Goals:
develop interest in mathematics, logic and ingenuity, the ability to prove and explain; communicative competence.
Preparation for the lesson:
tasks for mathematical combat are written down on album sheets in triplicate: for the teams and the teacher.
Progress 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 battle consists of two stages. The first stage is solving problems, the second is the battle itself. During the first stage, problem solving can occur jointly with 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 others have not solved must, during the first stage, tell his teammates the solutions he received.
- The second stage begins with a captain's competition. 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 call. This, as well as all other decisions of the team, is announced by the captain.
Captains competition:
A super-blitz is held on three questions, the captain who scores two or three points wins. The captain can earn a point by answering the question correctly. The first person to answer is the one who raises the signal card (prepared in advance) or hand faster.
- A chocolate costs 10 rubles and another half a chocolate bar. How much does a chocolate bar cost?
- Hares are sawing a log. 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 our opponents to task number...”. The other team may or may not accept the challenge. The team that accepted the challenge puts up speaker, the other team - opponent. After a meeting with the teams, the captains name the opponent and the speaker. The speaker’s task is to give a clear and understandable solution to the problem. The opponent’s task is to find errors in the report. During the report, the opponent has no right to object to the speaker, but may ask him to repeat an unclear point. 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 about all the unclear parts of the report. The discussion ends with the opponent’s conclusion: “I agree with the decision” or “I believe that there is no solution, since such and such was not explained.”
- After this, the jury (teacher) awards points according to following rules. Each task costs a different number of points, as they differ in difficulty level. First and second problems – 6 points. Third, fourth, fifth and sixth – 8 points. Seventh and eighth – 10 points. Ninth and tenth – 12 points. In case of an absolutely correct decision, the speaker's team receives all these points. For errors and inaccuracies, points will be deducted. The number of points deducted is determined by the proximity of the story told to the correct solution. If errors were found by the opponent, then the opposing team receives up to half of the points deducted. Otherwise, all selected points go to the jury. If the jury decides 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 makes an incorrect report nominates an opponent and can earn points by opposing.
- The team receiving 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 nominates a speaker, and the second team nominates an opponent. If there is no solution and this is proven by the opposing team, then they receive half the points for this problem, and the challenging team must repeat the challenge. This procedure is called checking the correctness of the call. In all other cases, calls alternate.
- During the bout, each team is entitled to six 30-second breaks. Breaks are made in cases where there is a need to help a student standing at the blackboard or replace him. The decision to take a break is made by the captain.
- A team that has received the right to a 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. Opposition is carried out according to the usual rules.
- At the end of the battle, the jury counts the points and determines the winning team. If the gap in the number of points does not exceed 3 points, then the battle is recorded as a draw.
- A team may be fined 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 Joseph Yakovlevich Verebeychik, a mathematics teacher at school No. 30 in Leningrad. Unlike Olympiads, Matboy is a team mathematical competition; it promotes the development of collective problem solving skills, which is especially valuable in modern science, when often one global problem is solved by a large team of scientists. Over the 40 years of its existence, mathematical battles have gained enormous popularity in various parts of our country. City and regional competitions are held in the form of math battles; not a single summer mathematics school takes place without math battles. Since 1993, the Ural tournaments of young mathematicians are held twice a year, where students in grades 6-8 compete. Despite the name, schoolchildren from all over Russia and even from neighboring countries gather for these tournaments. The spring tournament always takes place in Kirov, the autumn tournament 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 fall 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 Russian team championship in mathematics among schoolchildren. In November 2003, the “VII Cup in Memory of A.N. Kolmogorov” was held in Moscow; the VIII Cup will take place in the fall of 2004 in Yekaterinburg. In October 2002 and April 2004, the I and II All-Russian student tournaments were held in Tula mathematical battles, in which teams from universities and pedagogical institutes from various parts of Russia took part (Krasnodar, Rostov, Samara, Ryazan, Orenburg, Kazan, Chelyabinsk, Yekaterinburg, Kurgan, etc.). However, student matfights are held according to rules somewhat different from the classical ones ( "Leningrad"). The main difference is that in the “Leningrad” rules the team challenges the opponent to some task, but in the “Tula” rules the team itself volunteers to solve the problem that it “likes”. (These rules can be compared in more detail by studying the corresponding sections on our website.) But no matter what the rules of the match are, the truth is born in the dispute between the “speaker” and the “opponent” (however, the jury plays an important role in this dispute), who get the opportunity to demonstrate not only the power of their thoughts, but also their oratory skills. That is, matboy combines mathematics, sports game and theatrical performance. This is probably where its special appeal lies for everyone who is close to the great and wonderful science of mathematics.
Rules of mathematical combat
1. Order of battle. Math fight is a competition between two teams in solving mathematical problems. It consists of two parts. First, teams receive task conditions and a certain time to solve them. 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 or mobile phones. After this time, the actual battle begins, when the teams tell each other the solutions to the problems.
2. Start of the battle. 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, and he is not required to submit his answer. The winning team in the captain's competition decides whether it wants to challenge the opposing team to a report in the first round or be called.
3. Order of battle. The fight consists of several rounds. At the beginning of each round, one of the teams challenges the other team to one of the problems whose solutions have not yet been revealed. The calling command may also refuse further calls (§ 11). The called command can accept the challenge (§ 4) or perform a validation check (§ 9).
The team that made the challenge in the current round becomes 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 is accepted, the called team sets up a speaker, the calling team sets up an opponent. A team wishing to preserve access to the board (§ 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 with him a paper with drawings and calculations. But he has no right to take the text of the decision with him. The speaker tells the solution to the problem; the opponent, by agreement with the speaker, asks him questions either during the presentation or after the report. All calculations are usually carried out by the presenter on the board and without the use of a calculator. No more than 15 minutes are allotted for the report, and no more than 15 minutes for the subsequent discussion between the opponent and the speaker.
5. Rights of the speaker and opponent.
During the report, 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 facts that are obvious from the opponent’s point of view.
During the discussion, the speaker can: ask the opponent to clarify the question; refuse to answer your opponent’s question, citing 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 and not generally known statement (facts included in the school mathematics course are usually considered generally 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 the questions are asked and answered, the opponent makes a conclusion in one of three forms: (a) “I completely agree with the decision”; (b) “The solution is basically correct, but it has the following shortcomings...”; (c) “The solution is incorrect, the fundamental error is as follows...” The opponent should remember that the jury ultimately evaluates not his questions, but his conclusion, which must be motivated!
A 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 counterexample in writing, without revealing it to the speaker. If the jury accepts the counterexample, the speaker is given a minute to attempt to correct the decision. Similar actions are carried out based on the opponent’s statement “The solution is incomplete, not all cases have been considered.”
If the opponent agrees with the decision, he and his team no longer participate in this round; Then the jury asks questions to the speaker. Until the speaker's decision has been refuted, the opponent has no right to tell his solution, even if it is much simpler.
7. Accrual of points. In each round, 12 points are awarded, which are distributed between the presenter, opponent and jury. The speaker receives 12 points for an error-free solution. Otherwise, the jury deducts points from the speaker for holes contained in the solution. Each hole is worth an even number of points. If the speaker fills a hole after an opponent’s question asked before the end of the report, points are not deducted from the speaker. If the speaker repairs the hole after the opponent's question asked at the end of the report, the cost of the hole is divided equally between the opponent and the speaker. If the presenter fails to repair the hole, the opponent immediately receives half of its value. If the opponent did not notice the hole, and the jury pointed it out with their questions after making a conclusion, the jury receives half of the cost of the hole, and the other half goes to the speaker or the jury, depending on whether the speaker was able to repair the hole or not.
8. Role reversal. Having made a preliminary assessment of points, the jury asks the opponent if he would like to present a complete solution to the problem in the case where the opponent has proven that the speaker does not have one, or to fill in the remaining holes. If the opponent agrees to a partial or complete change of roles, he temporarily becomes the speaker and tries to earn the second half of the value of the holes he discovered. The former speaker, when opposing, can himself score half the points that the former opponent is trying to earn as a speaker. Secondary role changes cannot be made.
9. Validation check is that the called command refuses to tell the solution to the problem, but instead checks whether the calling command has solved it. In this case, the calling team sets up a speaker, and the called team sets up an opponent. If the calling team immediately admits that it does not have a solution, then the calling team receives 6 points. In this case, the speaker and opponent are not assigned and exits to the board are not counted. When checking for correctness, roles cannot be changed. If, during the correctness check, the opponent proves that the speaker does not have a solution, then he receives at least 4 points.
10. The order of the next call when checking for correctness And. If the call is recognized as correct (the calling team presented a solution, or the opponent could not prove that the presenter does not have a solution), then the called team makes the next call. If the call is recognized as incorrect (the calling team immediately admitted that it does not have a solution, or the opponent was able to prove that the presenter does not have a solution), then the next call is made again by the calling team.
11. Call Rejection. Starting from a certain round, one of the teams may refuse further challenges. In this case, opponents can nominate speakers for any previously unconsidered tasks, and the team that refuses the challenge nominates opponents. Once calls are rejected, roles can no longer be changed.
12. Time-out. Communication between the speaker and the team is allowed only during the 30-second break taken by the team. At this time, opponents can also deliberate, using up all 30 seconds of the break. A team may take no more than six 30-second breaks per bout. If the opponent begins to make a conclusion, his team can recall the opponent’s words within 10 seconds and take a time-out. If after the opponent’s conclusion there is no recall within 10 seconds, then the opponent’s conclusion is considered made and cannot be changed.
13. Number of exits to the board. Each player is allowed to come to the board (whether as an opponent or as a speaker) no more than twice per battle, regardless of the number of team members participating in this battle. If desired, the team may not field an opponent for the round, thereby saving the number of exits.
14. Replacement order. A team can change its speaker at any time, which is equivalent to using two breaks. When replacing, the exit is credited to 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 start of the break, makes a challenge in writing and submits it to the jury, who announces the call after the end of the break.
16. End of the fight. The battle ends when all problems have been considered or when one of the teams refused the challenge and the other team 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. If the difference is no more than 3 points, the battle is considered to end in a draw (except for specially specified cases).
18. General rules behavior I. During the battle, the team communicates with the jury only through the captain; if the captain is at the board - through his deputy. The speaker and opponent address each other only in a respectful manner, using the “you” form. If these rules are violated, the team is first warned and then penalized.
19.Jury. The jury is the supreme interpreter of the rules of combat. The decisions of the jury are binding on the teams. The jury can withdraw the opponent's question, stop the report or opposition if they are delayed. The jury keeps a record of the fight on the board. If one of the teams does not agree with the decision made by the jury on the task, it has the right to immediately demand an analysis of the situation with the participation of the league leader. Once the next round starts, the score of the previous round cannot be changed.
Combat structure.
Round I – Arithmetic mixture.
Round II – Historical.
Round III – Algebraic.
Stage IV – Fun tasks.
Stage V – Geometric.
Equipment.
2 tables for completing 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.
Preparation of the event.
Choose a team (class) captain, come up with a team name and motto, and prepare funny gifts for the opposing team. Place 2 tables on the stage, on which to place sheets for recording solutions to individual tasks. Choose a jury from high school students and mathematics teachers.
Progress of the event.
Leading.
Why is there solemnity around?
Do you hear how quickly the speech fell silent?
A guest appeared - the queen of all sciences,
And let us not forget 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.Mathematics, you give us
To overcome difficulties, harden yourself.
Young people study with you
Develop both will and ingenuity,And for the fact that in creative work
You help out in difficult moments,
We are sincere to you today
We send thunderous applause.
(Applause.)
Leading.
I open a mathematical battle,
I wish you all success,
Think, think, don't yawn,
Quickly calculate everything in your head!
– Now let’s get acquainted with the teams.
(Captains present the name, motto, exchange comic gifts.)
Leading.
One, 2, 3, 4, 5, 6, 7, 8, 9, 10 –
You can count everything
Count, measure, weigh.How many grains are in a tomato?
How many boats are there on the sea,
How many doors are there in the room?
There are lanterns in the alley,How many stones are there on the mountain?
How much coal is there in the yard?
How many corners are there in the room?
How many legs do sparrows have?How many fingers are there on your hands?
How many toes are there?
How many benches are there in the kindergarten?
How many kopecks are in a penny?
– I announce the beginning of the first round, which is called “Arithmetic mixture”.
Round I “Arithmetic mixture”
I. Two people per team complete the tasks on the cards:
1) Calculate:
II. For other participants, the following tasks are offered:
There are 8 people traveling on a stagecoach; at the first stop, five got off and three got on. We drove on, and at the next stops two people 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) Along the road along the bushes
There were 11 tails.
I was also able to count
That walked 30 legs.We were going somewhere together
Roosters and piglets.
And my question to you is this:
How many roosters were there?
Answer: 7.
III. One person from the team, each needs to count in order to thirty, only instead of numbers that are divisible by three and ending in three, say: “I won’t get lost.”
IV. The chessboard was invented in India. According to legend, the Indian Prince Sirom really liked this game, and he wanted to generously reward its inventor.
“Ask whatever you want, I am rich enough to fulfill your deepest desire,” the prince said to the inventor of chess, a scientist named Seta.
The inventor said that as a reward he would be given as many grains of rice as the total would be if the first square chessboard put one grain of rice, on the second - two grains, on the third - four, etc., doubling the number of grains each time. The prince laughed at such a cheap reward, in his opinion, and ordered the scientist to immediately give the scientist rice for all 64 squares of the chessboard.
But a reward of this size was not given to the inventor, since the prince did not have the 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 resulting 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 were scattered on the earth's land, it would form a layer of rice about 1 cm thick.
The jury sums up the results of the first round.
Music sounds (Symphony No. 40 by Mozart).
Leading. Wonderful music sounded. Music by a great composer who was passionate about mathematics. He covered the floor and walls, performing complex mathematical calculations. He had brilliant mathematical knowledge ( Appendix 2, Slide 1). It is with this music that we open the next round.
Round II “Historical”
I.
Assignment: 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 an urgent but extremely difficult calculation. The academicians took 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 a lot.”
Answer: Euler. After the calculations, his right eye leaked out, and by the end of his life he became blind.
2) The first manual on mathematics in Russia was the encyclopedia of mathematical knowledge. On the title page of this wonderful textbook are portraits of Pythagoras and Archimedes, and on the back is a bouquet of flowers, under which are the verses:
“Take, young one, flowers of wisdom,
It's nice to learn arithmetic,
There are different rules and things to follow…”
Mikhail Vasilyevich Lomonosov called this book “The Gates of His Learning.” Who is the author of this first in mathematics? What was it called?
Answer:“Arithmetic - that is, the science of numbers”, author - Magnitsky. Real name: Velyatin, a native of 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 the 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).
Task 2: Compare:
7th grade 2 2 and ((2 2) 2) 2
8th grade (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, so it will respond.” Two axes: the “hook axis” – horizontally, and vertically – the “response axis”. The response is equal to a whoop. The graph will be the bisector of the coordinate angle.
response axis graph proverbs
aucanya axis
You are invited to depict proverbs:
7th grade - “It shines, but does not warm.”
8th grade - “No stake, no yard.”
Answer: 7th grade – one of the semi-axes,
8th grade – point of intersection of 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.
Logical pause (miniature) (Appendix 1).
Leading. So, I announce the IV round of “Fun Challenges”.
IV round “Fun 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 goslings weigh 2 kg 500 g, and four ducklings and three goslings weigh 2 kg 400 g. How much does one gosling weigh?
III. The rest are offered tasks:
1) The guys are sawing logs into meter-long pieces. Sawing off one such piece takes one minute. How many minutes will it take them to cut a log 5 meters long?
Answer: 4 minutes.
2) A carriage drawn by three horses traveled 15 km in one hour. How fast was each horse going?
Answer: 15 km/h.
3) How much is 40 and 5 three times?
Answer: 4040405.
4) Two men have 35 sheep. One has 9 more sheep than the other. How many sheep does each person have?
Answer: 13 and 22.
5) A train left Moscow for St. Petersburg at a speed of 60 km/h, and a second train left from St. Petersburg for Moscow at a speed of 70 km/h. Which train will be further from Moscow at the time of the meeting?
Answer: the same.
6) What is the product of all the numbers?
Answer: 0.
7) Multiply two dozen by three dozen. How many dozens will there be?
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 it dried out, the water content became 98%. How long does it take for a watermelon to dry out?
Answer: initially - 1% 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) Using a computer, it was calculated that on average a child uses almost 3600 words, a teenager at 14 years old already uses 9000 words, an adult uses over 11000, A.S. Pushkin used 21,200 different words in his works. How many times is a teenager’s vocabulary larger than that of Ellochka the cannibal from the famous satirical novel “The Twelve Chairs” by Ilf and Petrov?
Answer: 450 times.
The jury sums up the results of the fourth round.
Leading. And now – a short pause. We present to your attention the poem “Again deuce” (Appendix 1).
Leading. I announce the V round “Geometric”.
Round V “Geometric”
I. One person per team:
Exercise: Cut a square sheet of paper into two unequal parts, and then form a triangle from them.
II. Blitz survey (time and correctness of answers are assessed).
Questions for the first team:
What is it called:
– A segment connecting a point on a circle to its center. (Radius).
– A statement that requires proof. (Theorem).
– The angle is less than a right angle. (Spicy).
– A rectangle with all sides equal. (Square).
– The ratio of the opposite side to the hypotenuse. (Sinus).
– The largest chord in the circle. (Diameter).
– Part of a straight line, limited on one side. (Beam).
– Device for measuring angles. (Protractor).
– The angle adjacent to the angle of the triangle at a given vertex. (External).
– Translated from Latin as “cutting into two parts.” (Bisector).
Questions for the second team:
What is it called:
– A segment connecting the vertex of a triangle with the middle of the opposite side. (Median).
- A statement that does not raise doubts. (Axiom).
– A line segment connecting two points on a circle. (Chord).
– The sum of the lengths of all sides of the rectangle. (Perimeter).
– The ratio of the adjacent leg to the hypotenuse. (Cosine).
– A device for constructing circles. (Compass).
– The magnitude of the rotated angle. (180º).
- A rhombus with all right angles. (Square).
– Part of a straight line, limited on both sides. (Segment).
– Translated from Latin as “spoke of the wheel.” (Radius).
III. Leading.
Even a preschooler often knows
What is a triangle?
How could you not know...But it’s a completely different matter -
Very quickly and skillfully
Count triangles.For example, in this figure
How many different ones? Take a look!
Examine everything carefully
Both on the edge and inside.
How many triangles are there in the picture?
Leading. While the jury sums up the results last round and the entire game, you are invited to watch the sketch “Arithmetic mean” performed by 7th grade students (Appendix 1).
The jury sums up the results of the fifth round and the entire fight.
The winning team is awarded, the losers receive a consolation prize.
Leading.
O wise men of the times!
You couldn't be more friendly.The fight is over today
But everyone should know:Knowledge, perseverance, work
They will lead to progress in life!
