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Showing posts from July, 2006

Life without answer keys?

Recently I've been exchanging emails with Alexander Givental, who has translated the famous Russian book Kiselev's Geometry into English.

He's raised some interesting points regarding answer keys. I will just quote him directly from his emails to me. The discussion concerns geometry problems (please see the earlier post for examples).

"What is actually accomplished by supplying any solution at all? Ideally a student should be able not only to find a solution himself, but also be able to check that his solution is correct. With the aid of a written colution, the exercise is rendered useless for both purposes. Furthermore, a beginning student who finds a more complex solution (or may be even the same solution but eplained differently) may decide - incorrectly! - that his solution was wrong.

So, comparing your solution with your classmate's, teacher's, or parent's one makes sense, but with the one written in a silent book - very little. That's why, I thin…

Prove two triangles are congruent

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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…

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 he…

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…

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 …

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 chang…

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 home…

Complex numbers

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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 ve…

Constructing an equilateral triangle

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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 gi…

Measuring sine

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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.


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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!

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 ever…