This article is "The Third Mathematical Culture Essay Contest."
"Approaching Yang Hui and Revealing the Secret Triangle" class transcript
Author: He Ping
Artwork no.: 057
1. Wen Ying Chishin
Teacher: Students, let's recall the polynomial and polynomial multiplication algorithms we learned earlier.
Raw (all): Multiply the polynomial and polynomial: The polynomial is multiplied by each term of one polynomial by each term of another polynomial, and then the resulting product is added. (PPT display)
Teacher: The students just answered the text language description of the polynomial and polynomial multiplication arithmetic, we pay attention to the expression of the three languages of text language, graphic language and symbolic language in junior high school, so what is the symbolic language it corresponds to?
Raw (all) :(a+b)(m+n) = am+an+bm+bn (PPT display)
Teacher: After learning the multiplication algorithm of polynomials and polynomials, what special formulas did we learn?
Raw 1: Complete square formula
Student 2: Square difference formula.
Teacher: These two types of formulas are very important, can students express the full square formula in symbolic language?
Raw (whole):
【Design Intent】On the basis of reviewing the polynomial multiplication algorithm, guide students to calculate the high power of (a + b), paving the way for extracting the expandable coefficients.
Second, the exploration of new knowledge
Activity 1:
Teacher: Before the class, I asked the students to complete the pre-study plan, and then we proofread it together.
Projection student works:
1. Calculate using the multiplicative formula of the integer (the results are arranged in order of the number of a from largest to smallest)
Teacher: The first question requires students to write out the expansion of 、、、 (a+b≠0) according to the power reduction of a.
Results of students completing exercises:
Projections:
2. Write out the expansion coefficients in the previous question by line.
Line 1: 1
Line 2: 1 1
Line 3: 1 2 1
Line 4: 1 3 3 1
Line 5: 1 4 6 4 1
Line 6: 1 5 10 10 5 1
Line 7: 1 6 15 20 15 6 1
Teacher: The students did a great job! We fill in the completed expandable coefficients in the table below.
PPT display:
Teacher: We took a group photo of these unfolding coefficients and changed the formation of the group photo.
Teacher: What kind of graphics are formed by numbers like this?
Raw (whole): Triangle, isosceles triangle.
Teacher: Very good, the figure composed of these numbers is a triangle, and we call this triangle coefficient table the Yang Hui triangle.
【Design intention】Through the students' self-starring calculation, the practice operation obtains the unfolding coefficient, and after transforming the unfolding coefficient into a formation, the triangle coefficient table is formed, and the first veil of Yang Hui's triangle is revealed from the perspective of "shape".
Activity 2:
Teacher: Students, can you continue to write the expanded coefficients of the 、、、...、?
Student 1: Yes! We can continue to operate using polynomial and polynomial multiplication algorithms.
Teacher: Students, how to solve problem types containing n in the past?
Student 2: According to the simple situation of the topic, gradually deduce, write out the law, summarize the law, and write out the nth case.
Teacher: That's a good point! Therefore, the expandable coefficient we find can not be used by analogy, according to the existing 、、、、、、 (a + b≠0) of the expandable coefficient, that is, the existing Yang Hui triangular data to explore the law, so as to obtain the expandable coefficient according to the law.
Teacher: Next, ask the students to observe Yang Hui's triangle and look for the law.
(Give students plenty of time to think, discuss, communicate, and summarize)
Student 3: Yang Hui triangle is 1 on both sides.
Student 4: Each number is equal to the sum of the two numbers above the head.
Teacher: Very good, the students observed the most important and basic law of Yang Hui's triangle.
Board Performance: Regularity 1. Two shoulder sum.
Teacher: Students, what other rules do you have? What does this shape look like?
Student 5: Triangle!
Raw 6: Isosceles triangle!
Teacher: Very good. The figures composed of these numbers can be seen as isosceles triangles, what is the essence of the isosceles triangle?
Student 7: The isosceles triangle is an axisymmetric figure.
Teacher: Awesome! Isosceles triangles are the most typical axisymmetric figures, : what about the axis of symmetry?
Student 8: A middle column of numbers.
Teacher: We can find that when we draw the axis of symmetry, the odd row is a number in the middle, and the middle of the even row is blank, which is equal to the two numbers of "equal distance" at the end of the first and last two ends. Therefore, we now find the second law of Yang Hui's triangle from the perspective of "shape".
Board Performance: Regularity 2. Symmetry.
Teacher: In the above activity, we looked at the relationship between Yang Hui's trigonometric numbers and numbers from the perspective of "numbers", so what about looking at some numbers as a whole? For example, the sum of the rows of Yang Hui's triangle is added up, is there any feature? (Give students enough time to calculate)
Sheng 9: After calculation, it is found that the sum of each row is related to the higher power of 2. For example, the second row sum is 2, the third row sum is 4, the third row sum is 8, the fourth row sum is 16, the fifth row sum is 32, and the sixth row sum is 64, these numbers are exactly the power of 2.
Teacher: The students answered very well, and the teacher and the students wrote it together.
(Board on the blackboard)
Board Acting: Regularity 3.
The Beauty of The Law 1: The two hypotenuses in Yang Hui's triangle are composed of the number 1, and each number is the sum of the two numbers on the shoulder.
Beauty of the Law 2: Equal to the two numbers of the "equidistant distance" at the end of the head, yanghui triangle has symmetry.
Beauty of the Law 3: The sum of n numbers in the nth row of Yang Hui's triangle is equal to the n-1 power of 2.
【Design Intention】Highlight the classroom mode with students as the main body, guide students to independently observe the Yang Hui triangle number table, intuitively feel the law of numbers, and students summarize the relationship between the summarized numbers through exploration. After analyzing the Yang Hui trigonometric table from the two angles of "number" and "shape", the students are guided to look at the Yanghui trigonometric table from the overall perspective, that is, the sum of the Yanghui trigonometric table is calculated "horizontally" and looking for the law. At the same time, it lays the groundwork for the subsequent observation of Yang Hui's triangle from the "oblique direction".
Teacher: Through the above activities, we look horizontally and vertically to find the law, and now we look obliquely, and ask the students to observe whether there are any characteristics of the oblique direction.
Student 10: A string of numbers diagonally on the left side adds up to equal the number in the lower right.
Student 11: Right, right, right! A string of numbers diagonally on the right side adds up to equal the number in the lower left.
Teacher: The students found out really fast!
Board Acting: Oblique row and.
Beauty of the Law 4: Starting from the "left (right) shoulder" of a definite number in Yang Hui's triangle, make a ray parallel to the left oblique side above the right (left), and the sum of each number on this ray is equal to this number.
Teacher: From the above 4 laws, different directions and different angles to see the Yang Hui triangle, "horizontally looking into the peak of the ridge side, far and near the height are different" in the Yang Hui triangle is vividly displayed, yang hui triangle is really beautiful!
Teacher: Now the teacher leads the students to continue to feel the charm of Yang Hui's triangle. Ask the students to add each diagonal line marked by the teacher, observe and characterize.
Raw 12: 1, 1, 2, 3, 5, 8, 13...
Teacher: What are the characteristics of this seemingly chaotic string of numbers?
Student 13: Growing.
Teacher: Yes, are there any more special laws?
Student 14: Each number is the sum of the first two numbers.
Teacher: Awesome! From the third number onwards, either number is equal to the sum of the first two numbers, which is the famous Fibonacci sequence, i.e. the rabbit sequence.
Board Acting: Regularity 5. Fibonacci sequence.
Beauty of the Law 5: Look obliquely at the sum of the numbers in Yang Hui's triangle, from the third number, any number is equal to the sum of the first two numbers.
【Design Intent】The Fibonacci sequence is an interesting representative example in the history of mathematical culture, which is worth promoting and learning, thus leading to fibonacci's discovery of the law of rabbit sequences.
Teacher: Next, let's take a look at the interesting Fibonacci sequence.
The medieval Italian mathematician Fibonacci's masterpiece The Law of Arithmetic raises an interesting question: suppose that a pair of newborn rabbits can grow into large rabbits in one month, and in another month they will begin to give birth to a pair of rabbits, and every month thereafter they will give birth to a pair of rabbits. The pair of rabbits born is a male and a female, and neither dies. Q How many pairs of rabbits can a pair of newborn rabbits breed into in a year? (Figure 1)
Figure 1
Rabbit breeding questions can also be answered from Yang Hui's triangle:
1，1，2，3，5，8，13，21，34，．．．
【Design Intention】By explaining the origin of the Fibonacci sequence, guide students to realize that mathematics comes from life and is applied to mathematics. At the same time, the introduction of mathematician Fibonacci loves life, is good at observing life, and records the phenomena in life to leave valuable spiritual wealth, guides students to establish the concept of research-based learning, and infiltrates the idea of mathematical modeling.
Teacher: In addition to the 5 laws we found in the Yang Hui Triangle, there are also some mysterious laws that need to be excavated by students after class, and students can continue to study when they enter high school.
Third, the time train
Teacher: Students, we have learned so many inherent laws of yanghui triangle, and found that Yanghui triangle is not only mysterious, but also very beautiful, who first discovered such an interesting Yanghui triangle? How did it develop? Let's walk into the YangHui Triangle through the form of micro-lessons, understand the story behind it, and unveil its mysterious veil.
Micro-lesson demonstration:
(Accompanied by the background music of "Alpine Flowing Water", the mathematical history of the origin and development of Yang Hui's triangle is told)
First of all, we will introduce the ancient mathematician Jia Xian, an outstanding mathematician of the first half of the eleventh century in China (Northern Song Dynasty). He wrote "The Yellow Emperor's Nine Chapters of Algorithm Fine Grass" (nine volumes) and "Ancient Collection of Algorithms" (two volumes), both of which have been lost. According to the "History of Song", Jia Xian studied astronomy and almanac from the mathematician Chu Yan, and wrote books such as "The Yellow Emperor's Nine Chapters Of Algorithm Fine Grass" and "Book of Interpretation and Lock Calculation". Jia Xian's main contribution was to create the "Jia Xian Triangle" and the "Method of Increasing Multiplication".
Then introduce the protagonist of this lesson, the ancient mathematician Yang Hui, born in the Southern Song Dynasty, in 1261,1261 in the book "Detailed Explanation of the Nine Chapters of The Algorithm", compiled a triangular number table, called the "origin of the opening method" diagram (figure 2), and explained that this table is quoted from the middle of the 11th century (about 1050 AD) Jia Xian's "Lock Interpretation Arithmetic", and painted the "Ancient Seven Multiplication Diagram". Yang Hui made Jia Xian's diagram of The Method of Opening the Fang in the chapters of "Detailed Explanation of the Nine Chapters Algorithm" and "The Origin of the Opening Method", and explained that "jia Xian used this technique to interpret the lock arithmetic book".
Figure 2
The famous French mathematician, physicist and philosopher Pascal discovered this law at the age of 13 (1654), so this table is also called Pascal's triangle. Pascal's discovery was 393 years later than Yang Hui's and 600 years later than Jia Xian's. Pascal published On Arithmetic Triangles in 1665, which talked about the construction and properties of "arithmetic triangles" and was the first to prove properties using mathematical induction.
The Discovery by the Italian mathematician Tartaglia predates Pascal by more than 100 years and about 500 years after Jason.
Finally, he quoted the famous words of the modern mathematician Hua Luogeng, who is well known to students, as a concluding remark: "Mathematics is a subject that our people are good at. The great people of our motherland have made incomparably wise achievements in the history of mankind. ”
【Design Intent】While describing the achievements of Jia Xian and Yang Hui, it details the ins and outs of Yang Hui's triangle, and informs that the Yang Hui triangle recognized today is actually Jia Xian's earliest discovery of this mysterious mathematical knowledge, a beautiful misunderstanding is full of mathematical culture interest, stimulating students' interest in learning. At the same time, the analogy leads to the historical facts of western countries about the Yanghui Triangle, which fully demonstrates the brilliant achievements of ancient Chinese mathematicians, stimulates students' national pride, and establishes the belief in learning mathematical knowledge well.
Teacher: Students, through taking the time train, we learned about the origin and development of Yang Hui's triangle, felt the great achievements made by our nation in the long river of mathematical culture, and we should have more confidence in learning mathematics well and inheriting and carrying forward our dominant position in mathematics!
Fourth, the past is used for the present
Teacher: Students, yanghui triangle is so wonderful, its use is certainly not limited to rabbit breeding problems, in life can often use the Yanghui triangle to solve practical problems, please look at this problem.
"Vertical and horizontal roadmap" is an interesting class of problems in mathematics. Figure 3 below is a partial street map of a city, each with three roads, if you walk from A to B (only from north to south, from west to east), then how many different ways of walking?
【Design Intent】Through interesting questions in life, students are triggered to think, hands-on practice, brainstorm, and students draw conclusions.
Teacher: After the examination is completed, please think about how many different moves you can get?
Raw 1: 5 species.
2: 6 species.
Teacher: Well, I invite the students with 5 types of answers to the blackboard to demonstrate, and the students with 6 kinds of answers can add the answers later.
Students take the stage to demonstrate. (After one classmate demonstrates 5 situations, another classmate comes on stage to add 1 situation)
Teacher: After the practical operation of the two students, we found that the answer was 6.
Teacher: As shown in Figure 3, turn the figure 45 degrees clockwise to Figure 4, so that A is directly above and B is directly below, and then mark the corresponding Yang Hui triangle number at the intersection point. What is the relationship between the Yang Hui triangle number at B and the A to B move?
Birth 3: The number 6 corresponding to the B place is exactly the answer 6.
Teacher: Students think, what secrets may be hidden in this?
Sheng 4: The number of Yang Hui triangles on each intersection is the number of methods used to reach that point from A.
Teacher: Great! We found that the original route problem in life is also related to Yang Hui's triangle, which is a great conclusion!
【Design Intention】The route problem in the actual life will be solved with the help of Yang Hui's triangle knowledge, and the new knowledge learned in this lesson will be naturally transitioned, reflecting the application value of Yang Hui's triangle.
Variant: "Vertical and horizontal roadmap" is an interesting class of problems in mathematics. Figure 5 below is a partial street map of a city, each with five roads, if you walk from A to B (only from north to south, from west to east), then how many different ways of walking?
Teacher: We have changed the three roads in the question into five roads, and students can try them after class. The teacher tried to demonstrate to the students with the Yang Hui Triangle, and we found that there were 70 kinds.
Teacher: Originally, the Yang Hui Triangle has a natural connection with the problem of the vertical and horizontal road map, so it can be seen that the Yang Hui Triangle is everywhere in life and the utilization value is quite high.
【Design Intent】The variant question uses the same method to transform the road map into Figure 6, with the help of Yang Hui Triangle Figure 7. With the help of students' travel experience, students are stimulated to be interested in this question, use their brains to think about the answers, and make hands-on operations to draw conclusions.
Fifth, improve thinking
Teacher: Is Yang Hui's triangle pure appreciation, and in the examination, will there be no Yang Hui triangle questions? Now the middle school entrance examination likes to bring mathematicians to join in the fun, so Yang Hui Triangle is also a frequent guest of the middle and high school entrance examination questions. Next, please ask the students to appreciate this question:
(2006 Middle School Examination Rizhao City Volume 17) German mathematician Leibniz discovered the following unit fraction triangle (the unit fraction is a fraction with a numerator of 1 and a denominator as a positive integer):
According to the law of the first 5 lines, it can be known that the number of the 6th line is: .
Teacher: Students, what was your first feeling and reaction when you saw this question?
Student 1: Yang Hui triangle, very similar in shape.
Teacher: Awesome! Does it look a bit like that? Let's look at the inside of this table of numbers together, is each number the same?
Student 2: The number of numbers is the same, but each number is different.
Teacher: Very good, as long as it is similar, we can solve it by analogy. We now have to find the same and different, what did the students find?
Student 3: The numbers have become fractions, and the numerators are all 1.
Teacher: Since they are all 1, what can we do?
Student 4: Remove 1 first.
Teacher: Very good! And then the numbers are the same?
Birth 5: The denominator seems to be regular and symmetrical.
Teacher: And then what?
Sheng 6: Can be seen as the deformed Yang Hui triangle.
Teacher: How did it change?
Student 7: After removing the molecule 1, the denominator number begins with 1, 2, 3, 4, 5 per line, while Yang Hui's triangle begins with 1, 1, 1, 1, 1.
Teacher: What can we do about it?
Raw 8: Can divide each row by the number of rows.
Teacher: Great!
Teacher: Through the deformation of these two steps, the familiar Yang Hui triangle is obtained. According to the laws learned today, what is the number in line 6 that can be obtained?
Sheng: The number in the sixth row of Yang Hui's triangle is 1, 5, 10, 10, 5, 1, and then multiplied by the number of rows to become 6, 30, 60, 60, 30, 6, and then count down each number separately.
【Design Intention】With the help of this question, guide students to understand the possible forms of Yang Hui Triangle in the examination, and guide students to grasp the idea of analogy and the method of transformation. Help students first observe the unit score triangle in the question, gradually guide the search for the difference between the unit score triangle and the Yang Hui triangle, step by layer, analyze in depth, find out the law, and solve the problem with the help of Yang Hui's knowledge of the triangle.
6. Class summary
Teacher: Students, today we feel the inner mystery of Yang Hui's triangle through the reading of Yang Hui's triangle story, experience the fun of mathematical culture, appreciate the beauty of mathematical culture, and be deeply impressed by the charm of mathematical culture. With a sense of national pride, the students stride forward and continue to explore the mysteries of mathematics!
7. Operation design
1. Consult the relevant information of "Yang Hui Triangle".
2. Review and consolidate the laws of "Yang Hui Triangle".
8. Board book design
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