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.
Wednesday, November 29, 2006
Square root problem
prove that
5√20 × √45 × √5 = 150√5
This is a quite easy problem. You use two basic "ideas" or properties relating to square roots:
* that √b × b is b - or that you can "pull out" a number times itself from under the root (this is just the definition of a square root of course).
* that √a √b = √ab - or you can combine the radicands under the same root when they are multiplied.
So √20 * √45 is equal to √20*45 = √4 × 5 × 5 × 9
= √2 × 2 × 5 × 5 × 3 × 3 = 2 × 5 × 3 = 30.
So that's the crux of that problem.
Square root problem
prove that
5√20 × √45 × √5 = 150√5
This is a quite easy problem. You use two basic "ideas" or properties relating to square roots:
* that √b × b is b - or that you can "pull out" a number times itself from under the root (this is just the definition of a square root of course).
* that √a √b = √ab - or you can combine the radicands under the same root when they are multiplied.
So √20 * √45 is equal to √20*45 = √4 × 5 × 5 × 9
= √2 × 2 × 5 × 5 × 3 × 3 = 2 × 5 × 3 = 30.
So that's the crux of that problem.
Math Mammoth Grade 5 Worksheets ready
I hope you're not tired of hearing this.... but this is what I've been busy with lately.
Just as of today, I got Math Mammoth Grade 5 worksheets ready and available for purchasing.
Like the others, there are two separate books, A and B, plus answer keys.
Price for the whole package is $10. And that includes 123 quality math worksheets all total.
Click the link to see sample worksheets.
And, I've also set up a volume discount for any of my math books:
For order totals at least $34 - a 20% discount.
For order totals at least $50 - a 25% discount.
For order totals at least $70 - a 30% discount.
Use coupon code 8A2301338 when ordering to get these discounts.
Just as of today, I got Math Mammoth Grade 5 worksheets ready and available for purchasing.
Like the others, there are two separate books, A and B, plus answer keys.
Price for the whole package is $10. And that includes 123 quality math worksheets all total.
Click the link to see sample worksheets.
And, I've also set up a volume discount for any of my math books:
For order totals at least $34 - a 20% discount.
For order totals at least $50 - a 25% discount.
For order totals at least $70 - a 30% discount.
Use coupon code 8A2301338 when ordering to get these discounts.
Math Mammoth Grade 5 Worksheets ready
I hope you're not tired of hearing this.... but this is what I've been busy with lately.
Just as of today, I got Math Mammoth Grade 5 worksheets ready and available for purchasing.
Like the others, there are two separate books, A and B, plus answer keys.
Price for the whole package is $10. And that includes 123 quality math worksheets all total.
Click the link to see sample worksheets.
And, I've also set up a volume discount for any of my math books:
For order totals at least $34 - a 20% discount.
For order totals at least $50 - a 25% discount.
For order totals at least $70 - a 30% discount.
Use coupon code 8A2301338 when ordering to get these discounts.
Just as of today, I got Math Mammoth Grade 5 worksheets ready and available for purchasing.
Like the others, there are two separate books, A and B, plus answer keys.
Price for the whole package is $10. And that includes 123 quality math worksheets all total.
Click the link to see sample worksheets.
And, I've also set up a volume discount for any of my math books:
For order totals at least $34 - a 20% discount.
For order totals at least $50 - a 25% discount.
For order totals at least $70 - a 30% discount.
Use coupon code 8A2301338 when ordering to get these discounts.
Tuesday, November 21, 2006
4th grade worksheets
I spent this last Saturday working on an idea that I probably should have done a long ago... making better use of my various worksheet generators.
I created a single page that has pre-made worksheets for 4th grade math. It's just long list of links, each of which generates a certain kind of worksheet. Should be handy for my visitors.
So these are still randomly generated, each time it's a new one. Just hit 'refresh from the browser and a new one is there.
And while building that page, I also fixed a few litte things in the generators, and added a few more options... For example, Fractions worksheet generator now has the option to make missing addend/factor/dividend sheets.
So hopefully with these changes they will better serve my visitors.
And yeah, I can hear some asking, "When are you going to do a similar page for 3rd or 5th grade?"
Well, I hope get such done before the year is over. We will see! I DO hope to get those done as well!
I created a single page that has pre-made worksheets for 4th grade math. It's just long list of links, each of which generates a certain kind of worksheet. Should be handy for my visitors.
So these are still randomly generated, each time it's a new one. Just hit 'refresh from the browser and a new one is there.
And while building that page, I also fixed a few litte things in the generators, and added a few more options... For example, Fractions worksheet generator now has the option to make missing addend/factor/dividend sheets.
So hopefully with these changes they will better serve my visitors.
And yeah, I can hear some asking, "When are you going to do a similar page for 3rd or 5th grade?"
Well, I hope get such done before the year is over. We will see! I DO hope to get those done as well!
4th grade worksheets
I spent this last Saturday working on an idea that I probably should have done a long ago... making better use of my various worksheet generators.
I created a single page that has pre-made worksheets for 4th grade math. It's just long list of links, each of which generates a certain kind of worksheet. Should be handy for my visitors.
So these are still randomly generated, each time it's a new one. Just hit 'refresh from the browser and a new one is there.
And while building that page, I also fixed a few litte things in the generators, and added a few more options... For example, Fractions worksheet generator now has the option to make missing addend/factor/dividend sheets.
So hopefully with these changes they will better serve my visitors.
And yeah, I can hear some asking, "When are you going to do a similar page for 3rd or 5th grade?"
Well, I hope get such done before the year is over. We will see! I DO hope to get those done as well!
I created a single page that has pre-made worksheets for 4th grade math. It's just long list of links, each of which generates a certain kind of worksheet. Should be handy for my visitors.
So these are still randomly generated, each time it's a new one. Just hit 'refresh from the browser and a new one is there.
And while building that page, I also fixed a few litte things in the generators, and added a few more options... For example, Fractions worksheet generator now has the option to make missing addend/factor/dividend sheets.
So hopefully with these changes they will better serve my visitors.
And yeah, I can hear some asking, "When are you going to do a similar page for 3rd or 5th grade?"
Well, I hope get such done before the year is over. We will see! I DO hope to get those done as well!
Friday, November 17, 2006
Multiplication trick
If you enjoy math 'tricks', here's one presented nicely in a slide show:
How to multiply numbers with 11-19 in less than 5 seconds
It explains how to multiply by 11, 12, etc.
And... here's a challenge (you can ask this of students, too):
After watching the slideshow, explain how it actually has very similar things going on as the standard algorithm.
In other words, how do the standard multiplication algorithm (in columns) and this one compare?
How to multiply numbers with 11-19 in less than 5 seconds
It explains how to multiply by 11, 12, etc.
And... here's a challenge (you can ask this of students, too):
After watching the slideshow, explain how it actually has very similar things going on as the standard algorithm.
In other words, how do the standard multiplication algorithm (in columns) and this one compare?
Multiplication trick
If you enjoy math 'tricks', here's one presented nicely in a slide show:
How to multiply numbers with 11-19 in less than 5 seconds
It explains how to multiply by 11, 12, etc.
And... here's a challenge (you can ask this of students, too):
After watching the slideshow, explain how it actually has very similar things going on as the standard algorithm.
In other words, how do the standard multiplication algorithm (in columns) and this one compare?
How to multiply numbers with 11-19 in less than 5 seconds
It explains how to multiply by 11, 12, etc.
And... here's a challenge (you can ask this of students, too):
After watching the slideshow, explain how it actually has very similar things going on as the standard algorithm.
In other words, how do the standard multiplication algorithm (in columns) and this one compare?
Thursday, November 16, 2006
Math Mammoth Grade 4 ready
Just tonight... I got my Math Mammoth grade 4-A and 4-B workbooks ready and available online. One of them is $5.50 and the other is $5...
I've truly put a lot of effort on those so I hope the worksheets serve well those who buy them.
They're mostly designed for teachers, but I've had homeschoolers buy the other grades so I guess they're for whoever needs a 4th grade math workbook or worksheets.
You can download sample sheets here.
I've truly put a lot of effort on those so I hope the worksheets serve well those who buy them.
They're mostly designed for teachers, but I've had homeschoolers buy the other grades so I guess they're for whoever needs a 4th grade math workbook or worksheets.
You can download sample sheets here.
Math Mammoth Grade 4 ready
Just tonight... I got my Math Mammoth grade 4-A and 4-B workbooks ready and available online. One of them is $5.50 and the other is $5...
I've truly put a lot of effort on those so I hope the worksheets serve well those who buy them.
They're mostly designed for teachers, but I've had homeschoolers buy the other grades so I guess they're for whoever needs a 4th grade math workbook or worksheets.
You can download sample sheets here.
I've truly put a lot of effort on those so I hope the worksheets serve well those who buy them.
They're mostly designed for teachers, but I've had homeschoolers buy the other grades so I guess they're for whoever needs a 4th grade math workbook or worksheets.
You can download sample sheets here.
Wednesday, November 15, 2006
All sorts of websites
I'm just going to drill through a few links people have sent me, or that I've taken notice of, lately. Hopefully you'll find something of interest!
1) JamesPoon.com is a teacher located in Singapore and his site has video tutorials (in English) and a forum getting started.
2) Weekly Math Problems from NASA - download free weekly math problems in the setting of space weather... quite interesting! You'll find topics such as radiation dosages at Mars, exposure calculations, background radiation on Earth... would be perfect for science-minded high school students.
3) This one you'll get to laugh at... Newton's laws revisited.
Some examples:
BATH THEOREM: When the body is immersed in water, the telephone rings.
LAW OF THE RESULT: When you try to prove to someone that a machine won’t work, it will!
LAW OF BIOMECHANICS: The severity of the itch is inversely proportional to the reach.
And so on... click on the link to read more.
4) Mathematical Moments - a series of one-page PDF posters or flyers... about the many various ways mathematics is used in modern society.
Teachers especially could use these to promote awareness of use of mathematics. It's just good to know in general how much mathematics is involved in various fields.. these flyers don't get into any deep stuff but are just real short 'snapshots'. Examples include:
Finding Oil
Solving Crimes
Unearthing Power Lines
Targeting Tumors
Beating Traffic
... and so on.
1) JamesPoon.com is a teacher located in Singapore and his site has video tutorials (in English) and a forum getting started.
2) Weekly Math Problems from NASA - download free weekly math problems in the setting of space weather... quite interesting! You'll find topics such as radiation dosages at Mars, exposure calculations, background radiation on Earth... would be perfect for science-minded high school students.
3) This one you'll get to laugh at... Newton's laws revisited.
Some examples:
BATH THEOREM: When the body is immersed in water, the telephone rings.
LAW OF THE RESULT: When you try to prove to someone that a machine won’t work, it will!
LAW OF BIOMECHANICS: The severity of the itch is inversely proportional to the reach.
And so on... click on the link to read more.
4) Mathematical Moments - a series of one-page PDF posters or flyers... about the many various ways mathematics is used in modern society.
Teachers especially could use these to promote awareness of use of mathematics. It's just good to know in general how much mathematics is involved in various fields.. these flyers don't get into any deep stuff but are just real short 'snapshots'. Examples include:
Finding Oil
Solving Crimes
Unearthing Power Lines
Targeting Tumors
Beating Traffic
... and so on.
All sorts of websites
I'm just going to drill through a few links people have sent me, or that I've taken notice of, lately. Hopefully you'll find something of interest!
1) JamesPoon.com is a teacher located in Singapore and his site has video tutorials (in English) and a forum getting started.
2) Weekly Math Problems from NASA - download free weekly math problems in the setting of space weather... quite interesting! You'll find topics such as radiation dosages at Mars, exposure calculations, background radiation on Earth... would be perfect for science-minded high school students.
3) This one you'll get to laugh at... Newton's laws revisited.
Some examples:
BATH THEOREM: When the body is immersed in water, the telephone rings.
LAW OF THE RESULT: When you try to prove to someone that a machine won’t work, it will!
LAW OF BIOMECHANICS: The severity of the itch is inversely proportional to the reach.
And so on... click on the link to read more.
4) Mathematical Moments - a series of one-page PDF posters or flyers... about the many various ways mathematics is used in modern society.
Teachers especially could use these to promote awareness of use of mathematics. It's just good to know in general how much mathematics is involved in various fields.. these flyers don't get into any deep stuff but are just real short 'snapshots'. Examples include:
Finding Oil
Solving Crimes
Unearthing Power Lines
Targeting Tumors
Beating Traffic
... and so on.
1) JamesPoon.com is a teacher located in Singapore and his site has video tutorials (in English) and a forum getting started.
2) Weekly Math Problems from NASA - download free weekly math problems in the setting of space weather... quite interesting! You'll find topics such as radiation dosages at Mars, exposure calculations, background radiation on Earth... would be perfect for science-minded high school students.
3) This one you'll get to laugh at... Newton's laws revisited.
Some examples:
BATH THEOREM: When the body is immersed in water, the telephone rings.
LAW OF THE RESULT: When you try to prove to someone that a machine won’t work, it will!
LAW OF BIOMECHANICS: The severity of the itch is inversely proportional to the reach.
And so on... click on the link to read more.
4) Mathematical Moments - a series of one-page PDF posters or flyers... about the many various ways mathematics is used in modern society.
Teachers especially could use these to promote awareness of use of mathematics. It's just good to know in general how much mathematics is involved in various fields.. these flyers don't get into any deep stuff but are just real short 'snapshots'. Examples include:
Finding Oil
Solving Crimes
Unearthing Power Lines
Targeting Tumors
Beating Traffic
... and so on.
Sunday, November 12, 2006
Worksheet generators
Someone pointed out to me how my addition worksheet generator had a bug: they couldn't make missing addend worksheets where the student is completing the next whole hundred.
Well I was able to fix it somewhat so now it works for that situation, as long as you don't choose step 2 for the value 2.
These generators... many people have mentioned how they like them. When I first made them, I didn't realize that putting so many options on them would make them not always work... not for every possible combination of options.
I figure that's why on most websites, math worksheet generators don't have many options!
But I made mine originally with as many options as I could possibly think of, because I figured teachers would like them that way! You know, you can choose the number of problems, how many rows, columns, font size, range of numbers including a 'step', missing addend or factor, switching value 1 and 2, etc.
So then it follows, that if you choose some step for values 1 and 2, AND for the answer, that quite likely the script won't find suitable problems. And similarly for other complicated combinations of various options.
Here's a direct link for generating a random worksheet like the lady wanted:
Missing addend worksheet - complete the next whole hundred.
Refresh the page to get a new one and new one and so on.
Here's the link to the addition worksheet generator itself.
Well I was able to fix it somewhat so now it works for that situation, as long as you don't choose step 2 for the value 2.
These generators... many people have mentioned how they like them. When I first made them, I didn't realize that putting so many options on them would make them not always work... not for every possible combination of options.
I figure that's why on most websites, math worksheet generators don't have many options!
But I made mine originally with as many options as I could possibly think of, because I figured teachers would like them that way! You know, you can choose the number of problems, how many rows, columns, font size, range of numbers including a 'step', missing addend or factor, switching value 1 and 2, etc.
So then it follows, that if you choose some step for values 1 and 2, AND for the answer, that quite likely the script won't find suitable problems. And similarly for other complicated combinations of various options.
Here's a direct link for generating a random worksheet like the lady wanted:
Missing addend worksheet - complete the next whole hundred.
Refresh the page to get a new one and new one and so on.
Here's the link to the addition worksheet generator itself.
Worksheet generators
Someone pointed out to me how my addition worksheet generator had a bug: they couldn't make missing addend worksheets where the student is completing the next whole hundred.
Well I was able to fix it somewhat so now it works for that situation, as long as you don't choose step 2 for the value 2.
These generators... many people have mentioned how they like them. When I first made them, I didn't realize that putting so many options on them would make them not always work... not for every possible combination of options.
I figure that's why on most websites, math worksheet generators don't have many options!
But I made mine originally with as many options as I could possibly think of, because I figured teachers would like them that way! You know, you can choose the number of problems, how many rows, columns, font size, range of numbers including a 'step', missing addend or factor, switching value 1 and 2, etc.
So then it follows, that if you choose some step for values 1 and 2, AND for the answer, that quite likely the script won't find suitable problems. And similarly for other complicated combinations of various options.
Here's a direct link for generating a random worksheet like the lady wanted:
Missing addend worksheet - complete the next whole hundred.
Refresh the page to get a new one and new one and so on.
Here's the link to the addition worksheet generator itself.
Well I was able to fix it somewhat so now it works for that situation, as long as you don't choose step 2 for the value 2.
These generators... many people have mentioned how they like them. When I first made them, I didn't realize that putting so many options on them would make them not always work... not for every possible combination of options.
I figure that's why on most websites, math worksheet generators don't have many options!
But I made mine originally with as many options as I could possibly think of, because I figured teachers would like them that way! You know, you can choose the number of problems, how many rows, columns, font size, range of numbers including a 'step', missing addend or factor, switching value 1 and 2, etc.
So then it follows, that if you choose some step for values 1 and 2, AND for the answer, that quite likely the script won't find suitable problems. And similarly for other complicated combinations of various options.
Here's a direct link for generating a random worksheet like the lady wanted:
Missing addend worksheet - complete the next whole hundred.
Refresh the page to get a new one and new one and so on.
Here's the link to the addition worksheet generator itself.
Thursday, November 9, 2006
Mental math tricks
Some children might be delighted to learn math 'tricks' - curious ways to do calculations such as multiplying 2-digit numbers.
The 'tricks' do not contain any magic but are based on solid mathematical principles.
For example, to subtract any number from 1000 or 10,000 or any power of ten... just subtract from 999 or 9999 etc. and add 1.
Subtracting 10,000 - 2,596 with the usual method in columns, you will get into lots of borrowing over zeros... and end up having a row of nines to subtract from - except in the ones column where you have 10.
So 10,000 - 2,596 is quickly done by looking at each digit's difference to 9 - except in the case of one's digit, when you will subtract from 10. The result: 7,404.
Another 'trick' is called vertically and crosswise and applies to multiplication.
It can easily be proven to work using simple algebra. But it is a nice little mental math method that can impress kids.
Read more about that trick here. You can also practice online.
(Please note: this so-called 'vedic' math probably is NOT from Indian vedas or anything. The original author of the book with that name just claimed that but it seems his claims have no base. It's just a system of mental math calculations. Learning a few tricks like that can be just some simple fun.)
The 'tricks' do not contain any magic but are based on solid mathematical principles.
For example, to subtract any number from 1000 or 10,000 or any power of ten... just subtract from 999 or 9999 etc. and add 1.
Subtracting 10,000 - 2,596 with the usual method in columns, you will get into lots of borrowing over zeros... and end up having a row of nines to subtract from - except in the ones column where you have 10.
So 10,000 - 2,596 is quickly done by looking at each digit's difference to 9 - except in the case of one's digit, when you will subtract from 10. The result: 7,404.
Another 'trick' is called vertically and crosswise and applies to multiplication.
It can easily be proven to work using simple algebra. But it is a nice little mental math method that can impress kids.
Read more about that trick here. You can also practice online.
(Please note: this so-called 'vedic' math probably is NOT from Indian vedas or anything. The original author of the book with that name just claimed that but it seems his claims have no base. It's just a system of mental math calculations. Learning a few tricks like that can be just some simple fun.)
Mental math tricks
Some children might be delighted to learn math 'tricks' - curious ways to do calculations such as multiplying 2-digit numbers.
The 'tricks' do not contain any magic but are based on solid mathematical principles.
For example, to subtract any number from 1000 or 10,000 or any power of ten... just subtract from 999 or 9999 etc. and add 1.
Subtracting 10,000 - 2,596 with the usual method in columns, you will get into lots of borrowing over zeros... and end up having a row of nines to subtract from - except in the ones column where you have 10.
So 10,000 - 2,596 is quickly done by looking at each digit's difference to 9 - except in the case of one's digit, when you will subtract from 10. The result: 7,404.
Another 'trick' is called vertically and crosswise and applies to multiplication.
It can easily be proven to work using simple algebra. But it is a nice little mental math method that can impress kids.
Read more about that trick here. You can also practice online.
(Please note: this so-called 'vedic' math probably is NOT from Indian vedas or anything. The original author of the book with that name just claimed that but it seems his claims have no base. It's just a system of mental math calculations. Learning a few tricks like that can be just some simple fun.)
The 'tricks' do not contain any magic but are based on solid mathematical principles.
For example, to subtract any number from 1000 or 10,000 or any power of ten... just subtract from 999 or 9999 etc. and add 1.
Subtracting 10,000 - 2,596 with the usual method in columns, you will get into lots of borrowing over zeros... and end up having a row of nines to subtract from - except in the ones column where you have 10.
So 10,000 - 2,596 is quickly done by looking at each digit's difference to 9 - except in the case of one's digit, when you will subtract from 10. The result: 7,404.
Another 'trick' is called vertically and crosswise and applies to multiplication.
It can easily be proven to work using simple algebra. But it is a nice little mental math method that can impress kids.
Read more about that trick here. You can also practice online.
(Please note: this so-called 'vedic' math probably is NOT from Indian vedas or anything. The original author of the book with that name just claimed that but it seems his claims have no base. It's just a system of mental math calculations. Learning a few tricks like that can be just some simple fun.)
Wednesday, November 8, 2006
Math Mammoth 3rd grade worksheets
... are ready as of today!
There are lots of sheets for multiplication concept and multiplication tables, division concept, geometry, measuring, for example.
Check out the sample sheets:
Math Mammoth Grade 3 Worksheets
There are lots of sheets for multiplication concept and multiplication tables, division concept, geometry, measuring, for example.
Check out the sample sheets:
Math Mammoth Grade 3 Worksheets
Math Mammoth 3rd grade worksheets
... are ready as of today!
There are lots of sheets for multiplication concept and multiplication tables, division concept, geometry, measuring, for example.
Check out the sample sheets:
Math Mammoth Grade 3 Worksheets
There are lots of sheets for multiplication concept and multiplication tables, division concept, geometry, measuring, for example.
Check out the sample sheets:
Math Mammoth Grade 3 Worksheets
Monday, November 6, 2006
Teaching integers
Recently I answered a question about teaching integers on an email list, and decided to post all that on my site as well.
Also, I spent a few hours making downloadable fact sheets about all integer operations - these are free.
The sheets try to include quite a bit about "why" the rules for adding, subtracting, multiplying, dividing integers work.
Teaching integers
Enjoy!
Also I will answer here a question left on my site... there was no email address left so I couldn't answer via email.
Prices per pound are always given in [dollars per pound] or [dollars / pounds ].
This gives you the idea: you need to take the dollar amount and DIVIDE it by the pounds.
$2.50 ÷ 3 1/3 lb = $2.50 ÷ 10/3 lb = $2.50 × 3/10 = $7.50/10 = $0.75 per lb.
Also, I spent a few hours making downloadable fact sheets about all integer operations - these are free.
The sheets try to include quite a bit about "why" the rules for adding, subtracting, multiplying, dividing integers work.
Teaching integers
Enjoy!
Also I will answer here a question left on my site... there was no email address left so I couldn't answer via email.
how do you do this problems: elizabeth bought 3 1/3 pounds of tomatoes for $2.50. how much did she pay per pound?
Prices per pound are always given in [dollars per pound] or [dollars / pounds ].
This gives you the idea: you need to take the dollar amount and DIVIDE it by the pounds.
$2.50 ÷ 3 1/3 lb = $2.50 ÷ 10/3 lb = $2.50 × 3/10 = $7.50/10 = $0.75 per lb.
Teaching integers
Recently I answered a question about teaching integers on an email list, and decided to post all that on my site as well.
Also, I spent a few hours making downloadable fact sheets about all integer operations - these are free.
The sheets try to include quite a bit about "why" the rules for adding, subtracting, multiplying, dividing integers work.
Teaching integers
Enjoy!
Also I will answer here a question left on my site... there was no email address left so I couldn't answer via email.
Prices per pound are always given in [dollars per pound] or [dollars / pounds ].
This gives you the idea: you need to take the dollar amount and DIVIDE it by the pounds.
$2.50 ÷ 3 1/3 lb = $2.50 ÷ 10/3 lb = $2.50 × 3/10 = $7.50/10 = $0.75 per lb.
Also, I spent a few hours making downloadable fact sheets about all integer operations - these are free.
The sheets try to include quite a bit about "why" the rules for adding, subtracting, multiplying, dividing integers work.
Teaching integers
Enjoy!
Also I will answer here a question left on my site... there was no email address left so I couldn't answer via email.
how do you do this problems: elizabeth bought 3 1/3 pounds of tomatoes for $2.50. how much did she pay per pound?
Prices per pound are always given in [dollars per pound] or [dollars / pounds ].
This gives you the idea: you need to take the dollar amount and DIVIDE it by the pounds.
$2.50 ÷ 3 1/3 lb = $2.50 ÷ 10/3 lb = $2.50 × 3/10 = $7.50/10 = $0.75 per lb.
Wednesday, November 1, 2006
Math learning and unhappiness
Recently Brown Center published a Report on American Education with a special sector about the happiness factor in learning.
The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data.
The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it.
American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America!
Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts.
The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable.
I can almost hear my homeschooling readers' anger raising...!
But wait a minute.
Mr. Loveless said, "We might want to focus on the math that kids are learning and just be a little less obsessed with the fact that they have to enjoy every minute of it."
"The implication is not 'Let's go make kids unhappy,"' he said. "It's 'Let's give kids better signals as to how they're performing, relative to the rest of the world."'
This effect in the U.S. may be due to the fact that by and large, mathematics instruction is delivered as easy, small, bite-size chunks that are easy for students to swallow.
Then, as they proceed in such a fashion from year to year, and never encounter problems that take more than X (X being a single-digit number) minutes to solve, they will obviously be confident of their mathematical abilities and think that they do well in mathematics.
In contrast, their peers in Singapore probably encounter challenging problems and frustration over those.
In the long run, those students don't feel so confident about math because they have gotten a glimpse about the fact there is a lot they don't know.
But in the process, they have learned the easier stuff better than their U.S. counterparts who seemingly don't often get beyond the simplest things in any mathematical topic.
Well, I certainly don't feel that we have to take the joy out of mathematics learning to get good results.
But on the other hand, the students need to encounter challenges if they are to be well proficient in the subject.
The truth, as always, must be somewhere in the middle.
When learning any topics - say fractions - we can give students easy bits at first. Then as they master those, go towards more difficult problems - AND not allow them to give up on these challenging problems so quickly.
Maybe group work could be used with those, as well.
It requires a good teacher that can do that - encourage and couch the students without giving them the answers, without letting them give up too easily.
I fathom that the frenzy on testing cuts down the time that would otherwise be used for deeper things and challenging problems. Teachers are probably in between a rock and a hard place as far as what they can devote the class time to.
What are your thoughts?
Several other bloggers have gotten on to this too:
Gooseania
Luboš Motl's reference frame
Mathematics and (un)happiness by Alexandre Borovik
Mathematics & (un)happiness at NeverEndingBooks
See also:
CNN news: Confident students do worse in math; bad news for U.S.
The 2006 Brown Center Report on American Education:
How Well Are American Students Learning? - With special sections on the nation's achievement, the happiness factor in learning, and honesty in state test scores
The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data.
The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it.
American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America!
Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts.
The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable.
I can almost hear my homeschooling readers' anger raising...!
But wait a minute.
Mr. Loveless said, "We might want to focus on the math that kids are learning and just be a little less obsessed with the fact that they have to enjoy every minute of it."
"The implication is not 'Let's go make kids unhappy,"' he said. "It's 'Let's give kids better signals as to how they're performing, relative to the rest of the world."'
This effect in the U.S. may be due to the fact that by and large, mathematics instruction is delivered as easy, small, bite-size chunks that are easy for students to swallow.
Then, as they proceed in such a fashion from year to year, and never encounter problems that take more than X (X being a single-digit number) minutes to solve, they will obviously be confident of their mathematical abilities and think that they do well in mathematics.
In contrast, their peers in Singapore probably encounter challenging problems and frustration over those.
In the long run, those students don't feel so confident about math because they have gotten a glimpse about the fact there is a lot they don't know.
But in the process, they have learned the easier stuff better than their U.S. counterparts who seemingly don't often get beyond the simplest things in any mathematical topic.
Is there a solution?
Well, I certainly don't feel that we have to take the joy out of mathematics learning to get good results.
But on the other hand, the students need to encounter challenges if they are to be well proficient in the subject.
The truth, as always, must be somewhere in the middle.
When learning any topics - say fractions - we can give students easy bits at first. Then as they master those, go towards more difficult problems - AND not allow them to give up on these challenging problems so quickly.
Maybe group work could be used with those, as well.
It requires a good teacher that can do that - encourage and couch the students without giving them the answers, without letting them give up too easily.
I fathom that the frenzy on testing cuts down the time that would otherwise be used for deeper things and challenging problems. Teachers are probably in between a rock and a hard place as far as what they can devote the class time to.
What are your thoughts?
Several other bloggers have gotten on to this too:
Gooseania
Luboš Motl's reference frame
Mathematics and (un)happiness by Alexandre Borovik
Mathematics & (un)happiness at NeverEndingBooks
See also:
CNN news: Confident students do worse in math; bad news for U.S.
The 2006 Brown Center Report on American Education:
How Well Are American Students Learning? - With special sections on the nation's achievement, the happiness factor in learning, and honesty in state test scores
Math learning and unhappiness
Recently Brown Center published a Report on American Education with a special sector about the happiness factor in learning.
The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data.
The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it.
American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America!
Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts.
The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable.
I can almost hear my homeschooling readers' anger raising...!
But wait a minute.
Mr. Loveless said, "We might want to focus on the math that kids are learning and just be a little less obsessed with the fact that they have to enjoy every minute of it."
"The implication is not 'Let's go make kids unhappy,"' he said. "It's 'Let's give kids better signals as to how they're performing, relative to the rest of the world."'
This effect in the U.S. may be due to the fact that by and large, mathematics instruction is delivered as easy, small, bite-size chunks that are easy for students to swallow.
Then, as they proceed in such a fashion from year to year, and never encounter problems that take more than X (X being a single-digit number) minutes to solve, they will obviously be confident of their mathematical abilities and think that they do well in mathematics.
In contrast, their peers in Singapore probably encounter challenging problems and frustration over those.
In the long run, those students don't feel so confident about math because they have gotten a glimpse about the fact there is a lot they don't know.
But in the process, they have learned the easier stuff better than their U.S. counterparts who seemingly don't often get beyond the simplest things in any mathematical topic.
Well, I certainly don't feel that we have to take the joy out of mathematics learning to get good results.
But on the other hand, the students need to encounter challenges if they are to be well proficient in the subject.
The truth, as always, must be somewhere in the middle.
When learning any topics - say fractions - we can give students easy bits at first. Then as they master those, go towards more difficult problems - AND not allow them to give up on these challenging problems so quickly.
Maybe group work could be used with those, as well.
It requires a good teacher that can do that - encourage and couch the students without giving them the answers, without letting them give up too easily.
I fathom that the frenzy on testing cuts down the time that would otherwise be used for deeper things and challenging problems. Teachers are probably in between a rock and a hard place as far as what they can devote the class time to.
What are your thoughts?
Several other bloggers have gotten on to this too:
Gooseania
Luboš Motl's reference frame
Mathematics and (un)happiness by Alexandre Borovik
Mathematics & (un)happiness at NeverEndingBooks
See also:
CNN news: Confident students do worse in math; bad news for U.S.
The 2006 Brown Center Report on American Education:
How Well Are American Students Learning? - With special sections on the nation's achievement, the happiness factor in learning, and honesty in state test scores
The report is based on 2003 Trends in International Mathematics and Science Study (TIMMS) data.
The TIMSS study found that countries in which kids report enjoying mathematics and feeling confident in it do worse in math than kids who report they don't like math and are not feeling confident in it.
American students are much more confident about their math abilities than Singapore students, yet they do far worse: Even the least confident students in Singapore outscore the most confident students in America!
Check the charts (slides) from the report... a PDF file. It's quick and easy to glance over for more details and charts.
The report author Tom Loveless was questioning the idea of teaching math so that students like it... that we don't need to always make math enjoyable.
I can almost hear my homeschooling readers' anger raising...!
But wait a minute.
Mr. Loveless said, "We might want to focus on the math that kids are learning and just be a little less obsessed with the fact that they have to enjoy every minute of it."
"The implication is not 'Let's go make kids unhappy,"' he said. "It's 'Let's give kids better signals as to how they're performing, relative to the rest of the world."'
This effect in the U.S. may be due to the fact that by and large, mathematics instruction is delivered as easy, small, bite-size chunks that are easy for students to swallow.
Then, as they proceed in such a fashion from year to year, and never encounter problems that take more than X (X being a single-digit number) minutes to solve, they will obviously be confident of their mathematical abilities and think that they do well in mathematics.
In contrast, their peers in Singapore probably encounter challenging problems and frustration over those.
In the long run, those students don't feel so confident about math because they have gotten a glimpse about the fact there is a lot they don't know.
But in the process, they have learned the easier stuff better than their U.S. counterparts who seemingly don't often get beyond the simplest things in any mathematical topic.
Is there a solution?
Well, I certainly don't feel that we have to take the joy out of mathematics learning to get good results.
But on the other hand, the students need to encounter challenges if they are to be well proficient in the subject.
The truth, as always, must be somewhere in the middle.
When learning any topics - say fractions - we can give students easy bits at first. Then as they master those, go towards more difficult problems - AND not allow them to give up on these challenging problems so quickly.
Maybe group work could be used with those, as well.
It requires a good teacher that can do that - encourage and couch the students without giving them the answers, without letting them give up too easily.
I fathom that the frenzy on testing cuts down the time that would otherwise be used for deeper things and challenging problems. Teachers are probably in between a rock and a hard place as far as what they can devote the class time to.
What are your thoughts?
Several other bloggers have gotten on to this too:
Gooseania
Luboš Motl's reference frame
Mathematics and (un)happiness by Alexandre Borovik
Mathematics & (un)happiness at NeverEndingBooks
See also:
CNN news: Confident students do worse in math; bad news for U.S.
The 2006 Brown Center Report on American Education:
How Well Are American Students Learning? - With special sections on the nation's achievement, the happiness factor in learning, and honesty in state test scores
Ejercicios de matemáticas
Recientemente he comenzado la tarea de traducir los generadores de ejercicios de matemáticas en español. Bueno, realmente es un amigo que hace la mayor parte de la traducción.
Ya tenemos algo de presentar: visita Mamutmatematicas.com y genera ejercicios de matemáticas gratis!
Los demás generadores de problemas siguen pronto...
Ya tenemos algo de presentar: visita Mamutmatematicas.com y genera ejercicios de matemáticas gratis!
Los demás generadores de problemas siguen pronto...
Ejercicios de matemáticas
Recientemente he comenzado la tarea de traducir los generadores de ejercicios de matemáticas en español. Bueno, realmente es un amigo que hace la mayor parte de la traducción.
Ya tenemos algo de presentar: visita Mamutmatematicas.com y genera ejercicios de matemáticas gratis!
Los demás generadores de problemas siguen pronto...
Ya tenemos algo de presentar: visita Mamutmatematicas.com y genera ejercicios de matemáticas gratis!
Los demás generadores de problemas siguen pronto...
Some math worksheets now in Spanish
I've had some of my math worksheet generators translated to Spanish, in case you're interested:
Math worksheets in Spanish. It's a new website... MamutMatematicas.com : )
Math worksheets in Spanish. It's a new website... MamutMatematicas.com : )
Some math worksheets now in Spanish
I've had some of my math worksheet generators translated to Spanish, in case you're interested:
Math worksheets in Spanish. It's a new website... MamutMatematicas.com : )
Math worksheets in Spanish. It's a new website... MamutMatematicas.com : )
Sunday, October 29, 2006
Elementary geometry: how much time should you devote to it?
A geometry question from a visitor:
During elementary mathematics, geometry plays more of a sideline role at first. It is intimately tied with measuring topics - and really, the word "geometry" means "measuring the earth", the science to measure the land.
The goal of elementary geometry seems to be that the student be able to find perimeters, areas, and volumes of common two and three dimensional shapes.
I would add to that the goal that the student can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes before entering 10th grade. (I've written about that before in the article Why is high school geometry difficult?.
According to the Curriculum Focal Points report recently released by National Council of Teachers of Mathematics, the following geometry topics play a major role in elementary grades:
Notice how the focal point below for grade 8 is different: no longer is the focus on area and volume of shapes, but on reasoning with lines and angles.
(Note: The absence of a geometry focal point for grades 2 and 6 does not mean that geometry is not studied on those grades. NCTM's focal points are only three per grade so on those grades there were other three topics that were in the focus.)
In a traditional textbook, how much time is spent on geometry? I checked a few books page counts to get an idea:
3rd 24/336 = 7%
4th 23/340 = 6.7%
4th 18/196 = 9.2%
6th 42/340 = 12.3%
6th 31/224 = 13.8%
7th 56/372 = 15.1%
These did not include measuring topics, but just geometry having to do with shapes, lines, angles, area, perimeter, volume.
Of course on lower grades, measuring topics are another 'slice', usually at least about as large as geometry.
So basically you would spend from 1/12 to 1/7 of the total time on geometry topics, increasing as you proceed to higher grades (while decreasing the amount of time devoted to measuring topics). Obviously various arithmetic topics take the bulk of time in elementary mathematics instruction.
1. How much time should be invested teaching geometry at an elementary level?
2. How much time is actually dedicated towards geometry in a tradicional textbook
Your guidance will be extremely appreciated!
During elementary mathematics, geometry plays more of a sideline role at first. It is intimately tied with measuring topics - and really, the word "geometry" means "measuring the earth", the science to measure the land.
The goal of elementary geometry seems to be that the student be able to find perimeters, areas, and volumes of common two and three dimensional shapes.
I would add to that the goal that the student can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes before entering 10th grade. (I've written about that before in the article Why is high school geometry difficult?.
According to the Curriculum Focal Points report recently released by National Council of Teachers of Mathematics, the following geometry topics play a major role in elementary grades:
| Grade | Explanations (from Curriculum Focal Points by NCTM) |
| Grade 1 Geometry: Composing and decomposing geometric shapes. | Children compose and decompose plane and solid figures (e.g., by putting two congruent isosceles triangles together to make a rhombus), thus building an understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine figures, they recognize them from different perspectives and orientations, describe their geometric attributes and properties, and determine how they are alike and different, in the process developing a background for measurement and initial understandings of such properties as congruence and symmetry. |
| Grade 3 Geometry: Describing and analyzing properties of two-dimensional shapes. | Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry. |
| Grade 4: Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes | Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle. |
| Grade 5: Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area | Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems. |
| Grade 7: Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity | Students also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. |
| Grade 7: and Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes | By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. ... Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders. |
| Grade 8: Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle | Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra. |
Notice how the focal point below for grade 8 is different: no longer is the focus on area and volume of shapes, but on reasoning with lines and angles.
(Note: The absence of a geometry focal point for grades 2 and 6 does not mean that geometry is not studied on those grades. NCTM's focal points are only three per grade so on those grades there were other three topics that were in the focus.)
In a traditional textbook, how much time is spent on geometry? I checked a few books page counts to get an idea:
3rd 24/336 = 7%
4th 23/340 = 6.7%
4th 18/196 = 9.2%
6th 42/340 = 12.3%
6th 31/224 = 13.8%
7th 56/372 = 15.1%
These did not include measuring topics, but just geometry having to do with shapes, lines, angles, area, perimeter, volume.
Of course on lower grades, measuring topics are another 'slice', usually at least about as large as geometry.
So basically you would spend from 1/12 to 1/7 of the total time on geometry topics, increasing as you proceed to higher grades (while decreasing the amount of time devoted to measuring topics). Obviously various arithmetic topics take the bulk of time in elementary mathematics instruction.
Elementary geometry: how much time should you devote to it?
A geometry question from a visitor:
During elementary mathematics, geometry plays more of a sideline role at first. It is intimately tied with measuring topics - and really, the word "geometry" means "measuring the earth", the science to measure the land.
The goal of elementary geometry seems to be that the student be able to find perimeters, areas, and volumes of common two and three dimensional shapes.
I would add to that the goal that the student can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes before entering 10th grade. (I've written about that before in the article Why is high school geometry difficult?.
According to the Curriculum Focal Points report recently released by National Council of Teachers of Mathematics, the following geometry topics play a major role in elementary grades:
Notice how the focal point below for grade 8 is different: no longer is the focus on area and volume of shapes, but on reasoning with lines and angles.
(Note: The absence of a geometry focal point for grades 2 and 6 does not mean that geometry is not studied on those grades. NCTM's focal points are only three per grade so on those grades there were other three topics that were in the focus.)
In a traditional textbook, how much time is spent on geometry? I checked a few books page counts to get an idea:
3rd 24/336 = 7%
4th 23/340 = 6.7%
4th 18/196 = 9.2%
6th 42/340 = 12.3%
6th 31/224 = 13.8%
7th 56/372 = 15.1%
These did not include measuring topics, but just geometry having to do with shapes, lines, angles, area, perimeter, volume.
Of course on lower grades, measuring topics are another 'slice', usually at least about as large as geometry.
So basically you would spend from 1/12 to 1/7 of the total time on geometry topics, increasing as you proceed to higher grades (while decreasing the amount of time devoted to measuring topics). Obviously various arithmetic topics take the bulk of time in elementary mathematics instruction.
1. How much time should be invested teaching geometry at an elementary level?
2. How much time is actually dedicated towards geometry in a tradicional textbook
Your guidance will be extremely appreciated!
During elementary mathematics, geometry plays more of a sideline role at first. It is intimately tied with measuring topics - and really, the word "geometry" means "measuring the earth", the science to measure the land.
The goal of elementary geometry seems to be that the student be able to find perimeters, areas, and volumes of common two and three dimensional shapes.
I would add to that the goal that the student can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes before entering 10th grade. (I've written about that before in the article Why is high school geometry difficult?.
According to the Curriculum Focal Points report recently released by National Council of Teachers of Mathematics, the following geometry topics play a major role in elementary grades:
| Grade | Explanations (from Curriculum Focal Points by NCTM) |
| Grade 1 Geometry: Composing and decomposing geometric shapes. | Children compose and decompose plane and solid figures (e.g., by putting two congruent isosceles triangles together to make a rhombus), thus building an understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine figures, they recognize them from different perspectives and orientations, describe their geometric attributes and properties, and determine how they are alike and different, in the process developing a background for measurement and initial understandings of such properties as congruence and symmetry. |
| Grade 3 Geometry: Describing and analyzing properties of two-dimensional shapes. | Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry. |
| Grade 4: Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes | Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle. |
| Grade 5: Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area | Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems. |
| Grade 7: Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity | Students also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects. |
| Grade 7: and Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes | By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. ... Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders. |
| Grade 8: Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle | Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra. |
Notice how the focal point below for grade 8 is different: no longer is the focus on area and volume of shapes, but on reasoning with lines and angles.
(Note: The absence of a geometry focal point for grades 2 and 6 does not mean that geometry is not studied on those grades. NCTM's focal points are only three per grade so on those grades there were other three topics that were in the focus.)
In a traditional textbook, how much time is spent on geometry? I checked a few books page counts to get an idea:
3rd 24/336 = 7%
4th 23/340 = 6.7%
4th 18/196 = 9.2%
6th 42/340 = 12.3%
6th 31/224 = 13.8%
7th 56/372 = 15.1%
These did not include measuring topics, but just geometry having to do with shapes, lines, angles, area, perimeter, volume.
Of course on lower grades, measuring topics are another 'slice', usually at least about as large as geometry.
So basically you would spend from 1/12 to 1/7 of the total time on geometry topics, increasing as you proceed to higher grades (while decreasing the amount of time devoted to measuring topics). Obviously various arithmetic topics take the bulk of time in elementary mathematics instruction.
Thursday, October 26, 2006
Word problem question
An interesting math teaching related question again from someone.
I have not written an ebook that would concentrate on those kind of problems. I do have some similar ones included in the 6th grade worksheets collection.
However, I wouldn't worry so much about those problems at 4th grade, especially the first one, because it requires quite many steps. She's still young.
You might be interested in Singapore math's word problem books; they might be helpful. They often employ diagrams that enable children to solve these kind of problems without the use of algebra.
But let's try to find the easiest way to solve these.
1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?
From the above you can see that $16 was the 'half of what was remaining'. Total is $32 + $4 = $36.
2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?
This one is easiest to solve with this reasoning:
If the two boys were of equal height, they'd both be 4 feet 4 inches. The DIFFERENCE in their heights is 4 inches, so just add half of that-or 2 inches- to the "midpoint" 4 ft 4 in to find the taller boy's height, and subtract that same 2 inches to find the shorter boy's height.
So Dan is 4 ft 6 in and Mike is 4 ft 2 in.
Hi Maria,
I have purchased some e-books from you and they are very helpful. My daughter is in 4th grade and I felt that she is weak in understanding the following problems. Would you please suggest which e-book we need to buy from you to help improve her basics.
Problems are as below:
1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?
2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?
I have not written an ebook that would concentrate on those kind of problems. I do have some similar ones included in the 6th grade worksheets collection.
However, I wouldn't worry so much about those problems at 4th grade, especially the first one, because it requires quite many steps. She's still young.
You might be interested in Singapore math's word problem books; they might be helpful. They often employ diagrams that enable children to solve these kind of problems without the use of algebra.
But let's try to find the easiest way to solve these.
1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?
|-- $4 --|----- half -----|----- half -----|
| $2 |----$14 ---|
From the above you can see that $16 was the 'half of what was remaining'. Total is $32 + $4 = $36.
2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?
|-------Dan ---------|----------- Mike -----------|
This one is easiest to solve with this reasoning:
If the two boys were of equal height, they'd both be 4 feet 4 inches. The DIFFERENCE in their heights is 4 inches, so just add half of that-or 2 inches- to the "midpoint" 4 ft 4 in to find the taller boy's height, and subtract that same 2 inches to find the shorter boy's height.
So Dan is 4 ft 6 in and Mike is 4 ft 2 in.
Word problem question
An interesting math teaching related question again from someone.
I have not written an ebook that would concentrate on those kind of problems. I do have some similar ones included in the 6th grade worksheets collection.
However, I wouldn't worry so much about those problems at 4th grade, especially the first one, because it requires quite many steps. She's still young.
You might be interested in Singapore math's word problem books; they might be helpful. They often employ diagrams that enable children to solve these kind of problems without the use of algebra.
But let's try to find the easiest way to solve these.
1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?
From the above you can see that $16 was the 'half of what was remaining'. Total is $32 + $4 = $36.
2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?
This one is easiest to solve with this reasoning:
If the two boys were of equal height, they'd both be 4 feet 4 inches. The DIFFERENCE in their heights is 4 inches, so just add half of that-or 2 inches- to the "midpoint" 4 ft 4 in to find the taller boy's height, and subtract that same 2 inches to find the shorter boy's height.
So Dan is 4 ft 6 in and Mike is 4 ft 2 in.
Hi Maria,
I have purchased some e-books from you and they are very helpful. My daughter is in 4th grade and I felt that she is weak in understanding the following problems. Would you please suggest which e-book we need to buy from you to help improve her basics.
Problems are as below:
1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?
2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?
I have not written an ebook that would concentrate on those kind of problems. I do have some similar ones included in the 6th grade worksheets collection.
However, I wouldn't worry so much about those problems at 4th grade, especially the first one, because it requires quite many steps. She's still young.
You might be interested in Singapore math's word problem books; they might be helpful. They often employ diagrams that enable children to solve these kind of problems without the use of algebra.
But let's try to find the easiest way to solve these.
1) Marge spent $4 for a magazine. She spent half of her remaining money on T-shirt. Then she spent $2 on a snack. Marge had $14 remaining. How much money did Marge begin with?
|-- $4 --|----- half -----|----- half -----|
| $2 |----$14 ---|
From the above you can see that $16 was the 'half of what was remaining'. Total is $32 + $4 = $36.
2) Dan is 4 inches taller than Mike. Together they are 8 feet 8 inches tall. How tall, in feet and inches, is each boy?
|-------Dan ---------|----------- Mike -----------|
This one is easiest to solve with this reasoning:
If the two boys were of equal height, they'd both be 4 feet 4 inches. The DIFFERENCE in their heights is 4 inches, so just add half of that-or 2 inches- to the "midpoint" 4 ft 4 in to find the taller boy's height, and subtract that same 2 inches to find the shorter boy's height.
So Dan is 4 ft 6 in and Mike is 4 ft 2 in.
Tuesday, October 24, 2006
Math Mammoth Grade 6 Worksheets now ready
This is what I've been working on behing the scenes so to speak... and now some of it is finally come to the fruition and is available to the public:
(now comes the sales pitch as was made up by my dear husband)
Meet Mrs. Maria Miller's Most Marvellous & Magnificent Math Mammoth Modified Modern Mathematics Meticulous Multiplication Methodology Major Madness - It's Mmm Mighty Majestic!
Ok, back to normal...
I have been making worksheets for SpiderSmart, Inc. tutoring company and the 6th grade ones are now available as two downloadable ebooks at:
Math Mammoth Grade 6 Worksheets
These aren't your run-of-the-mill worksheets, but more like carefully hand-crafted quality problem sheets, with varying problems that both emphasize understanding of concepts and computation.
The worksheets do NOT have explanations and therefore best suit math teachers or others who can explain the mathematical subject matter to the student(s).
I will later on (next year) be making more worktexts (with full explanations) using some of this material, and those will better serve the homeschool community.
(now comes the sales pitch as was made up by my dear husband)
Meet Mrs. Maria Miller's Most Marvellous & Magnificent Math Mammoth Modified Modern Mathematics Meticulous Multiplication Methodology Major Madness - It's Mmm Mighty Majestic!
Ok, back to normal...
I have been making worksheets for SpiderSmart, Inc. tutoring company and the 6th grade ones are now available as two downloadable ebooks at:
Math Mammoth Grade 6 Worksheets
These aren't your run-of-the-mill worksheets, but more like carefully hand-crafted quality problem sheets, with varying problems that both emphasize understanding of concepts and computation.
The worksheets do NOT have explanations and therefore best suit math teachers or others who can explain the mathematical subject matter to the student(s).
I will later on (next year) be making more worktexts (with full explanations) using some of this material, and those will better serve the homeschool community.
Math Mammoth Grade 6 Worksheets now ready
This is what I've been working on behing the scenes so to speak... and now some of it is finally come to the fruition and is available to the public:
(now comes the sales pitch as was made up by my dear husband)
Meet Mrs. Maria Miller's Most Marvellous & Magnificent Math Mammoth Modified Modern Mathematics Meticulous Multiplication Methodology Major Madness - It's Mmm Mighty Majestic!
Ok, back to normal...
I have been making worksheets for SpiderSmart, Inc. tutoring company and the 6th grade ones are now available as two downloadable ebooks at:
Math Mammoth Grade 6 Worksheets
These aren't your run-of-the-mill worksheets, but more like carefully hand-crafted quality problem sheets, with varying problems that both emphasize understanding of concepts and computation.
The worksheets do NOT have explanations and therefore best suit math teachers or others who can explain the mathematical subject matter to the student(s).
I will later on (next year) be making more worktexts (with full explanations) using some of this material, and those will better serve the homeschool community.
(now comes the sales pitch as was made up by my dear husband)
Meet Mrs. Maria Miller's Most Marvellous & Magnificent Math Mammoth Modified Modern Mathematics Meticulous Multiplication Methodology Major Madness - It's Mmm Mighty Majestic!
Ok, back to normal...
I have been making worksheets for SpiderSmart, Inc. tutoring company and the 6th grade ones are now available as two downloadable ebooks at:
Math Mammoth Grade 6 Worksheets
These aren't your run-of-the-mill worksheets, but more like carefully hand-crafted quality problem sheets, with varying problems that both emphasize understanding of concepts and computation.
The worksheets do NOT have explanations and therefore best suit math teachers or others who can explain the mathematical subject matter to the student(s).
I will later on (next year) be making more worktexts (with full explanations) using some of this material, and those will better serve the homeschool community.
Saturday, October 21, 2006
Free course materials for college level math
Some of you (if you're a mathematics professor in some college or university) might find this interesting to browse: Massachusetts Institute of Technology offers a LONG list of mathematics undergraduate AND graduate course materials as free downloads...
Some of them have lecture outlines or notes, some have student assignments and tests. I saw even Java applets for calculus with applications, but mostly they seemed to be PDF files.
So here's the link:
http://ocw.mit.edu/OcwWeb/Mathematics/index.htm
Some of them have lecture outlines or notes, some have student assignments and tests. I saw even Java applets for calculus with applications, but mostly they seemed to be PDF files.
So here's the link:
http://ocw.mit.edu/OcwWeb/Mathematics/index.htm
Free course materials for college level math
Some of you (if you're a mathematics professor in some college or university) might find this interesting to browse: Massachusetts Institute of Technology offers a LONG list of mathematics undergraduate AND graduate course materials as free downloads...
Some of them have lecture outlines or notes, some have student assignments and tests. I saw even Java applets for calculus with applications, but mostly they seemed to be PDF files.
So here's the link:
http://ocw.mit.edu/OcwWeb/Mathematics/index.htm
Some of them have lecture outlines or notes, some have student assignments and tests. I saw even Java applets for calculus with applications, but mostly they seemed to be PDF files.
So here's the link:
http://ocw.mit.edu/OcwWeb/Mathematics/index.htm
Wednesday, October 18, 2006
Homeschooling Carnival, week 42
I haven't made it to the carnival for a while, but this week I did because Shannon emailed me directly.
The theme of this 42th carnival is "42"! You might recognize that number from the hilarious sci-fi novel "The Hitchhiker’s Guide to the Galaxy". In it, the supercomputer Deep Thought computes the Ultimate Answer to Life, the Universe, and Everything - and spouts out the answer as "42". (Well, then they go to finding out the exact question... if the answer is 42, what is the question?)
Anyways, I enjoyed that book as a teenager. Wikipedia lets us know that the author Douglas Adams just had that there as a joke. (Well, that whole book is just full of jokes, as far as I can remember...)
But, the 'meat' of all this is of course the carnival itself... Is homeschooling the ultimate answer to life, the universe, and everything?
My submission was how teaching math in early grades does not have to cost anything.
The theme of this 42th carnival is "42"! You might recognize that number from the hilarious sci-fi novel "The Hitchhiker’s Guide to the Galaxy". In it, the supercomputer Deep Thought computes the Ultimate Answer to Life, the Universe, and Everything - and spouts out the answer as "42". (Well, then they go to finding out the exact question... if the answer is 42, what is the question?)
Anyways, I enjoyed that book as a teenager. Wikipedia lets us know that the author Douglas Adams just had that there as a joke. (Well, that whole book is just full of jokes, as far as I can remember...)
But, the 'meat' of all this is of course the carnival itself... Is homeschooling the ultimate answer to life, the universe, and everything?
My submission was how teaching math in early grades does not have to cost anything.
Homeschooling Carnival, week 42
I haven't made it to the carnival for a while, but this week I did because Shannon emailed me directly.
The theme of this 42th carnival is "42"! You might recognize that number from the hilarious sci-fi novel "The Hitchhiker’s Guide to the Galaxy". In it, the supercomputer Deep Thought computes the Ultimate Answer to Life, the Universe, and Everything - and spouts out the answer as "42". (Well, then they go to finding out the exact question... if the answer is 42, what is the question?)
Anyways, I enjoyed that book as a teenager. Wikipedia lets us know that the author Douglas Adams just had that there as a joke. (Well, that whole book is just full of jokes, as far as I can remember...)
But, the 'meat' of all this is of course the carnival itself... Is homeschooling the ultimate answer to life, the universe, and everything?
My submission was how teaching math in early grades does not have to cost anything.
The theme of this 42th carnival is "42"! You might recognize that number from the hilarious sci-fi novel "The Hitchhiker’s Guide to the Galaxy". In it, the supercomputer Deep Thought computes the Ultimate Answer to Life, the Universe, and Everything - and spouts out the answer as "42". (Well, then they go to finding out the exact question... if the answer is 42, what is the question?)
Anyways, I enjoyed that book as a teenager. Wikipedia lets us know that the author Douglas Adams just had that there as a joke. (Well, that whole book is just full of jokes, as far as I can remember...)
But, the 'meat' of all this is of course the carnival itself... Is homeschooling the ultimate answer to life, the universe, and everything?
My submission was how teaching math in early grades does not have to cost anything.
Sunday, October 15, 2006
Gender differences in math abilities
This is an interesting study about whether there are differences between men's and women's abilities in math.
The researchers first asked the participants (college students) a question, then they took he Vandenberg Mental Rotation Test, a standard test of visual-spatial ability.
One group was first asked whether they lived in a single-sex or co-ed dorm. That subtly triggers a person to think about their gender. Men in this group did 25 percent to 30 percent better than the women.
In the control group, the students were first asked about how it is to live in Northeastern United States. The results of the visual-spatial test were familiar: men performing 15-20 percent better. (That is the typical result whenever this test is given to men and women.)
BUT the surprise came in the group where students were first subtly made to think about their strengths. They were first asked about why they chose to attend a private liberal arts college.
In this group, there were no significant differences between men and women in the Vandenberg Mental Rotation Test!
Very interesting indeed.
The study was conducted by University of Texas psychologist Matthew S. McGlone and Joshua Aronson of New York University.
Read more details of the study here.
The researchers first asked the participants (college students) a question, then they took he Vandenberg Mental Rotation Test, a standard test of visual-spatial ability.
One group was first asked whether they lived in a single-sex or co-ed dorm. That subtly triggers a person to think about their gender. Men in this group did 25 percent to 30 percent better than the women.
In the control group, the students were first asked about how it is to live in Northeastern United States. The results of the visual-spatial test were familiar: men performing 15-20 percent better. (That is the typical result whenever this test is given to men and women.)
BUT the surprise came in the group where students were first subtly made to think about their strengths. They were first asked about why they chose to attend a private liberal arts college.
In this group, there were no significant differences between men and women in the Vandenberg Mental Rotation Test!
Very interesting indeed.
The study was conducted by University of Texas psychologist Matthew S. McGlone and Joshua Aronson of New York University.
Read more details of the study here.
Gender differences in math abilities
This is an interesting study about whether there are differences between men's and women's abilities in math.
The researchers first asked the participants (college students) a question, then they took he Vandenberg Mental Rotation Test, a standard test of visual-spatial ability.
One group was first asked whether they lived in a single-sex or co-ed dorm. That subtly triggers a person to think about their gender. Men in this group did 25 percent to 30 percent better than the women.
In the control group, the students were first asked about how it is to live in Northeastern United States. The results of the visual-spatial test were familiar: men performing 15-20 percent better. (That is the typical result whenever this test is given to men and women.)
BUT the surprise came in the group where students were first subtly made to think about their strengths. They were first asked about why they chose to attend a private liberal arts college.
In this group, there were no significant differences between men and women in the Vandenberg Mental Rotation Test!
Very interesting indeed.
The study was conducted by University of Texas psychologist Matthew S. McGlone and Joshua Aronson of New York University.
Read more details of the study here.
The researchers first asked the participants (college students) a question, then they took he Vandenberg Mental Rotation Test, a standard test of visual-spatial ability.
One group was first asked whether they lived in a single-sex or co-ed dorm. That subtly triggers a person to think about their gender. Men in this group did 25 percent to 30 percent better than the women.
In the control group, the students were first asked about how it is to live in Northeastern United States. The results of the visual-spatial test were familiar: men performing 15-20 percent better. (That is the typical result whenever this test is given to men and women.)
BUT the surprise came in the group where students were first subtly made to think about their strengths. They were first asked about why they chose to attend a private liberal arts college.
In this group, there were no significant differences between men and women in the Vandenberg Mental Rotation Test!
Very interesting indeed.
The study was conducted by University of Texas psychologist Matthew S. McGlone and Joshua Aronson of New York University.
Read more details of the study here.
Friday, October 13, 2006
Kindergarten math ideas
I recently talked with a friend who was concerned about the cost of math curricula for her soon-5-year-old, doing kindergarten math. Even regarding my ebooks which I've given her free access, she mentioned how even printing costs money and could get costly in the long run.
(And I know some people can print things real cheap, but not everyone. It depends on your printer.)
So I told her, teaching math doesn't have to cost anything in kindergarten or the early grades.
It's not of utmost importance to do worksheet work. You can largely just play games and explore various things. After all, playing is what that age kids do best anyway.
* Playing board games where you roll one die teaches them to recognize the dot patterns on the die.
* Later on, playing board games where you roll two dice can be used for addition practice.
* After learning the dot patterns on a die, use dominoes as "flash cards" for addition. Or better yet, make a simple game out of it: lay them right side down on table, then take turns turning one and if you can add the dots, you get to keep it.
* Try 10 out math card game for learning sums of 10. Adaptable for other sums as well.
* Game worth 1000 worksheets - this is a simple card game. Deal two cards to each player, each person adds those two, and the one with highest sum wins the cards to himself.
* Make cuisenaire rods out of cardboard and play with them.
* Let the child play freely with measuring tape, scales, and measuring cups.
* Make 'ten-bags' or 'ten-bundles' with marbles or sticks. Then you can learn place value with tens and ones with those.
And even with worksheets or written math problems, you can write those yourself in a notebook, and thus not spend a penny. You can write as few problems as you'd like.
I used to draw many empty number charts for my child in her notebook.
I also often made missing addend problems with balls for her. She had to draw more and then write the addition.
You can use colors freely, and let the child color in this notebook. For example, write a few problems on a page, and let the child draw a colored box around the problems when done.
You can draw geometric shapes, or make (colorful) patterns to be continued, such as
square square triangle
All this should be totally free.
See also:
My ebooks.
Math Lessons and Teaching Tips - scroll down the page to see addition, subtraction, multiplication, division, place value, fraction, geometry and decimals lessons taken from my ebooks. Most of those aren't for kindergarten but for elementary grades.
Note especially Teaching tens and ones.
(And I know some people can print things real cheap, but not everyone. It depends on your printer.)
So I told her, teaching math doesn't have to cost anything in kindergarten or the early grades.
It's not of utmost importance to do worksheet work. You can largely just play games and explore various things. After all, playing is what that age kids do best anyway.
* Playing board games where you roll one die teaches them to recognize the dot patterns on the die.
* Later on, playing board games where you roll two dice can be used for addition practice.
* After learning the dot patterns on a die, use dominoes as "flash cards" for addition. Or better yet, make a simple game out of it: lay them right side down on table, then take turns turning one and if you can add the dots, you get to keep it.
* Try 10 out math card game for learning sums of 10. Adaptable for other sums as well.
* Game worth 1000 worksheets - this is a simple card game. Deal two cards to each player, each person adds those two, and the one with highest sum wins the cards to himself.
* Make cuisenaire rods out of cardboard and play with them.
* Let the child play freely with measuring tape, scales, and measuring cups.
* Make 'ten-bags' or 'ten-bundles' with marbles or sticks. Then you can learn place value with tens and ones with those.
And even with worksheets or written math problems, you can write those yourself in a notebook, and thus not spend a penny. You can write as few problems as you'd like.
I used to draw many empty number charts for my child in her notebook.
I also often made missing addend problems with balls for her. She had to draw more and then write the addition.
You can use colors freely, and let the child color in this notebook. For example, write a few problems on a page, and let the child draw a colored box around the problems when done.
You can draw geometric shapes, or make (colorful) patterns to be continued, such as
square square triangle
All this should be totally free.
See also:
My ebooks.
Math Lessons and Teaching Tips - scroll down the page to see addition, subtraction, multiplication, division, place value, fraction, geometry and decimals lessons taken from my ebooks. Most of those aren't for kindergarten but for elementary grades.
Note especially Teaching tens and ones.
Kindergarten math ideas
I recently talked with a friend who was concerned about the cost of math curricula for her soon-5-year-old, doing kindergarten math. Even regarding my ebooks which I've given her free access, she mentioned how even printing costs money and could get costly in the long run.
(And I know some people can print things real cheap, but not everyone. It depends on your printer.)
So I told her, teaching math doesn't have to cost anything in kindergarten or the early grades.
It's not of utmost importance to do worksheet work. You can largely just play games and explore various things. After all, playing is what that age kids do best anyway.
* Playing board games where you roll one die teaches them to recognize the dot patterns on the die.
* Later on, playing board games where you roll two dice can be used for addition practice.
* After learning the dot patterns on a die, use dominoes as "flash cards" for addition. Or better yet, make a simple game out of it: lay them right side down on table, then take turns turning one and if you can add the dots, you get to keep it.
* Try 10 out math card game for learning sums of 10. Adaptable for other sums as well.
* Game worth 1000 worksheets - this is a simple card game. Deal two cards to each player, each person adds those two, and the one with highest sum wins the cards to himself.
* Make cuisenaire rods out of cardboard and play with them.
* Let the child play freely with measuring tape, scales, and measuring cups.
* Make 'ten-bags' or 'ten-bundles' with marbles or sticks. Then you can learn place value with tens and ones with those.
And even with worksheets or written math problems, you can write those yourself in a notebook, and thus not spend a penny. You can write as few problems as you'd like.
I used to draw many empty number charts for my child in her notebook.
I also often made missing addend problems with balls for her. She had to draw more and then write the addition.
You can use colors freely, and let the child color in this notebook. For example, write a few problems on a page, and let the child draw a colored box around the problems when done.
You can draw geometric shapes, or make (colorful) patterns to be continued, such as
square square triangle
All this should be totally free.
See also:
My ebooks.
Math Lessons and Teaching Tips - scroll down the page to see addition, subtraction, multiplication, division, place value, fraction, geometry and decimals lessons taken from my ebooks. Most of those aren't for kindergarten but for elementary grades.
Note especially Teaching tens and ones.
(And I know some people can print things real cheap, but not everyone. It depends on your printer.)
So I told her, teaching math doesn't have to cost anything in kindergarten or the early grades.
It's not of utmost importance to do worksheet work. You can largely just play games and explore various things. After all, playing is what that age kids do best anyway.
* Playing board games where you roll one die teaches them to recognize the dot patterns on the die.
* Later on, playing board games where you roll two dice can be used for addition practice.
* After learning the dot patterns on a die, use dominoes as "flash cards" for addition. Or better yet, make a simple game out of it: lay them right side down on table, then take turns turning one and if you can add the dots, you get to keep it.
* Try 10 out math card game for learning sums of 10. Adaptable for other sums as well.
* Game worth 1000 worksheets - this is a simple card game. Deal two cards to each player, each person adds those two, and the one with highest sum wins the cards to himself.
* Make cuisenaire rods out of cardboard and play with them.
* Let the child play freely with measuring tape, scales, and measuring cups.
* Make 'ten-bags' or 'ten-bundles' with marbles or sticks. Then you can learn place value with tens and ones with those.
And even with worksheets or written math problems, you can write those yourself in a notebook, and thus not spend a penny. You can write as few problems as you'd like.
I used to draw many empty number charts for my child in her notebook.
I also often made missing addend problems with balls for her. She had to draw more and then write the addition.
You can use colors freely, and let the child color in this notebook. For example, write a few problems on a page, and let the child draw a colored box around the problems when done.
You can draw geometric shapes, or make (colorful) patterns to be continued, such as
square square triangle
All this should be totally free.
See also:
My ebooks.
Math Lessons and Teaching Tips - scroll down the page to see addition, subtraction, multiplication, division, place value, fraction, geometry and decimals lessons taken from my ebooks. Most of those aren't for kindergarten but for elementary grades.
Note especially Teaching tens and ones.
Tuesday, October 10, 2006
Math misconceptions
It's very good to know something about the most common misconceptions students might have.
The website CountOn.org has 22 examples of them at http://www.counton.org/resources/misconceptions/.
Here are some examples:
#2. Multiplication always increases a number... is that really so?
Well, to small kids it appears to be so - but only if you just try whole numbers. Take 10 for example. If you multiply it by 2, 3, 4, 5, etc., it does get bigger.
But all you have to do is multiply it by a fraction less than 1, or by a negative number, and 10 does not get bigger... 1/2 x 10 is 5. 1/4 x 10 is 2.5. -3 x 10 is -30.
We must remember that repeated addition is not the only meaning or definition for multiplication. That's what it is for whole numbers.
For fractions, 1/3 x 12 is better understood as 1/3 of 12.
#3. As 1 x 1 = 1, then 0.1 x 0.1 = 0.1.
This one was new to me. Quite curious. A child might think of 0.1 as some kind of "unit" like 1.
But seriously, 0.1 is one tenth. Taking tenth part of a tenth is a hundredth, or 0.01.
#5. 3 divide 1/4 is same as 3 divide 4.
This must be due to forgetting the second part of the "invert and multiply" rule. If you remember to think of division as "how many times does the divisor fit into the dividend?", then it's easy and clear: 1/4 fits into 3 exactly 12 times.
#7. What is the value that satisfies -4 + ___ = -10 ? Many possible wrong answers, such as 6, 14, or -14.
It's a missing addend problem. We already have -4, and end up with much more negatives, namely -10. So the debt increased, by -6.
#20: 2.3 x 10 = 2.30 ? 20.3 ?
Well, multiply in parts: 10 x 2 and 10 x 0.3. You get 20 and 3, total 23.
These and more are presented at Counton.org/resources/misconceptions/. They also have a one single PDF file that contains them all.
Another site listing popular (or common) math mistakes is MathMistakes.info.
Tags: math, mistakes
The website CountOn.org has 22 examples of them at http://www.counton.org/resources/misconceptions/.
Here are some examples:
#2. Multiplication always increases a number... is that really so?
Well, to small kids it appears to be so - but only if you just try whole numbers. Take 10 for example. If you multiply it by 2, 3, 4, 5, etc., it does get bigger.
But all you have to do is multiply it by a fraction less than 1, or by a negative number, and 10 does not get bigger... 1/2 x 10 is 5. 1/4 x 10 is 2.5. -3 x 10 is -30.
We must remember that repeated addition is not the only meaning or definition for multiplication. That's what it is for whole numbers.
For fractions, 1/3 x 12 is better understood as 1/3 of 12.
#3. As 1 x 1 = 1, then 0.1 x 0.1 = 0.1.
This one was new to me. Quite curious. A child might think of 0.1 as some kind of "unit" like 1.
But seriously, 0.1 is one tenth. Taking tenth part of a tenth is a hundredth, or 0.01.
#5. 3 divide 1/4 is same as 3 divide 4.
This must be due to forgetting the second part of the "invert and multiply" rule. If you remember to think of division as "how many times does the divisor fit into the dividend?", then it's easy and clear: 1/4 fits into 3 exactly 12 times.
#7. What is the value that satisfies -4 + ___ = -10 ? Many possible wrong answers, such as 6, 14, or -14.
It's a missing addend problem. We already have -4, and end up with much more negatives, namely -10. So the debt increased, by -6.
#20: 2.3 x 10 = 2.30 ? 20.3 ?
Well, multiply in parts: 10 x 2 and 10 x 0.3. You get 20 and 3, total 23.
These and more are presented at Counton.org/resources/misconceptions/. They also have a one single PDF file that contains them all.
Another site listing popular (or common) math mistakes is MathMistakes.info.
Tags: math, mistakes
Math misconceptions
It's very good to know something about the most common misconceptions students might have.
The website CountOn.org has 22 examples of them at http://www.counton.org/resources/misconceptions/.
Here are some examples:
#2. Multiplication always increases a number... is that really so?
Well, to small kids it appears to be so - but only if you just try whole numbers. Take 10 for example. If you multiply it by 2, 3, 4, 5, etc., it does get bigger.
But all you have to do is multiply it by a fraction less than 1, or by a negative number, and 10 does not get bigger... 1/2 x 10 is 5. 1/4 x 10 is 2.5. -3 x 10 is -30.
We must remember that repeated addition is not the only meaning or definition for multiplication. That's what it is for whole numbers.
For fractions, 1/3 x 12 is better understood as 1/3 of 12.
#3. As 1 x 1 = 1, then 0.1 x 0.1 = 0.1.
This one was new to me. Quite curious. A child might think of 0.1 as some kind of "unit" like 1.
But seriously, 0.1 is one tenth. Taking tenth part of a tenth is a hundredth, or 0.01.
#5. 3 divide 1/4 is same as 3 divide 4.
This must be due to forgetting the second part of the "invert and multiply" rule. If you remember to think of division as "how many times does the divisor fit into the dividend?", then it's easy and clear: 1/4 fits into 3 exactly 12 times.
#7. What is the value that satisfies -4 + ___ = -10 ? Many possible wrong answers, such as 6, 14, or -14.
It's a missing addend problem. We already have -4, and end up with much more negatives, namely -10. So the debt increased, by -6.
#20: 2.3 x 10 = 2.30 ? 20.3 ?
Well, multiply in parts: 10 x 2 and 10 x 0.3. You get 20 and 3, total 23.
These and more are presented at Counton.org/resources/misconceptions/. They also have a one single PDF file that contains them all.
Another site listing popular (or common) math mistakes is MathMistakes.info.
Tags: math, mistakes
The website CountOn.org has 22 examples of them at http://www.counton.org/resources/misconceptions/.
Here are some examples:
#2. Multiplication always increases a number... is that really so?
Well, to small kids it appears to be so - but only if you just try whole numbers. Take 10 for example. If you multiply it by 2, 3, 4, 5, etc., it does get bigger.
But all you have to do is multiply it by a fraction less than 1, or by a negative number, and 10 does not get bigger... 1/2 x 10 is 5. 1/4 x 10 is 2.5. -3 x 10 is -30.
We must remember that repeated addition is not the only meaning or definition for multiplication. That's what it is for whole numbers.
For fractions, 1/3 x 12 is better understood as 1/3 of 12.
#3. As 1 x 1 = 1, then 0.1 x 0.1 = 0.1.
This one was new to me. Quite curious. A child might think of 0.1 as some kind of "unit" like 1.
But seriously, 0.1 is one tenth. Taking tenth part of a tenth is a hundredth, or 0.01.
#5. 3 divide 1/4 is same as 3 divide 4.
This must be due to forgetting the second part of the "invert and multiply" rule. If you remember to think of division as "how many times does the divisor fit into the dividend?", then it's easy and clear: 1/4 fits into 3 exactly 12 times.
#7. What is the value that satisfies -4 + ___ = -10 ? Many possible wrong answers, such as 6, 14, or -14.
It's a missing addend problem. We already have -4, and end up with much more negatives, namely -10. So the debt increased, by -6.
#20: 2.3 x 10 = 2.30 ? 20.3 ?
Well, multiply in parts: 10 x 2 and 10 x 0.3. You get 20 and 3, total 23.
These and more are presented at Counton.org/resources/misconceptions/. They also have a one single PDF file that contains them all.
Another site listing popular (or common) math mistakes is MathMistakes.info.
Tags: math, mistakes
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