Science,English,and math 4th

【A way of changing both of our hands into an electronic calculator】

《What should we do so as to decrease the people who think they are poor at math?》

But it is certain that students who are good at an applied question in which they need an ability of thinking have an ability of calculation more than ordinary people.

The author said he had never encountered any student who was poor at calculation but had a deep mathematical insight.

He hoped that the students should come to grip with a complicated calculation at least during a student of elementary school, and think it important for the...

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【A way of changing both of our hands into an electronic calculator】

《What should we do in order to decrease the people who think they are poor at math?》

...and thinks it important that they have a feeling for something complicated on the calculation, for the feeling makes them look for a device on the calculation, or characteristic number makes them feel easy when calculating.

After that they will be fond of numbers gradually and be good at numbers in the future.

It’s all right whether they learn the multiplication table by heart or use the list of multiplication table. It’s all right...

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【A way of changing both of our hands into an electronic calculator】

《What should we do in order to decrease the people who think they are poor at math?》

It’s all right either of them, the author said.

At any rate, he thinks students should handle lots of kind of numbers with their own hands at least in elementary school and should go through being familiar with numbers as more as possible then.

Without having a sensation of being familiar with numbers, numerical equation which is expressed with an alphabet which the students encounter in junior high school, they find it cold and distant.

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【A way of changing both of our hands into an electronic calculator】

《What should we do in order to decrease the people who think they are poor at math?》

It doesn’t always mean that the higher ability on calculation they have, the easier they feel when encountering a difficult question in which abstract thought is necessary, so the education on math is hard.

But if being familiar with numbers through the calculation, we should be able to have an image easily on numerical formula which is abstract and is expressed with alphabetical way. Then they won’t be the one who think they are poor at..

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【A way of changing both of our hands into an electronic calculator】

《What should we do so as to decrease the ones who think they are poor at math?》

Then they won’t be the ones who think they are poor at numerical formula at least, will they? The author said like that.

As for me, it’s too late?

Though this title has math, but to tell the truth, when expressing on math, I did it reluctantly, for math is too hard to understand, but once having its title, I hated to give it up, so I continued until now.

But I’m interested in math more than before due to this book. Too late? It’s all right.

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as area》

Can you calculate 16×13 mentally at once? It seems that they make children learn the multiplication table until 19×19 during elementary school in India, but we the Japanese do until 9×9, so it doesn’t always mean that lots of us can calculate the multiplication 16×13 at once, but if thinking it over with an illustration, we can do it immediately, the author said like that, and there is an illustration like the next in the book.

There is a rectangle of which vertical side is 13 and horizontal...

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【Calculating multiplication on two digit mentally at once】

《Thinking of the multiplication as the area of a rectangle》

The author said he wants us the readers to think of 16×13 as the area of a rectangle, and there is a figure on a rectangle of which horizontal length is 16 and of which vertical length is 13, and the rectangle is separated into a regular square of its side is 10 and three rectangles. How is situated the square and three other rectangles in the rectangle of 16×13？

The horizontal line on the rectangle of 16×13 is divided into 10 and 6, and there is a vertical line from the...

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

There is a rectangle of which horizontal length is 16 and a vertical line is drawn from the end of the regular square to the base of the rectangle of 16×13, as a result there is other rectangle of which horizontally length is 6 and vertical length is 13 next to the regular square.

The rectangle has the vertical line of which length is 19, and there is a horizontal line from the end of the regular square, and the horizontal line is parallel to the both sides of the rectangle.

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

I recommend you should write the figures actually.

As a result, the horizontal line starts from the end of the regular square in the rectangle of which vertical line is 13 and horizontal line is 19, there is another rectangle of which horizontal line is 19 and of which vertical line is three under the regular square.

Exactly there is two kinds of rectangle under the regular square. The one is just under the right square and the other is lower right from the regular square.

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

In other words, the vertical line from the regular square is expanded to the base of the rectangle which includes the regular square, so there is two kinds of rectangle. The one is just under the regular square, and the other is lower right from the regular square.

We cut out of the rectangle just under the regular square of which horizontal line is 10 and vertical line is 3, and turn its direction 90 degree, add it to right next to the figure which used to be the rectangle.

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

Though the rectangle just under the right square was cut out, so the rectangle of which horizontal length is 16 and vertical length is 13 doesn’t exist any more, we add the rectangle which is just under the right square to the right side of the figure which used to be the rectangle.

Its shape is like the next.

There is a new rectangle of which horizontal length is 19 and of which vertical length is 10, and there is other rectangle...

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

There is a new rectangle of which horizontal length is 19 and of which vertical length is 10 and there is other rectangle of which horizontal length is 6 and vertical length is 3.

Then the whole area on the figure is addition of the new rectangle of which horizontal length is 19 and of which vertical length is 10, 19×10=190and other that of the rectangle under the new one of which horizontal length is 6 and of which vertical length is 3, 6×3=18, 190＋18=208.

But as the ...

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

But as it’s just that the rectangle under the regular square was moved to the upper right, so the whole area is the same between before the rectangle is moved and after the rectangle is done, as a result, 16×13=190＋18＝208.

It means that （10＋a）×（10＋b）＝（10＋a＋b）×10＋ab.

If calculating, 12×14＝（12＋4）×10＋2×4＝168

In short, a multiplication on two digit until 19×19 is done like the next.

1 Adding the number in the first digit to the other number.

2 Ten times the value from the 1

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

3 Adding the product 積 by the number on the first digit.

We can calculate it easily with the order. We’ll be able to learn to calculate it mentally at once as long as we train a little, so let’s try, said the author like that.

If thinking of the multiplication as an area, it’s useful in various scenes.

For example dividing distance by speed is time, dividing total number by the number of thing is average, or dividing a portion by the whole is a proportion, there are ...

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

...there are lots of people who are poor at those formulas, but if changing into them in other form like, distance is multiplied speed by time, amount of numbers is multiplied a number of things by average, a portion is multiplied the whole by the proportion, and if using each of area and illustrating them, it is easy to understand.

So there are three kinds of illustration in the book. They are three kinds of rectangle, and each of the rectangles is shown as formula.

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of a rectangle》

In the illustration, the distance is the area of the rectangle and its vertical line is speed and horizontal line is time. The part which we want to ask is shown with oblique line, and there is other illustration on the distance.

The oblique line is cut out and added to under the rectangle or right side. The oblique line is what we want to ask, so intuition told us how we should use the horizontal and vertical line and calculate.

Other amount of number and the portion is...

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【Calculating multiplication on two digit mentally at once】

《Thinking of multiplication as the area of rectangle》

The amount of number and the portion is shown with the same way.

Needless to say, not only learning a formula by heart, but we should understand why it is done enough, but if there is the one who is poor at changing the numerical formula into other form, the way of thinking is useful, the author said like that.

《A way of solving what is called 鶴亀算》

There is a question like the next.

There are cranes and tortoises and their total number is 10. The total number of two kinds...

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【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

The total number of two kinds of legs are 26. As to the number of the cranes and tortoises, how many are there?

As to the 鶴亀算, if thinking of multiplication as the area of a rectangle, we can show it with an illustration.

The number of crane on leg is two, and the leg of tortoise is four, so the total number of legs on two kinds of creature is 2 × number of crane ＋ 4 × number of tortoise= 26, according to the question.

We show 2 × number of crane and ...

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【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

We show 2 × number of crane, and 4 × number of tortoise as the area of each rectangle which is different shape each other.

To my sorrow, I’m not good at showing the illustration in English alone, so I’m afraid it is hard to understand. I’m sorry for it.

There is a rectangle on the crane. Its vertical line is two, for the number of leg on a single crane is two, and its horizontal line is number of the cranes, but its number isn’t written yet. It’s expanded horizontally.

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【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

As to the tortoise, there is also a rectangle. Its vertical line is four, for the number of the leg on a single tortoise is 4 and its horizontal line on the number isn’t written as well.

While the rectangle on the crane is expanded horizontally, the one on the tortoise is both short on vertical and horizontal line, though the vertical line on the tortoise is longer than the one on the crane.

The rectangle on tortoise is right side to the one on the crane, so the shape which two rectangle is...

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【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

The shape of which two different shape of rectangles is added is uneven, it resembles L which turns its direction 90 degrees to left, so the author suggested we add other shape to the uneven rectangle and changes it into other rectangle which isn’t uneven.

The new rectangle on the vertical line is 4 and on the horizontal line is 10. As to the horizontal line is total number of the crane and tortoise.

The area on the new rectangle is 40. It means that an additional rectangle is added to the...

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【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

It means that an additional rectangle is added to the uneven shape and the new rectangle which isn’t uneven is completed, so the area on new rectangle is 4 ×10= 40, so the area on the additional rectangle is 40−26=14

As to the additional rectangle, its vertical line is 4−2=2

4 is the number of tortoise on the leg, and 2 is on the crane.

The horizontal line on the additional rectangle is the number of the crane.

2 × the number of the crane is equal to the area on the additional rectangle...

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【Calculating multiplication on two digit mentally at once】

《A way of solving what is called 鶴亀算》

...so dividing 14 by 2 is equal to 7, as a result it turned out that the number of the crane is 7, and the tortoise is 3.

Needless to say, without adopting the way, we can solve it as a question on simultaneous equations 連立方程式, but before establishing the way of solving an equation, in general we used to adopt the figure and solve the question.

By the way, the one who made the question on 鶴亀算 knew the each of total number of the tortoise and crane on their legs before the total number of...

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The one who made the question on the 鶴亀算 knew each of the total number of tortoise and crane on the leg before each of the total number on two of them, I’m wondering.

Counting their legs is harder than counting their number, I’m sure.

【Calculating multiplication on two digit mentally at once】

《From a linear equation to a quadratic equation》

With the same way, we can solve a question on a quadratic equation 二次方程式. As an example, let’s think over x squared ＋10x −75＝0. It’s x squared ＋2x＝75, and we think of x squared as the area of a regular square, and 10x as the area of a rectangle.

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【Calculating multiplication on two digit number mentally at once】

《From a linear to a quadratic equation》

As I’m not good at expressing on figure in English, so I recommend you actually write.

There is an illustration of the square of x squared and the rectangle of 10x which is added, and divided the rectangle of which area is 10x into two, and move the one of half of the rectangle to lower left under the square of which area is x squared.

Then adding a new regular square to one of half of the rectangle so as to complete a perfect regular square. The side of the new regular square is 5...

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【Calculating multiplication on two digit mentally at once】

《From a linear to quadratic equation》

As the new square on the area is 25, for its side is 5, and the new perfect regular square on the area is 25＋75 =100, so we can find the value on x.

By the way, immediately after when reading my English, I can understand, but after several says, when reading, I find it hard to understand, even if it’s my own sentence, so if others try to understand it, I’m afraid it’s very hard to understand. I’m sorry for it.

I have to work harder, but without showing illustration, it’s very hard to do.

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【When were the marks like ＋,−,×,÷ were born?】

《Origin on the mark on calculation which we don’t know unexpectedly》

The mark on calculation like ＋,−, ×, ÷, which we usually use every day life. Do you know the time when they began to use?

To tell the truth, the history on the marks isn’t so old. ＋ and − are at the end of 15th century, and × and ÷ are in the 17th century.

About 500 years ago from now on, Europe was in Age of Great Navigation, and commercial activity by ship was very active then.

As there was no radar, they had to do astronomical observation and to calculate a line for...

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【When were the marks like ＋,−,x,÷, born?】

《Origin of the marks on calculation like ＋,−,x,÷, which we don’t know unexpectedly》

As there was no radar then, they needed to do astronomical observation and to calculate a line for ship so as to sail farther safely. They had to calculate astronomical number literally.

They’d wanted to calculate a number which was on a grand scale easily, so the marks on calculating was born.

《＋and −》

As to ＋ and −, some people say both of them were born when taking down in shorthand, and it seems to be reliable.

It is said ＋ is originated from et which was...

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【When were the marks like ＋,−, ×,÷, born?】

《＋ and −》

It is said it's originated from et which was a Latin language and it means and , the and changed into ＋ gradually for a long long time.

On the other hand, − is m which is an initial on minus was abbreviated with a longhand 筆記体, so the mark on − has started.

There is other origin. It is said it was used by sailors for the first time. Sailors prepared for barrels in the ship and there were some water in barrels.

When using the water from the barrel, they drew the mark, −, on the barrel, and when adding water to the barrel, they drew a...

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【When were the marks like ＋,−, ×, ÷, born?】

《＋ and −》

...when adding water to the barrel, they drew a vertical line on the horizontal line, ＋. The one which was used when decreasing, −, and the other which was used when adding, ＋, two of them become the marks when adding and subtracting, it’s the origin from the sailors.

《×》

As to the × and ÷, the one who used them for the first time has been clear. As to the x,it was an English mathematician, William Oughtred. He used the × in his book for the first time in 1631, and it’s the origin on ×.

But its origin on the shape, there are some....

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【When were the marks, like ＋,−, ×, ÷, born?】

《×》

But its origin on the shape, there are some opinions.

The one is a cross on the Christianity. When the cross is inclined, it changes into x.

The other is it was taken from the national flag on Scotland.

By the way, William Oughtred is the one who used sin in trigonometric function 三角関数 for the first time.

Multiplication is expressed with other mark, • as well. In fact, x which is expressed with multiplication wasn’t popular among in European Continent so much.

A German mathematician, Leibniz used to say like the next.

He didn’t like....

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【When were the marks like ＋,−, ×, ÷, born?】

《×》

He didn’t like x as mark for multiplication, for it is easy to mistake it for the one in alphabet. When expressing a multiplication, he used • between two numbers.

It seemed that the way of thinking was the main stream then.

After a typewriter and a personal computer have been spread out among the people, the x has not learned to use when calculating on multiplication, especially in alphanumeric half-size font character, it’s misleading.

When calculating on multiplication with a keyboard for a personal computer at present, there is no...

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【When were the marks like ＋,−,×,÷, born?】

《×》

...there is no keyboard for x on multiplication. When calculating on multiplication with an excel, ＊ is used.

《÷》

A mathematician in Swiss used ÷ in his book in 1659 for the first time. A number which is expressed with a fractal number changed into ÷ during a long time.

When expressing a division, ：or ／ are used. ／　has an older history than ÷ and it has been used all over the world at present.

As to the ：, it is said Leibniz used it as a mark for division for the first time. ：has been used as the mark for the division in Germany and...

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【When were the marks like ＋, −, ×, ÷ born?】

《÷》
　
While ：has been used as a mark for division in Germany or in France at present, it’s usually used as a mark for proportion in other countries.

In fact there aren’t so many countries where ÷ is generally used. It’s limited in a part of the world like England, America, Japan, Thailand or Korea. In general ／ has been used in other countries.

In ISO it is described when expressing the division, it should be expressed with ／ or a fractal number. No one should use ÷ as the mark for the division. It is said clearly.

÷ may be going to disappear...

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【When were the marks like ＋, −, ×, ÷ born?】

《÷》

Possibly ÷ may be going to disappear from textbooks from all over the world in near future.

《A thing about which an almighty genius cared》

What is the first mark in arithmetic in childhood when starting to learn? If being asked like that, then lots of people may have answered it’s ＋ or =, but numbers like 1.2.3... are marks themselves.

An English mathematical philosophy said it needed a numberless years for us the humankind to notice that the number of 2 on February and two birds are the same two.

February is the second month. Two on...

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【When were the marks like ＋, −,×, ÷, born?】

《A thing about which an almighty genius cared》

February is second month. It has number of two. Two in two birds, two in two meters, and two in twenty thousand yen, it means that under the same unit they are the same in relation to its essence, so we express they are two.

In short, getting rid of extra information from each of concrete examples, and we make each of essence abstract.

Whenever a new mark was introduced, making a thing become abstract is done in math. The history on math is the history of making the thing become abstract.

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【When were the marks like ＋,−,x,÷ born?】

《A thing about which an almighty genius cared》

Though numerical formulas, including its numbers, have been expressed with marks alone which is abstract in high level, a part of people hate math so much.

Why has a new mark been sought with a new concept in math?

It has a side of character of wanting to an object on thought to be simple, but its biggest reason is preventing us from making a mistake.

When expressing, Leibniz appeared several times, and he cared about the marks so much.

He has been recognized as an authority who reached to essence ...

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【When were the marks like ＋,−,x, and ÷ born?】

《A thing about which an almighty genius cared》

Leibniz has frequently been recognized as an authority who reached to essence on calculus 微積分, and contending with Newton for its position, but neither he is well known nor is appreciated like Newton at present. We tend to think like that.

But he had been admired as almighty, and a giant on intelligence then so much that his reputation had been well known all over Europe. Actually he was a versatile mathematician and got results which got fame for coming generations in several fields not only in...

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【When were the marks like ＋,−,x, and ÷ born?】

《A thing about which an almighty genius cared》

His achievement reached not only in math but laws, history, logics and philosophy. Leibniz was a marvelous versatile.

He had put his heart and soul into a work from around age of 20 to just before being dead. His work is on marks. A general way of truth on all reason reduced to a single calculation and inventing the marks.

If it comes true, a reasoning in which we need to think it over with high level becomes a simple work like a calculation, and a wrong reasoning couldn’t have been occurred as...

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【When were the marks like ＋,−,x, and ÷ born?】

《A thing about which an almighty genius cared》

...and a wrong reasoning couldn’t have been happened as principle, but unfortunately he was dead before accomplishing his achievement and his work was taken over by other mathematician.

《The work is done so as to understand the world precisely》

The marks on calculus which Leibniz invented is excellent. Its calculation has a background which isn’t easy to understand in mathematical field at all, but everybody can calculate as if it handled a fractal number even if it doesn’t understand its principle

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【When were the marks like ＋,−,x, and ÷ born?】

《The work is done in order to understand the world precisely》

On the other hand, the marks which Newton invented are simple, those marks hardly lead the calculation anywhere, though it’s simple.

An mathematician in England who is well known as the founder on an electronically calculator said the marks which Newton contrived held up the proceeding the math in England by a century.

We use our language every day life and it has a variety of nuance in varied situation. Even if it has the same meaning, it has a different meaning in different scene.

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【When were the marks like ＋,−,x, and ÷ born?】

《The work is done so as to understand the world precisely》

Some of the words are hard to define with regard to simple ones like right and left in the first place.

When using the words and going ahead with a discussion, a misunderstanding absolutely happens. It is possible that a reasoning which has ability of logical thought is led in a wrong way.

On the other hand, the marks have been contrived so as to express mathematical concept alone, we aren’t confused with ambiguous words nor nuance.

It means that we don’t make a mistake as long as...

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【When were the marks like ＋,−,x, and ÷ born?】

《The work is done so as to make out the world exactly》

We don’t make a mistake as long as we understand the definition and rules on the marks, for the marks have been used in math there.

We may have felt the marks which are used in math is something inorganic and cold, but every mark is the fruit of effort and talent by people who had blood and at the same time, these marks are filled with ideal in which they tried to understand the world correctly.

The author said like that.

The math is over for the time being. I’ll start to the next topic.

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【Doubting is start from science】

《Do you know the meaning the probability of rain is 30%?》

I’m going to a new talk, being based on a book, はじめてのサイエンス, written by Akira Ikegami.

When hearing of a word on science, we tend to think we have nothing to do with it, as for me, when being a student, my school record on science was poor, I don’t feel like reading a book on physics and chemistry filled with difficult marks and numerical formulas, besides, I’m not spiritless for doing it. It’s all right as long as we leave it to experts on science.

Some people have may thought so, including me.

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【Doubting is start from science】

《Do you know the meaning that the probability of rain is 30%?》

But we’ve face to the way of watching scientifically every day life.

When going to the place for our job, we check the probability of rain today in weather forecast on the internet or TV, and we decide whether or not we should go with an umbrella.

Then let’s suppose that there was the probability of rain is 30%. As to the 30%, what does it mean?

A high pressure zone and a low pressure zone are written down in a weather chart. When forecasting the weather, the ones who are engaged in the...

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【Doubting is a start from science】

《Do you know the meaning that the probability on rain is 30%?》

When forecasting the weather, the ones who are engaged in the weather forecast searched from a large amount of data.

When being in the same weather chart, how often did it rain in the past? When there were the same weather of hundred of the weather chart and it rained 30th time, it means that the probability of rain is 30% then.

Both of the principle on weather and the concept on the probability are typical scientific way of thinking. In short scientific idea has been rooted deeply in our...

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【Doubting is a start from science】

《Do you know the meaning of the probability on raining is 30%?》

In short science has been rooted deeply in our every day life.

The author said he had never taken a science course in the past, and when looking at a technical book on science, he got cold feet, but he had to be in charge of a caster on the TV program in NHK, 週間こどもニュース, he couldn’t refuse it any more.

Lots of questions on weather were gathered to the 週間こどもニュース. It started from the meaning on the probability of raining to other question, what is the pressure at first?

The author had to...

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【Doubting is a start from science】

《Do you know the meaning that the probability on raining is 30%?》

The author had to clear up the problems and to explain them to children one by one.

When a Japanese accepted Nobel prize on physics or chemistry, he had to comment on the most advanced technology in order to make the children understand.

Not only the weather but every kind of scientific field, he read technical books and listened to experts and tried to understand basic principles and mechanism for himself, and to explain those things to the children with his own words.

Thus drawing...

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【Doubting is a start from science】

《Do you know the meaning that a probability on raining is 30%?》

Thus drawing science close to him, he was aware that scientific way or imagination has been related to not only the weather forecast in the scene of every day life but all kinds of situations from an international situation to the future on Japan closely.

A nuclear weapon has controlled the fate the international situation at present. Without progress on physics, it wouldn’t be developed.

It is said regenerative science 再生医療 decides the future on medical science, but without understanding...

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【Doubting is a start from science】

《Do you know the meaning that the probability on raining is 30%?》

Without knowing biology, we can’t understand the essence on the regenerative medicine.

When thinking over an earthquake just direct under the metropolitan area which breaks out within 30 years with the probability of 70%, the knowledge on earth science is essential for us.

Scientific world has spread out in the background on serious news. It has never nothing to do with us.

We need to understand news which is reported everyday from TV and newspaper enough. We have to understand how...

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