Rabu, 18 Juni 2008
One equal to one half
Theorem: 1 = 1/2:
Proof:
We can re-write the infinite series 1/(1*3) + 1/(3*5) + 1/(5*7) + 1/(7*9)
+...
as 1/2((1/1 - 1/3) + (1/3 - 1/5) + (1/5 - 1/7) + (1/7 - 1/9) + ... ).
All terms after 1/1 cancel, so that the sum is 1/2.
We can also re-write the series as (1/1 - 2/3) + (2/3 - 3/5) + (3/5 - 4/7)
+ (4/7 - 5/9) + ...
All terms after 1/1 cancel, so that the sum is 1.
Thus 1/2 = 1.
Proof:
We can re-write the infinite series 1/(1*3) + 1/(3*5) + 1/(5*7) + 1/(7*9)
+...
as 1/2((1/1 - 1/3) + (1/3 - 1/5) + (1/5 - 1/7) + (1/7 - 1/9) + ... ).
All terms after 1/1 cancel, so that the sum is 1/2.
We can also re-write the series as (1/1 - 2/3) + (2/3 - 3/5) + (3/5 - 4/7)
+ (4/7 - 5/9) + ...
All terms after 1/1 cancel, so that the sum is 1.
Thus 1/2 = 1.
Teachers Need More Knowledge of How Children Learn Mathematics
by Constance Kamii
*in indonesia often happened
Teachers need as much scientific knowledge about how children learn mathematics as physicians have about the causes of illnesses. Because of this need, teacher-preparation programs must change. Specific examples from classrooms illustrate this need.
I once wondered why some first graders were getting such answers as 3 + 4 = 4 (*1+1=11). By watching them, I found out that they were putting three counters out for the first addend and then four for the second addend, including the three that were already out.
Errors of this kind result from prematurely teaching a rule to follow. According to this rule, one must put counters out for the first addend, more counters for the second addend, and count all of them to get the answer. This rule works for children who already know that addition is the joining of two sets that are disjoint. However, the rule is superfluous for those who have constructed this logic, and it causes errors for those who have not constructed it.
Another example of imposing a rule that is either superfluous or premature is teaching counting-on to children who are counting-all. Counting-all refers to solving 3 + 4 by counting out three counters, then four other counters, and counting all of them again ("one-two-three-four-five-six-seven"). In counting-on, by contrast, children say "four-five-six-seven."
With scientific research replicated worldwide, Piaget showed that all children construct, or create, logic and number concepts from within rather than learn them by internalization from the environment (Piaget 1971; Piaget and Szeminska 1965; Inhelder and Piaget 1964; and Kamii 2000). Studying the research leads teachers to understand that addition involves part-whole relationships, which are very hard for children to make and which cannot be taught through practice and memorization. To add two numbers, children must put two wholes together ("three" and "four," for example) to make a higher-order whole ("seven") in which the previous wholes become two parts. When young children cannot think simultaneously about a whole and two parts, they count-all by changing both the "three" and the "four" into ones. Making them count-on is harmful when they cannot mentally make the part-whole relationship necessary to count-on.
When teachers study Piaget's theory and replicate the aforementioned research, they can understand why some first graders cannot count-on. When children have constructed their logic sufficiently to make the part-whole relationship of counting-on, they give up counting-all, just as babies give up crawling when they can walk. I hope that the day will come when teachers entering the classroom and those already in the classroom have as much scientific knowledge about how children learn mathematics as physicians have about the causes of illnesses. To reach this vision, the teacher-preparation programs must change.
References
Inhelder, Barbel, and Jean Piaget. The Early Growth of Logic in the Child. New York: Harper & Row, 1964.
Kamii, Constance. Young Children Reinvent Arithmetic. 2nd ed. New York: Teachers College Press, 2000.
Piaget, Jean. Biology and Knowledge. Chicago: University of Chicago Press, 1971.
Piaget, Jean, and Alina Szeminska. The Child's Conception of Number. New York: W. W. Norton & Co., 1965.
*in indonesia often happened
How to calculate square roots without a calculator
Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …
The square root of a number, n, written
is the number that gives n when multiplied by itself. For example,
because 10 x 10 = 100
Examples
Here are the square roots of all the perfect squares from 1 to 100.
Finding square roots of of numbers that aren't perfect squares without a calculator
1. Estimate - first, get as close as you can by finding two perfect square roots your number is between.
2. Divide - divide your number by one of those square roots.
3. Average - take the average of the result of step 2 and the root.
4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.
Example: Calculate the square root of 10 (
) to 2 decimal places.
1. Find the two perfect square numbers it lies between.
Solution:
32 = 9 and 42 = 16, so lies between 3 and 4.
2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer)
3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667
Repeat step 2: 10/3.1667 = 3.1579
Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623
Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001
If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.
Note: There are a number of ways to calculate square roots without a calculator. This is only one of them.
The Advantages of Organic Food
You Are What You Eat
Do you really know what goes into your food? Discover the advantages of organic food on this site and see exactly what producers have been adding to your fruit and vegetables to make it less healthy than a few years ago.
In the rush to produce more and more crops to satisfy growing demand producers have had to resort to using a lethal cocktail of pesticides to control disease and insect attack.
Good news for their bank balances perhaps but not good news for your health, this is why you need to be informed of the advantages of organic food.
Did you know that if you consumed an average apple you would be eating over 30 pesticides, even after you have washed it?
The quality of food has definitely gone down since the second world war. For instance, the levels of vitamin C in today's fruit bear no resemblance to the levels found in wartime fruit.
Organic food is known to contain 50% more nutrients, minerals and vitamins than produce that has been intensively farmed. Read more about this here.
You will have to eat more fruit nowadays to make up the deficiency, but unfortunately that means eating more chemicals, more detrimental affects on your health eating something that should be good for you!
Also don't forget about the cocktail of anti-biotics and hormones that cattle and poultry are force fed.
What happens to those chemicals when the animal dies?
Digested and stored in human bodies is the answer, have you seen pictures of animals in severly cramped conditions in battery farms?
It just does not make sense to state that any animal kept in these conditions is healthy and produces high quality food.
If you are as worried as I am about the health of your family then you need to read the articles on this and seriously consider converting your family to the organic lifestyle with the organic food information you are going to learn on this site.
Trust me, once you try some organic produce and taste an apple the way it should be, and perhaps how you recall it tasting in your youth, you will never go back to mass produced fruit again.
Sure there are issues with availability and cost but with a bit of research you should be able to find local stores who stock organic produce.
Also, don't forget about your local farmer, I'm sure you will be able to find one that has seen the light and opened up a farm shop to supply local residents.
You should be able to get some very keen prices from these shops, why not take a look around and see who is offering produce in your area?
Some more startling facts now. Pesticides in food have been linked to many diseases including:
Cancer
Obesity
Altzheimer's
Some birth defects
Not a nice list is it? There are probably others but if you think about it, how can it be okay for you to eat chemicals and not expect some form of reaction in your body. Our bodies are delicately balanced wonderful machines. Any form of foreign chemical is bound to cause irritation at the least.
Please take advantage of the organic food articles and information on this site and do consider taking a closer look at what you are eating. It's for your health after all!
Virginia Louise
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