Essentials of Mathematics Volume II Unit 1 Negative Numbers
1, the origin of negative numbers:
In order to express the two opposite quantities of profit loss and income and expenditure, it is not enough to learn 0 1 3.42/5 only. So there is a negative number, the profit is positive and the loss is negative; Take income as positive and expenditure as negative.
2. Negative number: the number less than 0 is called negative number without 0, and the number to the left of 0 on the number axis is called negative number.
If a number is less than 0, it is called a negative number.
There are countless kinds of negative numbers, including negative integers, negative fractions and negative decimals.
Negative numbers are written as:
The number is preceded by a minus sign "-"and cannot be omitted.
For example: -2, -5.33, -45, -2/5.
Positive number:
Numbers greater than 0 are called positive numbers, excluding 0, and numbers to the right of 0 on the number axis are called positive numbers.
If a number is greater than 0, it is said to be positive. There are countless positive numbers, including positive integers, positive fractions and positive decimals.
How to write a positive number: you can add a plus sign before the number or omit it.
For example: +2, 5.33, +45, 2/5
4,0 is neither positive nor negative, it is the dividing line between positive and negative numbers.
Negative numbers are less than 0, positive numbers are greater than 0, negative numbers are less than positive numbers, and positive numbers are greater than negative numbers.
5. Number axis:
6. Compare the size of two numbers:
(1) Use the number axis:
negative number
② Use the meaning of positive and negative numbers: the larger the positive number, the larger the number, and the smaller the number. Negative numbers are relatively large, the large number is small, and the small number is large.
1/3 > 1/6- 1/3 & lt； - 1/6
Essentials of Mathematics Volume II Unit 2 Percentage 2
I. Discounts and percentages
1, discount: used for goods, the current price is a few percent of the original price, called discount. Commonly known as "discount".
A few fold is a few tenths, that is, dozens of percent. For example, 20% = 8/ 10 = 20%,
65% discount = 6.5/10 = 65/100 = 65%
To solve the discount problem, the key is to convert the discount into percentage or score first, and then answer it according to the method of comparing a few percent with a few percent.
The goods are now 20% off: the current price is 20% off the original price.
The goods are now 50% off: the current price is 65% of the original price.
2, into the number:
A few percent is a few tenths, that is, dozens of percent. For example, ten percent =110 =10.
Eighty-five percent = 8.5/10 = 85/100 = 80%.
The key to solving a number problem is to first convert the number into a percentage or a fraction, and then solve it according to a problem-solving method of how many percent the number is.
The purchase price of clothes increased this time 10%: the purchase price of clothes increased this time 10%.
The wheat harvest this year is 85% of last year's.
Second, the tax rate and interest rate.
1, tax rate
1 tax payment: tax payment is to pay a part of the collective or individual income to the state according to the relevant provisions of the national tax law.
2 Significance of tax payment: tax payment is one of the main sources of national fiscal revenue. The state uses the collected taxes to develop economy, science and technology, education, culture and national defense security.
Taxes payable: Taxes payable are called taxes payable.
Tax rate: the proportion of tax payable to various incomes is called tax rate.
5. Calculation method of tax payable:
Taxable amount = total income × tax rate
Income = tax payable ÷ tax rate
2. Interest rate
1 Deposits are divided into demand deposits and lump-sum deposits.
Significance of saving: People often deposit temporarily unused money in banks or credit cooperatives, which can not only support national construction, but also make personal use of money safer and more planned, and increase some income.
Principal: Money deposited in the bank is called principal.
4 interest: the excess money paid by the bank when withdrawing money is called interest.
5 interest rate: the ratio of interest to principal is called interest rate.
6 Calculation formula of interest:
Interest = principal × interest rate× time
Interest rate = interest/time/principal × 100%
7 Note: If interest tax is to be paid, the interest of national debt and education savings is not taxed, then:
After-tax interest = interest-interest tax payable = interest-interest × interest tax rate = interest × 1- interest tax rate.
After-tax interest = principal × interest rate× time×1-interest tax rate
Shopping strategy:
Cost estimation: according to the actual problems, choose a reasonable estimation strategy and make an estimation.
Shopping strategy: according to the actual needs, analyze and compare several common preferential strategies, and finally choose the most favorable scheme.
Reflection after learning: the benefits of using strategies in doing things
Essentials of Mathematics Volume 2 Unit 3 Cylinders and Cones
I. Cylinder
1. Formation of a cylinder: rotate one side of a rectangle as an axis to form a cylinder.
Cylinders can also be obtained by curling rectangles.
Two ways:
1. The perimeter with the length of the rectangle as the base, with the width as the height;
2. Take the width of the rectangle as the perimeter of the bottom and the length as the height.
Among them, the cylinder volume obtained by the first method is larger.
2. The height of a cylinder is the distance between two bottoms. A cylinder has countless heights, and their values are equal.
3, the characteristics of cylinder:
The bottom feature of 1: The bottom of a cylinder is two completely equal circles.
2 side features: the side of the cylinder is a curved surface.
Characteristics of height: A cylinder has countless heights.
4, cylinder cutting:
① Crosscutting: the cross section is circular, and the surface area is increased by 2 times of the bottom area, that is, S increase =2πr?
② Vertical cutting diameter: If the cross section is rectangular and the cross section is square, the length of the rectangle is the height of the cylinder and the width is the diameter of the bottom of the cylinder, and the surface area is increased by two rectangles, that is, S is increased by h=2R 4RH.
5, the side of the cylinder:
① Spread along the height, and the spread diagram is rectangular. If h=2πr, the spread diagram is square.
(2) Do not expand along the height, and the expanded figure is a parallelogram or irregular figure.
③ You can't get a trapezoid no matter how you unfold it.
6, cylinder related calculation formula:
Bottom area: s bottom =πr?
Bottom circumference: C bottom =πd=2πr
Transverse area: S side =2πrh.
Surface area: s table =2S bottom +S side =2πr? +2πrh
Volume: V column =πr? h
Common test questions:
① Knowing the bottom area and height of a cylinder, find the lateral area, surface area, volume and bottom perimeter of the cylinder.
② Knowing the circumference and height of the bottom surface of the cylinder, find the lateral area, surface area, volume and bottom area of the cylinder.
③ Knowing the circumference and volume of the bottom surface of the cylinder, find the lateral area, surface area, height and bottom area of the cylinder.
④ Knowing the area and height of the bottom surface of the cylinder, find the lateral area, surface area and volume of the cylinder.
⑤ Given the lateral area and height of the cylinder, find the radius, surface area, volume and bottom area of the cylinder.
The solution to the above common problems is usually to find the radius and height of the bottom of the cylinder, and then calculate it according to the relevant calculation formula of the cylinder.
Surface area of uncovered oil drum = side area+surface area of oil drum with one bottom area = side area+two bottom areas.
Surface area of chimney ventilation pipe = transverse area
Right side area: lampshade, drain pipe, paint column, ventilation pipe, roller, toilet paper shaft, potato chip box packaging.
Side area+bottom area: glass, bucket, pen container, hat, swimming pool.
Side area+two bottom areas: oil barrel, rice barrel and tank.
Second, the cone
1, cone formation: rotate the right-angled side of the right-angled triangle as the axis to get the cone. A cone can also be obtained by sector curling.
2. The height of a cone is the distance between two vertices and the bottom. Unlike a cylinder, a cone has only one height.
3, the characteristics of the cone:
The bottom feature of 1: The bottom of the cone is a circle.
2 side features: the side of the cone is a curved surface.
3 high characteristics: the cone has a height.
4, cone cutting:
① Crosscutting: the section is circular.
② Vertically cut through the vertex and diameter: the section is an isosceles triangle, the height of the isosceles triangle is the height of the cone, and the bottom is the diameter of the bottom of the cone, and the area is increased by two isosceles triangles.
That is, s increase =2rh.
5, cone related calculation formula:
Bottom area: s bottom =πr?
Bottom circumference: C bottom =πd=2πr
Volume: V cone = 1/3πr? h
Common test questions:
① Given the area and height of the bottom of the cone, find the volume and circumference of the bottom surface.
② Given the circumference and height of the bottom of the cone, find the volume and bottom area of the cone.
③ Given the circumference and volume of the bottom of the cone, find the height and bottom area of the cone.
The solution to the above common problems is usually to find the radius and height of the cone bottom, and then calculate it according to the relevant calculation formula of the cylinder.
Third, the relationship between cylinder and cone
1, the height of cylinder and cone is equal, and the volume of cylinder is three times that of cone.
2. The volume of cylinder and cone is the same, and the height of cone is three times that of cylinder.
3. Cylinders and cones are both very large. Note that the bottom area of a cone is three times that of a cylinder, not the radius of the bottom surface.
4. The cylinder and the cone have equal bottoms and heights, and the volume difference is 2/3Sh.
Problem summary
① Direct use formula: the surface area, lateral area, bottom area and volume can be clearly obtained through analysis.
Obviously, the change of radius leads to the change of bottom perimeter, lateral area, bottom area and volume.
The radius, bottom area, bottom perimeter, lateral area, surface area and volume ratio of two cylinders or two cones are analyzed.
(2) Transformation of the relationship between cylinder and cone: including cutting the problem cube into the maximum volume, between the cuboid and the cylinder cone.
③ the problem of cross section
④ Immersion volume: The volume of the rising part of the water surface is the volume of the articles submerged in the water, which is equal to the bottom area of the water holding volume multiplied by the rising height. The volume is a cylinder or a cuboid, and it is also a cube.
⑤ Equal volume conversion problem: melting a cylinder and making it into a cone, or pouring the solution in the cylinder into a cone, this is a problem with constant volume. Be careful not to multiply by 1/3.
Essentials of Mathematics Volume II Unit 4 Proportion
The significance of 1 and ratio
1 The division of two numbers is also called the ratio of two numbers.
2 ":"is a comparative symbol, pronounced "than". The number before the comparison symbol is called the first item of comparison, and the number after the comparison symbol is called the last item of comparison. The quotient obtained by dividing the former term by the latter term is called the ratio.
Compared with division, the former term of ratio is equivalent to dividend, the latter term is equivalent to divisor, and the ratio is equivalent to quotient.
The ratio of 4 is usually expressed in fractions, decimals and sometimes integers.
The last term of the ratio of five to five cannot be zero.
According to the relationship between fraction and division, the former term of ratio is equivalent to numerator, the latter term is equivalent to denominator, and the ratio is equivalent to fractional value.
2. Basic properties of ratio: The first term and the second term of ratio are multiplied or divided by the same number 0 at the same time, and the ratio remains unchanged, which is called the basic properties of ratio.
3. Find the ratio and simplify the ratio:
The method of finding the ratio: divide the former term of the ratio by the latter term, and the result is that a numerical value can be an integer, a decimal or a fraction.
According to the basic properties of the ratio, the ratio can be reduced to the simplest integer ratio. Its result must be the simplest ratio, that is, the first term and the last term are prime numbers.
4. Proportional distribution:
In agricultural 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.
Methods: First, find out the scores of each part in the total, and then find out what the scores of the total are.
5. Meaning of proportion: Two expressions with equal proportion are called proportion.
The four numbers that make up a proportion are called proportional terms.
The two items at both ends are called external items, and the two items in the middle are called internal items.
6. Basic properties of proportion: In proportion, the product of two external terms is equal to the product of two internal terms. This is the basic nature of the so-called proportion.
7, the difference between ratio and proportion
The ratio of 1 represents the division of two quantities, which has two items, namely the former and the latter; Proportion refers to two equal formulas with four terms, namely, two internal terms and two external terms.
The ratio of 2 has basic properties, which is the basis of simplifying the ratio; Proportion also has a basic nature, which is the foundation of solution ratio.
8. Proportional quantity: two related quantities, one change and the other change. If the ratio of the corresponding two numbers in these two quantities is certain, these two quantities are called proportional quantities, and their relationship is called proportional relationship.
X/y with letters = K.
9. Inverse proportional quantity: two related quantities, one change and the other change. If the product of the corresponding two numbers in these two quantities is certain, these two quantities are called inverse proportional quantities, and their relationship is called inverse proportional relationship.
X×y=k in letters must be.
10, a method for judging whether two quantities are directly proportional or inversely proportional:
The key is to see that the quotient of two relative numbers in these two related quantities must still be a product, and if the quotient is certain, it is proportional; If the product is constant, it is inversely proportional.
1 1. scale: the ratio of the distance on a picture to the actual distance is called the scale of this picture.
12, classification of scale
1 Numerical scale and line scale 2 Reduced scale and enlarged scale
13, distance on the map:
Map distance/actual distance = scale
Actual distance × scale = map distance
Distance on the map/scale = actual distance
14, application steps of scale drawing:
1 Write the name of the drawing,
2. Determine the scale;
3. Calculate the distance on the map according to the scale;
Draw a picture to show the unit length.
Mark the actual distance and write down the name of the place.
6 mark scale
15. Magnification and reduction of graphics: same shape, different sizes.
16, solve the problem by proportion:
According to the invariants in the problem, find out two related quantities, correctly judge the proportional relationship between the two related quantities, and list the corresponding equations according to the positive and negative proportional relationship and solve them.
17. Common quantitative relationships: directly proportional or inversely proportional.
Unit price × quantity = total price
Single output × quantity = total output
Speed × time = distance
Efficiency × working hours = total amount of work
18, the distance on the known map and the actual distance can be scaled.
Given the scale and distance on the map, you can find the actual distance.
Given the scale and the actual distance, you can find the distance on the map.
When calculating, the units of drawing distance and real distance must be unified.
19, the total number of hectares sown is fixed. Is the number of hectares sown per day inversely proportional to the number of days to be used?
Answer: the number of hectares sown every day × days = the total number of hectares sown.
It is known that the total number of hectares sown is fixed, that is, the product of the number of hectares sown every day and the number of days to be used is fixed, so the number of hectares sown every day is inversely proportional to the number of days to be used.
Essentials of Mathematics Volume II Unit 5 Mathematical Wide Angle-Dove Nest Problem
1 and pigeon's nest principle is an important and basic combination principle, which plays a very important role in solving mathematical problems.
(1) What is the principle of pigeons? Let's start with a simple example. Put three apples in two boxes. There are four different ways to put * * *, as shown in the following table.
Release method
Box 1
Box 2
Either way, it can be said that there must be two or more apples in a box. This conclusion is the "inevitable result" in the case of "arbitrary release"
Similarly, if five pigeons fly into four pigeon coops, then a pigeon coop will certainly fly into two or more pigeons.
If there are 6 letters and put them into 5 mailboxes at random, there must be at least 2 letters in one mailbox.
We take "Apple", "Pigeon" and "Letter" in these examples as an object, and "Box", "Pigeon Cage" and "Mailbox" as a pigeon, and we can get the simplest expression of the pigeon principle.
② Using formulas to solve problems:
Number of objects = number of pigeons = quotient ... remainder
At least number = quotient+1
2. Calculation method of touching two identical colored balls.
(1) Make sure to touch two balls of the same color, and the number of balls touched is at least more than the number of colors 1.
Number of objects = number of colors × at least-1+ 1
② Extreme thinking: First touch two balls of different colors with the most unfavorable touch method, and then no matter what color you touch, you can guarantee that there must be two balls of the same color.
③ Formula:
Two colors: 2+ 1=3.
Three colors: 3+ 1=4.
Four colors: 4+ 1=5.
In 2022, the enrollment charter of Shaanxi Vocational College of Mechanical and Electrical Technology h