Prove two triangles are congruent

Here's an example geometry problem from Kiseleve's geometry book that I blogged about earlier. This problem is not difficult and is a good example of problem where you need to prove triangles congruent.

The reason I'm solving it here is simply to help those of you who will be homeschooling your child thru high school geometry. It helps enormously to be familiar with proofs and the type of reasoning before you actually need to write proofs yourself.

Problem:

80. Prove that if two sides and the median drawn to the first of them in one triangle are respectively congruent to two sides and the median drawn to the first of them in another triangle, then such triangles are congruent.

First draw a picture. We have two triangles with certain parts congruent, and we are supposed to prove that the triangles are congruent.



We know that AB = A'B', AC = A'C', and BM = B'M' (givens).

This proof in a nutshell is based on first proving that the two triangles AMB and A'M'B' and congruent (by SSS), from which follows that angle BAC = angle B'A'C'. Then the original triangles are congruent by SAS.

Consider now the two triangles AMB and A'M'B'. We already know two of their corresponding sides are congruent - AB = A'B', and BM = B'M'. But even their third sides are congruent, because by the fact M is a midpoint of AC, AM = 1/2AC and similarly A'M' = 1/2A'C'. Since AC and A'C' are congruent, it follows that half of those line segments, or AM and A'M' are congruent also.

By the SSS congruence theorem (or postulate; if your book calls it that) for triangles, the triangles AMB and A'M'B' are congruent.

Therefore their corresponding angles are congruent also. In particular, the angle BAC = angle B'A'C'.

So we have side AB, angle BAC, and side AC congruent to the side A'B', angle B'A'C', and side A'C', respectively, in the other triangle.

Now it follows by the SAS congruence theorem that the two triangles are congruent.

Tags: math, geometry, proof, triangles

Prove two triangles are congruent

Here's an example geometry problem from Kiseleve's geometry book that I blogged about earlier. This problem is not difficult and is a good example of problem where you need to prove triangles congruent.

The reason I'm solving it here is simply to help those of you who will be homeschooling your child thru high school geometry. It helps enormously to be familiar with proofs and the type of reasoning before you actually need to write proofs yourself.

Problem:

80. Prove that if two sides and the median drawn to the first of them in one triangle are respectively congruent to two sides and the median drawn to the first of them in another triangle, then such triangles are congruent.

First draw a picture. We have two triangles with certain parts congruent, and we are supposed to prove that the triangles are congruent.



We know that AB = A'B', AC = A'C', and BM = B'M' (givens).

This proof in a nutshell is based on first proving that the two triangles AMB and A'M'B' and congruent (by SSS), from which follows that angle BAC = angle B'A'C'. Then the original triangles are congruent by SAS.

Consider now the two triangles AMB and A'M'B'. We already know two of their corresponding sides are congruent - AB = A'B', and BM = B'M'. But even their third sides are congruent, because by the fact M is a midpoint of AC, AM = 1/2AC and similarly A'M' = 1/2A'C'. Since AC and A'C' are congruent, it follows that half of those line segments, or AM and A'M' are congruent also.

By the SSS congruence theorem (or postulate; if your book calls it that) for triangles, the triangles AMB and A'M'B' are congruent.

Therefore their corresponding angles are congruent also. In particular, the angle BAC = angle B'A'C'.

So we have side AB, angle BAC, and side AC congruent to the side A'B', angle B'A'C', and side A'C', respectively, in the other triangle.

Now it follows by the SAS congruence theorem that the two triangles are congruent.

Tags: math, geometry, proof, triangles

Ebooks summer sale ending soon

Just a reminder... the summer sale for my math e-books is ending soon. The offers are valid till end of July.

The e-books are basically PDF files that you print on your own. All the books have been written as worktexts, with explanations and problems in the same text.

Read what old customers say

Visit the ebooks page here!

Ebooks summer sale ending soon

Just a reminder... the summer sale for my math e-books is ending soon. The offers are valid till end of July.

The e-books are basically PDF files that you print on your own. All the books have been written as worktexts, with explanations and problems in the same text.

Read what old customers say

Visit the ebooks page here!

Planets

This blogpost is inspired and especially written for Homeschooling Carnival, as they are having a galaxy theme.

---

Astronomy and mathematics have always been closely related. Astronomers have always been using the latest mathematical knowledge and theories. Many mathematicians in the past have also done research in astronomy.

All sciences strive to help us understand the world we live in, but astronomy does it on the largest possible scale.

I have always found astronomy fascinating, and so have multitudes of other people, too. But I just wonder if mathematicians might have even a little bit stronger fascination or interest in that direction; somehow those two just seem to fit together very well.

It is part of well-rounded education, I feel, to know some basics of history of astronomy. And it sure is very interesting too! What I've written below is just some thoughts on the subject of planetary orbits.

---

Greeks believed that planets go around the earth, in circular orbits. This view was held true all thru many centuries and Dark Ages till the 1500s, though the idea of sun-centered solar system had been proposed by various people.

Nicolaus Copernicus was one of the astronomers who after the Dark Ages was firmly of the opinion that the Earth goes around the sun. But, it was the German astronomer Johannes Kepler who really worked on the idea with Tycho's observational data.

After years of calculations, Kepler realized - and found it hard to believe himself - that the planets do NOT orbit in circular orbits, but in elliptical ones. This is called Kepler's first law.

The planetary orbits are quite close to circles though. In other words, the ellipses are not very 'eccentric', and that explains why astronomers before him were able to believe them to be circles.

Please see this web page, Kepler's Laws of Planetary Motion, for an excellent explanation and diagrams about ellipses and planetary orbits. This topic is worth understanding, I feel.

Kepler also found that the planets don't travel around the sun at constant speed. Instead, they speed up when they are closest to the sun, and slow down when they are farthest from the sun. Kepler found a simple mathematical relationship that is called Kepler's second law: that a line between the planet and the Sun sweeps out equal areas in equal times.

I found a neat applet illustrating this speeding and slowing on an elliptical orbit. Go check it out! You can change the eccentricity of the ellipse; the traditional planets in our solar system all have eccentricity less than 0.3.

And, Kepler also settled the question whether all planets travel at the same speed. No, not at all. The planets that are furthest from the sun travel slower than the planets closer to sun. This is expressed in Kepler's third law of planetary motion.

The story of planetary motion does not end with Kepler, though. Astronomers kept finding irregularities in the orbits of planets - and this led to the discoveries of new planets.

For example, Neptune was found in 1840s because the orbit of Uranus didn't fit the predictions. In late 1800s, scientists started speculating that Neptune's orbit, too, was perturbated by some other then unknown planet. Pluto was finally discovered in 1930.

And the discoveries have not stopped there. In recent years there have been thousands of new "things" discovered that orbit our sun - they are called "trans-Neptunian objects" or "Kuiper-belt objects" based on their location.

One of them is larger than Pluto and has been called the tenth planet. Currently it only has the official number 2003 UB₃₁₃.

The International Astronomical Union (IAU) is scheduled to publish the definition of the term "planet" in early September 2006 and after that we will know whether 2003 UB₃₁₃ is going to be called the tenth planet or just a Kuiper-belt object.

And after that, the appropriate committee can go on deciding about its name.

At any rate, astronomical knowledge is currently expanding at an ever-increasing rate, it seems. It is one of the oldest sciences and still going strong!


See also:

Orbits and gravitation - a good article explaining the history of those topics.

Planets

This blogpost is inspired and especially written for Homeschooling Carnival, as they are having a galaxy theme.

---

Astronomy and mathematics have always been closely related. Astronomers have always been using the latest mathematical knowledge and theories. Many mathematicians in the past have also done research in astronomy.

All sciences strive to help us understand the world we live in, but astronomy does it on the largest possible scale.

I have always found astronomy fascinating, and so have multitudes of other people, too. But I just wonder if mathematicians might have even a little bit stronger fascination or interest in that direction; somehow those two just seem to fit together very well.

It is part of well-rounded education, I feel, to know some basics of history of astronomy. And it sure is very interesting too! What I've written below is just some thoughts on the subject of planetary orbits.

---

Greeks believed that planets go around the earth, in circular orbits. This view was held true all thru many centuries and Dark Ages till the 1500s, though the idea of sun-centered solar system had been proposed by various people.

Nicolaus Copernicus was one of the astronomers who after the Dark Ages was firmly of the opinion that the Earth goes around the sun. But, it was the German astronomer Johannes Kepler who really worked on the idea with Tycho's observational data.

After years of calculations, Kepler realized - and found it hard to believe himself - that the planets do NOT orbit in circular orbits, but in elliptical ones. This is called Kepler's first law.

The planetary orbits are quite close to circles though. In other words, the ellipses are not very 'eccentric', and that explains why astronomers before him were able to believe them to be circles.

Please see this web page, Kepler's Laws of Planetary Motion, for an excellent explanation and diagrams about ellipses and planetary orbits. This topic is worth understanding, I feel.

Kepler also found that the planets don't travel around the sun at constant speed. Instead, they speed up when they are closest to the sun, and slow down when they are farthest from the sun. Kepler found a simple mathematical relationship that is called Kepler's second law: that a line between the planet and the Sun sweeps out equal areas in equal times.

I found a neat applet illustrating this speeding and slowing on an elliptical orbit. Go check it out! You can change the eccentricity of the ellipse; the traditional planets in our solar system all have eccentricity less than 0.3.

And, Kepler also settled the question whether all planets travel at the same speed. No, not at all. The planets that are furthest from the sun travel slower than the planets closer to sun. This is expressed in Kepler's third law of planetary motion.

The story of planetary motion does not end with Kepler, though. Astronomers kept finding irregularities in the orbits of planets - and this led to the discoveries of new planets.

For example, Neptune was found in 1840s because the orbit of Uranus didn't fit the predictions. In late 1800s, scientists started speculating that Neptune's orbit, too, was perturbated by some other then unknown planet. Pluto was finally discovered in 1930.

And the discoveries have not stopped there. In recent years there have been thousands of new "things" discovered that orbit our sun - they are called "trans-Neptunian objects" or "Kuiper-belt objects" based on their location.

One of them is larger than Pluto and has been called the tenth planet. Currently it only has the official number 2003 UB₃₁₃.

The International Astronomical Union (IAU) is scheduled to publish the definition of the term "planet" in early September 2006 and after that we will know whether 2003 UB₃₁₃ is going to be called the tenth planet or just a Kuiper-belt object.

And after that, the appropriate committee can go on deciding about its name.

At any rate, astronomical knowledge is currently expanding at an ever-increasing rate, it seems. It is one of the oldest sciences and still going strong!


See also:

Orbits and gravitation - a good article explaining the history of those topics.

Russian geometry book

Recently I received the following note,

I would like to bring to your attention the following new
geometry textbook:

"Kiselev's Geometry / Book I. Planimetry" by A.P. Kiselev,
ISBN 0977985202 Publisher: Sumizdat

It is an English translation and adaptation of a classical Russian textbook in plane geometry, which has served well as to several generations of students of age 13 and up, and their teachers in Russia.

The English edition is intended for those students, homeschooled or not, who want to achieve a good command of elementary geometry, and learn to appreciate for its intellectual depth and beauty.

More information about the book and its author is available through the publisher's webpage: www.sumizdat.org.

The book is currently available at: www.sumizdat.org and Singaporemath.com.

I posted this note here because some of you might be interested - a classical Russian geometry book translated into English. You can browse quite many sample pages to get an idea of the book.
It was first published in 1892 has been revised and published more than forty times altogether.

The original author Kiselev wrote several math textbooks. QUOTING from the preface:

"...and a few years prior to Kiselev's death in 1940, his books were officially given the status of stable, i.e. main and only textbooks to be used in all schools to teach all teenagers in Soviet Union.

The books held this status until 1955 when they got replaced in this capacity by less successful clones written by more Soviet authors. Yet "Planimetry" remained the favorite under-the-desk choice of many teachers and a must for honors geometry students. In the last decade, Kiselev's "Geometry," which has long become a rarity, was reprinted by several major publishing houses in Moscow and St.- Petersburg in both versions: for teachers as an authentic pedagogical heritage, and for students as a textbook tailored to fit the currently active school curricula. In the post-Soviet educational market, Kiselev's "Geometry" continues to compete successfully with its own grandchildren."

Quite an accomplishment for a single book.

If you look at the type of exercises found in the book, I think you will easily see that there is a difference when comparing to modern American books (this book is meant for 7-9th graders).

For example (these are from the sample pages provided on the website)

79. Suppose that an angle, its bisector, and one side of this angle in one triangle are respectively congruent to an angle, its bisector, and one side of this angle in another triangle. Prove that such triangles are congruent.

80. Prove that if two sides and the median drawn to the first of them in one triangle are respectively congruent to two sides and the median drawn to the first of them in another triangle, then such triangles are congruent.

Prove theorems:

400. If a diagonal divides a trapezoid into two similar triangles, then this diagonal is the geometric mean between the bases.

401. If two disks are tangent externally, then the segment of an external common tangent between the tangency points is the geometric mean between the diameters of the disks.

402. If a square is inscribed into a right triangle in such a way that one side of the square lies on the hypotenuse, then this side is the geometric mean between the two remaining segments of the hypotenuse.

576. The altitude dropped to the hypotenuse divides a given right triangle into smaller triangles whose radii of the inscribed circles are 6 and 8 cm. Compute the radius of the inscribed circle of the given triangle.

577. Compute the sides of a right triangle given the radii of its circumscribed and inscribed circle.

578. Compute the area of a right triangle if the foot of the altitude dropped to the hypotenuse of length c divides it in the extreme and mean ratio.

Personally, I feel they are interesting sounding problems! (I will probably solve some in future blogposts, as examples.)

But how many US high school students would be willing and able to do them? Feel free to comment.

Now, this book could serve for a high school geometry course for sure. It does have one big disadvantage though if you're a homeschooler: there is no answer key. But the book appears to possess intellectual depth and beauty, just like its subject matter!


Tags: math, geometry, curriculum

Russian geometry book

Recently I received the following note,

I would like to bring to your attention the following new
geometry textbook:

"Kiselev's Geometry / Book I. Planimetry" by A.P. Kiselev,
ISBN 0977985202 Publisher: Sumizdat

It is an English translation and adaptation of a classical Russian textbook in plane geometry, which has served well as to several generations of students of age 13 and up, and their teachers in Russia.

The English edition is intended for those students, homeschooled or not, who want to achieve a good command of elementary geometry, and learn to appreciate for its intellectual depth and beauty.

More information about the book and its author is available through the publisher's webpage: www.sumizdat.org.

The book is currently available at: www.sumizdat.org and Singaporemath.com.

I posted this note here because some of you might be interested - a classical Russian geometry book translated into English. You can browse quite many sample pages to get an idea of the book.
It was first published in 1892 has been revised and published more than forty times altogether.

The original author Kiselev wrote several math textbooks. QUOTING from the preface:

"...and a few years prior to Kiselev's death in 1940, his books were officially given the status of stable, i.e. main and only textbooks to be used in all schools to teach all teenagers in Soviet Union.

The books held this status until 1955 when they got replaced in this capacity by less successful clones written by more Soviet authors. Yet "Planimetry" remained the favorite under-the-desk choice of many teachers and a must for honors geometry students. In the last decade, Kiselev's "Geometry," which has long become a rarity, was reprinted by several major publishing houses in Moscow and St.- Petersburg in both versions: for teachers as an authentic pedagogical heritage, and for students as a textbook tailored to fit the currently active school curricula. In the post-Soviet educational market, Kiselev's "Geometry" continues to compete successfully with its own grandchildren."

Quite an accomplishment for a single book.

If you look at the type of exercises found in the book, I think you will easily see that there is a difference when comparing to modern American books (this book is meant for 7-9th graders).

For example (these are from the sample pages provided on the website)

79. Suppose that an angle, its bisector, and one side of this angle in one triangle are respectively congruent to an angle, its bisector, and one side of this angle in another triangle. Prove that such triangles are congruent.

80. Prove that if two sides and the median drawn to the first of them in one triangle are respectively congruent to two sides and the median drawn to the first of them in another triangle, then such triangles are congruent.

Prove theorems:

400. If a diagonal divides a trapezoid into two similar triangles, then this diagonal is the geometric mean between the bases.

401. If two disks are tangent externally, then the segment of an external common tangent between the tangency points is the geometric mean between the diameters of the disks.

402. If a square is inscribed into a right triangle in such a way that one side of the square lies on the hypotenuse, then this side is the geometric mean between the two remaining segments of the hypotenuse.

576. The altitude dropped to the hypotenuse divides a given right triangle into smaller triangles whose radii of the inscribed circles are 6 and 8 cm. Compute the radius of the inscribed circle of the given triangle.

577. Compute the sides of a right triangle given the radii of its circumscribed and inscribed circle.

578. Compute the area of a right triangle if the foot of the altitude dropped to the hypotenuse of length c divides it in the extreme and mean ratio.

Personally, I feel they are interesting sounding problems! (I will probably solve some in future blogposts, as examples.)

But how many US high school students would be willing and able to do them? Feel free to comment.

Now, this book could serve for a high school geometry course for sure. It does have one big disadvantage though if you're a homeschooler: there is no answer key. But the book appears to possess intellectual depth and beauty, just like its subject matter!


Tags: math, geometry, curriculum

Being excited about math

I found Integer Jim's website two days ago, and I had to browse thru his whole website because it was quite interesting.

Here we have a math teacher who is very enthusiastic and excited about what he is teaching (you can sense that by reading his website).

Now, that is, I feel, one ingredient in what makes a good teacher: being enthused about your subject matter.

I realize not every homeschooler feels that way about math; but don't despair if you feel teaching math is a drudgery. All that CAN change... Head for the Living Math website for starters.

But back to Integer Jim. Besides being a math teacher, he's also an artist and has made some interesting projects with his students that tie art and math together.

I wanted to highlight one: The Math Journal project. Jim says,

The Math Journal is a comprehensive and in depth project. It requires a lot of time and effort on the part of the students. For that reason, I use it as the centerpiece of my curriculum; the textbook that I taught from previously, I now use as a resource book for problems and homework assignments. It is my observation that the students were never enthusiastic about the textbook anyway. Now they definitely have something that they are enthused about.

In other words, he has used the journal project as the core of the curriculum: students write in it lecture notes, charts, diagrams. Then they also write about interesting math topics from math history for example, and topics of their own choice such as puzzles and 'fab facts'.

The result is like an adventure.

I commend Jim for this; the students are creating a work of their own, learning math AND writing skills, being highly motivated. Sounds great all the way thru!

However, there are drawbacks, too: it will take some extra time compared to just using a workbook, and a broad knowledge of math from the teacher in order to guide the student(s) with the extra topics. Of course the internet is a fantastic source for interesting math topics, history, puzzles, and such.

Maybe the idea is adaptable on a smaller scale, somehow. (He's used it as a whole course curriculum.) Anyhow, it's definitely something I'd like to try some day.


On this page you can request a more detailed guidelines and typical contents of his algebra journal.

Tags: math, teaching

Being excited about math

I found Integer Jim's website two days ago, and I had to browse thru his whole website because it was quite interesting.

Here we have a math teacher who is very enthusiastic and excited about what he is teaching (you can sense that by reading his website).

Now, that is, I feel, one ingredient in what makes a good teacher: being enthused about your subject matter.

I realize not every homeschooler feels that way about math; but don't despair if you feel teaching math is a drudgery. All that CAN change... Head for the Living Math website for starters.

But back to Integer Jim. Besides being a math teacher, he's also an artist and has made some interesting projects with his students that tie art and math together.

I wanted to highlight one: The Math Journal project. Jim says,

The Math Journal is a comprehensive and in depth project. It requires a lot of time and effort on the part of the students. For that reason, I use it as the centerpiece of my curriculum; the textbook that I taught from previously, I now use as a resource book for problems and homework assignments. It is my observation that the students were never enthusiastic about the textbook anyway. Now they definitely have something that they are enthused about.

In other words, he has used the journal project as the core of the curriculum: students write in it lecture notes, charts, diagrams. Then they also write about interesting math topics from math history for example, and topics of their own choice such as puzzles and 'fab facts'.

The result is like an adventure.

I commend Jim for this; the students are creating a work of their own, learning math AND writing skills, being highly motivated. Sounds great all the way thru!

However, there are drawbacks, too: it will take some extra time compared to just using a workbook, and a broad knowledge of math from the teacher in order to guide the student(s) with the extra topics. Of course the internet is a fantastic source for interesting math topics, history, puzzles, and such.

Maybe the idea is adaptable on a smaller scale, somehow. (He's used it as a whole course curriculum.) Anyhow, it's definitely something I'd like to try some day.


On this page you can request a more detailed guidelines and typical contents of his algebra journal.

Tags: math, teaching

Pink Bunny

Have you ever seen a pink bunny? me neither, here's another cute little critter from our good friend Ryo to satisfy your papercrafting needs for the week. This lovely paper model wanted me to mention that her name was Briana, loves all things pink, and would definitely go out with anybody that has a Pink DS Lite (prefers Associate Editors of popular weblogs), hmmm? Anyways, go take a look.

Pink Bunny - Download

Adding spice to geometry studies

... with interactive resources online.

In my mind, geometry is lots of fun. It should involve lots of drawing, of course. But you can go beyond that: with modern technology, geometrical drawings can come 'alive' or dynamic. You can change one part of the picture and observe which things change with it, and which things don't. You can explore the situation, analyze, learn.

The best of it is that you don't even have to buy geometry software since the various math websites already offer so much.

I'd like to highlight today Mathsnet.net geometry section. This British site contains interactive courses and lessons for a wide range of geometry topics. In fact, the site is just HUGE. And, well-made too.

They offer three very comprehensive courses (totally free):

• Constructions:
  You will build points, line and circles on-screen just as you would on paper with a ruler and compasses. These constructions are then dynamic - you can move the objects and see the construction change.



• In Shape , you can get a clear and complete understanding of geometric shapes, their patterns and properties. You will explore, identify, create, recognize, describe... Even a crossword puzzle is included.



• And, the site also has a whole course on interactive transformations.

Other lessons include

Euclid's elements
Sacred geometry
The hidden world of triangles and circles
Circle only constructions,
An exploration into two circles: geometry & imaginary numbers
Pythagoras' theorem: proofs and problems
Is this a square?: how to decide
The egg project: how to construct Euclidian eggs
Circle theorems: six properties
Geometry in 3 dimensions: solid shapes
The incircle: unexpected properties
Feynman's lost lecture: motion of planets around the sun

I have been very impressed by the content of Mathsnet.net geometry section ever since I found it. I am sure it can help you, too.

Tags: math, geometry, lessons

Adding spice to geometry studies

... with interactive resources online.

In my mind, geometry is lots of fun. It should involve lots of drawing, of course. But you can go beyond that: with modern technology, geometrical drawings can come 'alive' or dynamic. You can change one part of the picture and observe which things change with it, and which things don't. You can explore the situation, analyze, learn.

The best of it is that you don't even have to buy geometry software since the various math websites already offer so much.

I'd like to highlight today Mathsnet.net geometry section. This British site contains interactive courses and lessons for a wide range of geometry topics. In fact, the site is just HUGE. And, well-made too.

They offer three very comprehensive courses (totally free):

• Constructions:
  You will build points, line and circles on-screen just as you would on paper with a ruler and compasses. These constructions are then dynamic - you can move the objects and see the construction change.



• In Shape , you can get a clear and complete understanding of geometric shapes, their patterns and properties. You will explore, identify, create, recognize, describe... Even a crossword puzzle is included.



• And, the site also has a whole course on interactive transformations.

Other lessons include

Euclid's elements
Sacred geometry
The hidden world of triangles and circles
Circle only constructions,
An exploration into two circles: geometry & imaginary numbers
Pythagoras' theorem: proofs and problems
Is this a square?: how to decide
The egg project: how to construct Euclidian eggs
Circle theorems: six properties
Geometry in 3 dimensions: solid shapes
The incircle: unexpected properties
Feynman's lost lecture: motion of planets around the sun

I have been very impressed by the content of Mathsnet.net geometry section ever since I found it. I am sure it can help you, too.

Tags: math, geometry, lessons

Comment on coherent math curriculum

I received this comment about my Coherent Math Curriculum article just recently, and I thought my readers here might find it interesting:

I am a 8th grade mathematics teacher that homeschools his four kids. There is no way in the world I am putting my kids back in public school for the exact reason I just read in your article. As a matter of fact, I am going into my 3rd year of teaching (I am 27) and I have been ranting and raving about precisely the same argument that is being posed within this article. There is too much stuff in a math curriculum over here in the States and we spend entirely too much time reviewing. I should not have 16 year olds in my 8th grade class who have no idea how to change a improper fraction into a mixed number, that's 3rd, 4th grade stuff. It's absolutely ridiculous and I commend you for this article.

M. S.

His rant about the math curriculum is is nothing new or surprising; I found it interesting though that he is working as a teacher and homeschools.

Anyway, what could a teacher do in this sort of situation? Not much alone. But, maybe one could get together with 6th and 7th grade teachers and discuss how to change the situation for better.

I know the constant testing makes it difficult to do radical changes, but maybe one could organize the math topics on grades 6-8 a little more coherently, with more time on each topic, so that fractions wouldn't need to be studied on each of those grades. (I find it amusing that 8th graders are still even studying fractions!) Say, organize most of the probability and statistics topics for 8th grade, most of the geometry and measuring for 7th, and fractions and decimals and other arithmetic mostly for 6th... And spread some topics such as percents, proportions, problem solving on all grades.

Maybe that would be possible locally, within just the school he teaches in.

What do all you teachers think?

Tags: math, curriculum

Comment on coherent math curriculum

I received this comment about my Coherent Math Curriculum article just recently, and I thought my readers here might find it interesting:

I am a 8th grade mathematics teacher that homeschools his four kids. There is no way in the world I am putting my kids back in public school for the exact reason I just read in your article. As a matter of fact, I am going into my 3rd year of teaching (I am 27) and I have been ranting and raving about precisely the same argument that is being posed within this article. There is too much stuff in a math curriculum over here in the States and we spend entirely too much time reviewing. I should not have 16 year olds in my 8th grade class who have no idea how to change a improper fraction into a mixed number, that's 3rd, 4th grade stuff. It's absolutely ridiculous and I commend you for this article.

M. S.

His rant about the math curriculum is is nothing new or surprising; I found it interesting though that he is working as a teacher and homeschools.

Anyway, what could a teacher do in this sort of situation? Not much alone. But, maybe one could get together with 6th and 7th grade teachers and discuss how to change the situation for better.

I know the constant testing makes it difficult to do radical changes, but maybe one could organize the math topics on grades 6-8 a little more coherently, with more time on each topic, so that fractions wouldn't need to be studied on each of those grades. (I find it amusing that 8th graders are still even studying fractions!) Say, organize most of the probability and statistics topics for 8th grade, most of the geometry and measuring for 7th, and fractions and decimals and other arithmetic mostly for 6th... And spread some topics such as percents, proportions, problem solving on all grades.

Maybe that would be possible locally, within just the school he teaches in.

What do all you teachers think?

Tags: math, curriculum

Complex numbers

what is the square root of -1?

Well, there is no real number that could be it, but there IS a solution when we go to imaginary numbers.

Square root of negative 1 is denoted by i. In other words, i is such a number that i² = -1. This i is called the imaginary unit.

Imaginary numbers are of the form a + bi, where a and b are real numbers. For example 2 + 9i, 7 - 82i, or -15.5 + 3/4i are imaginary numbers.

They have a real part and an imaginary part. For example, in 2 + 5i, the real part is 2 and the imaginary part is 5.

Imaginary (or complex) numbers are often plotted on the complex plane, which is just like your normal coordinate plane, except that the axes are different. Now the usual x-axis is the real part axis, and the usual y-axis is the imaginary axis.

You can plot them as points...



or as vectors.



It's easy to add or subtract complex numbers; you just add/subtract the real parts and imaginary parts separately.

For example,

(4 + 9i) + (-5 - 3i) = -1 + 6i

This corresponds to usual vector addition.

Multiplication is also very easy: you do it like you'd multiply algebraic expressions, just remembering that i² = -1. For example,

(3 - 4i)(2 + 5i) = 6 + 15i - 8i - 20i² = 6 + 7i + 20 = 26 + 7i.

But what does that sort of multiplication mean?


Let's first multiply some complex numbers by i:

i * (1 + i) = i + i² = - 1 + i.



i * (4 + i) = 4i + i² = -1 + 4i



i * (2 - 3i) = 2i - 3i² = 3 + 2i.



Look! It seems that the number, when multiplied by i, gets rotated 90 degrees counterclockwise.

That's something interesting: multiply a number, and it gets turned!

Here's some more:
(1 + i)*( 2 - 3i) = 2 - 3i + 2i - 3i² = 5 - i.


The multiplier (1 + i) is shown in the picture too. This time the number (2-3i) got rotated 45 degrees and it got longer.

And the last one:
(-0.5 + 0.5i)*(2 - 3i) = -1 + 1.5i + i - 1.5i² = 0.5



This time our number (2-3i) got rotated 135 degrees and got shorter.

It turns out there is an interesting geometric interpretation of multiplication of complex numbers.

If you multiply two complex numbers, (a + bi) and (c + di), you can get the result number by taking (c + di) and rotating it as many degrees as is between (a + bi) vector and the positive real axis (the usual x-axis). Also the length of the result vector is the product of the lengths of the two.

The calculations are much easier done when one represents complex numbers in the polar form. But that's beyond this blogpost for now.

You know, we are so used to multiplying real numbers. Sometimes it's good to expand one's horizons. In mathematics, there are all sorts of "multiplication" operations defined between all sorts of "objects" that aren't even always numbers.

Multiplication of complex numbers isn't the only example by far (multiplication of matrices, or dot and cross products of vectors come to mind as easy examples).

Grade school mathematics never gets to any of that. But believe me, there is a world of fascinating mathematics awaiting those students who get to study beyond arithmetic and beginning algebra.


And NOW... you can... get ready for Dave's Short Course on Complex Numbers or Introduction to the Mandelbrot Set!

---

P.S. If you wonder how I drew the pictures, I used the good ol' Microsoft Word with its 'gridlines' turned on and with the option "snap objects to grid", and took a screen capture using Irfanview.

Tags: math, geometry

Complex numbers

what is the square root of -1?

Well, there is no real number that could be it, but there IS a solution when we go to imaginary numbers.

Square root of negative 1 is denoted by i. In other words, i is such a number that i² = -1. This i is called the imaginary unit.

Imaginary numbers are of the form a + bi, where a and b are real numbers. For example 2 + 9i, 7 - 82i, or -15.5 + 3/4i are imaginary numbers.

They have a real part and an imaginary part. For example, in 2 + 5i, the real part is 2 and the imaginary part is 5.

Imaginary (or complex) numbers are often plotted on the complex plane, which is just like your normal coordinate plane, except that the axes are different. Now the usual x-axis is the real part axis, and the usual y-axis is the imaginary axis.

You can plot them as points...



or as vectors.



It's easy to add or subtract complex numbers; you just add/subtract the real parts and imaginary parts separately.

For example,

(4 + 9i) + (-5 - 3i) = -1 + 6i

This corresponds to usual vector addition.

Multiplication is also very easy: you do it like you'd multiply algebraic expressions, just remembering that i² = -1. For example,

(3 - 4i)(2 + 5i) = 6 + 15i - 8i - 20i² = 6 + 7i + 20 = 26 + 7i.

But what does that sort of multiplication mean?


Let's first multiply some complex numbers by i:

i * (1 + i) = i + i² = - 1 + i.



i * (4 + i) = 4i + i² = -1 + 4i



i * (2 - 3i) = 2i - 3i² = 3 + 2i.



Look! It seems that the number, when multiplied by i, gets rotated 90 degrees counterclockwise.

That's something interesting: multiply a number, and it gets turned!

Here's some more:
(1 + i)*( 2 - 3i) = 2 - 3i + 2i - 3i² = 5 - i.


The multiplier (1 + i) is shown in the picture too. This time the number (2-3i) got rotated 45 degrees and it got longer.

And the last one:
(-0.5 + 0.5i)*(2 - 3i) = -1 + 1.5i + i - 1.5i² = 0.5



This time our number (2-3i) got rotated 135 degrees and got shorter.

It turns out there is an interesting geometric interpretation of multiplication of complex numbers.

If you multiply two complex numbers, (a + bi) and (c + di), you can get the result number by taking (c + di) and rotating it as many degrees as is between (a + bi) vector and the positive real axis (the usual x-axis). Also the length of the result vector is the product of the lengths of the two.

The calculations are much easier done when one represents complex numbers in the polar form. But that's beyond this blogpost for now.

You know, we are so used to multiplying real numbers. Sometimes it's good to expand one's horizons. In mathematics, there are all sorts of "multiplication" operations defined between all sorts of "objects" that aren't even always numbers.

Multiplication of complex numbers isn't the only example by far (multiplication of matrices, or dot and cross products of vectors come to mind as easy examples).

Grade school mathematics never gets to any of that. But believe me, there is a world of fascinating mathematics awaiting those students who get to study beyond arithmetic and beginning algebra.


And NOW... you can... get ready for Dave's Short Course on Complex Numbers or Introduction to the Mandelbrot Set!

---

P.S. If you wonder how I drew the pictures, I used the good ol' Microsoft Word with its 'gridlines' turned on and with the option "snap objects to grid", and took a screen capture using Irfanview.

Tags: math, geometry

SpongeBob Squarepants

Here's an easy one for the kiddos, a SpongeBob Squarepants and Patrick Star paper model. These paper models are provided by Nickelodeon.nl (Dutch) for free download at their site. It comes in a single PDF file with two pages, I've also provided another link thru rapidshare.de for those who can't read Dutch. I can't say much about them (except that their very popular with the kids) because I've never seen the show, so here's a link if you want to know more about them.

SpongeBob - Wikipedia
SpongeBob pattern - [Download] - Nickelodeon.nl
SpongeBob pattern - [Download] - Rapidshare

Constructing an equilateral triangle

Constructing an equilateral triangle using a compass and straightedge is really simple, as you probably know.

I feel it is a good project for students to explore and figure out how to do it. But here's how:

First you choose your side length. Here it is marked with the two dots.





Then you draw two circles, using the side length as the radius:




Where the circles intersect, is the third vertex of the triangle.



The beauty of this construction, I feel, is in how it exemplifies and applies the definitions of both circle and equilateral triangle.

You know, a circle is the set of those points that are at a given distance from the center point.

And an equilateral triangle is a triangle whose all sides have the same length.

This construction can help your student to truly grasp what these are. And it's simple enough.

---

After that, you would naturally introduce this problem:

Draw a triangle with given side lengths. For example, suppose you have these three 'sticks' or line segments given to you. Now draw a triangle.





The idea is exactly the same as before. But let them think about it and find it out. Can you figure it out?

Tags: math, geometry

Constructing an equilateral triangle

Constructing an equilateral triangle using a compass and straightedge is really simple, as you probably know.

I feel it is a good project for students to explore and figure out how to do it. But here's how:

First you choose your side length. Here it is marked with the two dots.





Then you draw two circles, using the side length as the radius:




Where the circles intersect, is the third vertex of the triangle.



The beauty of this construction, I feel, is in how it exemplifies and applies the definitions of both circle and equilateral triangle.

You know, a circle is the set of those points that are at a given distance from the center point.

And an equilateral triangle is a triangle whose all sides have the same length.

This construction can help your student to truly grasp what these are. And it's simple enough.

---

After that, you would naturally introduce this problem:

Draw a triangle with given side lengths. For example, suppose you have these three 'sticks' or line segments given to you. Now draw a triangle.





The idea is exactly the same as before. But let them think about it and find it out. Can you figure it out?

Tags: math, geometry

Measuring sine

I get lots of questions, seemingly, about sine. It's because one of my pages with that topic ranks well in search engines and has the comment box in the end. Here's another one:

how to measure right triangle sine?

Well, you don't measure the sine per se. You measure certain sides of the triangle, and then calculate the sine.



For example, in this picture, if we want to find the sine of the angle α, we measure the opposite side and the hypotenuse. They are already given as 2.6 and 6 units. Then just take their ratio: 2.6/6 and that's your sin α.

You might also enjoy reading my lesson about sine in a right triangle.


Tags: math, trigonometry

Measuring sine

I get lots of questions, seemingly, about sine. It's because one of my pages with that topic ranks well in search engines and has the comment box in the end. Here's another one:

how to measure right triangle sine?

Well, you don't measure the sine per se. You measure certain sides of the triangle, and then calculate the sine.



For example, in this picture, if we want to find the sine of the angle α, we measure the opposite side and the hypotenuse. They are already given as 2.6 and 6 units. Then just take their ratio: 2.6/6 and that's your sin α.

You might also enjoy reading my lesson about sine in a right triangle.


Tags: math, trigonometry

Sunlight

As regular readers know, I keep this blog quite focused on math. But today I thought of veering a little. This post is mostly about sunlight and not so much about math.

After all, it's summertime!
But I will try to include some numbers :)



Why? Because I just feel for all the people who have been mis-educated along these lines in the past. Warning people to stay out of sunlight does MUCH MORE HARM than good.

It is now a well established fact that sunlight prevents cancer - most probably via multiple mechanisms, but vitamin D is appearing as the main one.

In fact, it has been estimated that Vitamin D deficiency is associated with more than 100,000 additional cases of cancer and 30,000 annual cancer deaths. The epidemic extends beyond cancer to Type I diabetes, heart disease and other chronic adult disease.

Just think, many things are done in order to prevent deaths. For example, people are told to wear seat belts, or not do this or that. You often see in news articles cited, "...400 lives could be saved if...", or 1,000, or whatever. But 30,000 lives could be saved yearly if people got enough sunlight!!! And estimated 100,000 cancer cases would be prevented yearly!!

Those are quite big numbers.

And it's not only cancer that's prevented by sunlight:

... study in Finland found that proper levels of vitamin D actually reduce the occurrence of Type 1 diabetes in children by about 80%.

Every body needs sunlight and vitamin D. Deficiency or insufficiency has been associated with:

* adrenal insufficiency
* Alzheimer's
* allergies
* autoimmune disorders including multiple sclerosis and rheumatoid arthritis
* cancers of the colon, breast, skin and prostate
* depression, seasonal affective disorder (SAD)
* diabetes, Type 1 and 2
* gluten intolerance, lectin intolerance
* heart disease, hypertension, Syndrome X
* infertility, sexual dysfunction
* learning and behavior disorders
* misaligned teeth and cavities
* myopia
* obesity
* osteopenia, osteoporosis, osteomalacia (adult rickets)
* Parkinson's
* PMS
* psoriasis
* rickets
* use of corticosteroids and more...

SunlightAndVitaminD.com

Besides all that, sunlight just feels so good!


But what about skin cancer?

The newest research has found that sunlight can also be preventive of melanoma, the deadliest form of skin cancer! This is very interesting. For example, melanoma is more common in inside workers than in outside workers. It seems to be associated with occasional sun exposure, sunburn, etc. and not so much with regular sun exposure:

For example, persons with the greatest risk of melanoma are not those with the greatest cumulative solar exposure; the anatomic areas that receive the most solar exposure are not preferentially affected; and not all light-skinned people suffer the same - albino Africans who have no pigmentation, are more likely to get sunburn and a number of other skin complaints as a result of exposure to the sun, but they don't get melanomas.
Sunlight, Skin Cancers and Vitamin D by Barry Groves

Sun exposure IS a risk factor for the other forms of skin cancer - those that are usually easy to treat. And, deaths from these types of cancers are far less than deaths from those cancers that vitamin D could prevent.

But then there is also evidence that diet is at fault even with skin cancer. It is appearing that too much omega-6 fats in the diet versus omega-3 fats is the culprit: In 2001, the National Academy of Sciences published a comprehensive review showing that the omega-3:6 ratio was the key to preventing skin cancer development.

Most Americans get WAY too much omega-6 fats and very little lomega-3 fats, and so the ratio of omega-6 to omega 3 is way off.

And what about skin damage and aging?

That's very true - but again it is largely preventable if your diet has lots of antioxidants and you are careful not to burn.


So the plan would go as this:
* Get sun exposure, but build up slowly so you don't burn.
* Avoid using sunscreens. After all, they prevent vitamin D formation. They also contain dangerous chemicals that are absorbed into the skin. Clothing is a safer sunscreen, while a good base tan is the best.
* Eat a diet rich in antioxidants. Might even help to take supplements on those days you sun.
* Don't consume too much omega-6 oils and margarine.
* Do consume omega-3 fats.


Note: I know many of my readers are female. Women need extra vitamin D during pregnancy and lactation because it affects calcium absorption, and deficiency makes the child softer-boned. And, obviously vitamin D deficiency is a major factor in osteoporosis in the elderly.

There are lots of internet articles and websites on the topics of sunlight, vitamin D, cancer, melanoma, and such. Here are some. You can easily find more.



Sunlight emerging as proven treatment for breast cancer, prostate cancer and other cancers


Sunlight, Skin Cancers and Vitamin D

Warning people to avoid sunshine causes more harm than good; lack of sunshine responsible for many diseases, says research


Does Sunshine Really Cause Skin Cancer?

Skin Biology Aging Reversal. Chapter 9.1 More Healthy Suntanning


http://www.greatestherbsonearth.com/articles/cancer_sunlight.htm

The Healing Power of Sunlight and Vitamin D: an exclusive interview with Dr. Michael Holick - a free download of an ebook.

---

(So where do you get omega-3 fats? In fish, fish oils, cod liver oil, flax seed, flax oil, walnuts. Also in tiny tiny amounts in green leafy veggies. If the animal you eat lived on grass, then the animal's meat has omega-3s too, such as grass-fed beef or all wild game. Eggs can have omega-3s, depending on what the chickens ate.)

(And where are omega-6 fats lurking? Just about everything else: sunflower oil, corn oil, and other seed oils, meats from grain-fattened animals, margarine, and all processed food made with these. But we need some omega-6 fats too. It's the ratio of the two that's important.)

Sunlight

As regular readers know, I keep this blog quite focused on math. But today I thought of veering a little. This post is mostly about sunlight and not so much about math.

After all, it's summertime!
But I will try to include some numbers :)



Why? Because I just feel for all the people who have been mis-educated along these lines in the past. Warning people to stay out of sunlight does MUCH MORE HARM than good.

It is now a well established fact that sunlight prevents cancer - most probably via multiple mechanisms, but vitamin D is appearing as the main one.

In fact, it has been estimated that Vitamin D deficiency is associated with more than 100,000 additional cases of cancer and 30,000 annual cancer deaths. The epidemic extends beyond cancer to Type I diabetes, heart disease and other chronic adult disease.

Just think, many things are done in order to prevent deaths. For example, people are told to wear seat belts, or not do this or that. You often see in news articles cited, "...400 lives could be saved if...", or 1,000, or whatever. But 30,000 lives could be saved yearly if people got enough sunlight!!! And estimated 100,000 cancer cases would be prevented yearly!!

Those are quite big numbers.

And it's not only cancer that's prevented by sunlight:

... study in Finland found that proper levels of vitamin D actually reduce the occurrence of Type 1 diabetes in children by about 80%.

Every body needs sunlight and vitamin D. Deficiency or insufficiency has been associated with:

* adrenal insufficiency
* Alzheimer's
* allergies
* autoimmune disorders including multiple sclerosis and rheumatoid arthritis
* cancers of the colon, breast, skin and prostate
* depression, seasonal affective disorder (SAD)
* diabetes, Type 1 and 2
* gluten intolerance, lectin intolerance
* heart disease, hypertension, Syndrome X
* infertility, sexual dysfunction
* learning and behavior disorders
* misaligned teeth and cavities
* myopia
* obesity
* osteopenia, osteoporosis, osteomalacia (adult rickets)
* Parkinson's
* PMS
* psoriasis
* rickets
* use of corticosteroids and more...

SunlightAndVitaminD.com

Besides all that, sunlight just feels so good!


But what about skin cancer?

The newest research has found that sunlight can also be preventive of melanoma, the deadliest form of skin cancer! This is very interesting. For example, melanoma is more common in inside workers than in outside workers. It seems to be associated with occasional sun exposure, sunburn, etc. and not so much with regular sun exposure:

For example, persons with the greatest risk of melanoma are not those with the greatest cumulative solar exposure; the anatomic areas that receive the most solar exposure are not preferentially affected; and not all light-skinned people suffer the same - albino Africans who have no pigmentation, are more likely to get sunburn and a number of other skin complaints as a result of exposure to the sun, but they don't get melanomas.
Sunlight, Skin Cancers and Vitamin D by Barry Groves

Sun exposure IS a risk factor for the other forms of skin cancer - those that are usually easy to treat. And, deaths from these types of cancers are far less than deaths from those cancers that vitamin D could prevent.

But then there is also evidence that diet is at fault even with skin cancer. It is appearing that too much omega-6 fats in the diet versus omega-3 fats is the culprit: In 2001, the National Academy of Sciences published a comprehensive review showing that the omega-3:6 ratio was the key to preventing skin cancer development.

Most Americans get WAY too much omega-6 fats and very little lomega-3 fats, and so the ratio of omega-6 to omega 3 is way off.

And what about skin damage and aging?

That's very true - but again it is largely preventable if your diet has lots of antioxidants and you are careful not to burn.


So the plan would go as this:
* Get sun exposure, but build up slowly so you don't burn.
* Avoid using sunscreens. After all, they prevent vitamin D formation. They also contain dangerous chemicals that are absorbed into the skin. Clothing is a safer sunscreen, while a good base tan is the best.
* Eat a diet rich in antioxidants. Might even help to take supplements on those days you sun.
* Don't consume too much omega-6 oils and margarine.
* Do consume omega-3 fats.


Note: I know many of my readers are female. Women need extra vitamin D during pregnancy and lactation because it affects calcium absorption, and deficiency makes the child softer-boned. And, obviously vitamin D deficiency is a major factor in osteoporosis in the elderly.

There are lots of internet articles and websites on the topics of sunlight, vitamin D, cancer, melanoma, and such. Here are some. You can easily find more.



Sunlight emerging as proven treatment for breast cancer, prostate cancer and other cancers


Sunlight, Skin Cancers and Vitamin D

Warning people to avoid sunshine causes more harm than good; lack of sunshine responsible for many diseases, says research


Does Sunshine Really Cause Skin Cancer?

Skin Biology Aging Reversal. Chapter 9.1 More Healthy Suntanning


http://www.greatestherbsonearth.com/articles/cancer_sunlight.htm

The Healing Power of Sunlight and Vitamin D: an exclusive interview with Dr. Michael Holick - a free download of an ebook.

---

(So where do you get omega-3 fats? In fish, fish oils, cod liver oil, flax seed, flax oil, walnuts. Also in tiny tiny amounts in green leafy veggies. If the animal you eat lived on grass, then the animal's meat has omega-3s too, such as grass-fed beef or all wild game. Eggs can have omega-3s, depending on what the chickens ate.)

(And where are omega-6 fats lurking? Just about everything else: sunflower oil, corn oil, and other seed oils, meats from grain-fattened animals, margarine, and all processed food made with these. But we need some omega-6 fats too. It's the ratio of the two that's important.)

Blog contest winners

Well, the contest is over, winners have been found, and prizes are given. The winners were:

• Webgraphing.com subscription:
  Anne Marie


• Multiplication 1 ebook:
  Melissa, lia, and Maureen.


• Fractions 1 ebook ebook:
  Theresa, Heidi, and Spunky.


• Geometry ebook:
  Brandi, Esther, and Teresa.

All in all, I thought it was fun. I got to give prizes to a good percentage of the participants, and several blogs spread the word. If I do it again, I won't choose the ending date to be so near 4th of July... it probably cut down on traffic. But all in all, I think this went well.

Thanks to all!

Blog contest winners

Well, the contest is over, winners have been found, and prizes are given. The winners were:

• Webgraphing.com subscription:
  Anne Marie


• Multiplication 1 ebook:
  Melissa, lia, and Maureen.


• Fractions 1 ebook ebook:
  Theresa, Heidi, and Spunky.


• Geometry ebook:
  Brandi, Esther, and Teresa.

All in all, I thought it was fun. I got to give prizes to a good percentage of the participants, and several blogs spread the word. If I do it again, I won't choose the ending date to be so near 4th of July... it probably cut down on traffic. But all in all, I think this went well.

Thanks to all!

Wicked Wench Papercraft

Ahoy Mateys! With Pirates of the Carribean: Dead Man's Chest showing in a few days, here is a paper model of one of the ships not used in the movie, the Wicked Wench. This paper model is a finely crafted beauty from its bow all the way to its dungbie, comes with 26 pages for the pattern and 16 pages for the instructions. It took me three and a half days to finish this, but I enjoyed it. We have Mr. Robert Nava to thank for this wonderful model and disneyexperience.com for hosting some of the great Disney paper models on the net. Now get to work ya land lubbers! o' me be showin you Davy Jones' Locker. Aaarh!

Complete Picture Set [Flickr]

Wicked Wench Papercraft [Disney Experience]

This and that

First a remainder... my blog contest with giveaways ends on Monday July 3rd! Soon we can announce the winners.

Here's just a bunch of interesting websites that I've taken note of lately... Hopefully something for everyone.

There are several subscription based websites with animated math tutorials (video), but now there is one where you find those free: HSTutorials.net has free animated and narrated math tutorials - pre-algebra, algebra 1, geometry. The collection is going to keep increasing little by little.


Simulations of Ruler and Compass Constructions - practice compass & ruler constructions interactively online. Hints and solutions included.


Textsavvy blog helps us learn the basics of what long division is based on.


Practice Problems for the California Mathematics Standards Grades 1-8. Solutions included. You could use these as tests, or for assessment of what grade level a student is in (approximately; I realize state standards vary).


Happy sunshiny days! (I hope everyone gets to get some sun as it is so integral to our well-being! :^)

This and that

First a remainder... my blog contest with giveaways ends on Monday July 3rd! Soon we can announce the winners.

Here's just a bunch of interesting websites that I've taken note of lately... Hopefully something for everyone.

There are several subscription based websites with animated math tutorials (video), but now there is one where you find those free: HSTutorials.net has free animated and narrated math tutorials - pre-algebra, algebra 1, geometry. The collection is going to keep increasing little by little.


Simulations of Ruler and Compass Constructions - practice compass & ruler constructions interactively online. Hints and solutions included.


Textsavvy blog helps us learn the basics of what long division is based on.


Practice Problems for the California Mathematics Standards Grades 1-8. Solutions included. You could use these as tests, or for assessment of what grade level a student is in (approximately; I realize state standards vary).


Happy sunshiny days! (I hope everyone gets to get some sun as it is so integral to our well-being! :^)