Visualizing numbers

Recently I added another math curriculum to the list of math curricula; it's called RightStart Mathematics.
They are using a little different abacus from the usual to help children visualize numbers: the abacus has five beads one color, and the next five another color.

Their idea is that using such abacus, children will eventually learn to recognize the amounts on abacus with a single glance, without counting the beads. They will then form a mental image and learn to 'see' seven as five and two.

After checking their website, just a few days later, when checking dead links, I stumbled upon this same abacus in electronic form on another website.
(However you can only do a few things with the applet, so it's not nearly as good as a real physical abacus.)

This Ten frame tool uses a similar idea - kids are supposed to recognize amounts 0-10 visually, without counting.

Obviously another tool for this is just a die, or dice.

Great ideas, great tools.

Visualizing numbers

Recently I added another math curriculum to the list of math curricula; it's called RightStart Mathematics.
They are using a little different abacus from the usual to help children visualize numbers: the abacus has five beads one color, and the next five another color.

Their idea is that using such abacus, children will eventually learn to recognize the amounts on abacus with a single glance, without counting the beads. They will then form a mental image and learn to 'see' seven as five and two.

After checking their website, just a few days later, when checking dead links, I stumbled upon this same abacus in electronic form on another website.
(However you can only do a few things with the applet, so it's not nearly as good as a real physical abacus.)

This Ten frame tool uses a similar idea - kids are supposed to recognize amounts 0-10 visually, without counting.

Obviously another tool for this is just a die, or dice.

Great ideas, great tools.

Online Math Resources link list updated

I have checked my online math resources pages for dead links (again). It's just inevitable, it seems, that some links go bad after a while. Anyway, hopefully it's all now working.

I am continually AMAZED at the wealth of interactive mathematics things that one can find on the internet. I compiled this list almost three years ago, and it has grown little by little ever since. The quality of many of these math activities or tutorials is just great. Various people have done a lot when publishing these interactive (or otherwise) math things online.

It's like you have a big collection of virtual manipulatives for about any math subject you can think of, at your fingertips!

I have divided my list into 9 different pages. Go and check; I'm sure you'll find something interesting to benefit your math teaching. I've concentrated on interactive tutorials and other interactive math activities but the list has other resources as well, such as tutorials and software.

Math Help and Tutoring

Math history, problem solving, and test preparation

Place value, basic operations, times tables, factoring etc. (elementary math)

Clock, money, measuring, and elementary geometry

Fractions, Decimals, Percent, Integers (middle school math)

Geometry, Measuring, Coordinate plane (middle school math)

Algebra, Graphing, Calculus

Geometry, Trigonometry, Statistics (high school math)

Sites with lots of math games and interactive activities

Online Math Resources link list updated

I have checked my online math resources pages for dead links (again). It's just inevitable, it seems, that some links go bad after a while. Anyway, hopefully it's all now working.

I am continually AMAZED at the wealth of interactive mathematics things that one can find on the internet. I compiled this list almost three years ago, and it has grown little by little ever since. The quality of many of these math activities or tutorials is just great. Various people have done a lot when publishing these interactive (or otherwise) math things online.

It's like you have a big collection of virtual manipulatives for about any math subject you can think of, at your fingertips!

I have divided my list into 9 different pages. Go and check; I'm sure you'll find something interesting to benefit your math teaching. I've concentrated on interactive tutorials and other interactive math activities but the list has other resources as well, such as tutorials and software.

Math Help and Tutoring

Math history, problem solving, and test preparation

Place value, basic operations, times tables, factoring etc. (elementary math)

Clock, money, measuring, and elementary geometry

Fractions, Decimals, Percent, Integers (middle school math)

Geometry, Measuring, Coordinate plane (middle school math)

Algebra, Graphing, Calculus

Geometry, Trigonometry, Statistics (high school math)

Sites with lots of math games and interactive activities

Subtraction stories

This was just so cute.

Background: We've been looking at some subtraction stories with pictures in a book called "MIMOSA Moving Into Math" (you might find it at eBay) with my three-year old daughter. Well one evening here recently, *somehow* her mind started working, and suddenly, totally spontaneously, she came up with this story:

"Mommy, I have a subtraction story: There were three potties at first. But then mommy broke one (which is very true! - I did manage to break a potty a few days ago), and so now there are only two left."

Then she proceeded to help me with rinsing the dishes and the stories continued: "I have four spoons to rinse. I rinse one, and now there's three left to rinse!"

I said how about one with bananas. She said, "YES! There were ten bananas originally. Then, we ate one. And there are 1,2,3,4,5,6,7,8,9,10... (she had to count up).. NINE left for DADDY!"

(My husband does like bananas quite well.)

After dishes she still made some more with the plastic bears and such.

What a moment of joy for the teacher - makes me thankful!

Subtraction stories

This was just so cute.

Background: We've been looking at some subtraction stories with pictures in a book called "MIMOSA Moving Into Math" (you might find it at eBay) with my three-year old daughter. Well one evening here recently, *somehow* her mind started working, and suddenly, totally spontaneously, she came up with this story:

"Mommy, I have a subtraction story: There were three potties at first. But then mommy broke one (which is very true! - I did manage to break a potty a few days ago), and so now there are only two left."

Then she proceeded to help me with rinsing the dishes and the stories continued: "I have four spoons to rinse. I rinse one, and now there's three left to rinse!"

I said how about one with bananas. She said, "YES! There were ten bananas originally. Then, we ate one. And there are 1,2,3,4,5,6,7,8,9,10... (she had to count up).. NINE left for DADDY!"

(My husband does like bananas quite well.)

After dishes she still made some more with the plastic bears and such.

What a moment of joy for the teacher - makes me thankful!

Math teacher's toolkit (elementary)

I was not going to post during these days when everyone is busy... But I just got this nice weblink and want to pass it along:

Math Teacher's Toolkit

It has...
- Place Value Calculator
- Virtual Hundrets/tens/units Place Value Cards
- Hundred Square
- Times tables tester
- 12 Hour Clock
- 24 Hour Clock
- Sequences
- 10 Digit Number Line
and a few more interactive math teaching tools for elementary grades (using Macromedia Flash).

Math teacher's toolkit (elementary)

I was not going to post during these days when everyone is busy... But I just got this nice weblink and want to pass it along:

Math Teacher's Toolkit

It has...
- Place Value Calculator
- Virtual Hundrets/tens/units Place Value Cards
- Hundred Square
- Times tables tester
- 12 Hour Clock
- 24 Hour Clock
- Sequences
- 10 Digit Number Line
and a few more interactive math teaching tools for elementary grades (using Macromedia Flash).

Some fun optical illusions

Here's 8 optical illusions for some fun:
www.grand-illusions.com/opticalillusions

Nothing short of amazing... little black balls that are yet aren't there, moving dots, moving cogs, Dr Angry and Mr Calm faces, and a few more.

Some fun optical illusions

Here's 8 optical illusions for some fun:
www.grand-illusions.com/opticalillusions

Nothing short of amazing... little black balls that are yet aren't there, moving dots, moving cogs, Dr Angry and Mr Calm faces, and a few more.

Mathematics: The science of patterns

I just started reading the book Mathematics : The Science of Patterns by Keith Devlin. I'm planning on writing a review of it for my site. I can already tell it's a good book, and I've just read the prologue, titled "What is mathematics?"

The author says that yes, mathematics WAS the study of number - up till about 500 B.C. Then it became the study of number and shape. After invention of calculus, it became the study of number, shape, motion, change and space.

But nowadays you can't say that any more, because the field of mathematics has expanded SO much: in 1900 it would have taken about eighty books to write all the world's mathematical knowledge. Devlin estimates that nowadays (or when he wrote the book about 10 years ago) it would take maybe 100,000 volumes to contain all knonw mathematics!

You know, your average person doesn't realize that. I got a glimpse of it while in university, when I saw all these lists of subcategories of mathematics, and when I studied a little bit of something like Abstract Algebra or Complex Analysis or Measure Theory - and all we did was touch the surface of such topics.

And so today, mathematics is often defined as the study of patterns - numerical patterns, patterns of shape, patterns of motion, and so on.

Devlin also touched on beauty in mathematics in his prologue - and I agree: mathematics is beautiful. That, too, I fear, most folks don't realize or understand. It's because you need to be able to understand the math first - then it kind of forms a picture in your mind of a structure that is beautiful, harmonious. I can't explain it.

So I want to recommend the book Mathematics : The Science of Patterns for all of you even before I've written complete review on it. It's written for a layman, and is about the essential history of mathematics.

Categories: philosophy, math

Mathematics: The science of patterns

I just started reading the book Mathematics : The Science of Patterns by Keith Devlin. I'm planning on writing a review of it for my site. I can already tell it's a good book, and I've just read the prologue, titled "What is mathematics?"

The author says that yes, mathematics WAS the study of number - up till about 500 B.C. Then it became the study of number and shape. After invention of calculus, it became the study of number, shape, motion, change and space.

But nowadays you can't say that any more, because the field of mathematics has expanded SO much: in 1900 it would have taken about eighty books to write all the world's mathematical knowledge. Devlin estimates that nowadays (or when he wrote the book about 10 years ago) it would take maybe 100,000 volumes to contain all knonw mathematics!

You know, your average person doesn't realize that. I got a glimpse of it while in university, when I saw all these lists of subcategories of mathematics, and when I studied a little bit of something like Abstract Algebra or Complex Analysis or Measure Theory - and all we did was touch the surface of such topics.

And so today, mathematics is often defined as the study of patterns - numerical patterns, patterns of shape, patterns of motion, and so on.

Devlin also touched on beauty in mathematics in his prologue - and I agree: mathematics is beautiful. That, too, I fear, most folks don't realize or understand. It's because you need to be able to understand the math first - then it kind of forms a picture in your mind of a structure that is beautiful, harmonious. I can't explain it.

So I want to recommend the book Mathematics : The Science of Patterns for all of you even before I've written complete review on it. It's written for a layman, and is about the essential history of mathematics.

Categories: philosophy, math

Geometry course for middle school

I was going to write about Euclid but this came up. Someone submitted a review of a math program called RightStart mathematics to my site. While browsing that site, I found out they offer a separate geometry program which looked really good:

See, many students have great problems with high school geometry. I've written an article Why is high school geometry so difficult? about that already.

One of the remedies for that is to do things right BEFORE high school: teach true geometry and DRAW - don't spend all your time calculating areas and volumes from grades 3 till 8.

So now I have stumbled upon a geometry program for middle grades that seems to be just that: emphasis is on drawing. Check it out:
RightStart Geometry program.

I haven't seen all of it, but based on the example pages, it looked good. It's $57 for the whole package.

Categories: geometry

Geometry course for middle school

I was going to write about Euclid but this came up. Someone submitted a review of a math program called RightStart mathematics to my site. While browsing that site, I found out they offer a separate geometry program which looked really good:

See, many students have great problems with high school geometry. I've written an article Why is high school geometry so difficult? about that already.

One of the remedies for that is to do things right BEFORE high school: teach true geometry and DRAW - don't spend all your time calculating areas and volumes from grades 3 till 8.

So now I have stumbled upon a geometry program for middle grades that seems to be just that: emphasis is on drawing. Check it out:
RightStart Geometry program.

I haven't seen all of it, but based on the example pages, it looked good. It's $57 for the whole package.

Categories: geometry

Pi's digits

Here's the answer as to how many decimal places Pi's decimal expansion has been calculated:

1,241,100,000,000 decimal digits

It's more than 1 trillion digits!

This was done in 2002 by Yasumasa Kanada and 9 other individuals in Tokyo.

See www.super-computing.org/pi_current.html for more details on this project.

Okay, and if you'd like to start memorizing it, here's a link to Pi trainer :)

Pi's digits

Here's the answer as to how many decimal places Pi's decimal expansion has been calculated:

1,241,100,000,000 decimal digits

It's more than 1 trillion digits!

This was done in 2002 by Yasumasa Kanada and 9 other individuals in Tokyo.

See www.super-computing.org/pi_current.html for more details on this project.

Okay, and if you'd like to start memorizing it, here's a link to Pi trainer :)

Lesson plan/article about Fibonacci numbers and golden ratio

I have combined the recent blogposts about Fibonacci numbers and golden ratio into one article on my site that you can use as a teaching guide or just to educate yourself:

Fibonacci numbers and golden section - lesson plan

I hope you enjoy my posts... you're welcome to leave commens about the blog - just mention in your comments it's about the blog.

Lesson plan/article about Fibonacci numbers and golden ratio

I have combined the recent blogposts about Fibonacci numbers and golden ratio into one article on my site that you can use as a teaching guide or just to educate yourself:

Fibonacci numbers and golden section - lesson plan

I hope you enjoy my posts... you're welcome to leave commens about the blog - just mention in your comments it's about the blog.

Hands-on with pi

Pi seems to be such a simple thing - it's just two letters, and after all, you just peek into any math book to find out its value. Everybody knows about it, "Pi? Ah, yes, they taught us that in school, so it's something pretty simple."

BUT have you ever given pi some more thought? And, if you let your students give pi some more thought, it might just make math a little more interesting and 'lively' for them, too.

Think thought about it - how come the ratio of any circle's circumference to its diameter can be an irrational number? After all, this is not something that would naturally come to mind if you've never heard of irrationals before... If you start measuring lots of circles and calculating ratios, yes, you would soon notice that in every circle the ratio is the same. You would notice it's a little over 3. But would you just GUESS on your own that this ratio isn't a rational number?

If you have the time (and homeschoolers might), give your student a project to find what is the ratio of a circle's circumference to its diameter by measuring. Tell them they can play an ancient mathematician now and let them work at it - don't even tell them it's called pi and hide all math books for a while...

Some kids might just love such a hands-on activity.

See if they get so far as to realize it's the same constant in all circles. And congrats if they get 3.14 or even get it between 3.1 and 3.2.

During all the measuring your student is not likely to think that this number would be irrational - he probably hasn't even heard the word before!

And mathematicians didn't find that out real soon either. Ancient cultures were aware of the fact that the ratio of circumference to diameter was a constant. The formulas they used for the area of circle indicate the equivalent value for pi as 377/120 or 3 1/8 or 256/81.

Archimedes found out that this ratio was between 223/71 and 22/7. Or, to be exact, he was finding an approximation for the area of the circle via this simple method: he drew a regular polygon inside the circle and outside the circle, and calculated the areas of the polygons-- obviously the area of the circle would have to be between those two numbers.

Archimedes wasn't afraid of calculations and used a polygon with 96 sides...! You might try giving 8th graders this as a project using just hexagons or octagons.

It makes for a nice exercise in geometry, plus it should be interesting since it ties in with history. Here's a nice slideshow about Archimedes' method.

Over time, mathematicians were able to calculate Pi more accurately - find better rational approximations to it. I'm sure they started to guess that maybe Pi is irrational - in fact, my guess is they suspected it to be so for a long time.

Proof of pi's irrationality came in 1768 by Johann Lambert. After that, mathematicians have been sure of that - but the quest for finding pi's decimals still continues.

Guess how many decimals they have found thus far?

Answer will be posted here in a coming blogpost. I'll leave you hanging - you can do the same in school. Why hand them all the answers on a plate?

P.S. Here's a link about history of pi if you want to learn more.

Hands-on with pi

Pi seems to be such a simple thing - it's just two letters, and after all, you just peek into any math book to find out its value. Everybody knows about it, "Pi? Ah, yes, they taught us that in school, so it's something pretty simple."

BUT have you ever given pi some more thought? And, if you let your students give pi some more thought, it might just make math a little more interesting and 'lively' for them, too.

Think thought about it - how come the ratio of any circle's circumference to its diameter can be an irrational number? After all, this is not something that would naturally come to mind if you've never heard of irrationals before... If you start measuring lots of circles and calculating ratios, yes, you would soon notice that in every circle the ratio is the same. You would notice it's a little over 3. But would you just GUESS on your own that this ratio isn't a rational number?

If you have the time (and homeschoolers might), give your student a project to find what is the ratio of a circle's circumference to its diameter by measuring. Tell them they can play an ancient mathematician now and let them work at it - don't even tell them it's called pi and hide all math books for a while...

Some kids might just love such a hands-on activity.

See if they get so far as to realize it's the same constant in all circles. And congrats if they get 3.14 or even get it between 3.1 and 3.2.

During all the measuring your student is not likely to think that this number would be irrational - he probably hasn't even heard the word before!

And mathematicians didn't find that out real soon either. Ancient cultures were aware of the fact that the ratio of circumference to diameter was a constant. The formulas they used for the area of circle indicate the equivalent value for pi as 377/120 or 3 1/8 or 256/81.

Archimedes found out that this ratio was between 223/71 and 22/7. Or, to be exact, he was finding an approximation for the area of the circle via this simple method: he drew a regular polygon inside the circle and outside the circle, and calculated the areas of the polygons-- obviously the area of the circle would have to be between those two numbers.

Archimedes wasn't afraid of calculations and used a polygon with 96 sides...! You might try giving 8th graders this as a project using just hexagons or octagons.

It makes for a nice exercise in geometry, plus it should be interesting since it ties in with history. Here's a nice slideshow about Archimedes' method.

Over time, mathematicians were able to calculate Pi more accurately - find better rational approximations to it. I'm sure they started to guess that maybe Pi is irrational - in fact, my guess is they suspected it to be so for a long time.

Proof of pi's irrationality came in 1768 by Johann Lambert. After that, mathematicians have been sure of that - but the quest for finding pi's decimals still continues.

Guess how many decimals they have found thus far?

Answer will be posted here in a coming blogpost. I'll leave you hanging - you can do the same in school. Why hand them all the answers on a plate?

P.S. Here's a link about history of pi if you want to learn more.

How do we know this is true? What is the proof?

Someone emailed me just recently with this question:

"How do we know that pi is indeed non-repeating? Do we have proof? What is that proof?"
(The person who emailed me this question is from Japan, based on the email address.)

I think it's an excellent question! And I hope every eight-grader is thinking about that when they are first told about pi.

You know, kids learn about pi when they are studying circles, and so they take several circles and measure the circumference and the diameter of them all, and they calculate the ratio and they get 3.3 or 3.1 or 3.45 or 3.274658 or whatever.

I tell you a truth: In all your measuring you will never stumble upon the fact that pi is irrational.

Proof: Because your measuring results are all rational numbers, and so you're diving a rational number by a rational number.

It's not something that would be found with observation or exploration of that sort. Students are just plain announced the fact that this ratio is called Pi and it's irrational. And what does that mean, irrational? they ask. And they're told it means the decimal expansion continues forever without repeating.

School mathematics does a lot of announcing facts. I hope it doesn't kill in the students that small wondering, questioning, interested voice that wants to know more:
But why? How do we know that? Is there a proof?

Do you know?

Now that I have you interested, I'll probably disappoint you by telling that in the case of pi, the proofs about its irrationality require understanding of calculus and "advanced math" - math advanced beyond basic algebra. But here's one anyway:
Proof that Pi is irrational

How do we know this is true? What is the proof?

Someone emailed me just recently with this question:

"How do we know that pi is indeed non-repeating? Do we have proof? What is that proof?"
(The person who emailed me this question is from Japan, based on the email address.)

I think it's an excellent question! And I hope every eight-grader is thinking about that when they are first told about pi.

You know, kids learn about pi when they are studying circles, and so they take several circles and measure the circumference and the diameter of them all, and they calculate the ratio and they get 3.3 or 3.1 or 3.45 or 3.274658 or whatever.

I tell you a truth: In all your measuring you will never stumble upon the fact that pi is irrational.

Proof: Because your measuring results are all rational numbers, and so you're diving a rational number by a rational number.

It's not something that would be found with observation or exploration of that sort. Students are just plain announced the fact that this ratio is called Pi and it's irrational. And what does that mean, irrational? they ask. And they're told it means the decimal expansion continues forever without repeating.

School mathematics does a lot of announcing facts. I hope it doesn't kill in the students that small wondering, questioning, interested voice that wants to know more:
But why? How do we know that? Is there a proof?

Do you know?

Now that I have you interested, I'll probably disappoint you by telling that in the case of pi, the proofs about its irrationality require understanding of calculus and "advanced math" - math advanced beyond basic algebra. But here's one anyway:
Proof that Pi is irrational

The Golden Section

Studying about Fibonacci numbers and the golden ratio makes an excellent project for high school to write a report on. Besides algebra, it ties in with geometry, botany and art at least. Students do projects and reports in history and English and other school subjects - why not do one or two in math too?

The discussion below covers the basics of golden section. I'll try to keep it short.

---

Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.

Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).

This is how we get this golden section or golden cut:


Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.

S:L = L:W

From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).

And the golden ratio is the ratio Phi:1.

Solving the equation - more details Skip this box if you so wish. Solving this simple-looking equation of golden cut requires using the formula for quadratic equations, so it is a nice exercise for high-schoolers. S:L = L:W is usually written in the form of S/L = L/W Since S+L = W, we can substitute that for W and get: S/L = L/(S+L) And another trick is, since this is just a general line, we can choose for the shorter part S to be 1. After that, the equation looks simple enough: 1/L = L/(1+L) Solving that using the quadratic formula, and discarding the negative root, you get L = (√5 + 1)/2

Golden rectangle

Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.

here is one golden rectangle

Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.

And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.

My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.

• The Golden section in architecture


• Dividing the line in golden section using compass and ruler


• Phi in pentagons and pentagrams


• Where Fibonacci numbers are found in nature


• Where golden section is found in nature

P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.

The Golden Section

Studying about Fibonacci numbers and the golden ratio makes an excellent project for high school to write a report on. Besides algebra, it ties in with geometry, botany and art at least. Students do projects and reports in history and English and other school subjects - why not do one or two in math too?

The discussion below covers the basics of golden section. I'll try to keep it short.

---

Last time we studied the ratios of a Fibonacci number to the previous Fibonacci number and how they approach a certain number as one continues the sequence - and this certain number is called Phi.

Phi is also called the golden section number. You might have heard about it. Even Euclid studied that in ancient times (he called it dividing the line in mean and extreme ratio).

This is how we get this golden section or golden cut:


Take a line and divide it into two parts, S (short part) and L (Long part). We want the ratio of short part to long part be the same as the ratio of long part to the whole line (W). In other words, as the short part is to the long part, so is the long part to the whole line.

S:L = L:W

From this can be solved that L = (√5 + 1)/2 × S or L ≈ 1.618 × S. This number (√5 + 1)/2 is Phi.
So if you divide the line so that longer part is Phi times (about 1.62) the shorter part, you've divided it in the golden section (or golden cut).

And the golden ratio is the ratio Phi:1.

Solving the equation - more details Skip this box if you so wish. Solving this simple-looking equation of golden cut requires using the formula for quadratic equations, so it is a nice exercise for high-schoolers. S:L = L:W is usually written in the form of S/L = L/W Since S+L = W, we can substitute that for W and get: S/L = L/(S+L) And another trick is, since this is just a general line, we can choose for the shorter part S to be 1. After that, the equation looks simple enough: 1/L = L/(1+L) Solving that using the quadratic formula, and discarding the negative root, you get L = (√5 + 1)/2

Golden rectangle

Golden rectangle is one where the length and the width of the rectangle are in the golden ratio... the length is approximately 1.62 times the width.

here is one golden rectangle

Some people say this shape is an especially aesthetic rectangle, or that humans prefer golden rectangle over others; it hasn't been proven true so think what you like! I like that kind of rectangle okay. Next time try crop a photograph in that ratio and see what you think.

And then you're ready to study where all golden section is found! The links below go to a fantastic website about Fibonacci numbers and golden ratio which is packed full of info - there is LOTS and LOTS more to study.

My list is just a suggestion of a few basic topics that could be included in a project in case you don't want to cover it all.

• The Golden section in architecture


• Dividing the line in golden section using compass and ruler


• Phi in pentagons and pentagrams


• Where Fibonacci numbers are found in nature


• Where golden section is found in nature

P.S. Some folks try to find golden ratio in everything in universe and make it some sort of mystical or sacred thing or "universal constant of design". It's true you can find it in nature in plant leaf arrangements and in seashells but not every statement you find on the internet about Phi or Fibonacci numbers has been confirmed scientifically. See for example this scientific study proving just the opposite: The Fibonacci Sequence: Relationship to the Human Hand.

Homeschooled math whiz wins competition

Michael Viscardi, 16 years, from San Diego, has won the 2005 - 06 Siemens Westinghouse Competition in Math, Science and Technology. The judges said they can't find the limits of his understanding!

He's been homeschooled since fifth grade but he takes math classes at the University of California three times a week. His father is a software engineer and his mother, who stays at home, has a Ph.D. in neuroscience. I read somewhere that he also plays piano and violin and has won music competitions too. Quite a whiz!

His competition entry was about a 19th century math problem, which he solved in a new way and was able to obtain several new results in his research. He says he liked the problem because it used complex analysis... That rang a bell for me, as I liked complex analysis, too, during my university studies and my master's thesis used it.

Read more...

See here his picture and read even more.

Now, he probably would have been taking extra math classes even if he had been in public school, but homeschooling is surely a great option for gifted students.

Homeschooled math whiz wins competition

Michael Viscardi, 16 years, from San Diego, has won the 2005 - 06 Siemens Westinghouse Competition in Math, Science and Technology. The judges said they can't find the limits of his understanding!

He's been homeschooled since fifth grade but he takes math classes at the University of California three times a week. His father is a software engineer and his mother, who stays at home, has a Ph.D. in neuroscience. I read somewhere that he also plays piano and violin and has won music competitions too. Quite a whiz!

His competition entry was about a 19th century math problem, which he solved in a new way and was able to obtain several new results in his research. He says he liked the problem because it used complex analysis... That rang a bell for me, as I liked complex analysis, too, during my university studies and my master's thesis used it.

Read more...

See here his picture and read even more.

Now, he probably would have been taking extra math classes even if he had been in public school, but homeschooling is surely a great option for gifted students.

Fibonacci numbers part 2

In a previous post, we studied about Fibonacci numbers :
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(add two consecutive numbers to get the next one).

I noted that those numbers appear in nature in many places, and asked if your child/student know about that.

My opinion is yes, he or she should know. But why? After all, that stuff is not needful in daily life.

I think it's important that youngsters learn a few math topics that show how math appears in nature. It is about math appreciation - (or better yet, appreciation of the Master Mathematician of the Universe...).

Kids learn Art Appreciation - so they can appreciate human works of art... Oh, how much better you can appreciate the "artworks" in nature such as flower petals or seedheads when you understand a little bit of the math behind them!

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Here's another amazing thing about these numbers:
Let's study the RATIOS when you take a Fibonacci number divided by the previous Fibonacci number, and make a list:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ...

So what's so great about that? you ask. (And your student might ask too when you do this with him.)
It's not so visible when you see the ratios written as fractions, but let's take the decimal expansions of these (and you should have the student or students do this on their own):

1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818...

Do you notice something about that sequence?

It's something special. If you continue calculating the decimal expansions of the ratios, they will keep getting closer and closer to a certain number... they never reach it totally but they keep getting closer and closer and closer each time.

The ratios keep approaching the number (√5 + 1)/2 which is approximately 1.6180339887... if you write out some of its decimal expansion. I couldn't write out all of the decimal expansion because this number in itself is IRRATIONAL and it has the name Phi.

(Oh I will some day write more about irrational numbers; I think the concepts of rational versus irrational numbers is very fascinating - they're kind of elusive and as if they weren't from this world)

So what we did was: take the ratios of a Fibonacci number per the previous one, look at decimal expansions, and notice they keep getting close to something - and I told you (without proving it) that something is Phi, and it's an irrational number and it's exactly (√5 + 1)/2.

See also The Ratio of neighbouring Fibonacci Numbers tends to Phi - this page has a graph and also a proof for this fact.

So far so good... I hope I'm not confusing you. I think I will stop here with pure math and continue some other time - yes, there's still more to come! This Phi is an exciting number in many mathematical senses that you may not know about, but you might have already heard about Phi as the GOLDEN RATIO... (to be continued)

Fibonacci numbers part 2

In a previous post, we studied about Fibonacci numbers :
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(add two consecutive numbers to get the next one).

I noted that those numbers appear in nature in many places, and asked if your child/student know about that.

My opinion is yes, he or she should know. But why? After all, that stuff is not needful in daily life.

I think it's important that youngsters learn a few math topics that show how math appears in nature. It is about math appreciation - (or better yet, appreciation of the Master Mathematician of the Universe...).

Kids learn Art Appreciation - so they can appreciate human works of art... Oh, how much better you can appreciate the "artworks" in nature such as flower petals or seedheads when you understand a little bit of the math behind them!

---

Here's another amazing thing about these numbers:
Let's study the RATIOS when you take a Fibonacci number divided by the previous Fibonacci number, and make a list:

1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, ...

So what's so great about that? you ask. (And your student might ask too when you do this with him.)
It's not so visible when you see the ratios written as fractions, but let's take the decimal expansions of these (and you should have the student or students do this on their own):

1, 2, 1.5, 1.6666..., 1.6, 1.625, 1.615384615..., 1.619047619..., 1.617647059..., 1.618181818...

Do you notice something about that sequence?

It's something special. If you continue calculating the decimal expansions of the ratios, they will keep getting closer and closer to a certain number... they never reach it totally but they keep getting closer and closer and closer each time.

The ratios keep approaching the number (√5 + 1)/2 which is approximately 1.6180339887... if you write out some of its decimal expansion. I couldn't write out all of the decimal expansion because this number in itself is IRRATIONAL and it has the name Phi.

(Oh I will some day write more about irrational numbers; I think the concepts of rational versus irrational numbers is very fascinating - they're kind of elusive and as if they weren't from this world)

So what we did was: take the ratios of a Fibonacci number per the previous one, look at decimal expansions, and notice they keep getting close to something - and I told you (without proving it) that something is Phi, and it's an irrational number and it's exactly (√5 + 1)/2.

See also The Ratio of neighbouring Fibonacci Numbers tends to Phi - this page has a graph and also a proof for this fact.

So far so good... I hope I'm not confusing you. I think I will stop here with pure math and continue some other time - yes, there's still more to come! This Phi is an exciting number in many mathematical senses that you may not know about, but you might have already heard about Phi as the GOLDEN RATIO... (to be continued)

Goals in math teaching

With my daughter, we've been concentrating on simple addition problems and the topic of tens and ones. Right now we are working on these goals:

• Understand subtraction concept - she partially does
• Tens and ones - she partially understands. (I realize mastering place value well takes often years.)
• Understand something + 1 type problems - she partially does
• Memorize answers to double 2, double 3, etc doubling facts

And, we're heading towards these goals that are further away, yes, but I am keeping them constantly in mind:

• Memorizing all basic addition facts
• Learn to use those to do subraction problems

Having definite goals is important. Good math teachers know where students are supposed to be heading, and also how to get there.

Make sure you know (and understand!) what the final and intermediate goals are. If you are simply following the math book page by page, it's like walking on a road while looking down at the road - not knowing exactly where the road is leading.

When you have the final goals down pat, you can often jump back and forth in your math book, skip pages or provide extra practice according to need.

Of course to do this, the teacher needs to understand the math beforehand and know how it's structured, how one concept leads to another. Soon I will be posting a review on a book that can help homeschoolers to do just that.

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I also often have my dd fill in 10x10 grid with numbers from 1 to 100. The other day, after filling it, I asked her to color yellow all whole tens, and then color pink all those that end in five. Okay, after those were done, she asked to continue coloring, and before I knew it, the little artist in her took over and the whole grid turned into a rainbow!
Well I thought you might enjoy it too so here it is:



P.S. You can make similar number grid worksheets (either partially filled in or not at all) for counting and skip-counting at www.homeschoolmath.net/worksheets/number-charts.php

Goals in math teaching

With my daughter, we've been concentrating on simple addition problems and the topic of tens and ones. Right now we are working on these goals:

• Understand subtraction concept - she partially does
• Tens and ones - she partially understands. (I realize mastering place value well takes often years.)
• Understand something + 1 type problems - she partially does
• Memorize answers to double 2, double 3, etc doubling facts

And, we're heading towards these goals that are further away, yes, but I am keeping them constantly in mind:

• Memorizing all basic addition facts
• Learn to use those to do subraction problems

Having definite goals is important. Good math teachers know where students are supposed to be heading, and also how to get there.

Make sure you know (and understand!) what the final and intermediate goals are. If you are simply following the math book page by page, it's like walking on a road while looking down at the road - not knowing exactly where the road is leading.

When you have the final goals down pat, you can often jump back and forth in your math book, skip pages or provide extra practice according to need.

Of course to do this, the teacher needs to understand the math beforehand and know how it's structured, how one concept leads to another. Soon I will be posting a review on a book that can help homeschoolers to do just that.

---

I also often have my dd fill in 10x10 grid with numbers from 1 to 100. The other day, after filling it, I asked her to color yellow all whole tens, and then color pink all those that end in five. Okay, after those were done, she asked to continue coloring, and before I knew it, the little artist in her took over and the whole grid turned into a rainbow!
Well I thought you might enjoy it too so here it is:



P.S. You can make similar number grid worksheets (either partially filled in or not at all) for counting and skip-counting at www.homeschoolmath.net/worksheets/number-charts.php

Math resources for gifted students

I added a few weblinks and books for gifted or talented students in math on the online resources list at my website.

Math resources for gifted students

I added a few weblinks and books for gifted or talented students in math on the online resources list at my website.