Understand the simple equation of number

1. Divisibility of numbers and operands of numbers. Basic knowledge of algebra

Operation ratio and quantity ratio

Length of general compound application questions

Typical application problem areas

Third, the application problem score, percentage application problem fourth, the measurement volume of quantity

Using column equation to solve the weight of application problems

Ratio and proportional application problem time

Renminbi (RMB)

Line statistics table

Understanding and calculation of plane figure angle VI. Statistics and probability

V. Spatial and graphic statistics

Cuboid, cube

Understanding and calculation of three-dimensional graphics

Cylinder, cone

A, number and number operation

(A) the understanding of quantity

Meaning of integers: Numbers such as …-3,-1, 0, 1, 2, 3, … are called integers.

Meaning of positive number and negative number: Numbers like 1, +5, 6, ... are called positive numbers; Numbers like -3, -2, -9, … are called negative numbers.

pole

0 is the smallest natural number, 0 is an even number, and the function of 0 indicates the starting point.

Mark the boundary

Natural number 1 is the smallest digit and the basic unit of natural number; 1 is neither prime nor composite.

Meaning of number: it is a part of an integer, which can represent radix or ordinal number.

Meaning: Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a score. The number representing one of them is a fractional unit.

mark

True fraction-numerator is less than denominator (less than 1)

Classification: Error score-numerator is greater than or equal to denominator (greater than or equal to 1)

With fraction-numerator is greater than denominator (greater than 1)

Meaning: divide the whole "1" into one or more of 10, 100, 1000 ...

Is a few tenths, a few percent, a few thousandths ... can be expressed in decimals.

Endless decimal

Divide infinite acyclic decimals by decimal parts.

Decimal Infinite Decimal Pure Cyclic Decimal

Classified pure decimal cyclic decimal

Divide the mixed cycle decimal by the integer part.

With decimals

Order table of integer and decimal places

Integer part decimal part

... billion, million, million.

Digital ... 100 billion bits, 10 billion bits, 1 billion bits.

Billions, tens of millions, millions, hundreds of thousands.

Hundred megabits

kilobit

The third number on the right

decade

One tenth, one percent, one thousandth, one thousandth ...

Counting unit ...100 billion

110 million

ten thousand

a thousand

one hundred

ten

one

One tenth, one thousandth, one thousandth ...

Percent: A number indicating that one number is a percentage of another number is called a percentage. (percentage or percentage)

Discount *: in business terms, a few percent is a few tenths, and a few percent is dozens.

Note: Percentages and discounts only represent the ratio of two numbers, and scores can represent specific quantities in addition to ratios.

Digital reading and writing:

1, integer reading: from high to low, read one level at a time, the zeros at the end of each level are not read, and only one zero is read for other digits.

2. Writing of integers: from high to low, writing step by step. If there is no unit on any number, write 0 on that number.

3. Decimal reading and writing: the integer part is read (written) as an integer, the decimal part is read as a "dot", and the decimal part reads (writes) the numbers on each bit in turn.

Digital rewriting

Write a number in units of "ten thousand" or "hundred million"

1, rewriting and ellipsis of multiple digits: mantissa after omitting "10,000" or "100 million" digits.

2. Exchange of fractions, decimals and percentages.

Use the rewrite components 10, 100, 1000 to lower the score again. ...

decimal

Divide the numerator by the denominator

Move the decimal point two places to the right and add%.

Decimal percentage

Delete% and move the decimal point two places to the left.

Write in fractional form, subtract points.

Percentage score

Write it as a decimal and then as a percentage.

Digital comparison:

1, integer size comparison: look at the number of digits first, the number with more digits is large: the number of digits is the same, and the number with the largest digit on the same digit is large from the high position.

2. Comparison of decimal sizes: first compare the integer parts of two numbers, and the number with larger integer parts will be larger; If the integer part is the same, look at the decimal part. Start from the high position and compare by digits.

3. Score comparison: the denominator is the same, and the score with large numerator is large; The fraction with the same numerator and small denominator is large; Denominators are different, divide first and then compare.

Basic attributes of numbers:

1, the basic property of the fraction: the numerator and denominator of the fraction are multiplied or divided by the same number at the same time (except 0), and the size of the fraction remains unchanged.

2. Basic properties of decimals: Add "0" or remove "0" at the end of decimals, and the size of decimals remains unchanged.

(2) the divisibility of numbers

Definition: (When learning "divisible numbers" in primary schools, numbers generally refer to non-zero natural numbers)

The quotient (b≠0) of the number A divided by B is just an integer without remainder, so we say that A can be divisible by B (or B can be divisible by A).

Multiple common multiple least common multiple

Divisible factor common factor maximum common factor

Prime Compound Prime Number (Deleted)

Prime Factor Decomposition Prime Factor (Delete)

Characteristics of multiples of 2: Units are 0, 2, 4, 6, 8.

Even and odd numbers (numbers divisible by 2 are called even numbers, and numbers divisible by 2 are called odd numbers. )

Characteristics of multiples of 3: the sum of the numbers on each bit is a multiple of 3.

Characteristics of multiples of 5: Numbers with 0 or 5 bits.

(3) The number of operations

The Meaning of 1 and Four Arithmetic Operations

digital

classify

Integer decimal part of operation name

The operation of combining two numbers into one number.

The operation of finding the sum of two addends and one of them by subtraction.

A simple operation to find the sum of several identical addends by multiplication. Decimal multiplication by integer has the same meaning as integer multiplication. Fractional multiplication of integers has the same meaning as integer multiplication.

Multiplying a number by a decimal is to find a few tenths and a few percent of this number. Multiplying a number by a fraction is to find a fraction of this number.

The operation of finding another factor by dividing the product of two factors and one of them.

2, four algorithms

Integer decimal part

Add and subtract the same number to align, and count from the low place.

Addition: Ten are full, one is forward and the other is forward.

Subtraction: If the subtraction is not enough, the previous digit will be retired and the decimal points will be aligned. Starting from the low order, the calculation will be made by integer addition and subtraction, and the decimal point in the result will be aligned with the decimal point of the addition and subtraction. 1, add and subtract fractions with the same denominator, denominator unchanged, numerator addition and subtraction.

2. Add and subtract the scores of different denominators, divide the scores first, and then add and subtract the scores of the same denominator.

3. Provide points that can be reduced as a result.

Multiplication 1. Starting from the unit, multiply the first factor by the number in each bit of the second factor in turn.

2. Multiply the number on which bit of the second factor, and the last bit of the number will be aligned with which bit of the second factor.

3. Add up the numbers multiplied by several times. 1, and calculate the product according to the integer multiplication rule.

2. Look at a factor * * *, how many decimal places there are, and count the decimal places from the right of the product. 1, the fraction times the fraction, the numerator is the product of the numerator multiplication, and the denominator is the product of the denominator multiplication.

2. If there is an integer, it is regarded as a false fraction with the denominator of 1.

3. If there is a score, generally turn the score into a fake score first.

Division and divisor are integers: from the high-order divisor, look at the first few digits of the dividend first. If the divisor is not enough, look at one more person and write the quotient on the divisor. The decimal point of quotient is aligned with the decimal point of dividend. Divider is decimal: first move the decimal point of the divisor to make it an integer, move the decimal point of the divisor to the right by several digits, and move the decimal point of the dividend to the right by the same digit (if the digit is not enough, make up 0), and then calculate it according to division where the divisor is an integer. The number A divided by the number B (except 0) is equal to the reciprocal of the number A multiplied by the number B.

3, the relationship between the four parts of the operation:

Appendix+Appendix = and minus-minus = difference

One addend = sum-another addend minus the minuend = minus+difference.

Subtraction = minuend-difference

Factor × factor = product dividend/divisor = quotient

One factor = product/another factor divided by dividend = quotient × divisor

Divider = Divider

4, operation law and operation nature

Additive commutative law: A+B = B+A.

Additive associative law: (a+b)+c=a+(b+c)

Multiplicative commutative law: a×b=b×a

Law of multiplicative association: (a×b)×c=a×(b×c)

Multiplication and distribution law: (a+b)×c=a×c+b×c

The operational nature of subtraction: a-b-c=a-(b+c)

The operational nature of division: a ÷ (b× c) = a ÷ b ÷ c

5, the order of the four operations:

In the formula without brackets, if it only contains operations at the same level, it should be calculated from left to right in turn; If there are two levels of operation, first calculate the second level of operation, and then calculate the first level of operation.

For the formula with brackets, calculate what is in brackets first, and then calculate what is out of brackets.

Second, the preliminary knowledge of algebra

(A) simple equation

1, alphanumeric:

The letters (1) can represent natural numbers, integers, decimals and percentages. ...

(2) Mathematical concepts, arithmetic rules and mathematical calculation formulas can be concisely expressed by formulas containing letters. The quantitative relationship can also be expressed concisely.

2. Simple equation

(1) equation: the expression of the equation.

(2) Equation: an equation with unknown numbers.

(3) Solution of the equation: the value of the unknown quantity that makes the left and right sides of the equation equal.

(4) Solving the equation: the process of solving the equation.

(5) the basis of solving the equation: the basic properties of the equation (balance principle)

(2) Proportion and specific gravity:

The meaning, nature, ratio and proportion of 1

Proportional ratio

The division of two numbers in the sense is also called the ratio of two numbers, and the expression that two ratios are equal is called proportion.

basis

The first item and the second item of the property ratio are multiplied or divided by the same number (except 0) at the same time, and the ratio remains unchanged. In proportion, the product of two internal terms is equal to the product of two external terms.

2. The relationship between ratio, fraction and division

The ratio of the first item to the second item.

Fractional value of numerator denominator of fractional line

Divider, divisor and quotient

3. The difference and connection between comparison and simplification.

Conventional method results

Find the ratio according to the meaning of the ratio, and divide the former by the latter. Is a quotient, which can be an integer, decimal or fraction.

Simplified ratio According to the basic properties of the ratio, the first term and the second term of the ratio are multiplied or divided by the same number (except 0). It is a ratio, and its first and last terms are integers.

4. Scale

The ratio of the distance on the picture to the actual distance is called the scale of this picture.

5. The difference and connection between positive proportion and inverse proportion.

similarities and differences

characteristic relation

In direct proportion, there are two related quantities, a change and a change. The ratio of two corresponding numbers in two quantities is determined.

The product of two corresponding numbers in two quantities of inverse proportional relation is certain.

ху=k (OK)

Third, the application problem.

(A) General compound application questions

1, the solution of general composite application problems

(1) Analysis: Start with the problem and analyze the known conditions in the problem step by step.

(2) Synthesis method: Starting from the known conditions of application problems, the unknown is gradually deduced.

(3) Analytical synthesis method: a method that combines analytical method and comprehensive method to use alternately. When there is an obvious calculation process in the known conditions, push forward with the comprehensive method, and then turn to the question raised in the original question when encountering difficulties. Use analytical methods to help, push back a few steps, push forward and push back together, and the problem will be solved.

2, general compound application problem solving steps:

(1) Examine the meaning of the question and find out the known conditions and problems;

(2) analyze the relationship between the quantities in the topic, so as to determine what to calculate first, then what to calculate, and finally what to calculate;

(3) Formulating and calculating the results;

(4) Test and write the answer.

(2) Typical application problems (application problems with certain answering rules)

1, general question

(1) The characteristic of the average problem is to combine all the "partial quantities" into a "total quantity", and then average according to the "total number of copies" to find out what one of them is.

(2) Method of solving the average problem: The key is to find out the "total amount" and "total number of copies" first, and then use "total amount ÷ total number of copies = average", and the special case can be solved by the method of "shifting more to make up less".

2, for a practical problem

The characteristic of (1) normalization application is to find a "single quantity" from the known conditions, and then calculate the required quantity according to this "single quantity". Normalization is usually divided into positive normalization and anti-normalization.

(2) Normalization problem solving method: first get a unit quantity, and then calculate the number of "unit quantities" by multiplication according to the requirements of the topic. This is the law to solve the normalization problem. Or calculate how many "units" the sum contains by division, which is the law of anti-normalization. The normalization problem can also be solved by solving the multiple ratio problem.

Step 3 encounter problems

(1) Features: A, two moving objects; B, the movement direction is opposite; C, exercise time at the same time.

(2) Law of solving problems: speed and × meeting time = distance.

Sum of distance/speed = meeting time

Distance ÷ Meeting time = speed and

(3) the application of scores and percentages

1, fractional multiplication application problem

Given a number, find its fraction (percentage) and multiply it. That is, "a number × a fraction (percentage)".

Known conditions: the quantity representing the unit "1"; Fraction (or percentage) of the unit "1" (also called fraction)

Features:

What is the fraction (percentage) of the unit "1" (also called partial quantity)?

The relationship between the three quantities is expressed by an equation: quantity of unit "1" × fraction = partial quantity.

consistent

2. The application of fractional division.

(1) What is the fraction (percentage) of a number? Find this number and divide it by it. That is, "What's the score?"

Known conditions: the fraction of the unit "1" (fraction); What is the score of the unit "1"

(Partial quantity)

trait

Question to ask: Number of units "1"

The relationship between the three quantities is expressed by an equation: the fractional rate of the partial quantity = the quantity in the unit of "1"

consistent

(2) Find the fraction (percentage) of one number to another by division. That is, "one number ÷ another number".

Known conditions: the quantity representing the unit "1"; What is the score of the unit "1" (partial quantity)?

trait

Question: What is the partial quantity of the unit "1"?

The relationship between the three quantities is expressed by an equation: partial quantity ÷ quantity of unit "1" = fraction.

consistent

3, the application of engineering problems

The total amount of work is expressed as "1", and the work efficiency is expressed as "a fraction" of the total amount of work done in unit time. According to the total amount of work and work efficiency, we can find out the working time of cooperation.

The relationship between the three quantities: working efficiency × working time = total work.

Total amount of work ÷ work efficiency = working hours

Total workload ÷ working time = working efficiency

(4) Using equations to solve application problems.

1, the thinking method of solving application problems by equations: replace the unknown in application problems with letters, and solve equations according to the equal relationship between quantity and quantity.

2. General steps to solve application problems with column equations

(1) Find out the meaning of the problem, and find out the unknown, which is represented by X.

(2) Find out the equal relationship between quantity and quantity, and list the equations.

(3) Solve the equation.

(4) Check and answer.

(5) Application of ratio and proportion

The application problems of ratio and proportion include: scale, proportion distribution, positive and negative proportion application problems.

1. Scale problem solving relationship: distance on the map: actual distance = scale.

2. Proportional distribution of application questions: total amount to be distributed × score of each part = each part.

3. Positive proportion у/χ=X/Y Inverse proportion χу=XY (positive and negative proportion application questions have been deleted)

Fourth, quantity and measurement.

(a) the meaning of quantity, measurement and measurement unit

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) Commonly used units of measurement and their entry rates.

1, length, area, plot, volume, volume, weight unit and its forward speed.

Length 1 km (km)= 1000 m (m)= 1 m (m)= 10 decimeter (dm)

1 decimeter (decimeter) = 1 0cm =1cm (cm) =10mm (mm)

Area1km2 =1million m2.

1 m2 = 100 square decimeter

1 square decimeter = 100 square centimeter

1 cm2 = 1 00mm2 plot1km2 =100ha.

1 ha = 1 10,000 m2

Volume 1 m3 = 1000 cubic decimeter.

1 cubic decimeter = 1000 cubic centimeter

1 cm3 = 1000 cubic millimeter volume 1 liter = 1000 ml.

1 cubic decimeter = 1 liter

1 cm3 = 1 ml

Weight1t =1000kg1kg =1000mg.

2. Common time units and their relationships

Century year month day hour minute second.

100 12 24 60 60

There are 3 1 day per month 1, 3, 5, 7, 8, 10, 12 months; There are April, June and September, 1 1 month, 30 days per month; Normal year is 365 days, February 28th; There are 366 days in leap year and 29 days in February.

3. RMB: 1 yuan = 10 angle = 1 angle = 10 minute.

(c) Conversion between similar units of measurement

Multiply by the forward speed

Number of high-level units, number of low-level units

Divided by the forward speed

Verb (abbreviation for verb) space and figure

(A) the understanding and calculation of plane graphics

1, line

Line segment: Connect two points with a ruler to get a line segment.

The length of a line segment is the distance between these two points. (There are two endpoints)

Straight line: two parallel lines extending infinitely at both ends of a line segment: two straight lines that do not intersect on the same plane are called.

You can get a straight line and a parallel line.

Vertical line: Two straight lines intersect at right angles, which are called mutual straight lines.

Vertical, one of which is called the perpendicular of the other.

Ray: you can get a ray by extending one end of the line segment indefinitely. (There is an endpoint)

2. Angle: A figure consisting of two rays drawn from a point.

Acute angle: an angle less than 90 degrees.

Right angle: an angle equal to 90 degrees

Oblique angle: an angle greater than 90 degrees and less than 180 degrees.

Flat angle: 180 degrees.

Fillet: 360 degree angle.

3, plane graphics

(1) triangle: a figure surrounded by three end-to-end line segments.

Acute triangle: all three angles are acute.

Divide right triangle by angle: one angle is right angle.

Obtuse triangle: One angle is obtuse.

triangle

Isosceles triangle: both sides are equal

Divide an equilateral triangle by the number of sides: three sides are equal.

Unequal triangle: three sides are not equal.

(2) Quadrilateral: a figure surrounded by four end-to-end line segments. department

Parallelogram Rectangular Square (3) Circular

Quadrilateral ring

Right trapezoid

trapeziform

isosceles trapezoid

(Draw line segment, angle, height, line segment, vertical line, circle and symmetry axis)

(4) Calculation formula of characteristics, perimeter and area:

Name, number, letter, meaning, characteristic perimeter area formula

square

A a: All four sides are equal in length, and all four corners are right angles C=4a.

S=a？

Rectangular b

A: Long.

B: The width is equal to the opposite side, and all four corners are right angles C=2(a+b).

S=ab

Parallelogram h

A: Bottom.

H: The two opposite sides of the upper group are parallel and equal respectively; S = ah.

Triangle h

A: Bottom.

H: There is a triangular height, and the sum of the internal angles is 180 degrees S=ah÷2.

trapeziform

h

Upper bottom

bottom

H: The height is only one set of parallel sides. S=(a+b)h÷2。

Circle d

Research and development: diameter

R: The radius is equal to the radius inside the circle, and the diameter is equal, and the diameter is twice the radius. C = π d = 2π r。

S=πr？

(B) the understanding and calculation of three-dimensional graphics

1, the difference and relation between cuboid and cube.

trait

Same name, different points.

Characteristic side length of vertex surface of face edge

Cubic

Article 6 12 8

Each of the six faces is usually rectangular (there may be two opposite faces that are square), and the areas of the opposite faces are equal. Each group (there are three groups, called length, width and height respectively) has four equal parallel sides.

cube

Article 6 12 8

Six faces are equal, 12 squares with equal sides.

2. Characteristics of cylinders and cones

Named graphic feature

circle

pillar; mainstay

The upper and lower bottom surfaces are circles with equal areas, and the distance between the two bottom surfaces is called height. When it is spread along the height, the edge is a rectangle (or a square). There are countless heights.

circle

awl

The bottom surface is circular, and the distance from the vertex to the center of the bottom surface is called height. There is only one height.

3, the calculation formula of surface area and volume of three-dimensional graphics

Name, number, letter, meaning, surface area s, volume v

cube

A: Side length S=6a? V=a？

Cubic

A: length b: width

H: Height S=(ab+ah+bh)x 2 V=abh.

cylinder

R: bottom radius h: height

C: the perimeter of the bottom s side =ch=πdh =2πrh.

S table =S surface +2S bottom surface V=sh=πr? h

cone

R: bottom radius

H: height V=sh÷3

=πr？ H3

Statistics and probability of intransitive verbs

Simple statistical table

Statistical table composite statistical table

Percentage statistics table

Statistical tables include: general title, column title, column title, data column, quantity unit and tabulation date.

Bar chart (single and composite)

Statistical chart broken line statistical chart (single type, double type)

Fan statistics chart

Methods and characteristics of making statistical charts

Characteristics of legal system

bar

Statistical chart 1. Organize data and plot horizontal and vertical axes. The unit length represents a certain quantity. 2. Draw straight lines according to the quantity.

3. Write down the name, tabulation date and legend, and it is easy to see the quantity.

dotted line

Statistical chart 1, sort out the data, draw the horizontal axis and the vertical axis, and the unit length represents a certain amount.

2. Draw points by number, and then connect the points with line segments in turn.

3. Write the name, tabulation date and legend, which can indicate both quantity and increase or decrease of quantity.

department

Statistical chart 1, calculate the percentage of each part in the total, and then calculate the degree of the central angle of the sector corresponding to each part. 2. Draw a circle with a proper radius, and measure the central angle of each sector with a protractor to make a sector. 3. Indicate the content and percentage of each sector, and distinguish them with different marks. 4. Write down the title and drawing date. Clearly show the relationship between each part and the whole and between each part.

Addition and deletion of mathematics knowledge in Beijing Normal University Edition and People's Education Edition

"Beijing Normal University Edition" is more knowledgeable than (People's Education Edition).

1, classification (classification according to a certain standard or different standards)

2. Position and sequence (front, back, left, right, up and down)

3. Location and direction (east, south, west and north)

4. Direction and route (southeast, northeast, southwest and northwest)

5. Observe the object (front, top, left or right)

6. Possibility (big or small; Possibility, impossibility and certainty; Fraction representation, several results)

7, reasoning in life (list solution)

8. Symmetry, translation or rotation (axisymmetric figure, direction, grid)

9, graphic transformation (around the point, direction, 90 rotation, translation)

10, determine the position (direction, north ×× degrees, distance; Digital pair)

1 1, the negative number in life (0 is neither positive nor negative)

12, digital graphics (digital angle, digital triangle, digital rectangle)

13, game formula (fairness)

14, graphic law (triangle, square, list solution)

15, try to guess (the law of the same cage and the same lattice of chickens and rabbits, solved by chart)

16, Numbers in Life (data world, digital usage, ID card)

17. Look at the pictures to find the relationship (the voice, behavior and relationship of the members on the football field)

18, median and mode

19, number, number of folds

20, factor, common factor, the greatest common factor

2 1, letter unit: m, dm, cm, mm, km; Grams, kilograms, tons, liters and milliliters

22, collocation knowledge (two or more)

23, the number of games (round robin)

24, combination graphics area (only two graphics)

25, observation range

26, equation (addition, subtraction or multiplication and division of the same number, the nature of the equation)

"Beijing Normal University Edition" deletes knowledge than "People's Education Edition"

1, divisor, common divisor, greatest common divisor

2. Prime numbers

3. Prime factorization

4. Use proportional knowledge to solve application problems