Click on the key points and difficulties

1. Understand the meaning of fractional multiplication and division, and master the calculation method of fractional multiplication and division.

2. Understand the meaning of ratio, the basic nature of ratio and the relationship between ratio and fraction and division, master the transformation of ratio, fraction and division, and apply the knowledge of ratio to solve practical problems.

3. Correctly answer practical questions such as "How much is the fraction of a number" and "How much is the fraction of a number, how to find this number".

Repeated appearance of required questions

Example 1 Which picture below shows it? Product of? ( )

The crowning touch of an idea

? The large rectangle is the unit 1. Divide the unit "1" into four parts, paint three parts, divide it into five parts and paint two parts, so Figure B is correct.

Reading dividing line

Example 2 Yonghe flour mill can grind 1000 tons of flour per hour. According to this calculation, how many tons of flour can be ground per hour?

How many tons of flour can be ground in an hour is the finishing touch. First, calculate how many tons of flour can be ground in 1 hour. Total work divided by working hours equals working efficiency, that is? = (tons). How many tons of flour can you grind in an hour? = 1 (ton).

Reading dividing line

Example 3 The school used 7000 kWh of electricity in September, 10, which was 7 1 less than that in September. /kloc-How many kWh was saved in October?

The finishing touch is that 10 saves 7 1 compared with September, that is, 10 saves 7 1 compared with September. Think of electricity consumption in September as "1". What is the electricity consumption in September? Electricity consumption saved in July1=10 compared with September. How much electricity is saved in 10 compared with September, that is, 7 1 in September? 7000? 7 1= 1000 (degrees)

Reading dividing line

Example 40.25? ( )=0.8? ( )=23? ( )=( )? 37= 1.5? ( )= 1

The crowning touch here is actually to find the reciprocal of a number. The reciprocal of a fraction only needs to change the position of the numerator and denominator. Other numbers turn it into fractions, and then switch the numerator and denominator. For example: 0.25=, the reciprocal is 4.

Reading dividing line

Example 5 is equipped with a kind of concrete, and the following figure shows the number of parts of materials used. If each of these three materials has 24 tons, how many tons of cement will be left when all the yellow sand is used up? How many tons of stones were added?

As can be seen from the figure, the share ratio of cement, yellow sand and crushed stone is 2:3:5, and the required cement tonnage is yellow sand, 24? = 16 (ton), and the remaining ton of cement is 24- 16=8 (ton). The tonnage of stone needed is yellow sand, 24? =40 (ton), and the increased stone tonnage is 40-24= 16 (ton).

Flowers, branches

Module 2 Graphics and Geometry

Click on the key points and difficulties

1. Understand the characteristics of cuboids and cubes, as well as their connections and differences.

2. Master the development diagrams of cuboids and cubes, and imagine the corresponding cuboids or cubes according to the development diagrams.

3. Grasp the meaning of surface area and volume of cuboid and cube, and use the calculation method of surface area and volume of cuboid and cube to solve practical problems in life.

4. Understand the dynamic changes of cuboids or cubes, and master the transformation between cuboids and cubes.

Repeated appearance of required questions

Example 1 Cut a cube with a volume of 1 cubic decimeter into a small cube with a volume of 1 cubic centimeter, which can be cut into () blocks. Arrange these cubes in a row, and the length is () decimeter.

The crowning touch of the idea is 1 cubic decimeter = 1000 cubic centimeter, so the cube with the volume of 1 cubic centimeter can be cut into small cubes with the volume of 1 cubic centimeter, which can be cut into 1000 pieces. 1000 cubes 1 cm3 are arranged in a row, and the length is 1000 cm, 1000 cm =1000 decimeter, so the length is1000 decimeter.

Reading dividing line

A classroom is 8 meters long, 6 meters wide and 4 meters high. To paint the walls and top of the classroom, excluding doors, windows and blackboards, covering an area of 24 square meters. How many square meters is the painting area?

The idea of painting the four walls and the top of the classroom is to paint five surfaces, and it is necessary to first find out the sum of the front, back, left and right areas of the classroom, (8? 4+6? 4)? 2+8? 6= 160 (square meter). You can also subtract the ground area from the sum of the areas of six faces, (8? 4+6? 4+8? 6)? 2-8? 6= 160 (square meter). Doors, windows and blackboards don't need painting. Finally, subtract the area of doors, windows and blackboards, which is 160-24= 136 (square meter).

Reading dividing line

Example 3 A section of square steel is 1 m long, and its cross section is a square with a side length of 5 cm. If the square steel weighs 7.8 grams per cubic centimeter, how many kilograms does this square steel weigh?

The crowning touch of the train of thought is "a section of square steel is 1m long, and its cross section is a square with a side length of 5cm". How many cubic centimeters is the volume of this section of square steel? 1m = 100 cm， 100 cm？ 5? 5=2500 (cubic centimeter). Because the square steel per cubic centimeter weighs 7.8 grams, the square steel of 2500 cubic centimeters weighs 7.8? 2500= 19500 (gram). Finally, we must pay attention to the conversion of units,19500g =19.5kg.

Reading dividing line

Example 4: How many square meters of iron sheets should be used to make a rectangular ventilation duct, with the bottom length and width of 15cm and the height of 0.4m?

The idea is to make a rectangular ventilation pipe with no upper and lower sides, only four sides, and here are four identical sides. Secondly, we should pay attention to the unity of the unit. 15cm = 0. 15m ... 0.15? 0.4? 4=0.24 (square centimeter)

Reading dividing line

Example 5: A cuboid with a length of 40 cm and a square cross section. If the length is increased by 5 cm, the surface area will increase by 80 cm. Find the surface area of the original cuboid.

The striking length of the train of thought increased by 5cm, adding five faces, but also covering one face. Actually, only four faces were added. Because the sides are square, the areas of the four sides are equal. With 80? 4=20 (square centimeter), and the length of the added surface is known to be 5 cm, using 20? 5-4 (cm), find the width of the added surface, that is, the width and height of the original cuboid. In this way, the surface area of the original cuboid can be obtained. (40? 4+40? 4+4? 4)? 2=672 (square centimeter).

Key points of final knowledge of sixth grade mathematics (Volume II) published by People's Education Press.

First unit negative number

1, the origin of negative numbers

In order to express two quantities with opposite meanings (such as profit and loss and income and expenditure), it is not enough to learn 0, 1 and 3.4, so negative numbers appear.

2. Positive and negative numbers

Numbers less than 0 are called negative numbers (excluding 0), and numbers to the left of 0 on the axis are called negative numbers.

Negative numbers are countless.

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.

Positive numbers are countless.

3. How to write positive and negative numbers?

Negative number: put a "-"sign before the number, and the negative sign cannot be omitted.

Positive number: add a "+"sign before the number, and the positive sign can be omitted.

4,0 is neither positive nor negative, it is the dividing line between positive and negative numbers.

5. Number axis:

Percentage of the second unit (2)

1, discount and percentage

(1) discount: used for goods, the current price is a few percent of the original price, which is called discount. Commonly known as "discount".

A few fold is a few tenths, that is, dozens of percent.

(2) Percent: A few percent is a few tenths, that is, dozens of percent.

(3) the discount problem

First, the number of hits is converted into a percentage or a fraction, and then it is solved according to the problem-solving method of finding a percentage (fraction) of a number.

Current price = original price? give a discount

Cheap money = original price-original price? Discount = original price? (1 discount)

(4) the problem of number.

First convert a number into a percentage or a fraction, and then solve it according to the problem-solving method of finding more (less) numbers than a number.

2. Tax rate and interest rate

(1) tax rate The ratio of taxable amount to various incomes is called tax rate. The tax paid is called the tax payable.

(2) Calculation method of tax payable:

Taxable amount = total income? tax rate

Income = tax payable? tax rate

(3) The money deposited in the bank is called the principal. The extra money paid by the bank when withdrawing money is called interest.

The ratio of interest to principal is called interest rate.

(4) interest calculation formula:

Interest = principal? Interest rate? time

Interest rate = interest? Time? Principal? 100%

(5) Note: If you want to pay interest tax (interest on national debt and education savings is not taxed), then:

Interest after tax = interest-taxable interest amount = interest-interest? Interest tax rate = interest? (1- interest tax rate)

Interest after tax = principal? Interest rate? Time? (1- interest tax rate)

3. Shopping strategy

(1) Cost estimation: According to the actual problems, select a reasonable estimation strategy and make an estimation.

(2) According to the actual needs, analyze and compare several common preferential strategies, and finally choose the most favorable scheme.

Unit 3 Cylinders and Cones

1, cylinder

(1) A cylinder is surrounded by two bottoms and one side.

Its bottom surface is two circles with the same size, and its side surface is a curved surface.

The side of the cylinder is rectangular (or square) after being unfolded along the height. One side of this rectangle (or square) is equal to the circumference of the bottom of the cylinder, and the other side is equal to the height of the cylinder.

(2) The height of a cylinder is the distance between two bottom surfaces.

(3) Characteristics of the cylinder

At the bottom of the cylinder are two completely equal circles.

The side of a cylinder is a curved surface.

Cylinders have countless heights.

(4) Relevant calculation formula of cylinder

Bottom area: bottom =? r？

Bottom circumference: C bottom =? d=2？ r

Lateral area: S side =2? right hand

Surface area: s table =2S bottom +S side =2? r？ +2? right hand

Volume: V column =? r？ h

Step 2: Cone

(1) A cone is surrounded by a bottom surface and a side surface. Its bottom surface is round and its side surface is curved.

(2) The distance from the apex of the cone to the center of the bottom surface is the height of the cone.

(3) the characteristics of the cone

The bottom of the cone is a circle.

The side of a cone is a curved surface.

The cone has only one height.

(4) Relevant calculation formula of cone

Bottom area: bottom =? r？

Bottom circumference: C bottom =? d=2？ r

Volume: V cone =? r？ h

Unit 4 Proportion

The significance of 1 and ratio

(1) The division of two numbers is also called the ratio of two numbers.

(2) "Bi:" is a comparative symbol, pronounced "Bi". 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.

(3) 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.

(4) The ratio is usually expressed in fractions, decimals or even integers.

(5) The latter term of the ratio cannot be zero.

(6) According to the relationship between fraction and division, we can know that 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, the basic nature of the ratio

The first term and the second term of a ratio are multiplied or divided by the same number (except 0) at the same time, and the ratio is unchanged, which is called the basic property of the ratio.

3. Find the ratio and simplify the ratio

(1) Method for calculating ratio

Divide the former item by the latter item, and the result is a numerical value, which can be an integer, a decimal or a fraction.

(2) Simplify the ratio

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, the meaning of proportion

Two expressions with equal ratios are called proportions.

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, the basic nature 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) indicates the division of two quantities, which has two terms (i.e. the former and the latter); Proportion refers to two formulas with equal proportions, with four terms (namely, two internal terms and two external terms).

(2) The ratio 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 (that is, quotient) of the corresponding two numbers in these two quantities is certain, these two quantities are called proportional quantities, and the relationship between them is called proportional relationship.

X/y=k (certain) is represented by letters.

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 in the letter? Y=k (ok)

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) digital scale and line scale

(2) Reduce the scale and enlarge the scale

13, distance on the map:

Map distance/actual distance = scale

Actual distance? Scale = distance on the map

Distance on the map? Proportion = actual distance

14, application steps of scale drawing:

(1) Write the name of the graph,

(2) determine the scale;

(3) Calculate the distance on the map according to the scale;

(4) Drawings (unit length of drawings)

(5) Mark the actual distance and write down the place names.

(6) mark the scale

15. Magnification and reduction of graphics: same shape, different sizes.

Unit 5 Mathematical Wide Angle-Pigeon Nest Problem

1, pigeon nest problem

(1) pigeon flight principle

Let's start with a simple example. Put three apples in two boxes. There are four different expressions.

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. Touching the ball

(1) In order to ensure that two balls of the same color are found, the number of balls found must be at least 1 more than the number of colors. That is, number of objects = number of colors? (at least-1)+ 1.

(2) Use extreme thoughts

With the most unfavorable touch method, first touch two balls of different colors, and then no matter what color you touch, you can ensure that there must be two balls of the same color.

(3) Calculation formula

Two colors: 2+ 1=3 (pieces)

Three colors: 3+ 1=4 (pieces)

Four colors: 4+ 1=5 (pieces)