Showing posts from May, 2006

I can guess your thoughts trick

I'm not a big fan of these various math tricks (such as how to multiply any two numbers between 11 and 19 or how to square some special numbers ending in certain digits) unless they increase students' understanding of the number system or math principles. Often they're just another thing to memorize that fades away quickly... at least with me.

But some people like them, and that's fine with me.

One particular kind of math "trick" is where you are told to think of a number, then do all sorts of manipulations with it, and then in the end... they know your number! How is that?

Here's one for you... it's neat, easy, amusing, and explains in the end how it works.

Scroll Down Slowly!

Math card game

Last few weeks I've been thinking I want to have a way to use normal playing cards to let my daughter practice addition.

I hadn't given it serious 'thought' to come up with any games yet, when just today I stumbled upon that exact thing at A Home for Homeschoolers Portal: The game that is worth 1,000 worksheets.

Definitely worth playing! The 'game' aspect can make practicing facts so much more fun, as we all know. And this game that Denise describes is so simple too.

If you have any other playing card ideas, feel free to comment here or there.

The exponential function

I just finished reading the third volume of Calculus Without Tears... worksheets designed to teach calculus concepts before high school, for those interested.

So I've gotten a refresher on calculus... which I've always loved, and so that's why in today's post I hope to explain to you ONE little point from calculus - a fascinating and important one - that is.... THE EXPONENTIAL FUNCTION and THE NUMBER E.

First you need to know a calculus concept, and something about e:
The steeper the graph of a function, the faster it's growing (and vice versa)
We can measure the steepness of a function at any point by drawing a tangent, and checking the slope of the tangent.
e is a VERY special irrational number with the approximate value 2.718281828459...

Simple enough, huh?
Okay, so here we go:

There is a function....
whose growth rate at any point
is always
its value at that point!

The function ex

The 'steepness' of the tangent shows us the growth rate of the function.
Tangent dr…

Website updates

I seem to get so involved in writing about math topics that I forget that I'm supposed to also write about my website updates. So here goes, some recent updates and additions:

1) I've reorganized the links and stuff on the reviews page and on the math curriculum reviews page, for easier navigating. (Not that my site couldn't use more improvement in that navigation... hopefully some day.)

2) Added a page for Thinkwell CDs. If you've used those, please leave a review.

3) I made the blogposts and discussions a few months back about word problems into an article: Teaching elementary word problems.

4) This month I've had some new advertisers: - individual audio, animated, secondary maths lessons with accompanying worksheets and automatic marking. - unlimited online math practice - an online curriculum with animated lessons for $19.99 per month. There is more info and reviews here.

I guess that's all that I can remember. Have …

Proving triangles congruent

My child is dealing identifying Triangles as SAS,SSS,AAS,ASA and HL Theroms. She is also having trouble with the Flow proofs and Column proofs on explaing why 2 triangles are congruent.

All of the theorems about proving that triangles (or other shapes) are congruent can be "translated" into a drawing problem:

If I have my 'secret' triangle and I give you THESE pieces of information, can you reproduce my triangle? Can you do that every time, no matter what my triangle?

We can describe a triangle using 6 pieces of information: the legths of the three sides, and the measures of the three angles. But you don't need all of those to be able to draw my secret triangle.

Can you draw a copy of my triangle if I tell you that....

my triangle has a side 5 cm long, another side 6 cm long, and the angle between those sides is 29 degrees (I've given you S - A - S)?
my triangle has a 30 degree angle, a 60 degree angle, and a 90 degree angle (I've given you A - A - A)?

In the…

Count On, Relax, Enjoy

Summer is approaching, and you all my readers probably have all kinds of different plans. If you're a homeschooler, maybe you continue homeschool, maybe not. Some people take a vacation, some stay at home.

Anyway, however it might fit into your plans, I'd like to recommend a nice (British) math website Count On (

In my mind, Count On website is almost like a mathematical vacation, full of enjoyment with mathematical themes. Even the logo looks like the site is supposed to make you feel happy.

It has various things on it: some very games of course, two interactive "museums" with galleries of mathematical paintings and explanations of their history (Matrix Museum and Pictures Interactive). A maths timeline for history geeks.

I especially like their "mathzines", or math related online magazines. There are two different ones: Meenie Minus for smaller kids, and Kaleidoscope for middle school age. The magazines have engaging, short games and activities,…

Carnival of Homeschooling

This week's carnival is online at Home Sweet Home. Melissa has arranged the contributions in an absolutely fascinating way - using animals and their characteristics! You have to go see it.

And, I also want to highlight a post I picked from there that fits well the theme of this blog: Number Bonds = Better Understanding.

Missing addend with x or a box?

Recently I've been writing worksheets for fourth grade... Later this year, I will make this collection available for whoever wishes to buy it. But for now, it's a work in progress.

One topic I was pondering a lot was whether to include problems such as

15 + x = 30 into the worksheets.
Or, 234 + x = 700
or x + 1,923 = 5,000.

I hope you get the idea; in elementary books you often see these kind of missing addend problems with a little box:

15 + = 30 or
234 + = 700 or
+ 1,923 = 5,000.

Well, I decided for the x over the box! I don't think solving missing addend problems with subtraction is too difficult to learn on fourth grade; after all, students have been working with addition and subtraction connection from 1st grade on, right?

In my own old schoolbooks I actually see x from 3rd grade on.

I made several problems with charts such as

Write a missing addend sentence
and a subtraction sentence to solve it.


x 346

Hopefully those h…

Examples of calculus use in medicine?

I got a question,
"I am supposed to teach my calculus class one lesson.
That lesson has to be on something that can be applied
to whatever I am hoping to major in. I am planning on
studying pre-med to become a doctor. Could you tell
me how doctors apply math learned in calculus 1?"
I suspect doctors don't actually use any calculus in their daily work with people. BUT, it is used in medical research and analysis.

For example, calculus concepts are applied in studying how medicines act in the body. I found an article called Half-life and Steady State that talks about how the patient might be taking a medication and all the same time the body is clearing the previous doses... Eventually there comes a "steady state" where the amount of "the amount of drug going in is the same as the amount of drug getting taken out."

Many drug effects occur primarily when the blood level of the drug is either going up or going down. When the drug reaches steady state, the…

I hate math but don't want to anymore...

I want to blog again on Julie's great Living Math website. You can really get help there if you're one of those who don't care for math, or even hate math, but want to get rid of that feeling.

She has suggestions on how to start teaching "living math" - teaching math in a way that makes connections to real life, takes away the dryness of it, takes away the 'kill' from "drill and kill" (note that drill in itself doesn't have to be always bad - drill is a tool amongst many), etc.

On this page, Julie has book suggestions to many different situations... Consider finding one or a few (from library or bookstore). Reading a math book that's not a school book can do SO much good! For your kids too!

As you probably know, I wish everyone would get to know some of the interesting, fascinating, fun, curious aspects of mathematics. Or, get to know a bit of math history. You don't get those in school books. Or, finally learn why things work.

These th…

Motivation to study math

I felt very inspired by the story I posted last time, about the 12 kids who wanted to have a class to study arithmetic, and then finished 6 years worth of school math in 20 weeks - meaning they had 20 contact hours with the teacher, and who knows how many hours spent on homework.

They studied the four basic operations, fractions, decimals, percent, and square root. I figure they didn't go into algebra - just arithmetic.

Like it said in the article, the material itself is not incredibly difficult, once your mind has developed to handle these concepts.

The story shows how much motivation (and subsequent hard work) can do. So how could we increase our students' motivation to study math?

I feel it is important to PREVENT the student's feeling of, "I hate math" or "I don't like math" that the traditional math instruction seems to produce. Little kids usually like learning about different things. Somehow we must keep that enthusiasm going strong.

For example:

How far you can go when you're REALLY motivated

Just found this... decided that it's good reading for all. When you have lots of motivation, like these kids did, you can learn and will learn - math or anything else. It may not be easy at all times, but the motivation will make you persistent so you will eventually learn.

A certain arithmetic class at Sudbury Valley School

"Because everyone knows," he answered, "that the subject matter itself isn't that hard. What's hard, virtually impossible, is beating it into the heads of youngsters who hate every step. The only way we have a ghost of a chance is to hammer away at the stuff bit by bit every day for years. Even then it does not work. Most of the sixth graders are mathematical illiterates. Give me a kid who wants to learn the stuff -- well, twenty hours or so makes sense."

The daily grind of math - making connections

I wanted to briefly touch some more on this topic of "grind" or daily grind that learning math might sometimes become that I mentioned briefly last time.

I didn't mean to imply that learning to add, subtract, multiply, and divide whole numbers, decimals, fractions, percents, numbers with exponents, and integers has to be such a boring task. You can avoid that type of feeling.

One way:
Make connections between the concepts. Don't make math appear as fairly separate compartments of "fractions" and "decimals" and "percents" and "geometry".

I sometimes wonder how students feel when every year (on 4th, 5th, 6th, 7th, and 8th grade) they have a chapter on fractions, a chapter on decimals, a chapter on whole numbers, a chapter on geometry... Almost the same thing over and over.

We need to make sure they can see how these things connect.

An example:
A jogging track is 3.5 km long. If you jog 2/5 of it, how far was that? How many percent of t…

Remainder in division

I was updating my Division 1 ebook and thought I'd share a lesson idea.

Have you ever tried this kind of exercise when studying remainder in division?

1 ÷ 3 = 0, R 1

2 ÷ 3 = 0, R 2

3 ÷ 3 = __, R __

4 ÷ 3 = __, R __

5 ÷ 3 = __, R __

6 ÷ 3 = __, R __

7 ÷ 3 = __, R __

8 ÷ 3 = __, R __

9 ÷ 3 = __, R __

10 ÷ 3 = __, R __

11 ÷ 3 = __, R __

12 ÷ 3 = __, R __

13 ÷ 3 = __, R __

14 ÷ 3 = __, R __

15 ÷ 3 = __, R __

16 ÷ 3 = __, R __

17 ÷ 3 = __, R __

18 ÷ 3 = __, R __

19 ÷ 3 = __, R __

20 ÷ 3 = __, R __

21 ÷ 3 = __, R __

22 ÷ 3 = __, R __

23 ÷ 3 = __, R __

24 ÷ 3 = __, R __

25 ÷ 3 = __, R __

26 ÷ 3 = __, R __

27 ÷ 3 = __, R __

You should do it with different divisors, and ask the student(s) to find a pattern. Do you know it?

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