First unit
(Chapter 1 Rich Graphic World)
Review target
1, learn more about the common cylinders, cones and spheres in life, and do some simple things about them.
Classification.
2, can understand the development diagram of simple geometric surfaces such as straight prism, pyramid, cylinder and cone, according to the development diagram.
Imagine, judge and make geometric models.
3. It can draw three views of three-dimensional graphics and judge the shape of three-dimensional graphics according to the three views.
4. Understand the section and imagine the shape of the section.
5. Experience the unfolding, folding and cutting of geometry, stimulate curiosity, accumulate experience in mathematical activities, and form and develop the concept of space.
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1. Fill in the blanks with basic knowledge
1. Graph is composed of points, lines and faces.
2. In a prism, the intersection of any two adjacent faces is called an edge, and the intersection of two adjacent sides is called a side. All sides of the prism are equal in length, and the top and bottom surfaces of the prism are the same in shape, and the shapes of the sides are all rectangular.
3. Cut a geometric figure with a plane, and the cutting surface is called a section.
We call the figure of the object seen from the front as the front view, the figure seen from the left as the left view, and the figure seen from above as the top view.
5. The part between point A and point B on the circle is called an arc, and the figure consisting of an arc and two radii passing through the end of this arc is called a sector, and the circle can be divided into several sectors.
6. The side development diagram of cylinder is rectangular, and the side development diagram of cone is fan-shaped.
Two. Typical example
Example 1: As shown in the figure, can the folded figure A form the prism of figure B? If it can be formed, answer:
(1) How many faces does this prism have? What is the relationship between the number of sides and the base?
(2) Which faces must be exactly the same in shape and size? If it cannot be formed, briefly explain the reasons.
Analysis and solution: fold the upper and lower pentagons to the same side of the rectangle in turn, and then fold them against the sides of the pentagon in turn to form the pentagon on the right.
(1) This prism * * * has five sides with the same number of sides as the bottom.
(2) The upper and lower sides of a pentagonal prism must be exactly the same, and both sides are rectangular, but not necessarily identical.
Note: the diagram obtained by folding the expanded diagram into a prism is unique, while the expanded diagram obtained by expanding the prism into a plane diagram is not unique.
Example 2: Cut the surface of the cube along some edges, can it be expanded into the figure shown below?
Analysis and solution: To solve this kind of problem, we should have certain spatial imagination ability and master some skills. In (2), there are five small squares connected in a line, and it is impossible for the surface of the cube to expand into this figure. There are seven small squares in (7), which is even more impossible. Generally speaking, there are four small squares connected into a line, and there are two small squares on both sides of this "line", which can be folded into a cube. Therefore, the surface of the cube can be developed into the graphs shown in (1) and (3). By developing spatial imagination or folding by hand, we can see that the surface of the cube can also be unfolded into the figure shown in (5) and (6), but it cannot be unfolded into the figure shown in (4). That is, (2), (4) and (7) are impossible, and the rest are possible.
Example 3: Please design a method to cut a cube with a plane so that the cross section is a triangle with three equal sides.
Analysis and solution: take a point from each of three adjacent sides of the cube, so that the distance between the point and the intersection of these three sides is equal, connect these three points to get three connecting lines, and cut along these three connecting lines with a plane to get a triangle with three equal sides. See below.
Note: when doing this kind of topic, you should fully imagine it first, then operate it to ensure it.
Prove correct.
Example 4: As shown in the figure, it is a top view of two geometric bodies A and B composed of several small cubes. The number in the small square indicates the number of small cubes in this position. Please draw their front and left views.
Analysis and solution: This question can determine the number of columns in the front view and the left view according to the top view, and then determine the number of squares in each column according to the numbers.
Note: When looking down at the front view and the left view of the picture, you should find the number of each column from left to right as the number of the row.
Example 5: As shown in the figure, it is three views of the geometry composed of several identical small cubes. Please fill in the number of cubes in this position in the small box in the top view.
Analysis and solution: According to the front view, the square in the second row and the 1 column in the top view have 1 cubes.
It makes sense to know that there are 1 cubes in the square in the upper right corner of the top view; As can be seen from the left figure, there are two small cubes in the two squares in the second column of the top view.
Second unit
(Chapter II Rational Numbers and Their Operations)
Review target
1, can flexibly use points on the number axis to represent rational numbers, understand opposites and absolute values, and can use the number axis.
Compare the size of rational numbers.
2. Skillfully use the arithmetic of rational numbers to calculate the addition, subtraction, multiplication, division and power of rational numbers.
Algorithms can simplify calculations.
3. Be able to use rational numbers and their operations to solve simple practical problems.
4, can use the calculator to add, subtract, multiply and divide, and solve complex problems in practical problems.
Count.
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First, fill in the blanks with basic knowledge.
1.0 is neither positive nor negative.
2. Integers and fractions are collectively called rational numbers. 、
4. The straight line defining the origin, positive direction and unit length is called the number axis.
5. There are only two numbers with different symbols, and we call one of them the antonym of the other.
6. The number represented by two points on the number axis, the number on the right is always greater than the number on the left; Positive numbers are all greater than 0, all less than 0, and all positive numbers are greater than negative numbers.
7. The distance between the point corresponding to a number and the origin on the number axis is called the absolute value of the number; The absolute value of a positive number is itself; The absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0; Comparing the sizes of two negative numbers, the absolute value is larger but smaller.
8. rational number addition rule: add two numbers with the same sign, take the sign of the addend, and add the absolute values. Two numbers with different signs are added, and the sum is 0 when the absolute values are equal; When the absolute values are not equal, take the sign of the number with larger absolute value and subtract the number with smaller absolute value from the number with larger absolute value; Adding a number to 0 still gets this number.
9. Subtracting a number is equal to adding the reciprocal of this number.
10. rational number multiplication rule: when two numbers are multiplied, the same sign is positive and the different sign is negative; Multiply any number by 0, and the product is 0.
1 1. Two rational numbers whose product is 1 are reciprocal.
12. The operation of finding the product of several same factors is called power, and the result of power is called power.
In 13, a is called the base and n is called the exponent.
14. The operation sequence of rational number mixing operation is: first calculate the power, then calculate the multiplication and division, and finally calculate the addition and subtraction; If there are brackets, count them first.
Second, typical cases
Example 1: use ">" to connect the following numbers: the reciprocal of -2.5, the absolute value of -3.8, 3 and -4.
Analysis and solution: When multiple rational numbers are relatively large, the number axis and the right number axis are often used.
This number is larger than the number on the left. Each number can be represented by letters, and then words can be represented on the number axis.
Numbers corresponding to mothers.
The absolute value of reciprocal c: -3.8d: 3e:-4 of a: 0b:-2.5.
So the absolute value of -4 >; The reciprocal of 3 > > >; 0 & gt-3.8
Note: Comparing the sizes of two or more numbers, we can take the number axis as an important tool, and use the points on the number axis to represent these five numbers, and the sorting from big to small will be completed naturally.
Example 2: Fill in the following figures in brackets indicating the corresponding set.
Positive array: {┄}, score group: {┄}
Negative integer set: {┄}, non-negative integer set: {┄}
Natural number set: {┄}, Rational Number Set: {┄}
Analysis and solution: It is the key to solve the problem to clarify the concepts of non-negative numbers, natural numbers, negative integers and rational numbers. Non-negative numbers include 0 and positive numbers, natural numbers include 0 and positive integers, and decimals in the stem can be regarded as fractions.
Note: The differences and connections between sets must be made clear to ensure the accuracy of the numbers in the set.
Example 3: Calculation:
Analysis and solution: this problem can first unify the mixed operation of addition and subtraction into addition, then write a simplified algebraic expression, and then simplify the operation by using the algorithm.
Note: When applying additive commutative law and the law of association, we must be careful not to change the property number of each number. According to the characteristics of the problem, it is the key to choose the appropriate solution flexibly.
Example 4: Calculation
Analysis and solution: After the division operation in the problem is converted into multiplication operation, we can find that this problem can be simplified by using the operational nature of multiplication.
Note: For calculation questions, we should carefully observe the characteristics of the questions and try to use simple methods.
Example 5: Calculate the value of (-0.25)2002×42004.
Analysis and solution: When a problem is found to be troublesome to calculate, you should observe it carefully, use your brains more, and try your best to find a simple solution to this problem. If it is difficult to directly solve (-0.25)2002 and 42004, we can find that it reminds us that it is easier to get the result of 16 by using multiplicative commutative law and associative law.
Example 6: Calculate with a calculator:
(-3)3-〔(-5)2+( 1-0.2× )÷(-2)〕
Third unit
(Chapter III Letters Representing Numbers)
Review target
1, to further explore the quantitative relationship between things, which can be expressed by letters and algebraic expressions.
2. Understand the meaning of numbers expressed by letters and algebraic expressions, analyze and explain the actual background or geometric meaning of some simple algebraic expressions, and experience the connection between mathematics and the real world.
3. Master the rules of merging similar items and removing brackets, and make calculations.
4, can find the value of the algebraic expression, can explain the practical significance of the value, can infer the law reflected by the algebraic expression according to the value of the algebraic expression.
Review content:
First, fill in the blanks with basic knowledge.
1, and the formula formed by connecting numbers or letters representing numbers with operation symbols is called _ algebraic formula; A single number or letter is also an algebraic expression.
2. In algebraic expressions, the number factor before the letter is called its _ _ _ _ _ _.
3. Items with the same _ letters and the same _ letter index are called similar items, and merging similar items into one item is called _ merging similar items _.
4. Rules for merging similar items: _ _ Add the coefficients of similar items, and the indexes of letters and letters remain unchanged.
5. Rules for removing brackets: add "+"before _ _ brackets. After removing the brackets and the "+"sign in front of them, the symbol of the original brackets remains unchanged; There is a "-"before the brackets. After removing the brackets and the "-"sign in front of them, the symbols of the original brackets will change.
Second, typical cases
Example 1: Use letters to indicate the following practical problems:
(1) The length, width and height of a cuboid are A, B and C, so what is the volume of the cuboid? What is the surface area?
(2) If a dress is priced at one yuan and sold at a 20% discount, what is the selling price?
(3) Each picture below is a triangular pattern composed of several potted flowers, with n (n >: 1) potted flowers on each side (including two vertices), and the total number of flowerpots in each pattern is S. According to this rule, the relationship between S and N is deduced.
Analysis and solution: (1) From the cuboid volume formula = length× width× height and surface area = the sum of six small areas, it can be concluded that the cuboid volume is abc and the surface area is 2 (AB+BC+AC); (2) The so-called 20% discount means selling at 20% of the marked price, so the selling price is 0.8a yuan; (3) Because there are n pots on each side, the total number of flowerpots on three sides is 3n, and the number of flowerpots on the vertex is counted repeatedly, so the total number of flowerpots should be 3n-3. Therefore, when n=2, the total number of flowerpots is 2× 3-3 = 3;
When n=3, the total number of flowerpots is 3× 3-3 = 6;
When n=4, the total number of flowerpots is 4× 3-3 = 9;
…
When there are n flowerpots on each side, the total number of flowerpots is S=3n-3.
Note: (1) When expressing an actual problem with a formula containing letters, the quantitative relationship in the actual problem must be clear;
(2) Multiplying numbers with letters, or multiplying numbers with formulas containing letters, generally omitting the multiplication sign and writing the numbers in front;
(3) When multiplying letters, you can write "x" as ",or you can not write it.
Example 2: Find the value of the following algebraic expression:
Analysis and solution: (1) Find the similar items first, and then add up the coefficients of the similar items to keep the letters and their indexes unchanged.
(2) This question can be directly enclosed in brackets, and then combined with similar projects for final evaluation, but careful observation shows that every
The formulas in brackets are all the same. You can merge these formulas directly just like merging similar items.
Note: Generally, when finding the value of an algebraic expression, we should first look at whether this algebraic expression can merge similar items. If so, it should be merged first and then evaluated.
Example 4: In the calendar of June 5438+ 10, 2003 as shown in the figure, circle any 3×3 number with a box.
Fourth unit
(Chapter IV Plane Graphics and Their Positional Relations)
Review target
1. Know the meaning of line segment, ray, straight line, angle, parallel line and vertical line, and give these examples in real life.
2. Can draw line segments and angles, line segments equal to known line segments and angles equal to known angles; Will compare the length of two line segments, will compare the size of two angles; Will draw parallel lines and vertical lines of known straight lines.
3. Understand the usage of Tangram and Tangram; Will design simple patterns according to actual needs.
View content
First, fill in the blanks with basic knowledge.
The 1. line segment has two endpoints. Extending a line segment to an endpoint indefinitely forms a ray with 1 endpoints. A straight line is formed by an infinite extension of a line segment to two endpoints, and the straight line has 0 endpoints.
2. In the connection between two points, the line segment is the shortest; The length of the line segment between two points is called the distance between these two points.
3. If point M divides line segment AB into two equal line segments AM and BM, then point M is called the midpoint of line segment AB. At this point, AM=BM=
AB .
4. An image composed of two rays with a common endpoint is called an angle.
5. 1 =60′=360″
6. A ray drawn from the vertex of an angle divides the angle into two equal angles. This ray is called the bisector of the angle.
7. On the same plane, two lines that do not intersect are called parallel lines.
8. After a point outside the straight line, there is one and only one straight line parallel to this straight line.
9. If two straight lines are parallel to the third straight line, then the two straight lines are parallel.
10. If two straight lines _ intersect at right angles, then these two straight lines are perpendicular to each other, and the intersection of these two vertical straight lines is called vertical foot.
1 1. In the plane, there is one and only one straight line perpendicular to the known straight line.
12. Let the vertical line of L pass through point A, and the vertical foot is B. The length of line AB is called the distance from point A to line L..
Second, typical cases
Example 1: The figure below * * *, how many straight lines, how many line segments and how many readable rays are there? Are they divided?
Analysis and solution: (1) A straight line has one Mn;
(2) The line segments are: line segment AB, line segment BC and line segment AC;
(3) rays include: ray AB, ray AM, ray BC, ray BA, ray CB and ray CN.
Note: In the process of solving problems, the principles of "classification", "order" and "classification" should be achieved.
That is, neither repetition nor omission; The "orderly" method refers to starting from a certain point and a certain line segment.
Orderly counting.
Example 2: (1) Turn 25 24? 36 "Chengdu (2) Looking for 80 2? 24"×6
Analysis and solutions:
(1) degrees, minutes and seconds should be converted into degrees. Starting from seconds, list 36 seconds separately.
Convert it into minutes, that is, 36 "60 = 0.6", and then convert 24'+0.6'=24.6' into degrees, that is, 24.6' 60 = 0.41? Finally.
25.45438+0? .
(2) The calculation of the number of times is the same as rational number, but it is only an operation.
The calculation order is different from decimal, as follows:
80 2? 24"×6=80? ×6+2′×6+24″=480? + 12′+ 144″=480? 14′24″
note:
(1) is the conversion from low-level units to high-level units, and the formula used is 1'= ().
1"=()′; (2) The calculation method of is similar to the distribution law of multiplication and addition in the rational number operation law, and the conversion of degrees-minutes-seconds is carried out step by step, and cannot be "skipped".
Example 3: As shown in the figure, straight line AB and CD intersect at point O, OE bisects AOD, AOC=38? , find the degree of DOE.
Analysis and solution: Because point C, point O and point D are on the same straight line.
Do you know that COD is a right angle with a degree of 180?
Because AOC=38?
So AOD= 142?
OE divides AOD equally again.
So =AOD=7 1?
Note: (1) has a hidden condition, that is, COD= 180? This is made up of
The straight line AB and CD intersect at point O.
(2) According to the definition of angle bisector and the sum and difference of angles, divide AOD by OE to obtain
DOE = doe =AOD
Example 4: How do PE teachers line up when the school holds an inter-school radio exercise competition?
1, all stand at attention, and the rows are aligned forward. What is this for?
2. What is the purpose of aligning rows left and right based on a certain behavior?
3. Based on a certain row, each row is dispersed into a radio exercise formation (keep a proper distance from front to back, left and right), so that the radio exercise formation is neat and beautiful. Why?
Analysis and solution: (1) Align each line forward to make each line a straight line;
(2) Align each line left and right to make each line a straight line;
(3) Keep a proper distance from left to right, so that each row is connected with each row's straight line.
Parallel, diagonal students all line up in parallel.
Note: Students can feel the beauty of geometry through their familiar personal experience, which is also helpful to understand the concept of "parallel lines".
Example 5: As shown in the figure, the points passing through O are the perpendicular lines of CB and AD respectively.
Analysis and solution: overlap one side of the triangular ruler with AB, and make the other side close to the O point. Draw a straight line along this side, which is the perpendicular to AB. Similarly, the vertical line of CD can be drawn through point O.
Note: When using a trigonometric ruler as the perpendicular of a known straight line, one side of the trigonometric ruler (understood as a straight line) must coincide with the known straight line.
Example 6: We are familiar with clocks, but have you noticed the quantitative relationship between the relative positions of the clock and the minute hand?
(1) The minute hand turns 6 degrees per minute and the hour hand turns 0.5 degrees per minute;
(2) In the same period of time, the angle of the minute hand is different from that of the hour hand.
The ratio is equal to12; Can you calculate the hour hand and minutes between 1 and 2 o'clock?
When do the needles coincide? When do the two needles make a 90-degree angle?
Note: The clock problem can be solved by using the above two laws (1) and (2).
Example 7: Using a jigsaw puzzle:
(1) Please use two pairs of identical puzzles to spell out the picture of two people meeting and saluting each other, as shown in the following figure (1).
(2) Please use three sets of identical puzzles to spell out the figure of two people playing table tennis, as shown in Figure (2).
Analysis and solution: The key to solve this problem is to correctly understand the various figures that make up the puzzle.
Third, the class summary
1. Based on the basic knowledge of geometry, this chapter further studies the meanings of line segments, rays, straight lines, angles, parallel lines and vertical lines in geometry, and gives some basic properties combined with common sense of life, so that we can have a deeper understanding of basic geometric figures.
2. Through the study of this chapter, students are required not only to develop the habit of hands-on operation, but also to cultivate the idea of combining numbers with shapes.
Fourth, homework
Fifth unit
(Chapter 5 One-dimensional Linear Equation)
Review target
1, understand the concept of linear equation with one variable and its solution;
2. Be able to skillfully solve the linear equation of one variable and use it to solve some practical problems;
3. The key to solving problems by equation is to grasp the equivalence relation and understand the importance of equation model.
View content
First, fill in the blanks with knowledge.
1, an equation with unknown numbers is called an equation.
2. An equation containing only one unknown quantity whose exponent is 1 is called a linear equation.
3. The result of adding (or subtracting) the same algebraic expression on both sides of the equation is still an equation; Multiply both sides of an equation by the same number (or divide by the same number that is not 0), and the result is still an equation.
4. After changing the sign of an item in the original equation, it moves from one side of the equation to the other. This deformation is called moving items.
5. The general steps of solving a linear equation with one variable are: removing denominator, removing brackets, shifting terms, merging similar terms, and converting unknown coefficients into 1. And "transform" the linear equation of one variable into a form.
6. Principal+interest = sum of principal and interest, and interest = principal × interest rate× number of periods.
Second, typical cases
Attention: ① When solving a linear equation with one variable, we should carefully observe its characteristics; ② When removing the denominator, items without denominator cannot be omitted; (3) Fractions not only represent division symbols and comparison symbols, but also act as brackets. Therefore, when removing the denominator, we should remove the fractional line, take the numerator as a whole, add brackets, and then remove brackets.
Example 3: A classmate covers five numbers of a month in the calendar with a cross-shaped box. Could the sum of these five numbers be 125? Why?
Analysis and solution: According to the arrangement law of numbers on the calendar: the difference between the upper and lower numbers is 7, and the difference between the left and right numbers is 1, so let the middle number be X and the other four numbers be: X- 1, X- 1, X-7, X+7, and get the equation (X-/kloc-).
Note: First, calculate the size of these five numbers according to the conventional method, and then check whether it is reasonable.
Example 4: There are two containers, A and B. Container A is a cuboid with a square with a side length of 2 and a height of 3 at the bottom. Container B is cylindrical, with a bottom radius of 1 and a height of 3. If container A is full of water, pour a part of water into container B to make the liquid level in the two containers the same, and find the liquid level at this time. (3. 14, accurate to 0.0 1)
Analysis and solution: ① volume of cuboid: v=abc, volume of cylinder: ② volume of container A = volume of water in container A+volume of water in container B. From the above two points, the equation can be listed. Let the liquid level at this time be x, and from the meaning of the question, x= 1.68.
Note: the key to solve this problem is to find the equivalence relation: the sum of the volumes of water in two containers is equal to the volume of container A.
Example 5: A city collects monthly gas charges according to the following regulations. Not more than 70m3, according to the 0.9 yuan per cubic meter. If it exceeds 70m3, the excess part will be charged per m3 1. 1 yuan. It is known that a user's gas bill in May is 0.95 yuan per cubic meter on average, so what should the user's gas bill be in May?
Analysis and solutions:
Because the average gas fee in May was 0.95 yuan per cubic meter, ranging from 0.9 yuan to 1. 1 yuan, it can be seen that the gas consumption of this user in May exceeded 70m3, and the gas fee should be composed of two parts. Therefore, it can be assumed that the user used xm3 gas in May, which is 70× 0.9+1.1(x-70) = 0.95x.
X≈93.3 ∴0.95x=89.
That is, in May, this user has to pay the gas bill in 89 yuan.
Third, the class summary
1, one-dimensional linear equation is the most basic content in equation knowledge, and it is to learn one-dimensional quadratic and one-dimensional multiple sum.
The cornerstone of other equations such as binary first order and binary second order;
2. The solution of one-dimensional linear equation is also the basis of other equations, and the solutions of other equations will eventually be transformed into the solutions of one-dimensional linear equations;
3. Some practical problems in life can be solved by establishing equation models.
Fourth, homework