Posts

Showing posts from September, 2006

Online math resources

Resources

These are some of the links I've added to my site recently. Maybe there's some that interest you.

Mathopenref.com
Free online textbook for high school geometry; not finished.

GapMinder
Visualizing human development trends (such as poverty, health, gaps, income on a global scale) via stunning, interactive statistical graphs. This is an interactive, dynamic tool and not just static graphs. Download the software or the reports for free.

How to write proofs
A 12-part tutorial on proof writing. Includes direct proof, proof by contradiction, proof by contrapositive, mathematical induction, if and only if, and proof strategies.

Money Math
Crystal clear tutorial on interest.

Graph Mole
A fun game about plotting points in coordinate plane. Plot points before the mole eats the vegetables.



All sorts of sites to explore! But if those didn't fit your bill, if you're in need of a game or tutorial about specific math topic, check my link lists of online math resources; they're orga…

Division of fractions

Image
This topic is often not understood real well by teachers or students. But we want them to learn, not only the rule, but also the meaning.

These ideas can help you to explain and understand division of fractions:


1) The rule of "invert and multiply" applies to division in general - not just to division of fractions. It is a general principle. For example:

20 ÷ 4

I can invert and multiply:

20 × 1/4 = 5.

With whole numbers, division can be thought of as making equal parts. When you divide something by 7, you're dividing it into 7 parts, so might as well just take 1/7 part - multiply by 1/7.

You can always change division into multiplication with this principle: 18 ÷ 2.51 = 18 × 1/2.51





2) Think of fraction division this way: how many times does the divisor fit into the dividend? You can use this to judge the reasonableness of your answer.

For example consider 1 3/5 ÷ 2/3. Clearly 2/3 can fit into 1 3/5 more than two times.

1 3/5 ÷ 2/3 = 8/5 × 2/3 = 16/15 = 1 1/15 - just a t…

Curriculum focal points

I'm going to talk again about the new report released by NCTM, Curriculum Focal Points.

For each grade, it describes THREE focal points, and also explains to which other mathematical topics these focal points connect to.

By familiarizing yourself with these few points, you can see the basics of school mathematics unfold before your eyes. Knowing the basic goals is essential for being a good teacher.

By the way, these aren't the only things kids might study on a given grade. They are the focus areas. Many of the other topics would connect with these.

Go look at the pics at textsavvy.blogspot.com about the way textbooks and programs typically present the topics, versus a "VISION OF CURRICULUM" laid out by the Focal Points document.


On grade 1, the focus is on
addition/subtraction: basic basic addition facts and related subtraction facts. Learning to add and subtract two-digit numbers.
Understanding whole numbers in terms of tens and ones
Composing and decomposing geometric s…

Rational or not?

is 9/56 rational? when converted to a decimal it seems to be never ending and it seems like there's no pattern (at least as far as the calculator shows)
Well, relying on a calculator is leading this person astray.

Obviously the number is rational - it's a fraction (!); it fits the definition of a rational number.

The calculator gives 0.160714286 (to nine decimal digits), but if you use long division and continue it till you get just a few digits more, you get 0.16071428571428..., or 0.160714285.

Curriculum Focal Points

National Council of Teachers of Mathematics (NCTM) has released a new report entitled "The Curriculum Focal Points".

It summarizes THREE focal points of math education for each grade, and also explains the major connections between those and other areas of math.

I feel this document can be of enormous help to homeschoolers. If you've ever felt "lost" in the jungle of math standards, objectives, goals, and such, then this document is especially good for you.

The teacher needs to know what the major goals are. Then, he/she can plan how to teach those topics, what tools to use, and so on. Without the goals clearly in one's mind, math instruction can become just mindless wandering from topic to topic.

I'll probably write more about this report later, but for now, just head on over the NCTM site and read:

The Curriculum Focal Points - the main page
or
Curriculum Focal Points by Grade.

Calculating percent with mental math

Would you say that students' understanding of percent is sometimes - or often - hazy?
Find the number of which 79.5% is 101.Often, solving these kinds of problems is taught with the idea that you "translate" certain words in the problem into certain symbols, and thus build an equation.

Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:

Y × 79.5% = 101.

0.795 Y = 101

Y = 101/0.795 = 127.044025157


I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?

So while this idea is great and works, it is also necessary for students to understand the concept of percent well.

In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 1…

Carnival and more

First of all, carnival of homeschooling is online at Principled Discovery.

Organized under a theme of homeschooling journey, it is shock full of good stuff.

Some picks of mine (though, I didn't really have time to read many of the posts, unfortunately):

One homeschooling parent has chosen to focus on music instruction, hoping it will translate to book learning. Well I don't know how it will go, but I do believe music is powerful, and learning music will help a child's brain grow and develop. (I myself studied piano for 13 years of my life). There are scientific studies that show how music instruction can greatly improve other academic areas.

Also from the carnival: Thinking Blocks has an interactive program that helps your child model or visualize word problems. The models use blocks but you could as well call them diagrams. The idea for the program has come from Singapore Math's usage of diagrams.

My entry was Let It Make Sense.

Living and Loving Math

X (however many) Habits of Highly Effective Math Teaching:

Part 4: Living and Loving Math
You are the teacher. You show the way - also with your attitudes, your way of life.

Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?

And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?

Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.

Math is not a drudgery, nor something just confined to math lessons.

Some ideas:
Let it make sense. This alone can usually make math quite a difference and kids will stay interested.
Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are even story books to teach math concepts - see a list h…

Writing some proportions

Got a question in today,
How can I solve the question, "write as many true proportions as you can with the numbers 3,6,9,12 and 18. Use a number only once".
This is a simple simple problem IF you know what the term "PROPORTION" means.

Proportion is simply an equation stating that one ratio is equal to another.

So to solve this, we play around and make ratios, and see if we can by happenstance come up with some equal ratios fitting the rule of using each number only once.

For example, 3:6 and 9:18 are two such ratios. So the PROPORTION then is (it needs an '=' sign)

3:6 = 9:18.

You can write the ratios using the fraction line, too.

Can you make others?

See also An idea of how to teach proportions.

Homeschooling Humor

Joyce Jackson just sent me a fun book filled with homeschooling humor, and said I can distribute it here.

The book is quite funny and inspirational. The stories, jokes, anecdotes, and other stories are by real homeschoolers that she simply compiled with background information.

You can download it here.

I liked it; I'm sure you will too. Enjoy!

Know Your Tools

Image
Math teacher's tools are quite numerous nowadays.

First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.

The choices are so numerous it's daunting. What's a teacher to do?

Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.

There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spi…