Here's a direct link in case you want to spread it around:
http://www.youtube.com/watch?v=-z4SUypJZxo
This is done by Dave Mitchell who also has a website Arithmecode.com.
Here's a direct link in case you want to spread it around:
http://www.youtube.com/watch?v=-z4SUypJZxo
This is done by Dave Mitchell who also has a website Arithmecode.com.
Monday, December 11, 2006
Multiplication, division, laser TVs, logs.
Well today I hopefully have something for everybody.
- The site DoubleDivision.org shows you an alternative long division algorithm, which takes the guessing away from estimating how many times the divisor goes into what needs divided. Also called 1-2-4-8 division.
This is a pretty cool way of dividing! The interactive tool shows you the steps right there for any problem you might come up with. - At MathLogarithms.com you can download an ebook by Dan Umbarger explaining logarithm how's, why's, and wherefore's in all detail for students.
Great resource for precalculus students. - You might also enjoy an alternative way to multiply called lattice multiplication. I did! It seems pretty simple.
- And lastly, if the math topics didn't interest you, how about my hubby's newest website called Laser-TVs.net ... It's about a totally new way of making TVs using lasers.
Multiplication, division, laser TVs, logs.
Well today I hopefully have something for everybody.
- The site DoubleDivision.org shows you an alternative long division algorithm, which takes the guessing away from estimating how many times the divisor goes into what needs divided. Also called 1-2-4-8 division.
This is a pretty cool way of dividing! The interactive tool shows you the steps right there for any problem you might come up with. - At MathLogarithms.com you can download an ebook by Dan Umbarger explaining logarithm how's, why's, and wherefore's in all detail for students.
Great resource for precalculus students. - You might also enjoy an alternative way to multiply called lattice multiplication. I did! It seems pretty simple.
- And lastly, if the math topics didn't interest you, how about my hubby's newest website called Laser-TVs.net ... It's about a totally new way of making TVs using lasers.
Friday, December 8, 2006
Nullity and dividing by zero by professor James Anderson
Recently this was highlighted at Slashdot and Digg.com both.
A math professor James Anderson has made a new 'number' or entity that he calls NULLITY, in order to solve problems such as 0/0, which traditionally is left undefined.
Basically he first defined 1/0 as infinity, -1/0 as negative infinity, and 0/0 as nullity, using Φ (Phi) as a symbol for it. He said this nullity lies outside our normal number line, not on it.
I watched the video shown on BBC news, and there he went on to show what happens with the "age-old" problem of 00, (zero to zeroth power) using just normal rules of arithmetic plus these definitions:
Anderson said in the comments following the main article on BBC that two other professors have helped him develop axioms for this new theory, and one of them checked them for consistency.
So.... a new theory. It doesn't sound that earthshaking to me; in fact I wonder if somebody in times past hasn't already tried this...?
(And yes, there exists a system for an "extended real number line".)
In the BBC 'divide by zero' article, you can leave comments, and lots of people have. Most of those seem to be on the mocking angle, putting down the theory.
That truly disappointed me! The same attitude seemed to prevail in people's comments at Digg.com.
Haven't we learned? The best I can remember from math history, negative numbers were certainly disliked and resisted for a long time, becore accepted by mathematicians.
And similarly when complex numbers came on the scene - Descarteds coined the term 'imaginary' as a derogatory term.
It seems that no one knows an application or use for this idea - at this time. But so what? It's no reason for ridiculing someone's theory. He has even consistent axioms written for it.
People have found uses for negative and complex numbers; they've proven very useful. Time will tell if this theory is of use or not; that's not my expertise or business at all. But I just wish people wouldn't be so quick to judge.
A math professor James Anderson has made a new 'number' or entity that he calls NULLITY, in order to solve problems such as 0/0, which traditionally is left undefined.
Basically he first defined 1/0 as infinity, -1/0 as negative infinity, and 0/0 as nullity, using Φ (Phi) as a symbol for it. He said this nullity lies outside our normal number line, not on it.
I watched the video shown on BBC news, and there he went on to show what happens with the "age-old" problem of 00, (zero to zeroth power) using just normal rules of arithmetic plus these definitions:
00 = 0(1 − 1) = 01 × 0-1 =(0/1)1 × (0/1)-1
Now, for every number, the 1st power is the number itself, while the -1 power is its reciprocal:
= (0/1) × (1/0) = (0 × 1) / (1 × 0 ) = 0/0 = Nullity.
Anderson said in the comments following the main article on BBC that two other professors have helped him develop axioms for this new theory, and one of them checked them for consistency.
So.... a new theory. It doesn't sound that earthshaking to me; in fact I wonder if somebody in times past hasn't already tried this...?
(And yes, there exists a system for an "extended real number line".)
In the BBC 'divide by zero' article, you can leave comments, and lots of people have. Most of those seem to be on the mocking angle, putting down the theory.
That truly disappointed me! The same attitude seemed to prevail in people's comments at Digg.com.
Haven't we learned? The best I can remember from math history, negative numbers were certainly disliked and resisted for a long time, becore accepted by mathematicians.
And similarly when complex numbers came on the scene - Descarteds coined the term 'imaginary' as a derogatory term.
It seems that no one knows an application or use for this idea - at this time. But so what? It's no reason for ridiculing someone's theory. He has even consistent axioms written for it.
People have found uses for negative and complex numbers; they've proven very useful. Time will tell if this theory is of use or not; that's not my expertise or business at all. But I just wish people wouldn't be so quick to judge.
Nullity and dividing by zero by professor James Anderson
Recently this was highlighted at Slashdot and Digg.com both.
A math professor James Anderson has made a new 'number' or entity that he calls NULLITY, in order to solve problems such as 0/0, which traditionally is left undefined.
Basically he first defined 1/0 as infinity, -1/0 as negative infinity, and 0/0 as nullity, using Φ (Phi) as a symbol for it. He said this nullity lies outside our normal number line, not on it.
I watched the video shown on BBC news, and there he went on to show what happens with the "age-old" problem of 00, (zero to zeroth power) using just normal rules of arithmetic plus these definitions:
Anderson said in the comments following the main article on BBC that two other professors have helped him develop axioms for this new theory, and one of them checked them for consistency.
So.... a new theory. It doesn't sound that earthshaking to me; in fact I wonder if somebody in times past hasn't already tried this...?
(And yes, there exists a system for an "extended real number line".)
In the BBC 'divide by zero' article, you can leave comments, and lots of people have. Most of those seem to be on the mocking angle, putting down the theory.
That truly disappointed me! The same attitude seemed to prevail in people's comments at Digg.com.
Haven't we learned? The best I can remember from math history, negative numbers were certainly disliked and resisted for a long time, becore accepted by mathematicians.
And similarly when complex numbers came on the scene - Descarteds coined the term 'imaginary' as a derogatory term.
It seems that no one knows an application or use for this idea - at this time. But so what? It's no reason for ridiculing someone's theory. He has even consistent axioms written for it.
People have found uses for negative and complex numbers; they've proven very useful. Time will tell if this theory is of use or not; that's not my expertise or business at all. But I just wish people wouldn't be so quick to judge.
A math professor James Anderson has made a new 'number' or entity that he calls NULLITY, in order to solve problems such as 0/0, which traditionally is left undefined.
Basically he first defined 1/0 as infinity, -1/0 as negative infinity, and 0/0 as nullity, using Φ (Phi) as a symbol for it. He said this nullity lies outside our normal number line, not on it.
I watched the video shown on BBC news, and there he went on to show what happens with the "age-old" problem of 00, (zero to zeroth power) using just normal rules of arithmetic plus these definitions:
00 = 0(1 − 1) = 01 × 0-1 =(0/1)1 × (0/1)-1
Now, for every number, the 1st power is the number itself, while the -1 power is its reciprocal:
= (0/1) × (1/0) = (0 × 1) / (1 × 0 ) = 0/0 = Nullity.
Anderson said in the comments following the main article on BBC that two other professors have helped him develop axioms for this new theory, and one of them checked them for consistency.
So.... a new theory. It doesn't sound that earthshaking to me; in fact I wonder if somebody in times past hasn't already tried this...?
(And yes, there exists a system for an "extended real number line".)
In the BBC 'divide by zero' article, you can leave comments, and lots of people have. Most of those seem to be on the mocking angle, putting down the theory.
That truly disappointed me! The same attitude seemed to prevail in people's comments at Digg.com.
Haven't we learned? The best I can remember from math history, negative numbers were certainly disliked and resisted for a long time, becore accepted by mathematicians.
And similarly when complex numbers came on the scene - Descarteds coined the term 'imaginary' as a derogatory term.
It seems that no one knows an application or use for this idea - at this time. But so what? It's no reason for ridiculing someone's theory. He has even consistent axioms written for it.
People have found uses for negative and complex numbers; they've proven very useful. Time will tell if this theory is of use or not; that's not my expertise or business at all. But I just wish people wouldn't be so quick to judge.
Monday, December 4, 2006
Blogging or a newsletter?
I've been blogging for over a year now.
It's been fun.
When I started, I wanted to have a means of connecting with my site's visitors, and I was leery of sending emails or establishing an email list because of potential problems when people report your newsletter as spam.
So I thought, hey, they can subscribe to the blog and it's going to be just like a newsletter to them that way.
This has been an experiment to me in that sense.
I put the "Subscribe to my blog" box on my site... and got a few subscribers a day, you know, between 0 and 3 daily. Subscriber numbers got between 250 and 300 and seemed to start slowing down.
I was thinking, these numbers aren't growing real fast. But I thought, well, that's just how many people want to listen to my blogging, so I wasn't worried about it either.
Then came last summer. I keep reading all sorts of search engine optimization and marketing stuff, and based on those, it was very important to have an email list.
So I decided that it was time to try do a newsletter. I went ahead and started it, using my blog writings as content for the newsletter.
And since last summer I've promoted a "Subscribe to my newsletter" on my site...
Guess what?
I think the proof is in the pudding.
I get 15 subscribers a day or so. The newsletter just starting accumulating subscribers in a much more rapid pace right from the start.
Most people just aren't ready for a blog... but embrace a newsletter easier!
I'm not sure why. You know, the stuff in my newsletters (see archives) comes from the blog writings. It's all the same stuff.
And in case you're wondering, no, I'm not planning to stop blogging. I think it can work great if some people read a blog and some people read all of that (or 'highlights') once a month in a newsletter.
But, there is one thing where a blog is greatly helpful, in terms of SEO. Google likes links from blogs. So promoting a new site in a blog greatly helps that site to get into the search engines.
So I'll keep blogging. People keep sending me links that I can blog about, and my hubby keeps finding interesting things on Digg.com that I sometimes blog about, and I keep having other ideas too.
For example, right now I have an idea about percent proportions that I want to blog about soon.
But all in all, December might be a little slower with blogging. I've gotten so busy with the Math Mammoth books.
But keep checking back when you can!
It's been fun.
When I started, I wanted to have a means of connecting with my site's visitors, and I was leery of sending emails or establishing an email list because of potential problems when people report your newsletter as spam.
So I thought, hey, they can subscribe to the blog and it's going to be just like a newsletter to them that way.
This has been an experiment to me in that sense.
I put the "Subscribe to my blog" box on my site... and got a few subscribers a day, you know, between 0 and 3 daily. Subscriber numbers got between 250 and 300 and seemed to start slowing down.
I was thinking, these numbers aren't growing real fast. But I thought, well, that's just how many people want to listen to my blogging, so I wasn't worried about it either.
Then came last summer. I keep reading all sorts of search engine optimization and marketing stuff, and based on those, it was very important to have an email list.
So I decided that it was time to try do a newsletter. I went ahead and started it, using my blog writings as content for the newsletter.
And since last summer I've promoted a "Subscribe to my newsletter" on my site...
Guess what?
I think the proof is in the pudding.
I get 15 subscribers a day or so. The newsletter just starting accumulating subscribers in a much more rapid pace right from the start.
Most people just aren't ready for a blog... but embrace a newsletter easier!
I'm not sure why. You know, the stuff in my newsletters (see archives) comes from the blog writings. It's all the same stuff.
And in case you're wondering, no, I'm not planning to stop blogging. I think it can work great if some people read a blog and some people read all of that (or 'highlights') once a month in a newsletter.
But, there is one thing where a blog is greatly helpful, in terms of SEO. Google likes links from blogs. So promoting a new site in a blog greatly helps that site to get into the search engines.
So I'll keep blogging. People keep sending me links that I can blog about, and my hubby keeps finding interesting things on Digg.com that I sometimes blog about, and I keep having other ideas too.
For example, right now I have an idea about percent proportions that I want to blog about soon.
But all in all, December might be a little slower with blogging. I've gotten so busy with the Math Mammoth books.
But keep checking back when you can!
Blogging or a newsletter?
I've been blogging for over a year now.
It's been fun.
When I started, I wanted to have a means of connecting with my site's visitors, and I was leery of sending emails or establishing an email list because of potential problems when people report your newsletter as spam.
So I thought, hey, they can subscribe to the blog and it's going to be just like a newsletter to them that way.
This has been an experiment to me in that sense.
I put the "Subscribe to my blog" box on my site... and got a few subscribers a day, you know, between 0 and 3 daily. Subscriber numbers got between 250 and 300 and seemed to start slowing down.
I was thinking, these numbers aren't growing real fast. But I thought, well, that's just how many people want to listen to my blogging, so I wasn't worried about it either.
Then came last summer. I keep reading all sorts of search engine optimization and marketing stuff, and based on those, it was very important to have an email list.
So I decided that it was time to try do a newsletter. I went ahead and started it, using my blog writings as content for the newsletter.
And since last summer I've promoted a "Subscribe to my newsletter" on my site...
Guess what?
I think the proof is in the pudding.
I get 15 subscribers a day or so. The newsletter just starting accumulating subscribers in a much more rapid pace right from the start.
Most people just aren't ready for a blog... but embrace a newsletter easier!
I'm not sure why. You know, the stuff in my newsletters (see archives) comes from the blog writings. It's all the same stuff.
And in case you're wondering, no, I'm not planning to stop blogging. I think it can work great if some people read a blog and some people read all of that (or 'highlights') once a month in a newsletter.
But, there is one thing where a blog is greatly helpful, in terms of SEO. Google likes links from blogs. So promoting a new site in a blog greatly helps that site to get into the search engines.
So I'll keep blogging. People keep sending me links that I can blog about, and my hubby keeps finding interesting things on Digg.com that I sometimes blog about, and I keep having other ideas too.
For example, right now I have an idea about percent proportions that I want to blog about soon.
But all in all, December might be a little slower with blogging. I've gotten so busy with the Math Mammoth books.
But keep checking back when you can!
It's been fun.
When I started, I wanted to have a means of connecting with my site's visitors, and I was leery of sending emails or establishing an email list because of potential problems when people report your newsletter as spam.
So I thought, hey, they can subscribe to the blog and it's going to be just like a newsletter to them that way.
This has been an experiment to me in that sense.
I put the "Subscribe to my blog" box on my site... and got a few subscribers a day, you know, between 0 and 3 daily. Subscriber numbers got between 250 and 300 and seemed to start slowing down.
I was thinking, these numbers aren't growing real fast. But I thought, well, that's just how many people want to listen to my blogging, so I wasn't worried about it either.
Then came last summer. I keep reading all sorts of search engine optimization and marketing stuff, and based on those, it was very important to have an email list.
So I decided that it was time to try do a newsletter. I went ahead and started it, using my blog writings as content for the newsletter.
And since last summer I've promoted a "Subscribe to my newsletter" on my site...
Guess what?
I think the proof is in the pudding.
I get 15 subscribers a day or so. The newsletter just starting accumulating subscribers in a much more rapid pace right from the start.
Most people just aren't ready for a blog... but embrace a newsletter easier!
I'm not sure why. You know, the stuff in my newsletters (see archives) comes from the blog writings. It's all the same stuff.
And in case you're wondering, no, I'm not planning to stop blogging. I think it can work great if some people read a blog and some people read all of that (or 'highlights') once a month in a newsletter.
But, there is one thing where a blog is greatly helpful, in terms of SEO. Google likes links from blogs. So promoting a new site in a blog greatly helps that site to get into the search engines.
So I'll keep blogging. People keep sending me links that I can blog about, and my hubby keeps finding interesting things on Digg.com that I sometimes blog about, and I keep having other ideas too.
For example, right now I have an idea about percent proportions that I want to blog about soon.
But all in all, December might be a little slower with blogging. I've gotten so busy with the Math Mammoth books.
But keep checking back when you can!
Rose petals puzzle - can you solve it?
This little puzzle is very intriguing... the answer is simple, SO simple that they claim more educated/intelligent folks have hard time figuring the answer.
So go try...!
Petals Around the Rose
So go try...!
Petals Around the Rose
Rose petals puzzle - can you solve it?
This little puzzle is very intriguing... the answer is simple, SO simple that they claim more educated/intelligent folks have hard time figuring the answer.
So go try...!
Petals Around the Rose
So go try...!
Petals Around the Rose
Friday, December 1, 2006
Mathematicians with tongue-in-cheek
Have you ever heard of the mathematician Niholas Bourbaki?
"Near the middle of the 20th century, Nicolas Bourbaki published several mathematics texts in areas such as Set Theory, Algebra, Topology, Functions of a Real Variable, etc. These texts had a profound impact on the mathematical landscape of the day and their affect is felt to this day."
So what's so important about Bourbaki?
Well, he never existed! It was just a tongue-in-cheek joke!
It's quite an interesting and funny story indeed... go read it at Natural Blogarithms.
"Near the middle of the 20th century, Nicolas Bourbaki published several mathematics texts in areas such as Set Theory, Algebra, Topology, Functions of a Real Variable, etc. These texts had a profound impact on the mathematical landscape of the day and their affect is felt to this day."
So what's so important about Bourbaki?
Well, he never existed! It was just a tongue-in-cheek joke!
It's quite an interesting and funny story indeed... go read it at Natural Blogarithms.
Mathematicians with tongue-in-cheek
Have you ever heard of the mathematician Niholas Bourbaki?
"Near the middle of the 20th century, Nicolas Bourbaki published several mathematics texts in areas such as Set Theory, Algebra, Topology, Functions of a Real Variable, etc. These texts had a profound impact on the mathematical landscape of the day and their affect is felt to this day."
So what's so important about Bourbaki?
Well, he never existed! It was just a tongue-in-cheek joke!
It's quite an interesting and funny story indeed... go read it at Natural Blogarithms.
"Near the middle of the 20th century, Nicolas Bourbaki published several mathematics texts in areas such as Set Theory, Algebra, Topology, Functions of a Real Variable, etc. These texts had a profound impact on the mathematical landscape of the day and their affect is felt to this day."
So what's so important about Bourbaki?
Well, he never existed! It was just a tongue-in-cheek joke!
It's quite an interesting and funny story indeed... go read it at Natural Blogarithms.
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