Unit 1 Understanding of Large Numbers
The counting units within 1 .100 million are: 1 (1), 10, 100, 10,000, 100,000, 1 million, 10 million, 100 million, and the progress rate between every two adjacent counting units is "10". According to our country's counting habit, every four digits are divided into one level, that is, one level, ten thousand levels and one hundred million levels, starting from the unit and from the low level to the high level.
100000 is one million,
10 1 million is 10 million,
10 10 million is 100 million.
Second, the reading method of more than 100 million numbers: (1) First read the billion level, then read the million level, and finally write the first level. (2) When reading the numbers of 100 million and 10,000 levels, we should read them according to the method of reading the numbers of individual levels, and then add the words "100 million" and "10,000" at the back. (3) No matter how many zeros are at the end of each level, don't read them; There is a 0 in the middle or in front of it, or several zeros in succession, all of which are read-only.
3. How to write numbers within 100 million: (1) Write 100 million levels first, then 10,000 levels, and finally one level. (2) Whoever doesn't have a previous unit will write 0 on that one.
Four, when the number of digits is different, the number of digits is greater than the number of digits; When the number of digits is the same, compared with the highest digit, the number on the highest digit is large, and this number is large; If the digits on the highest digit are the same, compare the next digit until the size is compared. When comparing multiple numbers, you should read the requirements clearly and don't lose the number. You can first group the numbers with the same number of digits and then compare them one by one.
5. How to rewrite an integer into a number with "ten thousand" as the unit: omit the four zeros after the ten thousand digits and replace them with a word "ten thousand". The method of rewriting an integer into a number with the unit of "100 million" is to omit the eight zeros after the hundred million digits and replace them with the word "100 million".
For example: 400000 = 4000120000000 =12 billion.
Sixth, to find the divisor of a number, we must first determine the mantissa after which bit is omitted, then look at the highest bit of the omitted part clearly, and finally find its divisor by "rounding".
567850 ≈ 57001356965487 ≈1400 million.
12756≈ 10000 1389000≈ 1390000
= 1 ten thousand = 1.39 million
If it is less than 5, it and the number on the right are discarded. If it is greater than 5, enter 1 in the previous position and add it to the number on the right.
Rewrite it to 0. Give it all away and rewrite it as 0.
Whether to "give up" or "enter" depends on whether the highest digit of the omitted mantissa part is less than 5 or equal to or greater than 5.
Seven, when people count objects, 1, 2, 3, 4, 5, 6, representing the number of objects are natural numbers. There is no object, which is represented by 0. 0 is also a natural number.
The smallest natural number is 0. There is no maximum natural number, and the number of natural numbers is infinite.
Eight, the ratio between every two adjacent counting units is ten. This counting method is called decimal counting method.
Measurement of the second unit angle
First, the characteristics of the line segment: there are two endpoints and the length is limited.
Characteristics of ray: there is only one endpoint, which can extend to one end indefinitely;
Features of a straight line: it has no end points and can extend to both ends indefinitely. You can draw countless rays after one o'clock, countless straight lines after one o'clock, and only one straight line after two o'clock.
Line segment ray straight line
Second, the figure composed of two rays drawn from a point is called an angle.
The edges and corners are usually represented by the symbol "∞".
pinnacle
edge
Third, the size of the angle has nothing to do with the length of both sides, but with the size of both sides. The bigger the fork, the bigger the angle.
The unit of measurement of angle is "degree" and the symbol is "?" Express delivery. Divide the semicircle into 180 equal parts, and the angle of each part is 1 degree, and record it as 1? .
Fourth, the method of measuring the angle: "double looking" 1, using the center of the protractor to coincide with the vertex of the angle. 2. The 0 scale line of protractor coincides with one side of the corner. 3. Look at the scale corresponding to the other side of the angle, which is the degree of the angle.
5. Angles can be divided into acute angles, right angles, obtuse angles, right angles and rounded corners.
The size of the (1) angle has nothing to do with the length drawn on both sides of the angle.
(2) The angle depends on the size of both sides. The bigger the opening, the bigger the angle.
6. Less than 90? This angle is called acute angle.
Equal to 90? This angle is called a right angle.
Greater than 90? And less than 180? This angle is called obtuse angle.
Equal to 180? This angle is called a right angle. The two sides of a right angle are on the same straight line.
Equal to 360? Trumpets are called rounded corners.
1 fillet =2 right angle =4 right angle.
⌒
Acute angle, right angle, oblique angle and fillet
7. Method of drawing the angle: 1, draw a ray, so that the center of the protractor coincides with the endpoint of the ray, and the 0 scale line coincides with the ray. 2. See clearly whether the 0 scale is the inner ring or the outer ring, and point a point on the required degree of the protractor. 3. Take the endpoint of the drawn ray as the endpoint, and draw another ray through the point just drawn.
Unit 3 Multiply three numbers by two numbers
First, the oral calculation method of multiplying two digits by one digit: first divide the two digits into ten digits and ten digits, then multiply them by one digit respectively, and finally add the products of the two multiplications.
16×3= 10×3+6×3=48
Two, hundreds of digits, tens of digits multiplied by one digit, first calculate the product according to the oral calculation method of multiplying two digits by one digit, and then add a 0 to the result.
160× 3 = 48016× 3 = 48 48 plus a 0 is 480.
Third, the pen calculation method of multiplying three digits by two digits: first multiply the three digits with the number on the two digits, and the last digit of the number is aligned with the single digit of the two digits; Then multiply the three digits by the number on the ten digits of the two digits to get the last digit of the number aligned with the ten digits of the two digits; Then add up the numbers multiplied twice.
1 4 5
× 1 2
2 9 0 Multiply 145 by 2 in the company first.
1 4 5 10th place 145 times 1.
1 7 4 0 Finally, add up the multiplied numbers.
Fourth, a simple algorithm for multiplication with zero at the end of the factor: multiply the numbers before zero first, then see how many zeros are at the end of the two factors, and add several zeros at the end of the product.
1 6 ┊ 0
× 3 ┊ 0
4 8 ┆ 0 0
┆
Calculate 16×3 first and then add two zeros.
5. Speed refers to the distance traveled per unit time. Its representation is the distance/time unit traveled.
Speed = distance/time
Example: The distance between A and B is 450 kilometers, and the bus can run in 5 hours. How many kilometers can it travel per hour?
450 ÷ 5 = 90km
The car travels 90 kilometers per hour.
Time = distance/speed
Example: The distance between A and B is 450 kilometers, and the car travels 90 kilometers per hour. How many hours can it be completed?
450÷90=5 (hours)
A: It will be completed in five hours.
Distance = speed × time
For example, the car travels 90 kilometers per hour, and it takes 5 hours to travel from A to B. How many kilometers is it between A and B?
90× 5 = 450km
A: The distance between A and B is 450 kilometers.
When the vehicle is running, if the distance is constant, the faster the speed, the shorter the time required.
Six, the product changes:
1, two numbers are multiplied, one factor is constant, the other factor is multiplied (or divided) by several (except 0), and the product is also multiplied (or divided) by several.
8 × 15 = 120
× ×
4 4
8 × 60 =480
2. Multiply two numbers, one factor multiplies (or divides) a number (except 0), and the other factor divides (or multiplies) the same number, and their products remain unchanged.
8 × 15 = 120
X- no
3 3 transformer
24 × 5 = 120
3. In multiplication, if you want to keep the product unchanged, the changes of the two factors are opposite. When one factor is multiplied by a number, another factor will be divided by the same number.
Seven, multiplication estimation, when to estimate the larger, when to estimate the smaller, should be based on the actual situation, can not mechanically use "rounding" to approximate the number, but the result must be close to the exact value.
Unit 4 Parallelogram and Trapezoid
1. There are only two positional relationships between two straight lines in the same plane: intersecting and non-intersecting (parallel), intersecting at right angles and not intersecting at right angles.
Second, parallelism: two straight lines that do not intersect on the same plane are called parallel lines, which can also be said to be parallel to each other.
3. Verticality: If two straight lines intersect at right angles, they are said to be perpendicular to each other, one of which is called the perpendicular of the other, and the intersection of these two straight lines is called vertical foot.
Four, in the same plane, if two straight lines are parallel to the third straight line, the two straight lines are parallel to each other.
A
B
C
Both a and b are parallel to c, and a and b are also parallel to each other.
Five, in the same plane, if two straight lines are perpendicular to the third straight line, the two straight lines are parallel to each other.
A b
C
Both a and b are perpendicular to c, and a and b are parallel to each other.
6. Draw the vertical line of the straight line at the point where it intersects: 1. Overlap the right-angled edge of the triangular ruler with a known straight line. 2. Move the triangular ruler along a straight line so that the right-angled vertex of the triangular ruler coincides with the known point on the straight line. Draw a ray from the vertex of a right angle along another right angle. Mark the vertical symbol at the vertical bottom.
The vertical line segment drawn from a point outside the straight line is the shortest, and its length is called the distance from that point to the straight line.
The opposite sides of a rectangle (square) are parallel to each other, and the two adjacent sides are perpendicular to each other.
7. Draw a vertical line of this straight line at a point outside the straight line: 1. Overlap the right-angled edge of the triangular ruler with a known straight line. 2. Move the triangle ruler along a straight line so that the other right-angled side of the triangle ruler is close to this point. 3. Draw a straight line along the other right angle of the triangle ruler. Mark the vertical symbol at the vertical bottom.
Eight, the drawing method of parallel lines: 1, fixed triangular ruler, draw a straight line along the right angle. 2, with a ruler close to the other right-angled edge of the triangular ruler, fix the ruler, and then translate the triangular ruler. 3. The first step is to draw a straight line along the right angle. The above method can be used to check whether two straight lines are parallel. )
Nine, the distance between parallel lines is equal everywhere. The line segment between two points is the shortest.
X. Parallelogram: Two groups of parallelograms with opposite sides are called parallelograms.
Characteristics of parallelogram: two groups of opposite sides are parallel and equal respectively; Diagonal degrees are equal.
The parallelogram is unstable. After the deformation of the parallelogram, the perimeter has not changed and the area has changed.
Rectangular and square can be regarded as special parallelograms.
parallelogram
Rectangular trapezoid
square
quadrilateral
Eleven, draw a vertical line from a point on one side of the parallelogram to the opposite side, and the line segment between this point and the vertical foot is called the height of the parallelogram. The side where the vertical foot is located is called the base of the parallelogram. Parallelogram can draw the height of two lengths.
high
high
bottom
12. Trapezoid: Only a set of quadrangles with parallel opposite sides is called trapezoid. bottom
A group of opposite sides in a trapezoid is called the bottom of the trapezoid, the shorter side is called the upper bottom of the trapezoid, the longer side is called the lower bottom of the trapezoid, and a group of non-parallel opposite sides are called the waist of the trapezoid respectively.
The isosceles trapezoid is called isosceles trapezoid; When a waist is perpendicular to the upper and lower soles, this trapezoid is called a right-angled trapezoid.
Draw a vertical line from a point on the upper bottom to the lower bottom of the trapezoid, and the line segment between this point and the vertical foot is called the height of the trapezoid.
The trapezoid is only one length high.
Upper base
Waist-high isosceles trapezoid right-angled trapezoid
Shadi
The divide of unit 5 is that division of two digits.
First, the oral division of labor.
1, there are two ways to divide an integer ten into integers ten and hundred: one is to divide by multiplication according to the multiplication and division relationship, and the other is to divide by table.
2. Dividers or divisors are numbers that are close to integers 10 or 100, and they should be estimated to be close to their integers 10 and 100 by "rounding".
Second, the calculation method of pen division is that the divisor is two digits:
1. Divide the first two digits of the dividend by the high divisor. If the first two digits are less than the divisor, divide the first three digits. (The top two figures of dividends are not enough, depending on the top three. )
2. Write the quotient except the dividend on the dividend.
3. For each quotient, the remainder must be less than the divisor.
Third, the changing law of quotient: 1. In the division formula, the dividend is unchanged. If the divisor is multiplied (or divided) by several (except 0), the quotient will be divided (or multiplied) by the same number.
48 ÷ 12 = 4
÷3 ×3
48 ÷ 4 = 12
48 ÷ 12 = 4
×2 ÷2
48 ÷ 24 = 2
2. In the division formula, if the divisor is constant, the divisor is multiplied (or divided) by several (except 0), and the quotient is multiplied (or divided) by the same number.
48 ÷ 12 = 4
÷2 ÷2
24 ÷ 12 = 2
48 ÷ 12 = 4
×2 ×2
96 ÷ 12 = 8
3. In the division formula, the dividend and divisor are multiplied (or divided) by the same number (except 0) at the same time, and the quotient remains unchanged.
48 ÷ 12 = 4
÷2 ÷2
24 ÷ 6 = 4
48 ÷ 12 = 4
×2 ×2
96 ÷ 24 = 4
Unit 6 Statistics
1. The method of making and representing vertical composite bar charts is basically the same as that of single bar charts, except that each group has two data, which need to be represented by two straight bars with different colors (or shading), and the legend should be indicated at the same time.
Second, use different methods such as horizontal, vertical, comprehensive and comparison to observe and analyze the composite bar chart, and get as much information as possible from it. You can ask questions and solve problems according to the obtained information.
Unit 7 Mathematics Wide Angle
First, the best plan for pancakes is to put as many cakes as possible in the pot at a time, which neither wastes resources nor saves time.
Second, the minimum time used for pancakes = the total number of pancakes, the maximum number of noodles that can be baked in the pot at a time × the time used for baking.
Total number of pancakes = number of pancakes to be baked ×2
Third, to solve the problem of reasonable time arrangement, we need to follow the following steps: 1. Think about what to do to finish a job; 2. How long does it take to analyze everything?
3, reasonable arrangement of work order, understand what to do first, then what to do, and what to do at the same time.
Four, to solve the same problem can have different strategies, to learn to find the best solution to the problem. When competing with each other, we should analyze ourselves and each other in detail, study various countermeasures repeatedly, and choose the best strategy with more advantages and fewer disadvantages among all possible strategies, so as to turn disadvantages into advantages and finally win.
1 1× 1 1= 12 1 25× 1=25 25×25=625 125× 10= 1250
12× 12= 144 25× 2=50 125× 1= 125
13× 13= 169 25× 3=75 125×2=250
14× 14= 196 25× 4= 100 125×3=375
15× 15=225 25× 5= 125 125×4=500
16× 16=256 25× 6= 150 125×5=625
17× 17=289 25× 7= 175 125×6=750
18× 18=324 25× 8=200 125×7=875
19× 19=36 1 25× 9=225 125×8= 1000
20×20=400 25× 10=250 125×9= 1 125