1, divide the integer 1 into 10, 100, 1000 ... These fractions can be expressed in decimals. For example, the record of110 is 0. 1, and the record of 7/ 100 is 0.07.
2. The first digit to the right of the decimal point is called ten digits, and the counting unit is one tenth (0.1); The second number is called the percentile, and the counting unit is one hundredth (0.0 1) ... The maximum counting unit of the decimal part is one tenth, and there is no minimum counting unit. There are several figures in the decimal part, which are called several decimals. For example, 0.36 is two decimal places and 3.066 is three decimal places.
3. Decimal reading: read the integer part, decimal point and decimal part in turn.
4. Decimal writing: the decimal point is written in the lower right corner of the unit.
5. The essence of decimals: adding 0 at the end of decimals will not change the size. Simplify the size change caused by the movement of decimal point position: move to the right to expand and contract to the left,1.1230,000 times.
6. Decimal size comparison: the integer part is large; If the integers are the same, ten digits will be big; And so on.
Second, scores and percentages.
(a) the meaning of score and percentage.
1, the meaning of the score:
Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction. In the fraction, the number indicating how many parts the unit "1" is divided into on average is called the denominator of the fraction; A number indicates how many copies have been made, which is called the numerator of the fraction; One of them is called fractional unit.
2, the meaning of percentage:
A number indicating that one number is a percentage of another number is called a percentage. Also called percentage or percentage. Percentages are usually not written in the form of fractions, but expressed in concrete "%". Generally, the percentage only indicates the multiple relationship between two quantitative relationships, and cannot be related to the company name.
3. Percent indicates the multiple relationship between two quantities, and the unit of measurement cannot be written behind it.
4, into a number: a few percent is a few tenths.
(2) the type of score.
According to the different conditions of numerator, denominator and integer, it can be divided into true fraction, false fraction and band fraction.
(3) The relationship between fraction and division and the basic properties of fraction.
1, division is an operation with operation sign; The score is a number. Therefore, it should generally be said that dividends are equivalent to a molecule, but it cannot be said that dividends are a molecule.
2. Because there is a close relationship between fraction and division, the basic properties of fraction can be obtained according to the properties of "constant quotient" in division.
3. The numerator and denominator of the score are multiplied or divided by the same number (except 0), and the size of the score remains unchanged. This is called the basic nature of fraction, which is the basis of divisor and total fraction.
(4) Approximate points and general points.
1, where both numerator and denominator are fractions of prime numbers, is called simplest fraction.
2. Turning a fraction into a fraction equal to it, but with smaller numerator and denominator, is called a reduced fraction.
3. Reduction method: divide the denominator by the common divisor of the denominator (except 1); Usually, we have to separate it until we get the simplest score.
4. Changing scores of different denominators into scores of the same denominator is equal to the original score, which is called the total score.
5. General division method: first find the least common multiple of the original denominator, and then turn each fraction into a fraction with this least common multiple as the denominator.
Third, the countdown.
Two numbers whose product is 1 are reciprocal.
To find the reciprocal of a number (except 0), just switch the numerator and denominator of this number.
3. The reciprocal of 1 is1,and 0 has no reciprocal.
Fourth, the comparison of scores.
1, the fraction with the same denominator, the larger the numerator, the greater the fraction.
2. The scores with the same numerator are larger with smaller denominator.
3. Fractions with different denominators and numerators are usually divided first, converted into fractions with a common denominator, and then compared.
4. If the scores to be compared have scores, compare their integer parts first, and the score with the larger integer part is larger; If the integer parts are the same, then compare their decimal parts, and the decimal part with the largest decimal part is the largest.
Five, percent and fold, into a number of mutual:
A 30% discount is a 30% discount, a 75% discount is a 75% discount, and the percentage is a few tenths. For example, a discount of 10% means poor quality. 0%, 65% is 65%.
Intransitive verbs tax and interest:
1, tax rate: the ratio of taxable amount to various incomes.
2. Interest rate: the percentage of interest and principal. Calculated by the bank on an annual or monthly basis.
3. Calculation formula of interest: interest = principal × interest rate × time.
Seven, the difference between percentage and score mainly has the following three points:
1 has different meanings.
Percent is "a number indicating that one number is the percentage of another number." It can only represent the multiple relationship between two numbers, not a specific quantity. For example, it can be said that 1 meter is 20% of 5 meters, and it cannot be said that "a rope is 20% meters long." Therefore, the percentage cannot be followed by the company name. The score is "divide the unit'1'into several parts on average, indicating the number of such parts or parts". Fraction can not only represent the multiple relationship between two numbers; It can also represent a certain amount.
2, the scope of application is different.
Percentages are often used for investigation, statistics, analysis and comparison in production, work and life. Fractions are usually used for measurement and calculation when integer results are not available.
3. Different writing forms.
Percentages are usually not expressed in fractional form, but in percent sign "%". Such as: 45%, writing: 45%; The denominator of percentage is fixed as 100, so no matter how many common divisors there are between the numerator and denominator of percentage, it is not irreducible; Percentages of molecules can be natural numbers or decimals. The numerator of a fraction can only be a natural number, and its expressions include true fraction, false fraction and banded fraction. The calculation result is not that simplest fraction is generally reduced to simplest fraction, but that the false score is converted into a banded score.
Number is divisible.
1, the meaning of divisibility.
(1) When the integer A is divided by the integer b(b≠0), the quotient is exactly an integer with no remainder, so we say that A is divisible by B (that is, B is divisible by A).
(2) The meaning of dividing A by B. When the quotient obtained is an integer or a finite decimal and the remainder is 0, we say that A can be divided by B (or B can divide A by B). Here a and b can be natural numbers or decimals (b cannot be 0).
2, divisor and multiple.
(1) If the number A is divisible by the number B, say A is a multiple of B and B, and say A is a divisor. ..
(2) The divisor of a number is finite, in which the smallest divisor is 1 and the largest divisor is itself.
(3) The number of multiples of a number is infinite, of which the smallest is itself, and it has no maximum multiple.
3. Odd and even numbers.
(1) Numbers divisible by 2 are called even numbers. For example: 0, 2, 4, 6, 8, 10 ... Note: 0 is an even number.
(2) Numbers that are not divisible by 2 are called cardinality. For example: 1, 3, 5, 7, 9 ...
4. The characteristics of separability.
(1) Features of a number divisible by 2: every bit of 0, 2, 4, 6 and 8.
(2) The characteristics of a number in a unit that can be divisible by 5: 0 or 5.
(3) Characteristics of numbers divisible by 3: The sum of' numbers on each bit of a number can be divisible by 3, and this number can be divisible by 3.
5. Prime numbers and composite numbers.
(1) A number has only 1 and its own two divisors. This number is called prime number.
(2) A number has other divisors besides 1 and itself, and this number is called a composite number.
(3) 1 is neither a prime number nor a composite number.
(4) Natural numbers can be divided into prime numbers and composite numbers according to the number of divisors.
(5) Natural numbers can be divided into odd and even numbers according to whether they are divisible by 2.
6. Prime factor decomposition.
(1) Every composite number can be written as the product of several prime numbers, which is called the prime factor of this composite number. For example, 18=3×3×2, and 3 and 2 are called prime factors of 18.
(2) Multiplying several prime factors to represent a composite number is called prime factor decomposition. Short division is usually used to decompose prime factors.
(3) The common factor of several numbers is called the common factor of these numbers. The largest one is called the greatest common factor of these numbers. Two numbers whose common factor is only 1 are called prime numbers. The common multiple of several numbers is called the common multiple of these numbers. The largest one is called the greatest common multiple of these numbers.
(4) The greatest common divisor and the least common multiple of several numbers under special circumstances.
(1) If among several numbers, the larger number is a multiple of the smaller number and the smaller number is a divisor of the larger number, then the larger number is their least common multiple and the smaller number is their greatest common divisor.
② If several numbers are pairwise coprime, their greatest common divisor is 1, and their least common multiple is the product of these numbers.
7, parity operation properties:
(1) The sum of two adjacent natural numbers is odd and the product is even.
(2) Odd+odd = even, odd+even = odd, even+even = even; Odd-odd = even, odd-even = odd, even-odd = odd, even-even = even; Odd× odd = odd, odd× even = even, even× even = even.
Nine, integer, elementary school, fractional elementary arithmetic.
(1) Four algorithms.
1, addition a, integer and decimal:
Alignment with the same number, starting from the low place, to 10 into1b. Fractions with the same denominator: denominator unchanged, numerator added; Fractions with different denominators: divide first, then add.
2. Subtract A, integer and decimal:
Align the same digits, starting from the low order, which digit is not enough to be reduced, and then subtract B when one is ten, and the same denominator score: the denominator is unchanged, and the numerator is reduced; Fractions with different denominators: divide first, then subtract.
3. Multiply a, integer and decimal:
Multiply the multiplicand by the number on each bit of the multiplier, and the last bit of the number will match the last bit. Finally, add the product, the factor is decimal, and the decimal places of the product are the same as those of the two-digit factor. B. Fraction: The product of numerator multiplication is numerator, and the product of denominator multiplication is denominator. What can be reduced should be reduced first, and the result should be simplified.
4. Part A, Integers and Decimals:
How many factors are there? Look at the first few digits of the dividend first (look at one digit if it is not enough). In addition to the dividend, which one is written by the Chamber of Commerce? The divisor is a decimal, which is converted into an integer and divided by it. The decimal point of quotient is aligned with the decimal point of dividend. B, the number A divided by the number B (except 0) is equal to the reciprocal of the number A divided by the number B.
(2) Operating rules.
1, additive commutative law: a+b = b+a.
2. Constraint law: (A+B)+C = A+(B+C)
3, the essence of subtraction:
( 1)a-b-c=a-(b+c)
(2)a-(b-c)=a-b+c
4. Multiplicative commutative law: a× b = b× a.
5. Constraint law: (a×b)×c=a×(b×c)
6. Distribution law: (a+A+B )× C = A× C+B× C
7. The nature of the division of labor:
( 1)a \(b×c)= a \b \c
(2)a \(b \c)= a \b×c
(3)(a+b)÷c=a÷c+b÷c
(4)(a-b)÷c=a÷c-b÷c
The quotient invariant property m≠0 a÷b=(a×m)÷(b×m) =(a÷m)÷(b÷m)
(3) Law of product change: In multiplication, if one factor remains unchanged and another factor is expanded (or reduced) several times, the product is also expanded (or reduced) by the same multiple.
Summary: One factor magnifies factor A, another factor magnifies factor B, and the product magnifies factor AB. One factor subtracts one factor, another factor subtracts B, and the product subtracts AB.
(4) Law of quotient invariance: In division, the dividend and divisor are expanded (or reduced) by the same multiple at the same time, and the quotient remains unchanged.
Generalization: the dividend enlarges (or reduces) a factor, and the quotient enlarges (or reduces) a factor, while the divisor remains the same. The dividend is constant, the divisor is enlarged (or reduced) by a factor, but the quotient is reduced (or expanded) by a factor.
(5) Using the changing law of product and the properties of quotient invariance law, some calculations can be simplified. But pay attention to the remainder in the division with remainder. For example: 8500÷200= Divider and divisor can be reduced by 100 times at the same time, that is, 85472 =, the quotient remains unchanged, but the remainder 1 minus 100, then the original remainder should be 100.
X. simple equation.
(1) Numbers are represented by letters.
Representing numbers by letters is a basic feature of algebra. It is not only simple and clear, but also can express the general law of quantitative relationship.
(2) Notes on using letters to indicate numbers.
1. When a number is multiplied by letters, letters and letters, the multiplication sign can be abbreviated as ""or omitted. Numbers are multiplied by numbers, and the multiplication sign cannot be omitted.
2. When 1 is multiplied by any letter, omit "1".
When a number is multiplied by a letter, write the number before the letter.
(3) Formulas containing letters and their evaluation.
Pay attention to the writing format when finding the value of a formula containing letters or evaluating it with a formula.
(4) Equations and equations.
The expression of equality is called equality. Equations with unknowns are called equations.
There are two conditions to judge whether a formula is an equation: first, it contains unknowns; The second is the equation. So the equation must be an equation, but the equation is not necessarily an equation.
(5) The solution of the equation and the solution of the equation.
The value of the unknown that makes the left and right sides of the equation equal is called the solution of the equation. The process of solving an equation is called solving an equation.
(6) When solving a text problem of a set of equations, if the unknown number required in the problem has been expressed in letters, then it is unnecessary to write it when solving, otherwise, the required unknown number is set to X first.
(7) Method of solving equations.
1, directly using the relationship between parts in the four operations to solve. For example, x-8 = 12
(1)① addend+addend = sum, ② one addend = and-another addend.
(2)① subtraction-subtraction = difference, ② subtraction = subtraction-difference, ③ subtraction = difference+subtraction.
(3)① Multiplier × Multiplier = product, ② One factor = product ÷ another factor.
(4)① Divider = quotient, ② Divider = Divider = quotient, ③ Divider = divisor × quotient.
2. First, consider the term containing the unknown x as a number, and then solve it. If 3x+20=4 1, consider 3x as a number first and then solve it.
3. Calculate according to the sequence of four operations, deform the equation, and then solve it. If 2.5×4-x = 4.2, first find the product of 2.5×4, transform the equation into 10-x = 4.2, and then solve it.
4. Transform the equation by algorithm or property, and then solve it. For example, 2.2x+7.8x = 20. Firstly, the equation is transformed into (2.2+7.8) x = 20 by using the algorithm or properties, then the equation is transformed into 10x = 20 by calculating the brackets, and finally the solution is made.
XI。 Ratio and proportion.
(1) ratio and the application of proportion.
In industrial production and daily life, it is often necessary to allocate a quantity according to a certain proportion. This distribution method is usually called "proportional distribution".
(2) Problem-solving strategies.
When doing exercises related to solution ratio distribution, we should be good at finding out the ratio of total and distribution, and then convert the ratio of distribution into components or copies to answer.
(3) Strategies for solving positive and negative proportion application problems.
1, examine the question and find out two related quantities in the question.
2. Analyze and judge whether the two related quantities in the problem are directly proportional or inversely proportional.
3. Set an unknown number and make a proportional formula.
4. Solution ratio style.
5. Test and write the answers.
(4) Sense of numbers and symbols.
1. Cultivating students' sense of numbers in mathematics teaching mainly means that students have the ability to express specific data and quantitative relations with numbers; Ability to judge and calculate different arithmetic operations, and experience in choosing appropriate methods (mental calculation, written calculation and using calculator) to implement calculation; You can infer from the data, and check the accuracy and reliability of the data and inference, and so on.
2. The purpose of cultivating students' sense of numbers is to enable students to learn mathematical thinking and understand and explain practical problems by mathematical methods.
3. The cultivation of sense of numbers is beneficial to the improvement of students' ability to ask and solve problems. When students encounter problems, they consciously and actively establish contact with certain mathematical knowledge and skills, so it is possible to construct mathematical models related to specific things. A certain sense of number is an important condition for completing this kind of task. For example, how to number all the athletes participating in the school sports meeting? This is a practical problem, there is no fixed solution, and different formulas can be used. Different arrangement schemes may be different in practicality and convenience. For example, you can distinguish between grades and classes numerically, distinguish between boys and girls, or quickly know what kind of events a player is participating in.
4. The concept of number itself is abstract, and the establishment of the concept of number is not completed at one time. It takes a process for students to understand and master the concept of numbers. In the process of understanding numbers, let students get in touch with situations and examples related to experience and feel and experience in the realistic background, which will enable students to grasp the concept of numbers more concretely and deeply and establish a sense of numbers. In the process of understanding numbers, let students talk about the numbers around them, the numbers used in life, and how to express things around them with numbers. It will make students feel that numbers are around them, and many phenomena can be expressed simply and clearly with numbers. Estimate the number of words in a page, how many pages there are in a book, how many grains there are in a handful of soybeans, and so on. These perceptions and experiences of specific quantities are the basis for students to establish a sense of numbers, which will be of great help to students to understand the meaning of numbers.
5. At any stage, students should be encouraged to express the quantitative relationship and changing law in specific situations in their own unique way, which is the decisive factor in developing students' sense of symbols.
6. The introduction of letter representation is an important step to learn mathematical symbols and learn to express the implied quantitative relationship and changing law in specific situations with symbols. Try to introduce from practical problems, so that students can feel the meaning of letters.
First of all, letters are used to represent arithmetic rules, arithmetic laws and calculation formulas. The generalization of the algorithm deepens and develops the understanding of logarithm.
Second, letters are used to express various quantitative relations in the real world and various disciplines. For example, the relationship between speed v, time t and distance s in uniform motion is s=vt.
Thirdly, using letters to represent numbers is convenient for abstracting quantitative relations and changing rules from specific situations and expressing them accurately, which is conducive to further solving problems with mathematical knowledge. For example, we use letters to represent the unknowns in practical problems, and use the equation relations in the problems to list the equations.
7. Letters and expressions have different meanings on different occasions. For example, 5=2x+ 1 represents a condition that X satisfies. In fact, x only occupies the position of a special number here, and its value can be found by solving the equation. Y=2x represents the relationship between variables, X is the independent variable, and you can take any number in the definition field, Y is the dependent variable, and Y changes with the transformation of X; (a+b) (a-b) = a-b stands for a generalized algorithm and an identity; If A and B represent the length and width of a rectangle and S represents the area of the rectangle, then S=ab represents the formula for calculating the area of the rectangle, which also means that the area of the rectangle changes with the change of length and width.
8. How to cultivate students' sense of symbol?
We should try our best to help students understand the meanings of symbols, expressions and relationships in practical problem situations, and develop their sense of symbols in solving practical problems. It is necessary to train the symbol operation and carry out a certain number of symbol operations appropriately and in stages. But it does not advocate too much formal operation training.
The development of students' sense of symbols cannot be achieved overnight, but should run through the whole process of mathematics learning and develop gradually with the improvement of students' mathematical thinking.
Twelve. Calculation of quantity.
1, quantity, length, size, weight, speed, etc. Among things, the characteristics of these measurable objective things are called quantity. Comparing the measured quantity with the standard quantity is called measurement. The quantity used as the standard of measurement is called the unit of measurement.
2. Number+company name = name number. Only one unit name is called a monomer. The number of high-level units with two or more unit names is called composite number, such as changing meters to centimeters and changing centimeters to meters for low-level units.
3.( 1) A number with only one company name is called a single number. Such as: 5 hours, 3 kilograms. Only one unit)
(2) Two or more unit names are called composite numbers. Such as: 5 hours and 6 minutes, 3 kilograms and 500 grams (using two units)
(3)56 square decimeter =(0.56) square meter, that is, a single number is converted into a single number.
(5)560 square decimeter =(5) Square meter (60 square decimeter) is an example of converting a single number into a compound number.
4. Senior units and lower units are relative. For example, "meter" is a high-level unit relative to decimeter and a low-level unit relative to kilometer.
5, commonly used calculation formula table.
(1) rectangular area = length × width, and the calculation formula is S = A B.
(2) Square area = side length × side length, and the calculation formula is s = a× a.
(3) The circumference of a rectangle is (length+width) × 2, and the calculation formula is s=(a+b)×2.
(4) Square perimeter = side length × 4, and the calculation formula is s= 4a.
(5) The area of a parallelogram = base × height, and the calculation formula is S = ah.
(6) Triangle area = base × height ÷2, and the calculation formula is s=a×h÷2.
(7) Trapezoidal area = (upper bottom+lower bottom) × height ÷2, and the calculation formula is s=(a+b)×h÷2.
(8) cuboid volume = length× width× height, and the calculation formula is v=abh.
(9) The area of a circle = π× radius square, and the calculation formula is S = л r 2.
(10) cube volume = side length × side length× side length, and the calculation formula is v = a 3.
(1 1) The volume of cuboids and cubes can be written as the bottom area × height, and the calculation formula is v=sh.
(12) cylinder volume = bottom area × height, and the calculation formula is v = s h.
6. 1 year 1 February (3 1 day includes1,3, 5, 7, 8, 10,1February, and 30 days includes 4, 6, 9,/February.
7. The leap year is a multiple of 4, and the whole hundred years must be a multiple of 400.
8. There are 365 days in a normal year and 366 days in a leap year.
9. A.D. 1- 100 is the first century, and A.D. 190 1-2000 is the twentieth century.
Thirteen, the understanding and calculation of plane graphics.
(1) triangle.
1, a triangle is a figure surrounded by three line segments. It has stability. Draw a vertical line from the vertex of a triangle to its opposite side. The line segment between the vertex and the vertical foot is called the height of the triangle. A triangle has three heights.
2. The sum of the internal angles of a triangle is 180 degrees.
3. Triangle can be divided into acute triangle, right triangle and obtuse triangle according to angle.
4. Triangle can be divided into isosceles triangle, equilateral triangle and equilateral triangle according to its sides.
(2) quadrilateral.
1, quadrilateral is a figure surrounded by four line segments.
2. The sum of the internal angles of any quadrilateral is 360 degrees.
3. Only one set of quadrilaterals with parallel opposite sides is called trapezoid.
4. Two groups of parallelograms with parallel opposite sides are called parallelograms, which are easy to deform. Rectangular and square are special parallelograms; A square is a special rectangle.
(3) circles.
A circle is a curved figure on a plane. The same circle or the same circle has the same diameter, and the diameter is equal to twice the radius. A circle has countless axes of symmetry. The center of the circle determines the position of the circle, and the radius determines the size of the circle.
(4) fan shape.
A figure enclosed by two radii of a central angle and the arc it subtends. The sector is an axisymmetric figure.
(5) Axisymmetric graphics.
1. If a graph is folded in half along a straight line, the graphs on both sides can completely overlap. This graph is called an axisymmetric graph. This suffocation is called symmetry axis.
2. Line segments, angles, isosceles triangles, rectangles, squares, etc. They are all axisymmetric figures, and the number of their symmetry axes is different.
(6) perimeter and area.
1, the length of a plane figure is called perimeter.
2. The size of a plane figure or the surface of an object is called the area.
3. Calculation formula of perimeter and area of common graphics.