Showing posts from June, 2006

Geometry constructions with software plus a sweet Tic-Tac-Toe game

This morning we played a few games of Tic-Tac-Toe with my dd. At one point I saw her do a move that absolutely didn't benefit her. I noted to her where she should have played. She said, "But I want to let you win."

Maybe she was copying my actions (I let her win often). But this was so sweet that I let go about learning strategy and just let her let me win!

But, back to math. If you've never heard about it before, I wish to draw you attention to geometry software.

Yes, there is software for learning geometry - and it's actually a pretty neat way to learn about geometrical constructions. In fact, it can be way better than just using compass and straightedge.

Just check for yourself. Refresh your memory about constructing a circle around a given triangle. After the text there is an interactive JavaSketchPad applet illustrating the construction.

You can move the points that define the initial triangle, and see how the perpendicular bisectors move, but how they alway…

Carnival of Homeschooling, week 26

... is online at The Homeschool Cafe.

Here are a few posts that interested me: how chess helps children learn to think, tips to mastering Miquon for those who use Miquon Math, and a story of how some homeschooled students learned to write well just by reading a lot. Enjoy these and other carnival posts!

Something interesting...

A parent and licensed math teacher who started week-end homeschooling her children in math... because of poor quality of math teaching in the kids' school. She chose Singapore Math as the home replacement curriculum.

Teens and Tweens: Singapore Math

Summer contest with giveaways

I just learned today of a great online graphing site called You can create graphs for free and analyze them too. It's the first site that I remember seeing that offers such broad range of options for analyzing the graphs.

Some functions are only available thru subscription. They offered me a free subscription, but I don't have a use for it right now, so I decided I will give it away with a blog contest!

Not only that, but I will also give away some free math ebooks of mine. So there are lots of prizes:

1) An A+ Membership to, good thru August, 2007. This is most useful for high school or beyond.

2) 3 copies of the ebook Multiplication 1 (multiplication concept and times tables)

3) 3 copies of the ebook Fractions 1 (Fraction operations such as addition, subtraction, equivalent fractions, mixed numbers)

4) 3 copies of the ebook Geometry (includes most topics of plane geometry for elementary grades).

YOU CAN ENTER the contest

1) Via email to . Please pu…

Sketching sine wave

Another (student?) question along trigonometry lines:

Sketch the sinusodial waveform given below over one complete cycle showing all essential values. i = 25sin(2t-30degress).

Use a graphic calculator or an online version, such as
Function Flyer at

There are more online graphing resources at my site.

You need to change the 30 degrees to radians first.

So compare 30 to 360; 30/360 is same as 1/12. So take 1/12th part of 2π, which is π/6.

Then enter the function. With the Shodor grapher, you need to use * for multiplication and x for variable, instead of t, and not use degrees but radians.

Seeing the sine graph will surely help you sketch yours on paper. Obviously the highest value is 25, the minimum value is -25.

To find its zeros, you need to think:

When does sin y = 0????

It's when y = _______ or y = _______ or when y = ______ (there are tons of these of course since it's cyclical).

Now take the argument 2t-π/6 and let it equal those numbers you found.

2t-π/6 = ___________…

Spiraling or mastery in a mathematics curriculum

Some math curricula are labeled as 'spiraling', and some are said to employ the 'mastery principle'. What does that mean, and which is better?

Spiraling mathematics curriculum introduces many topics in one grade level. It does not aim to teach those topics completely in one go, but revisits those topics the next year, the next year, and so on.

Mastery approach simply aims to teach to mastery any single topic, before going on to the next.

There is a lot of talk against spiraling math curricula, especially in regards to some reformist mathematics curricula.

So is the mastery principle then better?

Well, I think that dividing this matter into two opposite positions is a mistake. You CAN have good mathematics education employing parts of both principles.

For example, a student can learn to add 2-3 digit numbers on grade 2. She can revisit the topic on 3rd grade to learn to add 4-6 digit numbers. She can revisit the topic on 4th grade to learn to add even larger numbers.

At e…

Enjoying geometry proofs

Here recently I finished reading the two Dr. Math geometry books. (I will be writing a review of them, and I can say they're pretty good & inexpensive books!)

The one meant for high school geometry had as its last chapter circles and theorems about circles. Well circles never were the strongest part of my mathematical knowledge, for whatever reason... (I think it stems from the fact that so big part of school geometry concentrates on calculating areas and volumes, and not on properties of figures.)

So I wanted to brush up on circle theorems. I quickly read over the circle chapter on another geometry book I have, called "Geometry: A Guided Inquiry". (I can recommend that book as well. It often asks the student to explore and try find theorems. See more info here.)

I don't know about you, but to me, reading & learning theorems and proofs can be enjoyable. First you maybe struggle to grasp it all, but afterwards there is a great satisfying feeling and admiration o…

Adding 2-digit numbers - followup

So today I experimented, and gave my daughter a problem 15 + 25 for starters, to see what she'd do.

1 5
+ 2 5

3 10

She promptly wrote down 3 and 10. Then she looked at the numbers, kind of wondering, saying "It's ten three." I asked her how would we say the number, pointing with my finger and making a sweeping motion from 3 towards 10, and then she said, "Thirty-ten."

I said yes, it's thirty-ten, but we say it differently normally. Then I asked her to find it out using abacus. She easily found it was 40. Then we erased her numbers and wrote in 40.

She did two more similar problems on her own, first adding tens, then ones (as her custom seems to be), and then erasing the numbers and changing them to the right result. She seemed very happy for being able to understand the gimmick.

So we did all that without discussing 'carrying' or putting little number 1 up on top, or anything.

I figure next time I'll try a problem where the ones add up to 11 or 1…

Adding 2-digit numbers

Just today I made my dd some more math problems with adding 2-digit numbers - the easy type where you don't have to carry. She's getting it now pretty good.

I always talk about adding so many tens to so many tens so she won't forget it's about adding tens and adding ones. Somehow she's doing the tens first, then the ones, and she tends to say, after adding tens:

+ 37

"Oh, it's going to be eighty-something."

You know, that's the way many people add numbers mentally: 52 + 37, we add tens first and get 80. Then add ones.

Even with 46 + 36, you probably add tens first and get 70, then add ones and go over to 82.

So a thought hit my mind: one day I will present to her a problem where the ones make a ten, such as

+ 34

Is she going to get an answer fifty-ten?

Well, that's what it is, basically. We just need to recognize that's not the normal way of saying it but that we need to add that ten to the fifty so it becomes sixty.

Anyway, I really feel I…

Measurement projects

I have a feeling blogging might slow down over the summer...

I've been dealing with math daily, though (of course). I'm writing a collection of worksheets for a company, and teaching too.

The last few days have been spent with writing an answer key. I thought I'd post a few problems here for your enjoyment - sort of measuring projects. Maybe you can involve your kids in some hands-on splish splosh splash water experiments and learn geometry too. :)

1. You need: an empty cylinder (for example from inside a toilet paper roll), a small ball, and a tennis ball.
The small ball should fit inside the cylinder fairly snugly, and the tennis ball shouldn't.

a. What should you measure from the ball and the cylinder to know whether a ball fits, without actually trying it out?

b. Find how many balls you could stack inside the cylinder, if you had more. Think what measurements and calculations you should do to find that out.

c. How many of the small balls would fit inside a cylinder t…

Homeschooling Carnival

This week the Carnival of Homescooling is held at There were several math-related posts (including mine), again! I so delighted in reading and realizing that homeschoolers are appreciating mathematics and treating it 'positively'.

This is so important, as I've stressed many times. Our attitudes towards math (or history!) influence our students. An enthusiastic teacher is a big part of learning to enjoy a subject.

I am sincerely looking forward to re-learning geography, for example, with my kids. Here's a pick from the carnival: seems like US youngsters do not know geography very well... I think it is a fascinating subject, to learn about all the variety in the ways people live on this globe, and the variety of landscapes and land formations, or how volcanos or earthquakes work. I hope you can tell that I enjoy learning!

Math ebooks summer sale

I'm having a summer sale on my elementary math ebooks... They are not overpriced even normally and now you get some good discounts. Valid till end of July, I offer:

the 18 ebooks package for $21.90(before $27.90)!
the 18 ebooks package on CD for $27.90(before $33.90)!
order totals at least $10 - 20% discount
order totals between $15.10 and $21 - 30% discount

These ebooks cover topics in grades 1-5, such as addition facts, multiplication tables, division, long division, factoring, fractions, decimals, and geometry. They are 'worktexts' with explanations and problems in the same product. The individual ebooks start from $1.50.

Learn more about the ebooks and see preview pages.


Coming soon...

I'm planning on a SUMMER DISCOUNT on my ebooks package. It currently sells for $27.90, but I will soon lower the price for summer to $21.90.

There are 18 ebooks, covering topics from basic addition concept, mental math, subtraction, multiplication, times tables, division, place value, fractions, and basics of decimals.

They are worktexts, with explanations and problems. All total the ebooks have over 770 pages of instruction. That makes it... 2.8 cents per page.

You simply download the ebook, and print out the pages you want - and you can print them out over and over again if needed.

Just stay tuned... I will announce the actual summer sale soon. Meanwhile, feel free to browse the preview pages at

What jobs use Pythagorean theorem?

This is what someone asked me recently.

Well, obviously you'd need to know Pythagorean theorem if your job includes triangles or rectangles in some way, for example in land surveying, or planning farmland, or designing buildings (architects) or designing furniture or frames for TVs, or making parts for machines (machinist) etc.

Remember the diagonal for a rectangle is calculated using Pythagorean theorem, and TV screen size is actually the length of the diagonal in inches.

See also my article on similar lines: Where do you need square roots or algebra - why study math?.

10 Out - math card game

Today I told my dd to write down addition facts where the sum is 10 - "Which numbers make ten?" are the words I use with her. While she was writing them and using abacus, this math card came 'popped' to my mind...

I can't claim it as my own, because I have this 'feeling' that I've read about it somewhere, sometime, but since I can't remember when or where, I can't give credit to where credit might be due.

I will just name it "10 Out" - a math card game.

Anyway, this is how it goes:
Take away the 'picture' cards and joker from normal playing cards. Then deal 10 cards to each player; put the rest of the deck in the middle. The goal is to get rid of all cards in your hand.

Find all pairs of cards in your hand that add up to a ten (or single 10's) and discard them. Then you may ask the player left to you for one card, and if she has it, she has to give it to you.

For example, say you have 2, 3, 8, and 1 left in your hand. You ask …