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Showing posts from April, 2006

Free math materials

It's always exciting to find good quality free resources in the internet.

I got this site's address from an email list. It has free math workbooks to download (yes complete workbooks!) for most of the grades, and they're pretty good.

Mathematics Enhancement Programme - School curriculum materials

And, this link has Online mathematics tests for student assessment for grades K-11. Multiple choice.

I also wanted to start making a list of such great free or inexpensive curriculum-related materials. Please click here to see it.. just a start. If you have suggestions, feel free to email me.

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Distributive property

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A little bit of math for you today.

Usually kids encounter these words in algebra or maybe pre-algebra. But, they use the principle long before that, say in 2nd or 3rd grade. Did you know where?


The distributive property says (as it's stated in algebra class) that

a (b + c) = ab + ac.

It's true for all real numbers - including negative ones, so it follows it's true for subtraction too:

a (b - c) = ab - ac.


We use this principle often when multiplying numbers. For example, do in your head 7 × 21. Most people take 7 × 20 and 7 × 1 and add those. In other words, you 'broke down' 21 to two parts and multiplied the parts by 7 separately:

7 × (20 + 1) = 7 × 20 + 7 × 1.

How about 8 &times 98? Aren't you tempted to go 8 &times 100 and subtract 16?

8 × (100 - 2) = 8 × 100 - 8 × 2.

Studying multiplying 2-digit numbers is THE time to talk about this idea in detail.



37
×   4



Here, you should explain that this algorithm (or procedure) is based on the idea that you multiply 4 × …

Homeschooling Carnival

This time it's up and running at Common Room. Enjoy!

Fib

Fib
Is
A fad.
Write poems
Fibonacci style.
But what if I'm not a poet?

1 1 2 3 5 8

:^)

Reform math - we need a BALANCE

Recently Spunky wrote about reform math, and I feel I want to say something about it, too.

She was critizing the movement heavily. Some people call reform math 'new new math' or 'fuzzy math' (in a degrading way) - and like you can read in Spunky's post, sometimes when new ideas are implemented, it can cause math to become 'fuzzy' to the children.

But that is not necessarily a fault of the reform on a whole - maybe the teacher in question didn't understand how to implement the methods, or for some reason (unwisely) totally abandoned 'traditional' math fact drills.

I would say not all of the IDEAS that have come from this mathematics education reform are bad.

For example, encouraging kids to explore and investigate mathematical concepts, or discovering their own rules, can be a good thing. Group work CAN be a good thing.

The reformists promote understanding of concepts, problem solving, critical thinking. They try to distance themselves from the "…

Living math

I've been looking over the website LivingMath.net recently. Lots of good stuff there.

What is 'living math'? How do you make math "come to life"?

Well, I guess we could ask that when did math become 'dead' and who killed it?

Some people feel that it is worksheets, tests, drill, and 'scope and sequence' that killed it. But I don't feel it's that simple. Those things are tools you can use, and they don't necessarily make math 'all dead'.

There are many ways to make math 'living'. One important way, I feel, is to show your kids when you do use math in your daily life. I've written about it before.

When YOU use math, it conveys the idea that math and numbers are things we use, not some obscure school subject that one does for half an hour five times a week, and then tries to forget the rest of the time.

I have never understood mothers who say they didn't like math, hated it, etc. Don't say that! That attitude will …

The Math Goodies CD review

I've recently reviewed The Math Goodies CD - it is full of math lessons, interactive exercises, worksheets, crossword puzzles and an interactive Integer Football game. It's mostly for grades 5-8; however the lessons on symbolic logic better suit grades 7-10.

To summarize the lessons in one word, I'd use the word CLARITY. They are very clearly presented. The interactive exercises make it good for a student learning on his/her own.

Read more here!

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Comparing two third grade math books

As I mentioned, I have recently started on a new project in creating a worksheet package for third grade. So... I have had to take a real close look on what exact topics are usually taught in third-grade math.

I have been looking at two third-grade (public school) math books. One is from 1978, the other from 1992.

I have been quite amazed at the differences between the two! I knew about the 'trend' in mathematics curriculum where topics seem to get introduced on earlier and earlier grades. But it was eye-opening to see it "in action" in comparing the books.

I think it's good for homeschoolers to be aware of this trend, too.

The 1978 book is full-color, illustrated, very clear layout, and fairly little text. The word problems are extremely word-scarce. They've even put little symbols in the text instead of words (for sandcastle or beaver or squirrel or an LP record player, for example).

The 1992 book is still full color of course, less clear layout, and this time …

Clock worksheets

I've embarked on a new project in making worksheet collections for each grade from 3rd through 8th.

While doing that, I had to generate images of clock faces, and so I decided to just use a few hours to make a worksheet generator at the same time.

So if you're interested, click here to make clock worksheets. It's pretty simple, just two things you can do: either the student has to read the analog clock, or draw hands on it when time is given in digital form.

Multiplication Division 2 ebook updated

My ebook Multiplication Division 2 has been updated on the part of long division; it is more of an approach of smaller steps now: first divide numbers where there are no remainders at all (for example 684/2), then if there's a remainder in the ones (685/2), or in the tens (658/2). Then learn to look at two digits of the dividend if not enough hundreds (345/5), or place a zero in quotient (832/4) etc. Lastly the case with most steps.
Price: $3.

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Finding values of sine without a calculator

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How do I find the sin or inverse sin of 46 degrees (or any degree) without using a calculator?
Thanks.
Jackie

Finding an "inverse sin" doesn't apply to degrees by the way. We take sine of an angle and get a number; inverse sine is taken from a number and gives back an angle.

So is it possible to find values of sine without a calculator? I am sure there are various methods, but these two came to my mind.

1) We can go back to the definition of sine in a right triangle and using a protractor, DRAW a right triangle with that angle. Draw as accurately as you can.



Then from the picture again, we need to measure the two sides: the opposite side and the hypotenuse and then calculate their ratio (paper and pencil). Again, measure as accurately as you can.

Now, as far as the opposite problem, let's say you know that the sine of some angle is 0.86 or some other number (between -1 and 1). Can we find the angle without the calculator?

Draw a right triangle with hypotenuse 1, oppo…

Mathematics and internet security

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April is math awareness month - this year with the theme of internet security. Are you aware that mathematics is used extensively in that?



By now, we are used to shopping online and to "secure sites" where our credit card information is sent via encryption to the merchant.

But do you know what kind of mathematics is behind that? It is based on number theory - the branch of mathematics that looks at whole numbers and the special properties some of them have, such as being a prime.

In 1970s, a group of researchers developed a way to encrypt and decrypt data WITHOUT the need of the parties first sharing the encryption key.

You see, in traditional cryptography, the two parties wishing to exchange secret information first have to somehow let each other know the encryption/decryption 'key'. And that cannot be encrypted.

What if you can't do that? What if the parties haven't had previous contact with each other to share the key?

This exact problem led to the inventio…

Homeschool Blog Watch started

Gary at HomeschoolBuzz.com has started a Homeschool Blog Watch - it tracks headlines from selected homeschooling blogs and is updated as new posts appear. In other words, you can kind of check out several blogs all at once.

Distance = velocity * time, or Calculus Without Tears

You can learn calculus concepts starting from the formula distance = velocity * time.

Yes, that's true. That's what the book Calculus Without Tears is all about. It starts from the simple situation of a runner running with constant speed (velocity), and goes very step-by-step into actual calculus concepts, such as derivative, area under curve (integration), and differential equations.

The idea of the book is to make basic calculus concepts accessible to younger students, without need of much algebra.

Like I said earlier, calculus is the mathematics of change. It is usually studied as the last course in high school, or early in college studies. So should one study it earlier? I am sure people have varying opinions on that.

Certainly the aim of this book is NOT to further crowd the "mile wide" mathematics curriculum. But it provides something extra for gifted kids, or for students very interested in calculus and math (and physics). It also provides an alternative way to …

Math timeline, Math Doodles, and Figure This!

I thought I'd share some math links I've recently come across. The internet has SO much to offer when it comes to mathematics that it is amazing.

Maths Timeline is neat! It has pictures of major mathematicians on a timeline, and by clicking on the images you get to read an article about them. You have to kind of 'slow-click' on the arrow on the right. I first clicked and clicked in vain because I clicked too fast...
Math Doodles offers seven fun games that also make kids think and learn. Requires Shockwave player. The name of the site is easy to remember but the games aren't any 'doodles' - they are very well-done.
Figure This! Math Challenges for Families. This site has word problems related to real life. They don't always have all the information but you have to estimate and think. For each problem, there is a hint, other related problems, and interesting trivia. Website supported by National Council of Teachers of Mathematics. Kind of different problems…

What is calculus?

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I recently read a calculus book (actually two) that brought me to reflect what IS calculus.

First of all, let me state what calculus is not:
Calculus is NOT the epitome of math, the highest mathematics there is, or anything such like.
So what is calculus? It is basically the mathematics of change. Calculus allows us to study a thing that's changing (represented by some function), and the RATE of that change.

I'll give you an example.



Suppose you have a graph like the one below. Maybe this function (the red one) is depicting how temperature (or voltage or some other thing) changes over time.

The black straight line is a tangent to the function - in other words it "touches" the function in one point. The steepness of the tangent tells us how steep the function itself is going at that point.

Imagine that the tangent (the black straight line) was drawn into a point a little further on the red graph. The red function is increasing but it's coming to a 'hill'. When…