Showing posts from February, 2007

A little contest

In an effort to promote my newer site, I'm announcing this little "contest", in which it is quite easy to win a Math Mammoth book - or even two - for free, if you have a blog or website. The rules are as follows:

You write a little about site on your blog or website, and place within that text a link to the page of the book or books that you'd like to win!

For example, if you'd like to win the Geometry 1 book, then you'd link to the page

Instead of the individual book pages, you may link to the home page (, if that fits better your write up or your website. But I greatly appreciate the links pointing to the 'inside' pages.

Your link must be normal link, and not a scripted one or with nofollow tag. 

The link must point to or some subpage.

No adult, hate, or other weird or offensive blogs/websites/forums. 

No spammy sites, scraper sites, free-f…

Geometric patterns in islamic art

This was in the news recently so you might have seen it...

Physics student Peter J. Lu has discovered, after his trip to Uzbekistan, that the geometric patterns shown on the walls of old islamic shrines depicted a very complex mathematical pattern, one that was only found in the west during last century.

I feel the article at explains it all very well, and shows plenty of pictures of the patterns AND of the underlying tilings, plus has references and links for further study, so I'll just refer you there:

Ancient Islamic Penrose Tiles

Some links

First of all, the carnival of homeschooling #60 is online this week at Homeschool Hacks.

Secondly, recently I've learned of Squidoo lenses created by Rebecca Newburn on topics such as integers and fractions. She's collected some of the best resources about these topics, including games, articles, explanations, books and even free video clips.

Then I also wanted to include a link to some magic square addition worksheets that my daughter has greatly enjoyed recently (scroll down to item #5 on the page). We've been practicing adding single-digit numbers where the sum goes over 10, such as 7 + 7 and 9 + 5.

I always tell her that when it's 9 and other number, then one (dot) of the other number "jumps" to go with the nine and makes ten. That way she has learned to add 9 + 6 or any other such sum.

And if it's 6 + 7, I tell her to think of 6 + 6 which she knows by heart, and figure it out from there.

I hope she gets to remembering these by heart eventually, but at thi…

Printed (hardcopy) versions of Math Mammoth books

You can now buy a hardcopy (already printed version) of any of the 16 "BLUE" series Math Mammoth books.

These are found at - at the address

The drawback is, they're black-and-white (or grayscale to be exact). This is because the price per page for color printing is quite high and I figured no one would buy the books then. As it is, the b&w versions cost around $8-$11 each. provides on-demand printing for self-publishing folks (such as me). They have a nice service; you can even print calendars and other stuff and not just books.

How the four operations become two

Awhile back I posted about fraction division; that post made me think of reciprocals, algebra, and even properties of addition and multiplication.

You see, when you study abstract algebra, you get to concept of a group, or a ring, or a field - and all of those consist of a set of elements and one (group) or TWO (ring or field) operations.

Yet, in school math, we always study the FOUR basic operations.

Why are two operations enough in higher-level algebra? Where did they throw subtraction and division?

Addition and multiplication
Did you ever notice these similarities between addition and multiplication?

Both addition and multiplication are commutative:
a + b = b + a and ab = ba for all real numbers a, b.
Division and subtraction are not.
Boht addition and multiplication are associative:
(a + b) + c = a + (b + c).
Division and subtraction are not.

There exists an element so that adding it to a number doesn't change it (ADDITIVE IDENTITY). We call it zero:
0 + x and x + 0 both end up being just…

How to divide irrational numbers?

Many times students just accept what they are told in math class without much questions; it's perhaps boring, they don't want to take time to investigate and delve deeper, or whatever the reasons.

The teacher feeds them with knowledge and they take it in: "Oh, okay, there's such a thing as Pi. Oh, okay, some numbers are irrational."

But a question such as this shows that the person is wondering, wanting to understand the material more. So it's a good question!

I will divide the 'division' question to two parts.

1) When exact answer is desirable, often we do cannot technically divide. For example, let's take Pi ÷ 2. Well, we just write it as Pi/2 or π/2 and leave it at that.

If we can simplify the answer, we do that. For example, √15/√5 can be simplified to √3.

We could simplify √2/2 this way: since 2 = √2 √2, then √2/2 = √2/(√2 √2) = 1/√2.

But often the expression √2/2 is left as is, since there is a convention that we should consider a root in the …

An 8-year old boy finds error in science exhibit

Parker Garrison is an 8-year old math prodigy who recently discovered an error in the equations of a candy exhibit in Discovery Place.

The problem there asked to find the number of jellybeans in a half-pyramid, using the formula for the volume of the pyramid, and the known volume of one jellybean. The problem asked to multiply the volume of pyramid by 0.9 to account for the spaces between the jellybeans.

The boy's mom took the numbers home and there he calculated and found an error in the given measurements. They had already given dimensions of the half-pyramid in the problem, but still the problem asked to divide by 2...

(You can read more details here)

It's just an interesting example how adults can make simple mathematical mistakes and those can go unnoticed for years...'s special offer

I got word a little while ago that has a very special offer: you can get online access to their entire library of math videos for only $25 (for a full year).

That is indeed a very low price for such a service!

On their site, you can also buy those videos on a CD.

The videos cover basic mathematics, prealgebra, algebra 1, 2, and 3, word problems, and trigonometry

I have some of their CDs and found them very good. They show a math teacher solving problems on a board, step by step.

Multiplying decimals

Someone asked,
how can you use models to multiply decimals?
Learning to multiply decimals, I feel, is built on students' previous understanding of multiplying whole numbers and fractions.

So models wouldn't necessarily be the focus, but instead relating decimals to fractions first, and learning from that.

Multiply a decimal by a whole number
Of course, when multiplying a decimal by a whole number, you could use the same models as for fractions: say you have a problem

2 × 0.34

You can use little hundredths cubes, or draw something that's divided to 100 parts.

BUT you can also just use fractions, and justify the calculation that way:

2 × 0.34 = 2 × 34/100 = 68/100 = 0.68.

OR you can explain it as repeated addition:

2 × 0.34 = 0.34 + 0.34 = 0.68.

I employ that idea in these lessons:

Multiply mentally decimals that have tenths and
Multiply decimals that have hundredths

Multiply a decimal by a decimal
When students are learning to multiply a decimal by a decimal, they're on 5th or 6th …