Mathematics blackboard information content

Information content of the mathematics blackboard report

Interesting mathematics knowledge for junior high schools

1. Two boys each ride a bicycle. Two places that are 2o miles (1 mile equal to 1.6093 kilometers) apart start riding towards each other in a straight line. The moment they started, a fly on the handlebar of one bicycle started flying straight towards the other bicycle. As soon as it reached the handlebars of the other bike, it immediately turned and flew back. The fly flew back and forth between the handlebars of the two bicycles until the two bicycles met. If each bicycle moves at a constant speed of 10 miles per hour and the fly flies at a constant speed of 15 miles per hour, how many miles does the fly fly in total?

Answer

Each bicycle is moving at a speed of 10 miles per hour. The two will meet at the midpoint of a distance of 2o miles in 1 hour. The fly flies at a speed of 15 miles per hour, so in one hour it travels a total of 15 miles.

Many people try to solve this problem using complicated methods. They counted the fly's first trip between the handlebars of the two bicycles, then its return trip, and so on, working out those shorter and shorter distances. But this would involve what is called the summation of infinite series, which is very complex advanced mathematics. It is said that at a cocktail party, someone asked this question to John von Neumann (John von Neumann, 1903-1957, one of the greatest mathematicians of the 20th century.), and he gave the correct answer after thinking for a moment. The questioner looked a little frustrated and explained that most mathematicians always ignored the simple method of solving this problem and resorted to the complicated method of summing infinite series.

Von Neumann had a look of surprise on his face. "But, I use the method of summing infinite series." He explained.

2. There was a fisherman, wearing a big straw hat, sitting on a rowing boat and fishing in a river. The river was moving at 3 miles per hour, and his rowboat was moving down the river at the same speed. "I'll have to row a few miles upstream," he said to himself, "the fish won't take the bait here!"

As he started rowing upstream, a gust of wind knocked his straw hat off his head. Blown into the water next to the boat. However, our fisherman did not notice that his straw hat was missing and continued to paddle upstream. He didn't realize this until he rowed five miles away from the Straw Hat. So he immediately turned the bow of the boat and rowed downstream, finally catching up with his straw hat floating in the water.

In still water, a fisherman always rows at a speed of 5 miles per hour. He maintained this speed as he rowed upstream or downstream. Of course, it's not his speed relative to the bank. For example, when he paddles upstream at 5 miles per hour, the river is dragging him downstream at 3 miles per hour, so his speed relative to the bank is only 2 miles per hour; As he paddles downstream, his paddling speed and the current of the river will work together so that his speed relative to the bank is 8 miles per hour.

If the fisherman lost his straw hat at 2 p.m., when did he find it?

Answer

Since the flow speed of the river water has the same impact on the rowing boat and the straw hat, the flow speed of the river water can be completely ignored when solving this interesting problem. Although the river is flowing and its banks remain stationary, we can imagine that the river is completely still and its banks are moving. As far as we are concerned with rowboats and straw hats, this assumption is exactly the same as the above situation.

Since the fisherman rowed five miles after leaving the straw hat, of course he rowed back another five miles and returned to the straw hat. Therefore, relative to the water of the river, he rowed a total of 10 miles. The fisherman was rowing at a speed of 5 miles per hour relative to the water, so it must have taken him a total of 2 hours to row the 10 miles. So, he retrieved his straw hat that had fallen into the water at 4 p.m.

This situation is similar to calculating the speed and distance of objects on the earth's surface.

Although the Earth rotates through space, this motion has the same effect on all objects on its surface, so for most problems of speed and distance, this motion of the Earth can be completely ignored.

3. A plane flies from city a to city b, and then returns to city a. In calm conditions, its average ground speed (speed relative to the ground) for the entire round trip was 100 miles per hour. Suppose there is a continuous strong wind blowing in a straight direction from city a to city b. If the engine speed is exactly the same throughout the round trip, what effect will this wind have on the average ground speed of the round trip?

Mr. White argued: "This wind will not affect the average ground speed at all. When the plane flies from city A to city B, the strong wind will accelerate the speed of the plane, but during the return process "The wind will slow down the plane by an equal amount," Mr. Brown agreed. "But if the wind is 100 miles per hour, the plane will fly from City A at 200 miles per hour. City b, but its speed when returning will be zero! The plane cannot fly back at all!" Can you explain this seemingly contradictory phenomenon?

Answer

Mr. White said that the wind increases the speed of the aircraft in one direction by the same amount as it decreases the speed of the aircraft in the other direction. That's right. However, he was wrong when he said that the wind had no effect on the average ground speed of the aircraft during the entire round-trip flight.

Mr. White's mistake was that he failed to consider the time it took the aircraft to travel at these two speeds.

The return flight against the wind takes much longer than the outbound flight with the tailwind. As a result, the groundspeed-reduced flight takes more time, so the average groundspeed round trip is lower than when there is no wind.

The stronger the wind, the more the average ground speed decreases. When the wind speed equals or exceeds the speed of the aircraft, the average ground speed for a round-trip flight becomes zero because the aircraft cannot fly back.

4. "Sun Zi Suan Jing" is one of the famous "Ten Books of Suan Jing" that was used as a "numeracy" textbook in the early Tang Dynasty. It consists of three volumes. The first volume describes the system and counting system of arithmetic. The rules of multiplication and division, and the middle volume illustrates the calculation of fractions and the square root method with examples, which are all important materials for understanding calculations in ancient China. The second volume collects some arithmetic puzzles, one of which is the "chicken and rabbit in the same cage" problem. The original title is as follows: There are pheasants (chickens) and rabbits in a cage with thirty-five heads on top and ninety-four legs on the bottom.

Ask the geometry of the male and rabbit?

The solution in the original book is; suppose the head number is a and the foot number is b. Then b/2-a is the number of rabbits, and a-(b/2-a) is the number of pheasants. This solution is indeed wonderful. The original book probably used the equation method when solving this problem.

Suppose x is the number of pheasants and y is the number of rabbits, then we have

x+y=b, 2x+4y=a

Solution

Y=b/2-a,

x=a-(b/2-a)

Based on this set of formulas, it is easy to get the answer to the original question: 12 rabbits , 22 pheasants.

Mathematics Quotes

NO1. Learn mathematics as a language, learn the usage of every term, and be familiar with the meaning of every symbol.

NO2. Read "Mathematics Forms Thoughts", don't read "Mathematics Becomes the Form of Death".

NO3. Read "Language in Mathematics" and "Patterns in Mathematics (Question Types)".

NO4. Don’t miss any seemingly simple examples - they are often not that simple, or they can lead to a lot of knowledge points.

NO5. Being able to use mathematical formulas does not mean that you can do mathematics.

NO6. If you are not a genius, if you want to learn mathematics, don’t think about playing games - you think you have done it, but in fact your mathematics level has not improved together with your ability to pass levels - in fact, you can always Remember: Learning mathematics makes you better at playing the big game of "life"!

NO7. Impetuous people tend to say: Learning mathematics is useless, you should learn something useful; - it's you Is it useless!?

NO8. Impetuous people tend to ask: How should I learn? - Don’t ask, just learn.

NO9. Impetuous people tend to ask: Is it better to write down the teacher’s notes in class or follow the teacher’s thinking without taking notes? - Let me tell you, both are fine - as long as you learn.

NO10 There are two types of impetuous people: a) those who just watch without learning; b) those who only learn without persisting.

NO11 Please don’t be an impetuous person.

NO12 It is better to keep conventional problem-solving methods in mind than to talk about novel problem-solving methods.

NO13 Mathematics is not just about solving problems.

NO14 One of the best ways to learn to solve problems is to study examples.

NO15 Never think that you have solved enough problems at any time.

NO16 Please read the "Mathematics Textbook" to master the standard terms of mathematics.

NO17 Please read the examples carefully if you understand them; if you don’t understand the examples, please read them hard.

NO18. Don’t expect to remember and grasp anything after reading the book for the first time - please read it a second or third time.

NO19. Don’t stay at the cradle of basic question types. Learn to treat basic question types as comprehensive questions “assembled” from parts.

NO20. Don’t think that just because some words in mathematics look the same as words in natural language, they have exactly the same meaning.

NO21. The secret to learning mathematics is: solve problems, solve problems, and solve problems again.

NO22. Remember: the concepts and objects in mathematics are not exclusive to mathematics. Don’t forget to “use mathematics” in other subjects.

NO23. Please do the examples in the book yourself.

NO24. Please find some exercises and use the problem-solving methods you learned in the book!

NO25. Please pay attention to the detailed errors in problem-solving and remind yourself before the exam .

NO26. Always review the problems you have solved before, try new solutions, and apply the new knowledge you have learned.

NO27. Don’t miss any exercises in the book - please finish them all and record your ideas for solving the problems.

NO28. When you are halfway through a problem-solving idea but find that your method is clumsy, please do not throw it away immediately. At least come back after you have solved the problem with a new and better method. Reanalyze the previous ideas.

NO29. Never fail to follow some problem-solving rules that you are not proficient in just because the question is "small" - good habits are cultivated, not memorized once.

NO30. Whenever you learn a difficult mathematical point, try to explain this knowledge point to others and let them understand it - only if you can explain it clearly will you truly understand it.

NO31. Save all the exercises you have solved - that is one of your best accumulations.

NO32. Please love mathematics!