Goals:
develop an interest in mathematics, logic and ingenuity, the ability to prove and explain; communicative competence.
Preparing for the lesson:
tasks for mathematical combat are recorded on album sheets in three copies: for teams and for the teacher.
The course of the lesson:
- Two teams participate in a mathematical battle. Each team has a captain, who is determined by the team before the start of the battle. The fight consists of two stages. The first stage is solving problems, the second is the battle itself. During the first stage, problem solving can be done jointly by the whole team. Remember that none of the participants in the battle can go to the board more than twice. Therefore, a participant who has solved many problems that have not been solved by others must, during the first stage, tell his teammates about his solutions.
- The second stage begins with the competition of captains. By decision of the team, any member of the team can participate in the competition instead of the captain. The winning team decides which team makes the first challenge. This, like all other team decisions, is announced by the captain.
Captains Competition:
A super blitz is held on three questions, the captain who scores two or three points wins. A point can be earned by the captain by answering the question correctly. The first person to answer is the one who quickly raises the signal card (prepared in advance) or his hand.
- A chocolate bar costs 10 rubles and another half a chocolate bar. How much does a chocolate bar cost?
- Hares are sawing logs. They made 10 cuts, how many logs did they get?
- How much earth is in a hole 2 m deep, 2 m wide, 2 m long?
Answers: 20 rubles; 11 logs; not at all.
- The call is made as follows. The captain announces: “We challenge the opponents to task number…”. The other team may or may not accept the challenge. The team that accepted the challenge puts speaker, another command - opponent. After a meeting with the teams, the captains call the opponent and the speaker. The speaker's task is to give a clear and understandable solution to the problem. The task of the opponent is to find errors in the report. During the presentation, the opponent does not have the right to object to the speaker, but may ask him to repeat an unclear place. The main task of the opponent is to notice all the dubious places and not forget about them until the end of the report. At the end of the report, a discussion takes place between the speaker and the opponent. , during which the opponent asks questions on all obscure places in the report. The discussion ends with the conclusion of the opponent: “I agree with the decision” or “I think that there is no solution, since so-and-so was not explained.”
- After that, the jury (teacher) awards points according to the following rules. Each task is worth a different number of points, as different levels of difficulty. The first and second tasks - 6 points. Third, fourth, fifth and sixth - 8 points. Seventh and eighth - 10 points. Ninth and tenth - 12 points. In the case of an absolutely correct decision, all these points are received by the speaker's team. Points are deducted for errors and inaccuracies. The number of points taken is determined by the proximity of the story to the correct solution. If the errors were found by the opponent, then the opposing team receives up to half of the deducted points. Otherwise, all selected points go to the jury. If the jury decided that the report does not contain a solution to the problem, then the opposing team has the right to tell the correct solution. At the same time, to the points scored for opposing, she can add points for telling the solution to the problem. The team that made an incorrect report puts up an opponent and can earn points on the opposition.
- The team that received the call may refuse to report. In this case, the calling team must prove that it has a solution to the problem. To do this, she exposes the speaker, and the second team - the opponent. If there is no solution and this is proved by the opposing team, then they receive half the points of this problem, and the calling team is obliged to repeat the challenge. This procedure is called call validation. In all other cases, the calls are interleaved.
- During a bout, each team is entitled to six 30-second breaks. Breaks are made in cases where it became necessary to help a standing student at the blackboard or replace him. The decision to take a break is made by the captain.
- The team that has received the right to challenge may refuse it. In this case, until the end of the battle, only their opponents have the right to report, and the team that refused can only oppose. In this case, opposition is carried out according to the usual rules.
- At the end of the fight, the jury calculates the points and determines the winning team. If the gap in the number of points does not exceed 3 points, then a draw is recorded in the battle.
- A team may be penalized up to 6 points for noise, rudeness towards an opponent, failure to comply with the requirements of the jury, etc.
"Mathematical battle" is the second most popular form of mathematical competitions after classical Olympiads. Mathematical combat was invented in the mid-60s by a mathematics teacher at School No. 30 in Leningrad, Iosif Yakovlevich Verebeichik. Unlike olympiads, matboy is a team mathematical competition; it contributes to the development of the ability to collectively solve problems, which is especially valuable in modern science when one global problem is often solved by a large team of researchers. Over the 40 years of its existence, mathematical battles have gained immense popularity in various parts of our country. City and regional competitions are held in the form of mat-boys, not a single summer mathematical school passes without mat-boys. Despite the name, schoolchildren from all over Russia and even from neighboring countries gather for these tournaments. The spring tournament is always held in Kirov, the autumn one - in one of the cities of the Urals or Siberia. The XXII Tournament was held in Omsk, the next XXIV will be held in Nizhny Tagil. Since the autumn of 1997, in memory of the great mathematician and wonderful teacher Andrei Nikolaevich Kolmogorov, mathematical tournaments for high school students have been held annually. These tournaments traditionally bring together the strongest participants and are rightfully recognized as the unofficial team championship of Russia in mathematics among schoolchildren. In November 2003, the "VII Kolmogorov Memorial Cup" was held in Moscow, the VIII Cup will take place in autumn 2004 in Yekaterinburg. In October 2002 and April 2004, I and II All-Russian Student Tournaments were held in Tula math fights, in which teams of universities and pedagogical institutes from various parts of Russia took part (Krasnodar, Rostov, Samara, Ryazan, Orenburg, Kazan, Chelyabinsk, Yekaterinburg, Kurgan, etc.). The main difference lies in the fact that in the "Leningrad" rules the team challenges the opponent to some task, while in the "Tula" rules, the team itself is called to solve the problem that it "likes". (In more detail, these rules can be compared by studying the relevant sections on our website.) But no matter what the rules of the matboy are, the truth is born in the dispute between the "speaker" and the "opponent" (however, the jury plays an important role in this dispute), which get the opportunity to demonstrate not only the power of their thought, but also oratory. That is, matboy combines mathematics, sports game and theatrical performance. Probably, this is its special attraction for everyone who is close to the great and beautiful science - mathematics.
Math Fight Rules
1. Order of battle. Math fight is a competition between two teams in solving mathematical problems. It consists of two parts. First, the teams receive the conditions of the tasks and a certain time for their solution. When solving problems, the team can use any printed literature, non-programmable calculators, but has no right to communicate with anyone except the jury. Also, teams do not have the right to use the Internet, any electronic media and mobile phones. After this time, the actual battle begins, when the teams tell each other how to solve problems.
2. Start of the fight. The fight starts with captains competition. The captain who first solved the proposed task raises his hand and presents the answer. If his answer is correct, he wins, if incorrect, his opponent wins, who is not required to submit his answer. The team that wins the captains' competition decides whether it wants to call the opposing team for a report in the first round or be called.
3. Fight order. The fight is made up of several rounds. At the beginning of each round, one of the teams challenges the other team to one of the problems, the solutions of which have not yet been told. The calling team may also refuse further calls (§ 11). The invoked command may accept the call (§ 4) or perform a validation check (§ 9).
The team that made the challenge in the current round becomes the challenged in the next round, except in the case of an incorrect challenge (§ 10), when it is forced to repeat the challenge in the next round.
4. Accepted call. If the challenge was accepted, the called team puts up a speaker, the calling team - an opponent. A team wishing to keep boardwalks (§ 13) may refuse to field an opponent. Then she does not participate in this round. The speaker, with the permission of the jury, can take paper with drawings and calculations. But he has no right to take the text of the decision with him. The speaker tells the solution of the problem; the opponent, by agreement with the speaker, asks him questions either in the course of the presentation or after the report. All calculations, as a rule, are carried out by the speaker on the board and without the use of a calculator. No more than 15 minutes are allotted for the report, no more than 15 minutes for the subsequent discussion of the opponent and the speaker.
5. Rights of speaker and opponent.
During the presentation, the opponent can: ask questions to the speaker with his consent; ask the speaker to repeat any part of the report; allow the speaker not to prove any obvious facts from the point of view of the opponent.
During the discussion, the speaker can: ask the opponent to clarify the issue; refuse to answer the opponent's question, motivating his refusal by the fact that (a) he does not have an answer, (b) he has already answered this question, (c) the question, in his opinion, is not relevant to the task.
During the discussion, the opponent can: ask the speaker to repeat any part of the report; ask the speaker to clarify any of his statements; ask the speaker to prove the formulated non-obvious not well-known statement (the facts included in the school mathematics course are usually considered well-known).
The speaker is not obliged: to state the method of obtaining the answer, if he can prove the correctness and completeness of the answer in another way; compare your solution method with other possible methods.
6.Opponent's Conclusion. When questions are asked and answers are received, the opponent makes a conclusion in one of three forms: (a) "I fully agree with the decision"; (b) "The solution is basically correct, but it has the following shortcomings..."; (c) "The solution is wrong, the fundamental error is as follows...". The opponent should remember that in the end the jury evaluates not his questions, but his conclusion, which must be motivated!
The conclusion on an incorrect decision can be made in the form: "The decision is incorrect, I have a counterexample." In this case, the jury asks the opponent to present a written counterexample without disclosing it to the speaker. If the jury accepts a counterexample, the speaker is given a minute to try to correct the solution. Similar actions are taken at the request of the opponent "The decision is incomplete, not all cases have been considered."
If the opponent agreed with the decision, he and his team no longer participate in this round; further questions to the speaker are asked by the jury. Until the speaker's decision has been refuted, the opponent has no right to tell his decision, even if it is much simpler.
7. Scoring. In each round, 12 points are awarded, which are distributed between the speaker, the opponent and the jury. The speaker for an error-free solution receives 12 points. Otherwise, the jury deducts points from the speaker for the holes contained in the decision. The cost of each hole is estimated by an even number of points. If the speaker patched up the hole after the opponent's question, asked before the end of the report, points from the speaker are not deducted. If the speaker patched up the hole after the opponent's question was asked at the end of the report, the cost of the hole is divided equally between the opponent and the speaker. If the speaker fails to close the hole, the opponent immediately receives half its cost. If the opponent did not notice the hole, and the jury pointed to it with their questions after the conclusion, the jury receives half of the cost of the hole, and the second half goes to the speaker or the jury, depending on whether the speaker managed to close the hole or not.
8. Role reversal. After preliminary scoring, the jury asks the opponent if he would like to present a complete solution to the problem in the case when the opponent has proved that the speaker does not have it, or to close up the remaining holes. If the opponent agrees to a partial or complete change of roles, he temporarily becomes a speaker and tries to earn the second half of the cost of the holes he discovered. A former speaker, while opposing, can himself gain points in half of those that the former opponent is trying to earn as a speaker. Secondary role reversal cannot be performed.
9. Validation consists in the fact that the called command refuses to tell the solution of the problem, but instead checks whether the calling command has solved it. In this case, the calling team will nominate a speaker, and the called team will nominate an opponent. If the calling team immediately admits that they do not have a solution, then the called team receives 6 points. The speaker and the opponent in this case are not appointed and the exits to the board are not counted. During validation, role reversal cannot be performed. If, when checking the correctness, the opponent proved that the speaker does not have a solution, then he receives at least 4 points.
10. The order of the next call when checking the correctness And. If the call is recognized as correct (the calling team presented a solution, or the opponent could not prove that the speaker did not have a solution), then the next call is made by the called team. If the challenge is recognized as incorrect (the calling team immediately admitted that it did not have a solution, or the opponent managed to prove that the speaker did not have a solution), then the calling team again makes the next call.
11. Rejection of calls. Starting from a certain round, one of the teams may refuse further challenges. In this case, the opponents can nominate speakers for any tasks not previously considered, and the team that refused the challenge nominates opponents. Once the calls have been abandoned, role reversal can no longer be performed.
12. Time-out. Communication between the speaker and the team is allowed only during the 30-second break taken by the team. Opponents at this time can also confer, spending all 30 seconds of the break. A team may take no more than six 30-second breaks per fight. If the opponent proceeded to issue a conclusion, his team within 10 seconds can withdraw the words of the opponent and take a timeout. If after the conclusion of the opponent within 10 seconds there was no withdrawal, then the conclusion of the opponent is considered made and it can no longer be changed.
13. Number of exits to the board. Each player is allowed to come to the board (whether as an opponent or speaker) no more than two times per battle, regardless of the number of team members participating in this battle. If desired, the team may not put the opponent in the round, thus saving the number of exits.
14. Substitution order. The team can replace its speaker at any time, which is equivalent to using two breaks. When replacing, the exit is counted for both participants.
15. 10 minute breaks. Team captains have the right to ask the jury for a 10-minute break during the fight (approximately every two hours). A break may only be granted between rounds. In this case, the calling team, before the break, makes a call in writing and hands it over to the jury, which announces the call after the end of the break.
16. End of the fight. The battle ends when all the problems have been considered or when one of the teams has refused the challenge, and the other team has refused to tell the solutions to the remaining problems.
17. Determination of the winner. The team with the most points is considered the winner of the battle. With a difference of no more than 3 points, the fight is considered to have ended in a draw (except in special cases).
18. General rules behavior I. During the fight, the team communicates with the jury only through the captain; if the captain is at the board - through his deputy. The speaker and the opponent address each other only in a respectful manner, on "you". If these rules are violated, the team is first warned, and then punished with penalty points.
19.Jury. The jury is the supreme interpreter of the rules of the fight. The decisions of the jury are binding on the teams. The jury can remove the opponent's question, stop the report or opposition if they are delayed. The jury keeps the protocol of the fight on the board. If one of the teams does not agree with the decision made by the jury on the problem, it has the right to immediately demand an analysis of the situation with the participation of the senior in the league. After the start of the next round, the score of the previous round can no longer be changed.
Combat structure.
I round - Arithmetic mixture.
II round - Historical.
III round - Algebraic.
Stage IV - Fun tasks.
Stage V - Geometric.
Equipment.
2 tables for individual tasks; task cards; blank sheets for completing assignments, 2 sheets with coordinate axes; 2 calculators; posters with drawings of triangles, with the number 18446744073709551615.
Event preparation.
Choose a team (class) captain, come up with a name, team motto, prepare comic gifts for the opposing team. Put 2 tables on the stage, on which to put sheets for writing down the solution of individual tasks. Choose a jury from high school students and mathematics teachers.
Event progress.
Leading.
Why solemnity around?
Do you hear how quickly the speech fell silent?
A guest appeared - the queen of all sciences,
And do not forget us the joy of these meetings.There is a rumor about mathematics
That she puts her mind in order,
Because good words
People often talk about her.You give us mathematics
To overcome hardship hardening.
Youth is learning with you
Develop both will and ingenuity,And for the fact that in creative work
Help out in difficult times
Today we sincerely to you
We send thunderous applause.
(Applause.)
Leading.
Math fight I open
I wish you all success
Think, think, don't yawn,
Quickly count everything in your mind!
Now let's get to know the teams.
(Captains present the name, motto, exchange comic gifts.)
Leading.
One, 2, 3, 4, 5, 6, 7, 8, 9, 10 -
Everything can be counted
Count, measure, weigh.How many seeds are in a tomato
How many boats on the sea
How many doors are in the room
In the lane - lanterns,How many stones are on the mountain
How much coal is in the yard.
How many corners are in the room
How many legs do sparrows haveHow many fingers are on the hands
How many toes are on the feet
How many benches in the garden
How many kopecks in a patch?
- I announce the beginning of the first round, which is called "Arithmetic mixture".
I round “Arithmetic mixture”
I. Two people from the team complete tasks on the cards:
1) Calculate:
II. For the rest of the participants, the tasks are:
Eight people are traveling in a stagecoach, at the first stop five got off, three got on. We drove on, at the next stops two got off, then five, and finally three more. Then the stagecoach arrived at the final stop, where everyone got off. How many stops were there?
Answer: 5.
2) On the road along the bushes
There were 11 tails.
I could also count
That walked 30 feet.It's going somewhere together
Roosters and piglets.
And my question for you is this:
How many roosters were there?
Answer: 7.
III. One person from the team, each needs to count in order up to thirty, but instead of numbers that are divisible by three and end in three, say: “I won’t go astray.”
IV. The chessboard was invented in India. According to legend, the Indian prince Sirom liked this game very much, and he wanted to generously reward its inventor.
“Ask whatever you want, I am rich enough to fulfill your most cherished desire,” the prince said to the inventor of chess, a scientist whose name was Seta.
The inventor said that he should be rewarded with as many grains of rice as the total would be if one grain of rice was placed on the first square of the chessboard, two grains on the second, four on the third, etc., doubling the number of grains each time. The prince laughed at such, in his opinion, a cheap reward and ordered the scientist to immediately give out rice for all 64 squares of the chessboard.
But the award in this amount was not given to the inventor, since the prince did not have such an amount of grain that the joker-scientist asked for.
The calculation shows that the inventor had to issue:
2 +2 2 + 2 3 + 2 4 + … + 2 64 = 18446744073709551615 grains.
(Open three digits from the end and the teams take turns reading the received numbers.)
Answer: 18 quintillion 446 quadrillion 744 trillion 73 billion 709 million 551 thousand 615.
Leading. Mathematicians have calculated that all this grain will have a mass of about 700 billion tons. If it was scattered over the earth's land, a layer of rice about 1 cm thick would form.
The jury sums up the results of the first round.
Sounds of music (symphony No. 40 by Mozart).
Leading. There was great music. Music of the great composer who was fond of mathematics. He painted the floor, walls, performing complex mathematical calculations. He had excellent mathematical knowledge ( Appendix 2, slide 1). It is with this music that we open the next round.
II round “Historical”
I.
Task: write down the names of famous mathematicians and physicists.
II. The rest are asked questions on a historical topic:
1) An amazing fact happened in 1735. The St. Petersburg Academy of Sciences received a proposal from the Government to carry out a hasty but extremely difficult calculation. The academics required several months to complete this task. However, one of the mathematicians of this Academy ( Appendix 2, slide 2) undertook to carry out these calculations in three days, and indeed, to the great amazement of this Academy, he did it. But this work cost him dearly.
Name this mathematician and explain what it means: "this work cost him dearly."
Answer: Euler. After the calculations, his right eye leaked out, and by the end of his life he was blind.
2) The first mathematics textbook in Russia was an encyclopedia of mathematical knowledge. On the title page of this wonderful textbook are portraits of Pythagoras and Archimedes, and on the back there is a bouquet of flowers, under which are the verses:
“Accept, young, wisdom flowers,
Kindly learn arithmetic,
Stick to different rules and pieces in it ... ”
Mikhail Vasilyevich Lomonosov called this book “The Gates of His Learning”. Who is the author of this first in mathematics? What was his name?
Answer:“Arithmetic - that is, the science of numerals”, the author is Magnitsky. Real surname - Telyatin, a native of the Tver province ( Appendix 2, slide 3).
3) Which of the ancient Greek mathematicians took an active part in the Olympic Games and was the winner in the pentathlon?
Leading. You probably already guess that the next round is “Algebraic”.
III round “Algebraic”.
I. Two people per team:
1 task: Mark points on the coordinate plane and connect them sequentially:
(-2;3), (-3;4), (-1;6), (5;7), (3;5), (1;5), (1;3), (6;2), (8;-4), (8;-6), (-3;-6), (-1;-4), (0;-4), (-1;-1), (-1;-3), (-2;0), (-1;1), (-1;2), (-2;3) and (-1.5; 5 ).
2 task: Compare:
7 cells 2 2 and ((2 2) 2) 2
8 cells (cos 60º) 2 and (cos 60º) 3
II. Leading: algebra can be applied to non-mathematical fields. For example, you can graphically depict proverbs and sayings.
Let's take the proverb: "As it comes around, it will respond." Two axes: “acoustic axis” – horizontally, and vertically – “response axis”. The response is a whoop. The graph will be the bisector of the coordinate angle.
response axis proverb graph
aucana axis
You are invited to depict proverbs:
7 cells - "Shines, but does not warm."
8 cells “No stake, no yard.”
Answer: 7 cells one of the half shafts
8 cells is the point of intersection of the coordinate axes.
III. One person per team.
Task: calculate on a calculator
((14628.25 + 4: 0.128) : 1.011 0.00008 + 6.84) : 12.5
Answer: 0,64.
The jury sums up the results of the third round.
Logic pause (thumbnail) (Appendix 1).
Leading. So, I announce the IV round of "Funny Problems".
IV round "Funny tasks".
I.
Exercise: Draw a person using numbers and mathematical symbols.
II. Two people per team:
Exercise: Solve the problem in different ways.
Three ducklings and four caterpillars weigh 2 kg 500 g, and four ducklings and three caterpillars weigh 2 kg 400 g. How much does one gosling weigh?
III. The rest of the tasks are:
1) The guys saw logs into meter-long pieces. Sawing one such piece takes one minute. In how many minutes will they cut a log 5 meters long?
Answer: 4 minutes.
2) A crew of three horses traveled 15 km in one hour. How fast was each horse traveling?
Answer: 15 km/h.
3) How much will be three times 40 and 5?
Answer: 4040405.
4) Two men have 35 sheep. One has 9 more sheep than the other. How many sheep does each have?
Answer: 13 and 22.
5) A train left Moscow for Petersburg at a speed of 60 km/h, and a second train left Petersburg for Moscow at a speed of 70 km/h. Which of the trains will be further from Moscow at the time of the meeting?
Answer: equally.
6) What is the product of all the digits?
Answer: 0.
7) Two dozen times three dozen. How many dozens?
Answer: 72.
8) Alyosha and Borya together weigh 82 kg, Alyosha and Vova weigh 83 kg, Borya and Vova weigh 85 kg. How much do Alyosha, Borya and Vova weigh together?
Answer: 125 kg.
9) A freshly split watermelon contained 99% water. After drying, the water content was 98%. How many times has the watermelon shrunk?
Answer: initially - 1% of dry matter by weight, and after drying - 2%. This means that the proportion of dry matter in the watermelon has doubled, and the mass of the watermelon itself has halved.
10) With the help of a computer, it was calculated that on average a child uses almost 3600 words, a teenager at 14 already 9000 words, an adult over 11000, A.S. Pushkin used 21200 different words in his works. How many times greater is the vocabulary of a teenager than that of Ellochka the cannibal from the well-known satirical novel by Ilf and Petrov “The Twelve Chairs”?
Answer: 450 times.
The jury sums up the results of the fourth round.
Leading. And now - a small pause. Your attention is invited to the poem "Again deuce" (Annex 1).
Leading. I announce the V round "Geometric".
V round “Geometric”
I. One person per team:
Exercise: Cut a square sheet of paper into two unequal parts, and then make a triangle out of them.
II. Blitz Poll (time and correctness of answers are estimated).
Questions for the first team:
What is the name of:
A line segment that connects a point on a circle to its center. (Radius).
- A statement that requires proof. (Theorem).
- The angle is less than right. (Spicy).
- A rectangle with all sides equal. (Square).
- The ratio of the opposite leg to the hypotenuse. (Sinus).
- The largest chord in the circle. (Diameter).
A part of a straight line bounded on one side. (Ray).
- A device for measuring angles. (Protractor).
– The angle adjacent to the angle of the triangle at the given vertex. (External).
- Translated from Latin, "cutting into two parts." (Bisector).
Questions for the second team:
What is the name of:
A line segment that connects the vertex of a triangle with the midpoint of the opposite side. (Median).
- A statement beyond doubt. (Axiom).
A line segment that connects two points on a circle. (Chord).
The sum of the lengths of all sides of a rectangle. (Perimeter).
- The ratio of the adjacent leg to the hypotenuse. (Cosine).
- A device for constructing circles. (Compass).
- The value of the expanded angle. (180º).
A rhombus with all right angles. (Square).
A part of a straight line bounded on two sides. (Line segment).
– Translated from the Latin language “the spoke of the wheel”. (Radius).
III. Leading.
Often a preschooler knows
What is a triangle.
And how could you not know...But it's quite another thing -
Very fast and skillful
Count triangles.For example, in this figure
How many different? Consider!
Explore everything carefully
Both on the edge and inside.
How many triangles are in the picture?
Leading. While the jury is summing up last round and the whole game, you are invited to watch the scene “Arithmetic Mean” performed by students of the 7th grade (Annex 1).
The jury summarizes the results of the fifth round and the entire fight.
The winning team is awarded, the losers receive a consolation prize.
Leading.
Oh wise men of the times!
You can't find friends.The fight is over today
But everyone should know:Knowledge, perseverance, work
Lead to progress in life!
