Teaching research 2010-01-1506: 54 Reading 28 comments 0 font size: chapter 11 linear functions of large, medium and small schools.

1, function: Generally speaking, in a changing process, if there are two variables X and Y, and for each certain value of X, Y has a unique definite value corresponding to it, then it is said that X is an independent variable and Y is a function of X.

Understanding: ① Change process, ② Two variables, ③ Every fixed value of X, ④y is unique, ⑤x is an independent variable, and ⑤ Y is a function of X (you can't say that Y is a function).

2. Method of expressing function: ① List method. (Can't list all, limitations) ② Mirror method. List → Tracking Point → Connecting Line. (inaccurate, but intuitive and vivid) ③ Analysis method. (One-to-one correspondence is intuitive, but it is extremely difficult to see its changing trend)

3. linear function: y=kx+b(k and b are constants, k≠0). When b=0, it is the proportional function y=kx, so the proportional function is a special case of linear function.

4. Image: straight line.

Y=kx is a straight line that passes through (0,0) and (1, k) and is symmetrical about the origin. When k > 0, the image passes through one or three quadrants; When k < 0, the image passes through two or four quadrants.

Y=kx+b is a straight line passing through point (0, b) and parallel to the straight line y = kx. When k > 0 and b > 0, the image passes through one, two or three quadrants; When k > 0 and b < 0, the image passes through the first, third and fourth quadrants; When k < 0 and b > 0, the image passes through the first, second and fourth quadrants; When k < 0 and b < 0, the image passes through 234 quadrants.

5. Increase or decrease: when k > 0, y increases with the increase of x; When k < 0, y decreases with the increase of x.

6. undetermined coefficient method: establish a relationship, substitute two-point coordinates, and solve binary linear equations.

7. Relationship between linear function and linear equation of one variable: the solution of kx+b=0 is the abscissa of the intersection of straight line y=kx+b and X axis.

8. Relationship between linear function and binary linear equations: The solution of binary linear equations is the coordinates of the intersection of two straight lines.

9. the relationship between linear function and linear inequality in one variable: the solution of KX+B > 0 is the range of independent variables corresponding to the part of straight line y=kx+b above the x axis; The solution of kx+b < 0 is the value range of the independent variable corresponding to the part of the straight line y=kx+b below the X axis; Y 1 > Y2 is the range of independent variables corresponding to the straight line y1above Y2.

10, practical application of linear function: construct a linear function model, list the equations by solving application problems with column equations, and then sort them into a general form.

Chapter 12 Data Description

Several common statistical charts:

⑴ Histogram: the horizontal axis is the group type and the vertical axis is the group frequency. There is a gap between the bars. The sum of each set of frequencies is equal to the total data. The ratio of frequency to total data is frequency. Features: ① The specific data of each group can be displayed. ② Specific differences in data are relatively easy.

⑵ Fan chart: represents the percentage of each group in the whole. The sum of the percentages of each group is equal to 1. Features: ① Sector area represents the percentage of this part in the whole. ② It is easy to display the size of each group of data relative to the total. Application: ① The greater the sector area, the greater the degree of the central angle. ② Degree of central angle = percentage × 360.

⑶ Line chart: It is convenient to display the changing trend of data. Basically, the values on the horizontal axis are not taken.

⑷ Histogram: Calculate the difference between the maximum value and the minimum value, determine the group distance and group appropriately. Generally, the more data, the more groups. When the data is within 100, it is usually divided into 5 to 12 groups according to the number of data. There is no gap between the strips. Features: ① The frequency distribution of each group can be displayed. ② It is easy to display the frequency difference between groups.

5. Frequency distribution line chart: take the group median of each group (the average of two endpoints in each group) and take the group median outside the two endpoints on the horizontal axis.

Chapter 13 congruent triangles

1, the property of congruent triangles: the corresponding edges of congruent triangles are equal; Congruent triangles's corresponding angles are equal.

2. congruent triangles's judgment: ①SSS. (Note the implicit condition: male * * * side)

②SAS。 (Note that the angle is the included angle between two sides)

③ASA。 (Note that an edge is an edge between two corners.)

④ Atomic absorption spectrometry. (Pay attention to the correspondence and difference with ASA)

⑤HL。 (Be careful to determine the right triangle)

3. Angular bisector: ① The two parts are equal, both equal to half of the original angle (note the three writing methods).

② Property: The distance from the point on the bisector of the angle is equal to both sides of the angle.

③ Judgment: The points with equal distance to both sides of the angle are on the bisector of this angle.

Chapter 14 Axisymmetric

1, axisymmetric figure: If a figure is folded along a straight line, the parts on both sides of the straight line can overlap each other, and this figure is called an axisymmetric figure. (Note that this is the nature of the graph itself)

2. Axisymmetric: Fold the graph along a straight line. If it can overlap with another graph, then the two graphs are said to be symmetrical about this line. (Note that this is the relationship between two pictures. )

Properties: ① If two figures are symmetrical about a straight line, then the symmetry axis is the middle perpendicular of the line segment connected by any pair of corresponding points.

(2) The symmetry axis of an axisymmetric figure is the median vertical line of a line segment connected by any pair of corresponding points.

3. The essence of axisymmetric transformation: ① The two figures before and after axisymmetric transformation are conformal.

(2) Every point on the new graph is a symmetrical point of a point on the original graph about the straight line L..

③ The line segment connecting any pair of corresponding points is vertically bisected by the symmetry axis.

4. Midline: ① Definition: It passes through the midpoint of a line segment and is perpendicular to this line segment.

② Property: The distance between the point on the vertical line of the line segment and the two endpoints of the line segment is equal.

③ Judgment: The point with the same distance between the two ends of the line segment is on the middle vertical line of the line segment.

5. Isosceles triangle: ① Properties: a. The two base angles of isosceles triangle are equal (equilateral and equilateral). B the bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide (the three lines are one). 2 judgment: equilateral and equilateral.

6. equilateral triangle: ① properties: three sides are equal, three angles are equal, and each inner angle is 60. ② Judgment: A triangle with three equal angles is an equilateral triangle. An isosceles triangle with an angle of 60 is an equilateral triangle.

7. Right triangle: In a right triangle, if an acute angle is equal to 30, then the right side it faces is equal to half of the hypotenuse.

Chapter 15 Algebraic Expressions

1, algebraic expression: (1) monomial: (1) Definition: A formula that only contains the product of numbers or letters is called a monomial. (A single letter or number is also a monomial) ② Coefficient: the numerical factor in the monomial.

③ Times: the index sum of all the letters in a single item.

⑵ Polynomial: ① Item: each item (note the symbol).

② Degree: the degree of the term with the highest degree in the polynomial.

2. Similar items: items with the same letters and the same letter index.

3. Merge similar items: add their coefficients as new coefficients, and the letter part remains unchanged.

4. Same base power multiplication: same base power multiplication and same base exponential addition. am * an=am+n

5. Power of power: power of power, constant radix, exponential multiplication. (am)n=amn

6. Power of the product: the power of the product is equal to the power obtained by multiplying each factor of the product and then multiplying it. (ab)n=anbn

7. Multiplication of monomials: Multiply their coefficients and the same letters respectively. For letters contained only in monomials, they are used as a factor of the product together with their exponents.

8. Polynomial multiplied by monomial: that is, each term of polynomial is multiplied by monomial, and then the products are added.

9. Polynomial multiplication: Multiply each term of one polynomial by each term of another polynomial, and then add the products.

10, multiplication formula: ① square difference formula: (a+b)(a-b)=a2-b2.

② Complete square formula: (A B) 2 = A2AB+B2

1 1, and the rule of parenthesis: "plus positive invariance, plus negative covariation".

12, same base powers's division: same base powers's division, the base is unchanged, and the exponent should be reduced. am÷an=am-n，a0= 1(a≠0)，a-p= 1/ap

13. Division of single item: divide the coefficient and the same base by the factor of quotient respectively, and take the letters only included in the division formula as the factor of quotient together with their indices.

14. Polynomial divided by monomial: first divide each term of this polynomial by this monomial, and then add the obtained quotients.

15. Factorization: "One Mention" → "Two Sets" → "Trident Multiplication".

Chapter 16 Scores

1, Fraction: ① Definition: Generally speaking, if A and B represent two algebraic expressions and B contains letters, then the formula A/B is called a fraction.

(2) Conditions for the establishment of the score: the denominator is not zero. Condition of meaningless score: denominator is zero.

(3) The condition that the fractional value is zero: the numerator is zero and the denominator is not zero.

2. The basic properties of the fraction: the numerator and denominator of the fraction are multiplied (or divided) by a non-zero to get an algebraic expression, and the value of the fraction remains unchanged.

① Division → Finding the simplest common denominator: take the product of the highest power of all factors of each denominator as the common denominator.

② reduction → finding common factor: take the product of the lowest power of the same factor in the numerator denominator as the common factor.

3. Multiplication and division of fractions: ① multiplication rule: the fraction is multiplied by the fraction, the product of molecules is the numerator of the product, and the product of denominator is the denominator of the product.

2 division rule: fractional division, the numerator and denominator of division are multiplied by divisor in turn.

③ Fractional power: numerator and denominator should be multiplied by fractional power respectively.

4. Addition and subtraction of fractions: ① Addition and subtraction of fractions with the same denominator and addition and subtraction of molecules with the same denominator.

(2) Addition and subtraction of fractions with different denominators, first divided by fractions with the same denominator, and then added and subtracted.

5. Fractional equation: denominator → integral equation → equation solution → test (whether the simplest common denominator is zero is the solution of the equation).

Chapter 17 Inverse proportional function

1. Definition: A function in the form of y = k/x (where k is a constant and k≠0) is called an inverse proportional function, where x is an independent variable and its range is all real numbers not equal to 0.

Note: The commonly used formats are y=kx- 1 (commonly used in the definition of the second survey) and xy=k (commonly used in the calculation of the third proportional coefficient).

2. Picture: hyperbola. This is a figure with a symmetrical center. Does not intersect with the coordinate axis.

When k > 0, the image is in one or three quadrants; When k < 0, the image is in two or four quadrants.

3. properties: when k > 0, y decreases with the increase of x in each quadrant; When k < 0, y increases with the increase of x in each quadrant. (Always answer questions according to pictures)

4. Practical application: constructing inverse proportional function model.

Chapter 18 Pythagorean Theorem

Pythagorean Theorem: In Rt△ABC, where two right angles are A and B and the hypotenuse is C, then a2+b2=c2.

Note: The application of Zhao Shuang's string diagram.

2. Inverse Pythagorean Theorem: If three sides A, B and C of a triangle satisfy a2+b2=c2, then the triangle is a right triangle.

Chapter 19 Quadrilateral

1, parallelogram: ① Definition: A parallelogram with two groups of opposite sides parallel to each other is called a parallelogram.

② Properties: A. The opposite sides are parallel and equal. B, diagonally equal. C, diagonal bisection.

③ Judgment: A. Two groups of parallelograms with opposite sides are parallelograms.

B, two groups of quadrilaterals with equal opposite sides are parallelograms.

C. Quadrilaterals whose diagonals bisect each other are parallelograms.

D. Two groups of quadrangles with equal diagonal angles are parallelograms.

E. A set of quadrilaterals with parallel and equal opposite sides is a parallelogram.

2. The midline of the triangle: the midline of the triangle is parallel to the third side of the triangle and equal to half of the third side.

3. Rectangle: ① Definition: A parallelogram with a right angle is a rectangle.

② Properties: A. The opposite sides are parallel and equal. B, all four corners are right angles. C is divided diagonally and equal to each other.

③ Judgment: A parallelogram with a right angle is a rectangle.

B. Parallelograms with equal diagonals are rectangles.

A quadrilateral with three right angles is a rectangle.

4. The midline of hypotenuse of right triangle is equal to half of hypotenuse.

5. Diamond: ① Definition: A set of parallelograms with equal adjacent sides is a diamond.

② Properties: A and four sides are equal. B, diagonally equal. C. Diagonal lines are divided vertically, and each diagonal line divides a set of diagonal lines equally.

③ Judgment: A. A set of parallelograms with equal adjacent sides is a diamond.

Parallelograms with diagonal lines perpendicular to each other are diamonds.

A quadrilateral with four equilateral sides is a diamond.

6. Square: A. A group of rectangles with equal adjacent sides is a square.

A diamond with a right angle is a square.

7. Trapezoid: A set of quadrangles with parallel opposite sides and another set of quadrangles with non-parallel opposite sides are called trapeziums.

① Properties of isosceles trapezoid: A. The two bottom angles on the same bottom are equal. B, the two diagonals are equal.

② Determination of isosceles trapezoid: A. A trapezoid with two equal angles on the same base is an isosceles trapezoid.

B. The isosceles trapezoid is an isosceles trapezoid.

A trapezoid with equal diagonal lines is an isosceles trapezoid.

8. Center of gravity: A. The center of gravity of a line segment is the midpoint of the line segment. B, the center of gravity of parallelogram is the intersection of diagonal lines.

C, the center of gravity of a triangle is the intersection of three midlines. D, use the line hammer method to find the center of gravity of any figure.

9. Midpoint quadrilateral: ① Shape: a. The midpoint quadrilateral of any quadrilateral is a parallelogram.

B the midpoint quadrilateral of a quadrilateral with equal diagonals is a diamond.

C the midpoint quadrilateral of a quadrilateral with vertical diagonal is a rectangle.

D the midpoint quadrilateral of a quadrilateral with vertical and equal diagonals is a square.

Chapter 20 Data Analysis

1, representing data: ① average value: x = 1/n(x 1+x2+…+xn).

Weighted average: x = (f1x1+f2x2+…+fkxk)/(f1+f2+…+fk).

Note: f 1+f2+…+fk=n, and fk is the weight.

② Median: Arrange a set of data in the order from small to large (or from large to small). If the number of data is odd, the middle number is the median of this set of data; If the number of data is even, the average of the middle two data is the median of this set of data.

The median value is the position representative value. If the median of a set of data is known, then the data less than or greater than this median is half.

③ Mode: The data with the highest frequency in a group of data is the mode of this group of data.

Note: A, the calculation of the average uses all the data, which can make full use of the information provided by the data, so it is often used in real life. But it is greatly influenced by extreme values. B, when some data in a set of data appear repeatedly, the mode is often the quantity that people care about and is not affected by extreme values. C, the median only needs a little calculation and is not affected by the extreme value.

2. Data fluctuation: ① Extreme value range: the difference between the largest data and the smallest data in a group of data. (Fluctuation range of response data)

② Variance: Measure the volatility of data. The greater the variance, the greater the data fluctuation; The smaller the variance, the smaller the data fluctuation.

S2= 1/n[(x 1-x pull) 2+(x2-x pull) 2+…+(xn-x pull) 2]