Mathematics Wide-angle Chickens and Rabbits in the Same Cage The first teaching goal: 1. Let the students understand? Chicken and rabbit in the same cage? Problems, master the trial-and-error method and hypothetical substitution method to solve problems, and initially form a general strategy to solve such problems.
2. Let students experience different solutions through independent exploration, cooperation and exchange? Chicken and rabbit in the same cage? The process of problem, in the process of solving problems, cultivate students' thinking ability.
3. Let students feel the interest of ancient math problems and realize it? Chicken and rabbit in the same cage? The extensive application of problems in life can improve the interest in learning mathematics.
Teaching emphasis: solving with hypothesis? Chicken and rabbit in the same cage? question
Preparation of Teaching Tools: Computer Courseware
First, introduce questions and assign tasks. Each person sends an envelope with a question card and learning tools. )
? There are five yuan and two yuan denominations of RMB1* *10, totaling 32 yuan. How much are the two kinds of RMB?
Second, cooperation and exploration, showing high. (Take a lifetime to go on stage and replace them one by one, the teacher records)
1. Inspirational demonstration:/Let the students assume that these 10 sheets are all binary. So I took out 10 binary ones (one * * * 20 yuan, which obviously failed to meet the requirements)//and then replaced them one by one. I took out 1 binary and replaced it with 1 five-yuan, so there were more 3 yuan, and it became 20+3=23 yuan. //I took out 1 binary one. It becomes 23+3 = 26//,and then 1 binary is taken out and replaced with 1 five-yuan, so there are more 3 yuan, and it becomes 26+3 = 29////and then 1 binary is taken out and replaced with/kloc-0.
2. Method research: 32-20= 12 yuan, and 12 yuan was changed exactly four times, indicating that there are four yuan and five yuan. 5 yuan replaced 2 yuan with an extra 3 yuan, 12/3=4. Only by changing four tickets can the missing 12 yuan be changed back.
The same demonstrations are all from 5 yuan, which can be changed to binary.
3. The abstract algorithm (formation strategy):
(32-2? 10)/(5-2)=4 fives or (5? 10-32)/(5-2)=6 binary blocks.
Third, classification consolidation (independent practice).
Show me question 2. ? There are *** 100 RMB five yuan and two yuan denominations, totaling 365 yuan. How much RMB is there in each one?
First, students discuss in groups and show the algorithm on the stage:
Assuming that 100 is five yuan, how many are there in a * * *? 100=500 yuan, which means 500-365= 135 yuan. How much 2 yuan should I exchange? One 2 yuan for 5 yuan is 5-2=3 yuan, 135/3=45 2 yuan. 5 yuan has 100-45=55.
Similarly, if 100 is binary, then a * * * has 2? 100=200 yuan, which means 365-200= 165 yuan is missing. How much 5 yuan should I exchange? One 5 yuan for 2 yuan is 5-2=3 yuan, 165/3=55 5 yuan. 2 yuan has 100-55=45.
Ask your own questions and exchange answers.
Show the question given by student A: 42 people went boating and one person rented a boat of 10. Each big boat takes five people and each small boat takes three people. How many big boats and small boats are there?
Show student B's analysis process: (Hint: suppose all 10 articles are chartered. 10*3=30 people, 42-30= 12 people didn't sit on it, so they changed to a big ship. Replacing a big boat with a small boat will increase 5-3=2 people, 12/2=6 big boats have just been replaced. Boat: 10-6=4 or (5? 10-42 = 8,8/(5-3) = 4 ships)
Fourth, induction and improvement:
Problem-solving strategies: ① make a problem-solving plan, assume and replace (both conditions are met, assuming the first condition is met) ② guess and try. (Try on the basis of thinking) ③ Reverse deduction. (Verify that the assumptions are correct).
Verb (abbreviation of verb) knowledge expansion.
In fact, as early as ancient times, many mathematicians have also done research on this kind of problem we have just studied, you see. Slide show.
? Chicken and rabbit in the same cage? It is a well-known mathematical problem in China's ancient arithmetic "Sunzi Suanjing". Its content is: Today there are chickens and rabbits in the same cage, with 35 heads on the top and 94 feet on the bottom. Find the geometry of chicken and rabbit
Six, solve life problems (standard test):
1. Required questions: ① Our class sent 12 students to plant trees. Each male student planted three trees, while each female student planted two trees. A * * * planted 32 trees. How much do you ask each male and female student? (Students finish independently, teachers patrol and guide) Roll call board.
(2) Xiao Ming spent 5 yuan money to buy stamps with 60 cents and 80 cents. How many stamps did he buy respectively?
2. Selected as the theme:
(1) 5 yuan and 2 yuan have RMB 100, how many are there in 290 yuan, 2 yuan and 5 yuan?
②2 large boxes, 5 small boxes, 100 balls. Each big box contains 8 more balls than the small box. How many balls do you want in the big box and the small box?
Self-examination/introspection
The Outline of Basic Education Curriculum Reform (Trial) clearly requires that teachers should actively interact with students in the teaching process, develop together, handle the relationship between imparting knowledge and cultivating ability, pay attention to individual differences and meet the learning needs of different students.
First of all, I start with the problem and use the method of autonomous learning to let students think independently. In the inspiring demonstration, I spent my whole life on stage and replaced them one by one. The rest of the students took out the demonstration coins in the envelope to exchange them, and then asked the students to discuss in groups: what hasn't changed and what has changed in this process? (The number of sheets has not changed, but the amount of money has changed. This process embodies the learning style of group learning and cooperative inquiry. Practice has proved that students learn easily and clearly, which also reflects the way of efficient classroom-the core: autonomy, cooperation and inquiry.
In the process of inquiry, I let students be small teachers, solve problems by themselves and exchange answers, which improves students' interest in learning and enables them to develop actively to meet different needs.
In the assignment, I adopted compulsory exercises and optional exercises, aiming at improving every student and embodying the teaching principle of teaching students in accordance with their aptitude. At the same time, the design of test questions is closely combined with reality, so that students can learn to solve problems in life and solve mathematical problems in life, so that mathematics is no longer isolated and unfamiliar.
I tried to do three movements in this class: physical movement, mental movement and mental movement.
With the development of teaching forms, it is imperative to create an efficient classroom and teach students the correct learning methods. ? It is better to teach people to fish than to teach them to fish. I think we should cultivate students from the following aspects to create an efficient classroom: 1. Cultivate good study habits. 2. Master efficient learning methods: ① Preview. Adopt effective preview methods. Take notes and practice writing when previewing. Important mathematical concepts and formulas should be marked for easy memory and discussion. Please listen carefully when the teacher explains. ② Effective review. Confucius said:? It is a pleasure to study from time to time and review in time. Step-by-step memory method: half a day, one day, three days, seven days and half a month after study, step by step. Systematic review in stages-weekly review, mid-term review and regular review. You can recall before reading, read before doing the questions, and review before taking notes. (3) extrapolate. Don't be satisfied with the answer. Math problems can be synthesized step by step, and it is easier to use equations to see if arithmetic is simpler. 4 learn to organize knowledge points.
Mathematical wide-angle chicken and rabbit in the same cage? Chicken and rabbit in the same cage? In question teaching, teachers usually attach a brief introduction to China's ancient work "Sun Zi Shu Jing" in the teaching process, aiming at reflecting the historical development of mathematics and infiltrating the cultural factors in the history of mathematics into students. That's a good idea, but what about this? Additional? It is difficult for the introduction of the formula to play a substantial role in achieving such a goal. For change? Additional? For what? Blend in? In order to make the knowledge and culture in the history of mathematics play a better role in educating people, teachers need to have a broader and deeper understanding of the relevant contents of the history of mathematics.
? Chicken and rabbit in the same cage? The problem has a long history in ancient China. From the description of the problem to the algorithm of the problem, it has undergone different forms of changes. Understanding these contents will be beneficial to the compilation of course content and teaching design.
1. What's in Sun Tzu's calculation? Are pheasants and rabbits in the same cage?
? Chicken and rabbit in the same cage? The problem first appeared in Sunzi Suanjing in the 3rd and 4th centuries, and the author of this book is unknown. Division integration from the Qing Dynasty? Technology? Mathematical chemistry? Sun Tzu calculates classics? Sun Tzu's Art of War (Song Engraving Edition)? Get down, see? Chicken and rabbit in the same cage? The question is:? Today, there are pheasants and rabbits in the same cage, with 35 heads above and 94 feet below. Ask pheasant rabbit geometry. ? [1] (see figure 1)
One of them? Hey? what's up Pheasant? Meaning? Geometry? what's up How much/how much? The meaning of. In today's language, this problem can be described as:? Chicken and rabbit are in the same cage, with 35 heads and 94 feet. Asking everyone how many chickens and rabbits there are is divided into the following four steps:
Step 1: Put 35 heads at the top and 94 feet at the bottom.
In ancient China, it was calculated by calculation. Calculate support? It is a small stick for calculation, a tool used by the ancients for calculation. What does it say here? Thirty-five heads up and ninety-four feet down? Is to put the number of heads in the title? 35? How about counting feet? 94? Put the stick in the upper position (upper position) and the lower position (lower position) respectively. (See Figure 2)
When the ancients used counting chips to represent numbers, they were placed vertically and horizontally. Usually, vertical sticks are used to put single digits, horizontal sticks are used to put ten digits, and then they are alternately placed vertically and horizontally. Like what? 35? It appear in that form of figure 3.
If the number placed horizontally is greater than 5, 5 is represented by a vertical bar, for example, in Figure 2, 5+4=9.
Step 2: Half foot is 47.
This means that half of the total number of bottoms 94 is equal to 47. Figure 2 becomes the form of Figure 4.
The upper horizontal bar in Figure 4 indicates? 5? , the following two vertical sticks said? 2? , so it means 5+2=7.
Step 3: divide the upper three by the lower three, and divide the upper five by the lower five.
Here? Except? what's up Remove? Or? Reduce? Meaning? Divide the top three by the bottom three? Is it? Remove the same 30 from the following 47? ,? Divide the top five by the bottom five? Is it? Remove the same five from the following 47? . (See Figure 5)
In today's language, 47 minus 35 is 12, which gives the number of rabbits. Is this process in sun Tzu's The Art of War? Surgery? It's called. Less and less, what about life? (see figure 1), that is to say, after reducing more with less, it is lower? Total feet? The meaning of has changed and needs to be renamed, that is, Total feet? Renamed? How many rabbits are there? . (See Figure 5)
Step 4: divide the next one by the above one, and divide the next two by the above two.
Similar to the previous sentence, this sentence means that the number of chickens is obtained by subtracting the number of rabbits from the total of 35. Superior? Total number? Do I need to change my name? How many heads are there? . (See Figure 6)
The rationality of the above algorithm is not difficult to understand. The total number of feet is 94, and half of them become 47. At this time, it is equivalent to all chickens becoming golden cocks. One-legged chicken All the rabbits stood up and became? Biped rabbit? . At this time, the number of heads and feet of each chicken is 1, and the number of heads and feet of each rabbit is 1 and 2, so subtract the total number of heads from 47 and the number of rabbits is 12. Finally, subtract 12 from the total number of 35 to get the number of chickens. This algorithm is summarized as:? Raise your head, put down your feet, half your feet, head off your feet, head off your feet. ? Call this method? The table at the top right of the half-foot method can show this process more clearly.
Second, the "unified family of algorithms"? Chicken and rabbit in the same cage?
? Chicken and rabbit in the same cage? The question was later included in Cheng Dawei's The Arithmetic Unity, Volume 8 of Ming Dynasty (1533 ~ 1606)? Where is Shao? . [2] (See Figure 7)
What is the statement of the problem? Hey? Changed? Chicken? , so? Chicken and rabbit in the same cage? This statement has been used to this day. In the unified algorithm, there are two algorithms that can solve this problem. These two algorithms are different from those in Sun Tzu's calculation, which are equivalent to what we are talking about now? Hypothetical method? . The process of the first algorithm is:
Step one:? Seventy times the total? Twice the total number of heads of 35, that is, multiply by 2 to get 70.
Step two:? And the total foot is reduced by more than 70? That is, subtract 70 feet from the total of 94 feet to get 24 feet.
Step 3:? Half price 12. Is it a rabbit? Divide 24 by two (that is, divide 24 by 2) to get 12, which is the number of rabbits.
Step 4:? Four feet by 48 feet? Multiply the number of feet of each rabbit by 12, and the total number of feet of rabbits is 48.
Step 5:? After the total foot is reduced, 46 feet are chicken feet? The total number of feet is 94, and the total number of feet of rabbits is 48 to get 46, which is the total number of feet of chickens.
Step 6:? Half price 23? Divide the total number of chickens by two (46 divided by 2), and the number of chickens is 23.
Another algorithm is to find the number of chickens first, which is basically the same as the previous procedure of finding the number of rabbits first. The algorithm can be presented in the form of the following table.
About "algorithm unification"? Chicken and rabbit in the same cage? The two algorithms of the problem are summarized in two sentences:? Is the double-headed foot shrinking a rabbit? And then what? Four heads and a half feet are chickens? (See Figure 7). The first sentence means that the process of finding the number of rabbits is divided into three steps: doubling the head, doubling the feet and folding in half. Double head? Is to double the total number of heads from 35 to 70; ? Foot reduction? Is to subtract 70 from the total number of heads 94 to get 24; ? Half? That is, taking half of 24, the number of rabbits is 12. This process is written as today's formula:
(94-35? 2)? 2= 12 (only)
The second sentence means that the process of finding the number of chickens is divided into three steps: four heads, feet reduced and half folded. Four heads? Is to multiply the total number of heads by 4 to get140; ? Foot reduction? 140 is used to subtract 94 from the total number of feet to get 46; Similar to the process of counting rabbits. Half price? Just take half of 46 and get only 23 chickens. Written formula is:
(35? 4-94)? 2=23 (only)
This process is obviously the same as Sun Tzu's calculation? The half-foot method is different. The half-foot method first halved the total number of feet. The first step here is to multiply the total number of heads by the number of feet per chicken or rabbit (2 or 4), so we might as well call this method? Double-headed method? . It's not hard to find. Double-headed method? The reason behind it is actually what we are talking about now? Hypothetical method? .
The problem of chickens and rabbits in the same cage appeared in the eighth volume of the book, and actually appeared in the previous fifth volume. Chicken and rabbit in the same cage? About the same amount of questions? Mimi problem? :? Today, rice and wheat are 500 stones, and the silver price is 405.27 yuan. Only the price of rice and wheat is eight yuan and six cents per stone, and the price of wheat and wheat is seven yuan and two cents per stone. Ask m wheat each a few. ?
The third part is the summary of China's famous traditional mathematical problems, which have their own mathematical ideas and background culture. This paper mainly studies the famous problem of China's traditional mathematics-chicken and rabbit in the same cage, and the mathematical thought permeated in it, so as to improve everyone's emotional attitude, thinking ability and values, and enhance their mathematical literacy.
Keywords chicken and rabbit in the same cage; Problem-solving thinking; Solution; Mathematical thought
Chickens and rabbits are in the same cage, which is one of the famous interesting topics in ancient China. About 1500 years ago, this interesting question was recorded in Sun Tzu's calculation. That's what the book says: Today, there are pheasant rabbits in the same cage, with 35 heads above and 94 feet below. These four sentences mean: there are several chickens and rabbits in the same cage, counting from the top, there are 35 heads; It's 94 feet from the bottom. How many chickens and rabbits are there in each cage?
Thinking of solving the problem: Assuming all chickens, then we can calculate how many feet there are under the assumption according to the total number of chickens and rabbits. Compare the number of feet obtained by this method with the number given in the question to see how much difference there is. A difference of every two feet means 1 rabbit. Divide the foot difference by two, and we can figure out how many rabbits there are. To sum up, the basic formula to solve the problem of chickens and rabbits in the same cage is: number of rabbits = (actual number of feet-number of feet per chicken? The total number of chickens and rabbits) the number of feet per rabbit-the number of feet per chicken. Similarly, it can be assumed that all rabbits.
Solution: suppose all chickens: 2? 35=70 (only) is less than the total number of feet: 94-70=24 (only) The difference between their legs: 4-2=2 (strips) 24? 2= 12 (only)-rabbit 35- 12=23 (only)-chicken
Equation:
Solution: suppose there are x rabbits, then there are 35-x chickens. 4x+2(35-x)= 94 4x+70-2x = 94 2x = 24 x = 12 35-x = 35- 12 = 23
A: Rabbits 12, 23 chickens.
We can also use the method of column equation: let the number of rabbits be x and the number of chickens be y, then: X+Y=35, then 4X+2Y=94. After solving this equation, we can get that there are 65438 rabbits +02 chickens, and 23 chickens have solved the hypothesis.
For this problem, we give the following solutions and corresponding formulas;
Solution 1: (How many feet does a rabbit have? Total-total number of feet)? (The number of rabbits' feet-the number of chickens' feet) = the total number of chickens-the number of chickens = the number of rabbits.
Solution 2: (total number of feet-the number of feet of a chicken? Only)? (number of rabbit feet-number of chicken feet) = total number of rabbits-number of rabbits = number of chickens.
Solution 3: the total number of feet? 2- total number of rabbits = total number of rabbits-number of rabbits = number of chickens
Solution 4: number of rabbits = total number of feet? 2- Total number of rabbits = number of chickens
Solution 5 (Equation): X= (Total number of feet-number of chicken feet? Only)? (number of rabbits-number of chickens) (X= number of rabbits) Total number-number of rabbits = number of chickens.
Solution 6 (Equation): X=: (How many feet does a rabbit have? Total-total number of feet)? (number of feet of rabbits-number of feet of chickens) (X= number of chickens) Total number-number of chickens = number of rabbits.
Number of chickens =(4? Total number of chickens and rabbits-total number of feet of chickens and rabbits)? Number of rabbits = total number of chickens and rabbits-number of chickens
Solution 8 Total number of rabbits = (total number of chickens and rabbits -2? How many chickens and rabbits are there? Number of chickens = total number of chickens and rabbits-total number of rabbits
Solution 9 Total Legs /2- Total Head = Total Rabbit-Rabbit = Chicken
? Chicken and rabbit in the same cage? Mathematical thinking method lies in
First, the idea of conversion
Transformation is a basic and typical mathematical thought. Transformation refers to the problem to be solved, which is reduced to a kind of solved or easily solved problem through transformation, in order to find a solution. We often use this way of thinking, such as turning the unknown into the known, turning the difficult into the easy, simplifying the complex, and turning the song into the straight. ? Chicken and rabbit in the same cage? The data in the original question is relatively large, which is not conducive to students who are exposed to this kind of problem for the first time to explore. According to the idea of simplifying, first arrange the questions with small data, such as? There are some chickens and rabbits in the cage. From the top, there are 7 heads, and from the bottom, there are 18 feet. There are several chickens and rabbits (all named after this). It will be easier for students to find a general method to solve this kind of problem and then apply it to solve the original problem of large amount of data in the calculation of Sun Tzu's Art of War. ? Chicken and rabbit in the same cage? There are many varieties of problems in life, such as? Turtle crane problem? 、? The boat problem? Wait, these questions boil down to? Chicken and rabbit in the same cage? Problems, and then further solve, so that students feel? Chicken and rabbit in the same cage? Variants of problems and their wide application in life, experience? Conversion method? The charm of solving problems.
Second, hypothetical thinking.
Hypothesis is an important mathematical thinking method. Hypothesis method is a method that first assumes a situation or result, and then solves the problem through deduction and verification. Reasonable application of hypothesis method can often make problems difficult, and finding new solutions is conducive to cultivating students' flexible problem-solving skills and developing their logical reasoning ability.
There are many ways to solve the above problems through assumptions. You can assume that all chickens or rabbits are chickens, then calculate the difference between the actual and hypothetical total number of feet, and finally infer the number of chickens and rabbits. For example, if all seven are chickens, then the rabbit has (18-7? 2)? (4-2)=2 (only), and the chicken has 7-2=5 (only). It is difficult to solve the problem by hypothetical methods. The teacher can let the students use the above methods first. Painting? Students establish the appearance of thinking through the combination of numbers and shapes in intuitive operation activities, and then further abstract it, which is helpful for students to really understand? Hypothetical method? Form an orderly and rigorous consciousness of thinking. Teachers can also introduce the ancients to students? Chicken and rabbit in the same cage? Question? Foot lifting method? , this also applies? Hypothetical method? .
Third, the idea of equation.
Equation is an effective model to describe the real world. Translation? Algebraic language establishes the equation between known number and unknown number according to the equivalence relationship between known number and unknown number in the problem, which is the origin of equation thought. Are you online? Chicken and rabbit in the same cage? The problem can be assumed to have X chickens or rabbits, and then solved according to the relationship between the number of chickens and rabbits and the total number of feet. For example, if there are x rabbits, there are (7-X) chickens. The equation can be listed as 4X+2(7-X)= 18, and the solution is X=2, so there are: 7-2=5 chickens. The idea of equation solution is simple and general. In teaching, we should highlight the advantages of equation solution and constantly infiltrate the idea of equation.
Fourth, modeling ideas.
Friedenthal believes that students should study mathematicization, not mathematics. In the primary school stage, it is to abstract some characteristics of mathematical research objects, express them with mathematical language, graphics or patterns, and establish mathematical models. Are you working hard? Chicken and rabbit in the same cage? After the question, students can be guided to observe and think, and the problem-solving model can be summarized and refined: the number of rabbits = (the actual number of feet-the total number of chickens and rabbits? 2)? (4-2), number of chickens = (total number of chickens and rabbits? 4- actual feet)? (4-2)。 Then guide students to consolidate and expand this model in application, and put? Chicken? With what? Rabbit? There are turtles and cranes. What is the variant? Turtle crane problem? 、? The boat problem? 、? Planting trees? 、? Answer the question? And with whom? Chicken and rabbit in the same cage? The connection between the problems makes? Chicken and rabbit in the same cage? Become the model of these problems, and apply the model to solve the problems, and constantly promote the internalization of the model. Teachers should pay attention to the cultivation of students' modeling ideas in teaching, so that mathematical modeling can become an idea and method for students to think and solve problems.
What's on it? Chicken and rabbit in the same cage? From the above discussion, we can see that a solution can contain different mathematical ideas, and different solutions can contain the same mathematical ideas.
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