arithmetic
One: Mixed operation without brackets
Key points: master the two-level operation sequence.
Difficulty: Using mixed operation to solve practical problems.
Knowledge point 1: the operation order of addition and subtraction mixed operation without brackets.
In the formula without brackets, if there is only addition and subtraction, it should be calculated from left to right.
Knowledge point 2: the operation order of mixed operation of multiplication and division without brackets.
In arithmetic without brackets, if there is only multiplication and division, it should be calculated from left to right.
Knowledge point 3: the sum of product quotient (the mixed addition and subtraction of difference, the multiplication and division method must be calculated first and then the addition and subtraction method.
Two: the operation sequence with brackets and the operation on O.
Key point: master the operation order of bracketed expressions.
Difficulty: Understand why O can't be divided.
Knowledge point 1: mixed operation with brackets.
The operation order in parentheses should be calculated in parentheses first, and then outside parentheses.
Knowledge point 2: Operation sequence of elementary arithmetic.
The operation order of elementary arithmetic, without brackets.
If there is only addition, subtraction or multiplication and division, it should be calculated from left to right. If there are multiplication and division and addition and subtraction, calculate the multiplication and division first, and use the calendar to calculate the addition and subtraction. If there are brackets, count the inside of the brackets first, then the outside.
Knowledge point 3: about the operation of O.
The operation letter about o can be expressed as: a+0=a a-0=a 0×a=0 0÷a=0(a≠0).
Students' common problems and mathematical guidance: 1: In elementary arithmetic, students often forget the rules of multiplication and division, addition and subtraction, multiplication in brackets and calculation in brackets. Teachers should always remind them.
2. The investigation of elementary arithmetic is not limited to simple formulas, but pays more attention to students' problem-solving ability, that is, the way to apply problems.
The knowledge points that cannot be divided at 3:0 must be clearly explained by the teacher (not participating in the complete solution of P 17).
Three algorithms and simple calculation
One: addition and subtraction
Key points: Understand the operation rules and be able to perform simple operations.
Difficulties: Flexible application of algorithms to solve problems.
Knowledge point 1: additive commutative law
The two addends are interchanged, and the sum is a constant, expressed in letters: a+b = b+a.
Knowledge point 2: law of additive association
When three numbers are added, two numbers are added first, or two numbers are added first, and the total remains the same. Use letters: (a+b)+c=a+(b+c)
In an addition expression, when some addends can be added to an integer+an integer hundred, using additive commutative law's law of addition to change the calculation order can make the calculation simple.
Teaching guidance:
1: The combination law of transformation and addition often appears in the same problem.
2. In the use of simple operations, "addition of benchmark numbers" and "rounding method" are sometimes used. These two methods require students with a good foundation to master, while students with a general foundation do not. See the complete solution P48-49 for details.
Two: the law of multiplication:
Key points: Understand the law of multiplication and be able to perform simple calculations.
Difficulties: Flexible application of algorithms to solve practical problems.
Knowledge point 1: Multiplicative commutative law:
Swap the positions of two factors, and the product is constant, which is expressed in letters: a× b = b× a.
Knowledge point 2: the law of multiplicative association
The first two numbers are multiplied, or the last two numbers are multiplied, and the product is a constant, which is expressed by letters: (a×b)×c=a×(b×e).
Knowledge point 3: Multiplication and distribution law
Multiplication distribution law is a law between multiplication and addition, and associative law is only a law in multiplication operation, which is expressed by letters:
(a+b)×c=a×c﹢b×c
The application of the law of multiplication can be found in the problem.
Find a friend: 25× 4 =100125× 8 =1000 When you see 25% 125, you should think of 25,125; If you encounter multiples of 4 or 8 such as 32 and 72, and there is 25% 125 in the question, fold the multiples of 4 or 8 into 4× () or 8× ().
Zero folding: for example, the distribution law of 75×101= 75 (100+1) multiplication.
Flexible application of multiplicative distribution: for example, 37× 29+37+37× 70 = 37× (29+1+70) and learning to use the positive and negative forms of distribution law are the difficulties for students.
Three: Simple calculation
Key points: master the simple and convenient methods of continuous subtraction, continuous division and regression mixed operation.
Difficulties: The calculation method can be flexibly selected according to actual needs.
Knowledge point 1: Simple calculation of continuous decrement
The nature of subtraction: (1) A number subtracts two numbers continuously, and the sum of these two numbers can be subtracted by this number, that is, A-B A-B-C = A-C-B C-B.
Knowledge point 3: Simple algorithm used in multiplication and division.
In multiplication, if one factor is 25 or (125) and the other factor is a multiple of 4 or (8), it is unnecessary to analyze the multiple of 4 or (8), and 25 or (125) can also be written as 100÷4 (or/kloc).
For example: 12×25
Method 1: 12×25 Method 2: 12×25 Method 3: 12×25.
=3×4×25 = 12×( 100÷4) =( 12÷4)×(25×4)
=3×(4×25) = 1200÷4 =3× 100
=300 =300 =300
Teaching guidance: 1: There cannot be multiple simple methods for a problem in practical operation, so students must use the methods they have learned flexibly.
2. The investigation of simple operation will also appear in the problem solving.
The Significance and Properties of Four Decimals
The meaning of decimals and their reading and writing methods
One: the generation and significance of decimals.
Key point: Understand decimals and their meanings.
Difficulty: Know the calculation unit of decimals and master the progress between them.
Knowledge point one: the generation of decimals.
In measurement and calculation, it is often impossible to get integer results, so it is necessary to divide the average value of a unit into smaller units such as 10, 100, 1000, etc., thus generating decimals.
Knowledge point 2: the meaning of decimal and the counting unit of decimal.
Meaning of decimals: Divide the unit 1 into 10, 100, 1000, and how many units are there? It can be expressed in fractions of denominators 10, 100, 1000, or in decimals. Decimals are counted in tenths, hundredths and thousandths ... Write 0, 1, 1 and 0.0068 respectively.
Two: decimal reading and writing.
Important: Be able to read and write decimals correctly.
Difficulties: Understand the numerical order of decimals.
Knowledge point 1: organize decimal order table.
Digital sequence table
Integer part decimal part decimal part
Tens of thousands, tens of thousands, hundreds of millions
Position, position, position, position
Position position
count
Calculate thousands of points.
One of the few
Yi Shan yiyiyi
place
Knowledge point 2: How to read decimals
When reading decimals, read the integer part first, then read the decimal point. The decimal point is read as "dot", and the decimal part is read last. The decimal part should read the numbers on each bit in turn. (Note: the integer part is a decimal of 0, and the integer part is read as zero; If the decimal part has 10 zeros, read a few zeros)
Knowledge point 3: How to write decimals?
Write the integer part first, according to the integer writing method, if the whole part is zero, write 0 directly, and then the decimal point in the lower right corner of the unit; Finally, write the decimal number and the number on each bit in turn.
Comparison of the nature and size of decimals
Key points: Understand the nature of decimals and master the comparison method between large numbers and decimals.
Difficulty: Override decimals by applying their attributes.
Knowledge point 1: the nature of decimals
Add "0" or remove "0" at the end of the decimal, and the size of the decimal remains the same.
Knowledge point 2: the method of simplifying decimals
According to the nature of decimal, removing the zero at the end of decimal will not change the size of decimal.
Knowledge point 3: increase the number of decimal places and rewrite the decimal size. Just add "0" at the end of the decimal to rewrite the integer into a decimal. First place the decimal point in the lower right corner of the integer, and then add the corresponding "0" knowledge point as needed.
Four: Comparison of decimal size
Compare the integer parts first, and the big ones will be big. If the integer parts are the same, compare the tenth digit, and the number with the tenth digit is larger. One tenth of the numbers are the same, so comparing the numbers in the percentile, the number with the largest number in the percentile is larger; One tenth of the figures are the same, so comparing the figures in the percentile, the largest figure in the percentile is nine; And so on.
Movement of decimal point
Key point: master the law of decimal size change caused by decimal position movement.
Difficulty: how to use "0" to make up when the number of digits is not enough.
Knowledge point 1: the law of decimal size change caused by decimal point movement.
If the decimal point moves one place to the right, it will be expanded to 10 times the original number; If the decimal point moves two places to the right, the decimal point will be expanded to 100 times; If the decimal point is moved three places to the right, it will be expanded to 1000 times the original number.
If the decimal point moves one place to the left, the decimal point will be reduced to110 of the original number; If the decimal point is moved by two places, the decimal point will be reduced to1100 of the original number. If the decimal point is moved three places to the left, the decimal point will be reduced to1100 of the original number.
Knowledge point 2: the application of the law of large number change caused by decimal point movement.
Expanding a number to 10 times, 100, 1000, ... is to multiply this decimal by 10, 100, 1000 respectively, that is, to move the decimal point one place to the right. ...
Reducing a number to 10 times, 100 times and 1000 times means multiplying this decimal by 10, 100, 1000 ... that is, multiplying the decimal by/kloc respectively.
Reduce a number to110,1100,11000. .................................................................................................
Decimals in life.
Knowledge point 1: Decimals are widely used in daily life, so they represent quality, height, performance, price, temperature difference, body temperature and so on.
Knowledge point 2: Rewrite the meaning of names and numbers.
In real life, sometimes it is necessary to rewrite the data of different units of measurement into the same unit of measurement for calculation or comparison.
Knowledge point 3: the method of rewriting the single number or composite number of low-level units into the single number of high-level units expressed in decimals.
The method of rewriting the single number of low-level units into the single number of high-level units: divide this number by the propulsion rate between two single names. If the propulsion rate between two units is 10, 100, 1000…
The method of rewriting composite numbers into decimals: the number of high-order units of composite numbers is fixed, and as the integer part of decimals, the number of low-order units of composite numbers is changed into the number of high-order units as the decimal part.
Knowledge point 4: the method of rewriting the single number of high-level units expressed by decimals into the single number or composite number of low-level units.
Multiply this number by the forward speed between two units. If the propulsion rate between two units is 10, 100, 1000 ... you can directly move the decimal point to the right by corresponding digits.
Approximate the decimal number.
Key point: master the method of finding decimal divisor.
Difficulties: the method of rewriting large numbers into decimals in units of "ten thousand" or "hundred million".
Knowledge point 1: the method of finding decimal divisor
You can use the rounding method. Keep one place after the decimal point, which means that it is accurate to one place after the decimal point. Whether to carry it or not depends on the size of the value on the tenth place; Keep one place after the decimal point, indicating that it is accurate to ten places, and whether to carry it or not should be judged according to the size of the numerical value in the percentile; When two decimal places are reserved, it means that it is accurate to one hundredth, and whether to carry it should be judged according to the value on one thousandth. .....
Knowledge point 2: the method of rewriting an integer that is not 10 thousand or 100 million into a number with "10 thousand" or "100 million" as the unit.
Place the decimal point in the lower right corner of "10,000" or "100 million" and add the words "10,000" or "100 million" after the decimal point. If the demand is similar, the decimal number can be retained as required.
Pentagonal triangle
Triangular quota characteristics
Knowledge point 1: the definition of triangle and the names of its parts.
pinnacle
Corner edge
Angle height angle
Vertex edge vertex
Definition of triangle: a figure surrounded by three line segments (the endpoints of each adjacent line segment are connected).
It is called a triangle. Draw a vertical line from one vertex of the triangle to its opposite side. The line between the vertex and the vertical foot is called the height of the triangle, and this side is called the bottom of the triangle. Triangles can be represented by letters, forming triangle ABC.
Knowledge point 2. The characteristics of a triangle.
Triangles are stable and widely used in life.
Knowledge point 3: the relationship between the three sides of a triangle.
The sum of any two sides of a triangle is greater than the third side.
Classification of triangles.
Key points: master the different classifications of triangles. Difficulties: Understanding the relationship between equilateral triangle and isosceles triangle.
Knowledge point 1: triangles are classified by angle.
Triangle can be divided into acute triangle, right triangle and obtuse triangle. Because a triangle has at least two acute angles, we can judge the type of triangle according to the largest angle in the right angle and what kind of angle is the largest angle.
It calculates those triangles.
Knowledge point 2: Triangle bank classification.
Triangles are classified by sides: equilateral triangles and isosceles triangles, which include equilateral triangles.
Unequal triangle isosceles triangle
equilateral triangle
Sum of internal angles of triangle
Key point: the sum of the internal angles of the triangle is 180.
Difficulty: Using the internal angle of triangle to solve practical problems.
Knowledge point 1: The sum of the internal angles of the triangle is 180.
The three internal angles of a triangle form a right angle. Because the right angle is 180, the sum of the internal angles of the triangle is 180.
Knowledge point 2: The sum of the internal angles of a triangle is the application of 180.
Application 1: Know the degree of two angles in a triangle and find the degree of the third angle.
Application 2: Know the degree of a triangle and find the degrees of the other two angles. (mainly used for isosceles triangles)
Graphical components:
Knowledge point 1: the relationship between triangle and quadrilateral.
Any two identical triangles can be combined into a parallel quadrilateral; Two identical right triangles can be combined into a rectangular parallelogram; Two identical isosceles triangles can be combined into a square or parallelogram; Three identical triangles can form a trapezoid.
Six: addition and subtraction of decimals
Addition and subtraction of decimals (1)
Key points: master the calculation method of decimal addition and subtraction.
Difficulties: Understand the management of decimal point alignment.
Knowledge point: the method of adding and subtracting decimal points with a pen
Note: (1) decimal point alignment, that is, the same digit alignment; (2) When adding from the last digit, pay attention to which camera will advance by one digit 1 when it is full of ten digits, and when subtracting, pay attention to which digit is not enough to reduce the waist 1(3) from the previous digit. There is a 0 at the end of the number (referring to the decimal part), which is generally removed.
Mixed calculation of decimal addition and subtraction
The operation order of decimal addition and subtraction mixed operation is the same as that of integer addition and subtraction mixed operation, in the formula without brackets. If there is only addition and subtraction, it is calculated in the order of 4 from 1 10,000 to the right. If there are brackets in the formula, count the brackets first.
Addition and subtraction of decimals (3)
Knowledge points: Simple calculation of decimals by applying the law of integer operation.
The law of integer operation also applies to decimal operation. Therefore, in fractional elementary arithmetic, we should carefully observe the characteristics of each number, whether the direct relationship between any number and each number is consistent with the operation in front of each number, and properly use the operation properties of addition and conversion law, association law and subtraction to carry out simple operations.
Additive commutative law: (a+b) = b+a.
Additive associative law: (a+b)=a+(b+c)
The operational nature of subtraction: a-b-b=a-(b+c)
Seven statistics
Key points: be able to understand simple broken-line statistical chart, complete broken-line statistical chart and analyze it.
Difficulties: Solve the curved pond according to the statistical chart and make a reasonable guess.
Knowledge points 1 folded statistical chart features.
The characteristic of broken-line statistical chart is that it can reflect both quantity and change of quantity. In practical problems, if you need to know the change of quantity, it is more reasonable to choose the broken line statistical chart.
Knowledge point 2: draw a broken line statistical chart and make a reasonable guess according to the statistical chart data.
Steps to complete the statistical chart of broken lines: (1) tracing points; (2) Connect points into line segments (3) to represent data. When drawing points, you should first find the points on the horizontal axis, and then find the corresponding points on the vertical and horizontal lines to draw the horizontal axis. The vertical line of the vertical axis and the intersection of the two vertical lines are the points to be drawn.
Application of statistical chart: according to statistical chart, problems can be found, solved and simply predicted.
Eight mathematical wide angles
Key points: Understand and master the characteristics and solutions of "planting trees".
Difficulties: the ability to apply mathematical methods to solve practical problems.
Knowledge point: the problem of planting trees at both ends of a closed route.
Planting trees at both ends of a line segment: the total distance is kept at two intervals, and the number of trees is two intervals+1.
Knowledge point 2: the problem of not planting trees at both ends of the closed line.
On the problem that both ends of a straight line are not planted: the interval number of two trees is-1.
Knowledge point 3: the problem of planting trees on the closed graphic route.
The number of intervals between trees.
Location and direction (1)
Key point: master the method of determining the position of an object according to the direction and distance.
Difficulties: the method of marking the position of objects on the plan according to the description.
(1) Determine the direction and measure the azimuth of the measured point with a protractor.
(2) Measure the distance between the measured point and the observation point with a ruler, and calculate the actual distance according to the proportion.
(3) Accurately judge or describe the position of the measured object according to the direction (angle) and distance.
The second knowledge point is the method of marking the position of objects on the plan.
First determine the direction, then determine the distance according to the selected unit length, and finally draw the specific position of the object and mark the name.
Position and direction (2)
Key point: understand the relativity of object position relationship.
Difficulty: the change of observation point makes the object reposition.
Relativity of knowledge point-position relationship
Describe the position of an object in relation to the observation point. Different observation points will describe the position of objects at the same distance in different directions.
Knowledge point 2 describes and draws a simple road map.
When describing the road map, we should first determine each observation point according to the walking route, then take each observation point as a reference, and then describe the direction and route to the next target.