We split in half
2019. is not a prime number. The sum of the digits is 2 + 0 + 1 + 9 = 12, which means that the number is divisible by 3. A prime number will have to wait a long time, until 2027. Yet very few readers of this episode will live into the twenty-second century. But they certainly are like that in this world, especially the fair sex. I'm jealous? Not really... But I have to write about mathematics. Lately, I've been writing more and more about primary education.
Can a circle be divided into two equal halves? Definitely. What are the names of the parts you will receive? Yes, half circle. When dividing a circle with one line (one cut), is it necessary to draw a line through the center of the circle? Yes. Or maybe not? Remember that this is one cut, one straight line.
Justify your faith. And what does "justify" mean? Mathematical proof is different from "proof" in the legal sense. The lawyer must convince the judge and thus force the Supreme Court to find that the client is innocent. It has always been unacceptable for me: how much the fate of the defendant depends on the eloquence of the “parrot” (this is how we characterize the lawyer a little disparagingly). Are you convinced that every straight line passing through the center of the circle divides them into equal parts? Are you convinced that in order to divide the circle into equal parts of one straight line, you need to draw it through the center?
For a mathematician, faith alone is not enough. The proof must be formal, and the thesis must be the last formula in the logical sequence from the assumption. This is a rather complex concept, which is almost impossible to implement in everyday life. Perhaps this is true: lawsuits and sentences based on "mathematical logic" would be just ... soulless. Apparently, this is happening more and more often. But all I want is math.
Even in mathematics, formal proof of simple things can be problematic. How to prove both these beliefs about dividing the circle? Simpler than the first is that each line passing through the center divides the circle into two equal parts. You can say this: let's flip the figure from fig. 1 180 degrees. Then the green box will turn blue and the blue box will turn green. Therefore, they must have equal squares. If you draw a line not through the center, then one of the fields will be clearly smaller.
Triangles and squares
So let's get on square. Do we have the same as:
- each line passing through the center of the square divides it into two equal parts?
- If a straight line divides a square into two equal parts, should it pass through the center of the square?
Are we sure of this? The situation is different than for the wheel (2-7).
let's go to equilateral triangle. How do you cut it in half? Easy - just cut off the top and perpendicular to the base (8). I remind you that the base of a triangle can be any of its sides, even the inclined ones. The cut passes through the center of the triangle. Does any line passing through the center of a triangle bisect it?
Not! look at fig. 9. Each of the colored triangles has the same area (why?), so the top of the big triangle has four parts, and the bottom part has five. The ratio of fields is not 1:1, but 4:5.
What if we divide the base into, say, four parts, and divide the equilateral triangle with a cut through the center and a point at one quarter of the base? Reader, you see that fig. 10 the area of the "turquoise" triangle is 9/20 of the area of the entire triangle? You can not see? Too bad, I'll leave that to you to decide.
First question - explain how it is: I divide the base into four equal parts, draw a straight line through the division point and the center of the triangle, and on the opposite side I get a strange division, in a ratio of 2: 3? Why? can you calculate it?
Or maybe you, Reader, are a high school graduate this year? If yes, then determine at what position of the rows the ratio of fields is minimal? You do not know? I'm not saying that you should fix it right now. I give you two hours.
If you don't solve it, then... well, good luck with your high school finals anyway. I will return to this topic.
Wake up independence
- Can you be surprised? This is the title of a book published a long time ago by Delta, a monthly mathematical, physical and astronomical journal. Take a look at the world around you. Why are there rivers with a sandy bottom (after all, the water should be immediately absorbed!). Why do clouds float through the air? Why is the plane flying? (should fall immediately). Why is it sometimes warmer in the mountains at the peaks than in the valleys? Why is the sun in the north at noon in the southern hemisphere? Why is the sum of the squares of the hypotenuse equal to the square of the hypotenuse? Why does the body seem to lose weight when immersed in water, since it displaces water?
Questions, questions, questions. Not all of them are immediately applicable to everyday life, but sooner or later they will be. Do you realize the importance of the last question (about water displaced by a submerged body)? Realizing this, the elderly gentleman ran naked around the city and shouted: "Eureka, I found it!" He not only discovered the physical law, but also proved that the jeweler of King Heron was a counterfeiter!!! See details in the depths of the Internet.
Now let's look at other shapes.
Hexagon (11-14). Does any line passing through its center bisect it? Should the line that bisects the hexagon go through its center?
What about pentagon (15, 16); Octagon (17)? And for ellipses (18)?
One of the shortcomings of school science is that we teach "in the nineteenth century" - we give students a problem and expect them to solve it. What's bad about it? Nothing - except that in a few years our student will have to not only respond to commands that he “received” from someone, but also see problems, formulate tasks, navigate in an area where no one has yet reached.
I am so old that I dream of such stability: "Study, John, make shoes, and you will work as a shoemaker for the rest of your life." Education as a transition to the highest caste. Interest for the rest of your life.
But I'm so "modern" that I know that I have to prepare my students for professions that ... don't exist yet. The best thing I can and can do is show students: WILL YOU CHANGE YOURSELF? Even at the level of elementary mathematics.
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