Notice:
Added a note for the face-to-face class, because the face-to-face class content is the content of the final exam
The first meeting class: the hierarchy and value of mathematical thinking
Question about eating eggs: Mom bought 8 eggs and eats at least 2 eggs a day. How many different ways are there to eat them?
Basic principles: principle of addition (categorical addition), principle of multiplication (multiplication step by step)
1. Classification - from vague to clear:
| one day | A sort of |
| two days | Five ((6,2), (5,3), (4,4)) |
| three days | Six ((2,2,4), (2,3,3)) |
| four days | A sort of |
| total | 13 types |
(1) Classification criteria: order in chaos, certainty in change, extremes in general
(2) Thinking refinement
Method: Carry out scientific classification and study by category according to the different attributes of the research objects.
Principle: The classification criteria are clear and identifiable (scientific), and the classification results are neither redundant nor omitted (ideal).
Value: Turn difficult into easy, complex into simple, whole into parts, general into special (return to normal).
Examples: collections; introduction of concepts; classification of figures; classification of numbers; classification of methods.
Related thoughts: reduction thought; extreme principle; specialization; standardization.
(3) Special emphasis
Classification is based on certain criteria
Types of questions that are suitable for discussion in categories: general, complex, extreme, exploratory
Categorical thinking has its limits
2. Induction - from unknown to possible
Question about eating eggs: Mom bought 100 eggs and ate at least 2 eggs a day. How many different ways are there to eat them?
(1) Start with the simplest case
| total eggs | number of methods |
| 2 | 1 |
| 3 | 1 |
| 4 | 2 |
| 5 | 3 |
| 6 | 5 |
| 7 | 8 |
| 8 | 13 |
| …… | …… |
Obtain sequence: 1,1,2,3,5,8,13,21, 34, 55, 89, 144,… → Fibonacci sequence
(2) Refinement of mathematical thinking
Method: A method of reasoning that draws a general conclusion from a number of individual premises.
Features: expansibility, divergence, and innovation.
Value: from particular to general, from finite to infinite, from part to whole.
Examples: Number laws; Figure laws; Counting problems; Recursion problems.
(3) Special emphasis
The conclusion drawn by inductive thinking may not be correct, but it provides a possible correct direction:
Based on different perspectives and different experiences, the results of induction may be different.
Different inductive results may all be correct, or they may all be wrong, and it is more likely that there are right and wrong.
Inductive thinking is perceptual divergent thinking and creative thinking—this is where the value of inductive thinking lies.
3. Deduction - from possible to certain
Question about eating eggs: Mom buys n eggs and eats at least 2 eggs a day. How many different ways are there to eat them?
(1) Remember that there are an ways to eat n eggs
Goal: Divide the eating methods of n eggs into two categories:
One class is equivalent to the number of ways to eat n-2 eggs, an-2 kinds;
The other category is equivalent to the number of ways to eat n-1 eggs, an-1.
(2) Refining mathematical thinking
Method: According to certain known properties of things, according to strict logic rules, deduce the unknown properties of things. Its structure is a syllogism:
major premise (some general conclusion)
Minor premise (a known conclusion about a particular object)
conclusion (new conclusion for this particular object)
Features: Rigorous, reliable, restrained.
Value: Establish conclusions.
Examples: proofs of various mathematical theorems, formulas, rules, etc.
4. Analogy - from one thing to all things
Question about eating eggs: Mom bought 8 eggs and ate at least 3 eggs a day . How many different ways are there to eat them?
(1) Three eggs to m eggs
(2) Extract the essence:
Fibonacci sequence
Climbing stairs: If a person climbs stairs in two ways: one step at a time or two steps at a time. How many different ways are there to climb 12 steps?
There are two ways to climb up the nth step:
From the n-1th step, there are an-1 kinds of climbing 1st step;
From the n-2th step, there are an-2 ways to climb 2 steps.
(3) Refinement of mathematical thinking
Method: By comparing the sameness or similarity of two things in some aspects, it is speculated that they are also the same or similar in other aspects, or results, or methods.
Features: Transplantability, expandability, divergence, and creativity.
Value: Inspiring ideas, providing clues, and understanding by analogy.
Example: Discovery of many mathematical theorems, formulas, laws, etc.
(4) Jiaxian Triangle
Second Encounter Lesson: Cultural Elements in the Fibonacci Sequence
1. Recursion: Contains invariance and cycles
(1) Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
Features: Starting from the third item, each item is the sum of the previous two items
(2) Two basic perspectives for studying sequence: difference and summation
差分：an+1 – an = an-1
Summation: Sn = an+2 - 1
(3) Duality of partial sums of sub-columns
Part of the Fibonacci Sequence's Odd-Numbered Subsequence and the Even-Numbered Subsequence
The sum of the part of the even-numbered sub-sequence of the Fibonacci sequence is the odd-numbered sub-sequence minus 1
(4) fragments and:
① The unity of sum and difference
②Mean value
In fact, the sum of any 10 consecutive items satisfying an+2 = an+1+an is equal to 11 times the seventh item of the 10 items.
(5) Quasi-periodical
1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , …
①Remainder law: Odd-even-Odd-even-Odd-even-Odd-even-…
②Remainder when divided by 3: 1120-2210-1120-2210-……
Remainder of division by 5:, 11230-33140-44320-22410-11230-……
2. General item: Seemingly unreasonable but reasonable
The general term of the Fibonacci sequence can be expressed by three irrational numbers
3. The combination of numbers and shapes: the attachment of lines and surfaces, and the integration of square moments
(1) Product series of related items
The square number sequence of the Fibonacci sequence is the core of the product sequence of related items of this sequence.
① The product of any adjacent two terms is equal to the sum of the squares of the smaller term and its predecessor.
② The difference sequence of the adjacent product sequence of the Fibonacci sequence is its square sequence
③The partial sum of the product sequence of the Fibonacci sequence interval term is the partial sum of its square sequence
④ The sum of the squares of any adjacent two terms
(2) The asymptotic stability of the quotient of adjacent two terms
4. Discrete and continuous similarity, local overall matching
(1) Jiaxian triangle and Fibonacci sequence
(2) Continued fractions and Fibonacci sequence
(3) Exponential function, Fibonacci sequence and geometric sequence
5. Integration of nature and society, unity of mathematics and art
Third meeting class: Mathematics and magic
1. Chaos and Order, Change and Persistence
1. Calendar guessing
Note that the upper right corner of the box is n, and the result is 4n+48.
The difference between the numbers in each row is 1.
The difference between the numbers in each column is 7.
Each row and column in the box has one and only one number selected.
The sum of the four numbers is 4n+6+42 = 4n+48
2. Poker guessing
3. The magical 1089
A hundred different three-digit numbers
B A write backwards
C |A - B|
D C write backwards
E C + D
E = 1089
2. Odd and even 1 and 0
1, 4 Kings
2. Which card was moved?
3. Guess the last name (binary game)
3. Sequence and number, division and combination
1. Poker Magic
(1) Pigeon cage principle: There must be two cards of the same suit among the five cards → take out one card and put the other card first
(2) There are three left → 1 ~ 6
Different points depend on the size, and the same points depend on the suit color (spades > hearts > clubs > squares)
| Large, medium and small |
6 |
| medium size | 5 |
| medium, large and small | 4 |
| small, medium, large | 3 |
| small big medium | 2 |
| Small, Medium, Large | 1 |
(3) Which one to keep?
When it does not exceed 6, keep the small one and show the difference in points
When it exceeds 6, keep the big one and put out "small + (13 - big)".
2, 4 Aces
3. Count to your favorites
4. Perfect in every way
The fourth meeting class: Thinking outside the box
1. Three ancient mathematical problems (5th century BC - 1837 - 1882)
1. Problems: squaring a circle, doubling a cube, trisecting an angle
2. Breakthrough: Descartes' Analytic Geometry (1637)
Algebraic Interpretation of Geometry Problems (1837)
The problem of drawing with ruler and compasses is transformed into the problem of the intersection between straight lines and circles
The problem of intersection point is transformed into the problem of root of equation of one degree and two times
The root problem is transformed into the four arithmetic operations and the square root operation problem of known quantities
3. Reduction : Rulers and compasses can only make the sum and difference product quotient and square root of known quantities
4. Solve: French mathematician Wan Qier noticed the algebraic interpretation of the problem of geometric construction, and answered the problems of trisecting angles and doubling cubes
Lindemann proved that π is a transcendental number and answered the question of squaring a circle
2. Fermat's Conjecture (1637-1994)
1. Problem: When the integer n>2, there is no positive integer solution for x^n + y^n = z^n
2. Breakthrough: Reduction to algebraic problems → the solvability of positive rational solutions to the equation x^n + y^n = 1
Reduction geometry problems → the existence of positive rational points on the plane curve x^n + y^n = 1
(Curve classification: rational curve, elliptic curve, other curves)
Modal's conjecture: there are at most finitely many rational points on each curve of the third kind
Shimura-Taniyama-Waiyi Conjecture: Every curve of the second kind over the field of rational numbers is a modular curve
3. Solve:
In 1983, the French mathematician Fatins proved the Modal conjecture
In 1985, German mathematicians proved that if the Fermat conjecture is not true, then the Shimura-Taniyama-Waiyi conjecture is not true
In 1994, the British mathematician Wiles proved the Shimura-Taniyama-Waiyi conjecture
3. Goldbach’s Conjecture (1742——)
1. Question: Every even number greater than or equal to 6 is the sum of two odd prime numbers
2. Method: Analytical Number Theory
3. Development: factor Goldbach problem
4. Sprint: In 1966, Chen Jingrun used the sieve method to prove (1+2): 2n = p + q or 2n = p + q1 * q2
4. Four-color conjecture (1852-1976)
1. Problem: To draw any map, as long as four colors can effectively distinguish different regions
2. Development: Kemp proposed the concept of a regular map (if no country surrounds other countries, or no more than three countries meet at one point, such a map is said to be "regular")
Prove that any map can be modified into a regular map without adding cartographic colors
3. Solve
(1) Four-color theorem: In July 1976, two mathematicians Kenneth Appel and Wolfgang Hanken from the University of Illinois in the United States proved the establishment of the "four-color conjecture" by computer
(2) Five-color theorem: In 1890, the 29-year-old British mathematician Sheawood proposed
5. The Poincaré Conjecture (1904-2004)
Question: Any simply connected three-dimensional closed manifold must be homeomorphic to the unit sphere of R4: x^2 + y^2 + z^2 + t^2 = 1
6. Riemann Hypothesis (1859——)
