Maths_速读笔记
初一下数学笔记 济南
初一下数学笔记济南Studying mathematics in the first year of junior high school can be both exciting and challenging. For many students, it serves as their first real introduction to more complex mathematical concepts beyond basic arithmetic. While some may find it intimidating at first, with practice and determination, it can become a subject that they excel in.在初中一年级学习数学可能既令人兴奋又具有挑战性。
对很多学生来说，这是他们第一次真正接触到比基础算术更复杂的数学概念。
虽然一开始可能觉得很吓人，但通过练习和决心，数学可以成为他们擅长的科目。
One of the key aspects of studying mathematics in the first year of junior high school is building a strong foundation for future learning. Concepts such as algebra, geometry, and statistics are introduced at this level, laying the groundwork for more advanced topics in later years. Developing a solid understanding of these fundamental concepts early on can greatly benefit students as they progress through their mathematical education.在初中一年级学习数学的关键之一是为将来的学习打下坚实的基础。
mathsbook趣味记忆法
mathsbook趣味记忆法
【原创实用版】
目录
1.趣味记忆法简介
2.趣味记忆法在数学书中的应用
3.趣味记忆法的实际效果
4.如何使用趣味记忆法提高数学学习兴趣
5.总结
正文
【1.趣味记忆法简介】
趣味记忆法是一种通过将信息与有趣、生动的图像或情境联系起来，从而提高记忆效果的方法。
这种记忆法可以帮助学习者更容易地记住知识点，尤其是在数学这类需要大量记忆的学科中。
【2.趣味记忆法在数学书中的应用】
在数学书中，趣味记忆法可以通过以下方式应用：
- 将抽象的数学概念转化为具象的图像，如将数字与具体的物品联系起来，帮助记忆。
- 通过有趣的故事或情境，将数学知识点融入其中，使学习者在理解故事的同时记住知识点。
- 利用游戏或竞赛的方式，让学习者在玩乐中学习，提高记忆效果。
【3.趣味记忆法的实际效果】
研究表明，趣味记忆法能够显著提高学习者的记忆效果，尤其是对于小学生和初中生来说，这种方法能够有效提高他们对数学的兴趣和成绩。
【4.如何使用趣味记忆法提高数学学习兴趣】
想要使用趣味记忆法提高数学学习兴趣，可以尝试以下几种方法：- 寻找有趣的数学故事或情境，将知识点融入其中。
- 制作数学卡片，每张卡片上既有知识点，又有与之相关的图像或情境。
- 利用游戏或竞赛的方式进行数学学习，如数学接龙、数学谜语等。
【5.总结】
趣味记忆法是一种有效的记忆方法，尤其在数学学习中，能够帮助学习者提高记忆效果，增强学习兴趣。
七年级上册数学笔记math
、有理数的加减乘除混合运算，如无括号指出先做什么运算，则与小学所
、求n个相同因数的积的运算，叫做乘方，乘方的结果叫做幂。在a?中，
叫底数，n叫指数，当a?看作a的n次方的结果时也可读作“a的n次幂”。
26、除0外，互为相反数的两个数的偶次幂相等。除0外，互为相反数的两
、根据有理数的乘法法则可以得出:负数的奇次幂是负数，负数的偶次幂是
分三类 ⑵ 分二类
正整数 整数 正整数
正分数 有 0
理 零 理 负整数
注意:等式里包含了方程。
、 列方程时，要先设字母表示未知数，然后根据问题中的相等关系，写出含有
、
、 上面各方程都含有一个未知数(元)，未知数的次数都是1，等号两边都是整
、 解方程就是求出使方程中等号左右两边相等的未知数的值，这个值就是方程
、 （1)等式两边加或减同一个数或式子，结果仍相等。
、 两点的所有连线中，线段最短。简单说:两点之间，线段最短。连接两点间的
(最后一句是说明两点的距离的定义)
、 有公共端点的两条射线组成的图形叫做角，这个公共端电视角的顶点，这两
、 一度有六十分，一分有六十秒。也就是说:60″＝1′;60′＝1°。
、 一般地，从一个角的顶点出发，把这个角分成两个相等的角的射线，叫做这
、一般地，我们有有理数乘法法则:两数相乘，同好得正，异号得负，并把
0相乘，都得0.
、正数的倒数正数，负数的倒数是负数，乘积是1的两个数互为倒数。
、几个不是0的数相乘，负因数的个数是偶数时，积是正数；负因数的个
、几个数相乘，如果其中有因数为0，那么积等于0.
、一般地，有理数乘法中，两个数相乘，交换因数的位置，积相等。乘法
IGCSE Maths 英语数学词汇知识点总结(配图)
1.∅: The empty set (proper subset)2.∈: is an element of {set}3.∉: is not an element of {set}4.0.7 recurring as a fraction: 10f = 7.777..., 10f-f=7, 9f=7, f=7/95.0.3181818 recurring as a fraction: 100f =31.818181..., 100f-f=31.5, 99f=31.5, f=315/990=7/226.1/2(a+b)h: Area of a trapezium7.1 centiliter =: 10 ml8.1 cm cubed =: 1 ml9.2-²: 1/4 (The reciprocal of the number without the negative)10.(2x⁵)³ =: 8x to the power of 1511.-3²=-9: (-3)²=912.1000 cm cubed =: 1L13.1000 kg =: 1 tonne14.1000 Litres =: 1 metre cubed15.1000 milliliters =: 1L16.A ∩ B: Intersection of A and B (items in both)17.A ∪ B: Union of A and B (items in A or B or both)18.A ⊂ B: A is a proper subset of B19.A ⊆ B: A is a subset of B20.a/b + c/d: (ad + bc)/bd21.a/b - c/d: (ad - bc)/bd22.a/b x c/d: ac/bd23.Adding and subtracting fractions: For fractions to beadded or subtracted, the denominator MUST be the same. 24.Adding and subtracting indices: Powers and variablesmust be the same25.Adding and subtracting matrices: For matrices to beadded or subtracted they must be of the same order.26.Angles in a circle - centre, circumference:If an angle is extended to the centre of a circle from arc ABand another angle is extended to the circumference from arc AB then the angle at the circumference will be half the angle atthe centre.27.Angles in a circle - cyclic quadrilaterals: Opposite anglesin a cyclic quad add to 180 degrees.28.Angles in a circle - same arc at circumference:All angles extending out to the circumference from arc AB willhave the same angle.29.Angles in a circle - two semi-circles:If a diameter is drawn and an angle in the semi-circle uses it as its base (with its third point touching the circumference) then the angle at the circumference is 90 degrees.30.Angles in a quadrilateral...: add to 360 degrees31.Angles in a triangle...: add to 180 degrees32.A polygon is regular if...: all interior angles are equal andall its sides are the same length. e.g. square, pentagon,hexagon, octagon.33.A power to a power (x²)³: If a power is raised a variable thatis already being squared, you multiply the indices. Theexample will equal x to the power of 6.34.A ratio as a fraction: You can express particular numbers inthe ratio as fractions by finding what the whole ratio adds up to and using that as the denominator. e.g. 3/5 is the first part of the ratio 3:235.Arc: An arc is a section of the circumference of a circleIGCSE Maths英语数学词汇知识点总结（配图）36.Area and volume scale factor: Used in similar shapes. Thisapplies the same principles as the linear scale factor but issquared or cubed. To find the linear scale factor from these, find the square or cube root.37.area of cross-section x length: volume of a prism38.a/sinA=b/sinB=c/sinC: The sine rule (can be inverted sosines are on top)39.Average speed: Distance/time40.ax to the power of -5: a/x to the power of 541.Bar charts:All bars are the same width, the heights represent frequency. A dual chart is sometimes useful to compare two bar charts withsimilar data.42.base x height: Area of a square, rectangle and parallelogram43.Calculating probability: Probability can be found with aprobability fraction. The fraction of P(outcome) is the number of ways the outcome can happen/the total number of possible outcomes. This can then be changed to a decimal orpercentage.44.Calculating the angles of a pie chart: To find what angle asector will be, take the frequency divided by the total number and then times that by 360. These angles do not have tolabelled on the chart45.Choosing the right rule - two angles and a side: Sinerule46.Choosing the right rule - two sides and included angle:Cosine rule followed by sine rule47.Chords:A chord is a line in a circle where both ends touch thecircumference, cutting a section of the bined frequency of two events: You can illustrate thepossibilities of two combined events on a possibility diagram where all the possible outcomes for one variable is on oneaxis and for the other on the other axis. Every possibleoutcome for both variables is then represented on the diagram and the probability can be found.posite function expressions: To make an expressionfor the composite function fg(x), put the expression for g, in the place of x in the expression for f. "Take x, apply g, thenapply f"posite functions: When functions are put together, readthem backwards. e.g. fg(2) means start with 2, apply g thenapply fpound interest: amount x 1. percentage to number ofyearspound interest formula: P(1+r/100)^n where r is thepercentage, n is the time and P is the amount invested.53.Congruent shapes: Shapes with the same size and shape aseach other are congruent, even if they are placed in different positions.54.Construct a perpendicular from a point to a line: Withthe point as the centre, draw an arc that intersects the line on either side. Using these points as the centre, follow the same directions as constructing a perpendicular.55.Constructing a line bisector:Open compass to abut three quarters the length of the line and draw a semi-circle arc from either point on the line. The bisector is the point where the two arcs intersect.56.Constructing an accurate trapezium: To construct atrapezium accurately, use first a protractor and a ruler andthen a set square to find the parallel side.57.Constructing an angle bisector:With the vertex of the angle as the centre, draw an arc through both lines and then, with centres where the arc intersects both lines, draw two more arcs. The angle bisector extends from the vertex to the point where the two arcs intersect.58.Constructing a triangle: Draw the longest side as the base.Draw the next longest side with a compass as an arc and do the same with the shortest side. The draw the straight line to where the two arcs intersect.59.Continuous data: Data that can have any value within a rangeof values, for example mass, height, time.60.CosA= b squared+c squared - a squared/2bc:The cosine rule61.Cosine rule notes: b and c in the rule are the sides makingthe angle being calculated and a is the opposite side. Is not restricted to right-angled triangles62.Cubic sequences: Compare to cube number first. Differencebetween the second difference is the same (3rd difference). 1/6 of the 3rd difference will be the coefficient of n cubed. Take that away from the original sequence and find the rule for the remaining numbers.63.Cumulative frequency diagrams: A frequency table wherethe next the value has been added on to the previous value so that the number is always rising. This can then be plotted with the top value of the class group against the cumulativefrequency. Plots can be joined with a freehand curve.64.Currency conversions: Use value that is not one for bothexchanges (either times by for currency represented by 1 ordivide by for currency represented by point something)65.Decreases in ratios: Decrease 450 in the ratio 5:3. 450=5parts, need to find 3 parts. Decreased amount = (3/5) x 450 66.Describing translations with vectors: As shapes that havebeen translated are congruent, all points have moved thesame amount. The translation is how any one point movedfrom its original place to its new place with its movementacross on top and up and down on the bottom of the fraction without a line.67.Different form of function notation - f:x→3x-5: This isthe same as f(x)=3x-568.Discrete data: This is data which contains separate numberslike goals scored, number of children or shoe size.69.Dividing a quantity in a ratio: To divide a quantity in a ratio,you must first find the total number of parts. Divide thequantity by the total number of parts to find one part and then multiply this to find parts of the ratio.70.Dividing fraction: first term x reciprocal of second term71.Dividing fractions: To divide fractions, times the first fractionby the reciprocal (turned upside down) of the second.72.Dividing with indices: Subtract indices, numbers stay73.ℰ: Universal set74.Enlarging a shape: An enlargment will occur about a pointand the distance away from this point should be taken intoconsideration as well as the size. The lengths of each side can be easily found by multiplying the original length by the scale factor.75.Enlarging a shape - ray method: If you need to find thepoint from which the shape was enlarged, the ray method ishelpful. Draw lines through corresponding points on the shape.The point where these lines meet is the point from which the shape was enlarged.76.Equilateral triangular prism: It has four planes of symmetry.One from each of the points lengthways and one widthways halfway down. It also has two axis of rotational symmetry. 77.Estimating gradients: You can estimate the gradient of acurved line by drawing a tangent to it and finding the gradient of that line.78.Estimating the median from a cumulative frequencydiagram: The median can be found by looking at the mark on the vertical axis which is half of the total. Follow this along to meet the curve and the value on the horizontal axis down from this is an estimate of the median. The median is also the 50th percentile79.Exponential graphs: This graph will have a changing gradient80.Exponential growth: Use same formula as compoundinterest81.Exponential growth and decay: Use compound interestprinciples82.Exterior angles: The sum of the exterior angles of anypolygon is 360 degrees83.Factorising quadratic equations with coefficients(ax²+bx+c): Write out the factors of a and c. Find the pair ofa factors that multiply with the pair of c factors to make twonumbers that add to b.84.Factorising simple quadratic equations (x²+bx+c):Factorise in the morning - AM stands for add, multiply. Findtwo numbers that add to the second number and multiply to the third number.85.Factorising tips: Factor out ALL common factors even ifthese are brackets, numbers and multiple variables.86.F angles on parallel lines:Called corresponding angles. They are equal87.Finding a matrix transformation: To do this, you only needto find the vector for two points. Find where 1 0 (over eachother) and 0 1 (over each other) will end up and put thevectors for these together in a matrix. Put the image vector of the vector that started on the x-axis first.88.Finding the area of a triangle using sine: 1/2 ab sinC (twosides with included angle)89.Finding the interquartile range.: The interquartile range canbe found by taking the upper quartile (75th percentile, 75%)and taking away the lower quartile (25th percentile, 25%) togive the middle 50%. The upper and lower quartile can befound in the same way as the median but by finding 1/4 and3/4 of the data instead of 1/2.90.Finding the inverse of a 2 x 2 matrix: Of matrix abcd,swap the positions of a and d and change the signs of b and c.Then divide all the numbers by the determinant.91.Finding the multiplier: A multiplier is the percentage dividedby 100 to find the decimal. However to decrease a quantity by 7%, the multiplier used will be 0.93. If a quantity has beendecreases by 7%, the amount left will be 93% of the total 92.Finding the points of a quadratic graph: Draw out a tableand in each row put a separate component of the equation and the answer, then add these all together to find the points. 93.Frequency table:A tally chart of the all the options. The first column is theoptions or numbers you are tallying, the second is the tallys and the third is the numerical value of the number of tallys inthe second column.94.Function notation: f(x)=3x-695.Histograms: Histograms are similar to bar charts except,there are no gaps between bars, the horizontal axis has acontinuous scale, the area of each bar represents thefrequency.96.Histograms with bars of unequal width: Frequency isrepresented by area. THe vertical axis should read frequency density. To find the height of a bar where class widths aredifferent, use the formula: frequency density=frequency of class interval/width of class interval.97.Histograms with unequal width - proportionality: Areasare proportional to frequencies98.If the determinant of a matrix is 0...: It is a singular matrixand the inverse does not exist99.Increase or decrease by a percentage: Find the multiplier(24%=0.24)100.Increases in ratios: Increase 450 in the ratio 5:3. 450 = 3 parts, need to find 5 parts. Increased amount = (5/3) x 450 101.Integers: Whole, positive, negative102.Interior angles: The sum of the interior angles of any polygon can be found using 180 x (n-2) where n is the number of sides103.Inter-quartile range: Bases the measure of spread on the middle 50% of the data removing extreme values104.Inverse functions - f-¹(x): To find the inverse of a function, replace f(x) with y and rearrange the equation to make x the subject. In the final answer, change the y to x.105.Kite:Two pairs of equal adjacent sides. Has one long diagonal and one short diagonal that bisect each other at right angles. The opposite angles between the sides of different lengths areequal.106.length x width x height: volume of a cuboid107.Linear scale factor: Used in similar shapes. To find it, find two corresponding sides and put the bigger side as thenumerator of a fraction. Simplify the fraction.108.Linear sequences: Look for: difference between each number in sequence, difference between that and the first term 109.Line of symmetry (mirror line): A line that can be drawn through a shape so that what can be seen on one side is amirror image of what can be seen on the other side.110.Locus (loci): A locus is a shape or a line which is always the same distance from a given point or line. You may be alsoasked to draw a locus that is equidistant from two points. This will usually be a straight line.111.Map scales: 1:900000 (1cm on map = 900000cm on ground).1cm = 9km. Actual distance km = distance on map cm x 9km 112.Mean: The mean is the sum of all the values in a set divided by the total number of values in the set. IT uses all the values although extreme values can affect it and it has to becalculated.113.Median: The middle value when put in order of size. It is not heavily affected by extreme values and is easy to find forungrouped data. But, it doesn't use all the values and can be hard to understand.114.Mensuration: The part of geometry concerning lengths, areas and volumes115.Mode: The value that occurs most often in a set of data. It can be used for qualitative data, is easy to find and is not affected by extreme values. But, it doesn't use all the values and may not exist.116.Multiplying fractional indices: Fractional indices can be added in multiplication in the same way as normal indices.They should be left as improper fractions not mixed numbers. 117.Multiplying fractions: Put the product orf the numerators over the product of the denomiators and then simplify ifneeded.118.Multiplying matrices together: When you multiply two matrices together, each row in the first matrix combines with each column of the second to give a single number. (The first to the left will combine with the uppermost, the second to the left will combine with the second down)119.Multiplying with indices: Add indices, numbers stay120.n(A): Number of elements in set A121.Natural numbers: Whole, positive122.Negative indices: If something is raised to a negative indices, find the reciprocal.123.Obtuse cosine: The cosine of an obtuse angle is equal to the negative of its supplementary angle (cos100 = -cos80)124.Obtuse sine: The sine of an obtuse angle is equal to that of its supplementary angle (sin100 = sin80)125.Order of multiplied matrices: You can only multiply matrices if the number of columns in the first is equal to the number of rows in the second. The other two values willcombine to give the order of he product.126.Order of operations: BEDMAS127.Order of the matrix: The number of rows and columns in a matrix. Written as rows x columns128.Parallel lines on a graph: For a line to be parallel it must have the same gradient, so variable m must be the same inboth equations in y=mx + c 129.Parallelogram:Opposite sides are parallel and equal. Opposite angles areequal.130.Percentage change: change/original x 100131.Perpendicular lines on a graph: If a line is perpendicular to the other line, the gradient will be the negative reciprocal of it.Also, the product of both gradients will be 1132.Pictograms:Pictograms are frequency tables which use pictures to represent frequency. Each picture represents a number of items.It helps people to understand it more quickly but doesn'tallow for fractions of pictures.133.Pie charts:These do not show individual frequencies but instead compare the frequencies. Each is represented as a sector of the circle where the angle of each sector is proportional to the frequency it represents. The sectors should always be labelled. 134.pi x diameter: Circumference of a circle135.pi x radius squared: area of a circle136.Prime factorisation: Writing a number as a product of its prime factors (use a factor tree, until all bottom numbers are prime)137.Proper subset: All subsets except the original set (P)138.Quadratic graphs - parabolas: This will have either a u or upside down u shape and will be a smooth, continuous curve with no straight lines139.Quadratic sequences: Compare to square numbers first.Difference between the difference between each number (2nd difference) is the same. 1/2 of the 2nd difference will be thecoeffecient of n squared. Take that away from originalsequence and find the rule for the remaining numbers.140.Range: The range is the highest value in the set minus the lowest value in the set. It shows the spread of the data, it can help to compare data and consistency.141.Rates: Speed applied more generally to show changes in quantities. e.g. temperature goes down by 5 degrees everyminute142.Ratio: Ratio is a way of comparing the sizes of two or more quantities. and will be set out in the form x:y. It can also begiven in the form x/y. All the units in a ratio must be the same 143.Ratios when only some information is known: Find one part of the ratio. If 30 is 2 parts of the total ratio, divide it by 2 to find one part.144.Real numbers: Integers including decimals - two subsets 145.Real numbers - irrational: Decimals or numbers that cannot be written as a fraction146.Real numbers - rational: Integers or fractions147.Reciprocal graphs - y=a/x: Values should be rounded to two decimal points. x will never equal zero. It will besymmetrical. The graph never touches either axes.148.Reflecting a shape over an axis: The easiest and most effective methods is counting how many squares away each individual point is from the mirror line and then making thepoint that many squares away on the opposite side. Thisshould be done using the most direct route (even diagonally through squares)149.Relative frequency: Also known as experimental probability, it is given by frequency of the outcome of event/total number of trials. This relative frequency is an estimate of theprobability150.Reverse percentages - multiplier method: 110%=x. Find the original amount: x/1.1 (the multiplier) = the original amount.88%=x. Find the original amount: x/0.88 (the multiplier) = the original amount.151.Reverse percentages - unitary method: 110%=x. Find the original amount: 1%=x/110 original price = 1% x 100 (Used for decreases as well)152.Rhombus:All sides are equal, opposite sides are parallel. Diagonals bisect at right angles. Opposite angles are equal 153.Rotating a shape: The easiest and most effective way torotate a shape about a point is to count how many squaresacross and up and down each individual point is away from the axis, then turn your paper the amount required and count the squares using the new axes.154.Rotational symmetry: A 2D shape has rotational symmetry if it can be rotated about a point to look exactly the same in a new position. It can be represented by the order of rotational symmetry or number of times it can be rotated to look exactly the same until it is back to its original position. e.g. anequilateral triangle has a rotational symmetry of three155.Rotational symmetry of 3D shapes: 3D shapes have axes of symmetry. A cubiod has three axes of symmetry. Each axis has its own order of rotational symmetry.156.Scatter diagrams:Compares two variables by plotting corresponding values ona graph. They can have positive, negative or zero correlationbetween the points.157.Sector: Part of a circle bounded by two radii and one of the arcs.158.Similar shapes: Shapes are similar if one is an enlargement or reduction of the other. The corresponding angles of similar shapes are equal. Their sides can be found using ratios orlinear scale factors159.Simplifying algebraic fractions: Always fully factorise and cancel common brackets. Brackets that are the same withopposite sides are equal to -1 on the numerator. Two negatives over each other cancel each other out. Variables should be in brackets before they are cancelled.160.sine/cosine/tangent of known angle = side/side is used to: calculate the length of a side in a right-angledtriangle161.Sine rule notes: When you are calculating a side, use the rule with sides on top. When you are calculating an angle, use the rule with sines on top. Is not restricted to right angledtriangles162.Sketching an inequalities graph - linear programming: Sketch a broken line if the symbol is not equal to. Sketch asolid line if it is. Shade the unwanted region, e.g. if the wanted is y is less than x, shade the region where y is more than x. 163.Solve Direct proportion (a ∝ b): Divide to find the constant of proportionality (a=kb, k=constant)164.Solve Inverse proportion (a ∝ the reciprocal of b): Find total by multiplying values (a=k/b, k=constant)165.Solve x²+6x-7 by completing the square: (x+(6/2))^2 -(6/2)^2 - 7 = (x+3)^2 -3^2 - 7 = (x+3)^2 - 16.x=-2 +or- the square root of 16166.Solving problems of angles of elevation and depression: Use the trigonometric ratios(sine/cosine/tangent)167.Solving quadratic equations by completing the square: Solves quadratic equations by rewriting x squared + px +q in the form (x+a)^2 +b. Found by putting it into the form (x+(p/2))^2 -(p/2)^2 + q and rearranging to solve.168.Symmetry in circles - bisector of chords: The perpendicular bisector of a chord passes through the centre of the circle169.Symmetry in circles - chords: Two chords of equal length are the same distance from the centre of the circle.170.Symmetry in circles - tangents: Two tangents extending from a point to touch the circle at any two points will be the same length171.Symmetry of 3D shapes:3D shapes have planes of symmetry. One is a reflection of the other half. A cuboid has 3 planes of symmetry 172.Tangent: A tangent is a straight line that touches a circle or curve as one point only. It is drawn perpendicular to the radius 173.The determinant of a matrix: The determinant helps find the inverse. It can be represented by two lines around theletter representing the matrix. It is found using the formula ad -bc of matrix abcd 174.The identity matrix: A 2 x 2 matrix with 1 0 as its top numbers and 0 1 as its bottom numbers. This matrix is thesame as multiplying it by 1.175.The inverse sine/cosine/tangent of known side/known side is used to: Find the angles in a right-angled triangle 176.The magnitude of a vector: the square root of x square plus y squared - written with two vertical lines on either side of the vector177.Theoretical probability vs experimental probability: The way of calculating a probability is actually only theoreticaland in practice, this probability is only an estimate of the real results. However, the more times you take data, the closer the experimental probability will get to the theoretical probability 178.The probability of an event occurring =: 1 - the probability of the event not occurring.179.The probability scale: The probability scale goes from 0 being impossible to 1 being certain. Any probability values fall in between 0 and 1.180.the quadratic formula:181.The zero matrix (Z): The zero matrix is a 2 x 2 matrix which is just zeros and acts like a normal zero in equations.182.Three sides: Cosine rule followed by sine rule183.Transformations using matrices: You can transform a shape using matrices, the matrix for a shape is the vector for each point put together e.g. a triangle will form a 2 x 3 matrix.You can then multiply it by a specific 2 x 2 matrix to find the vectors of the transformed shape.184.Trapezium:Two parallel sides. The sum of the interior angles at the ends of each non-parallel side is 180 degrees.185.Tree diagrams: Probabilities of sequences of events can be shown on a tree diagram. Each brach represents an outcome which branches into more outcomes. When doing this type of diagram, whether the outcome is put back into the total should be taken into consideration.186.U angles on parallel lines:Called interior angles. They add up to 180 degrees187.Upper and lower bounds: Minimum and maximum any rounded number could possibly be188.Upper and lower bounds e.g. 27.5≥x>28.5: The upper bound is the maximum possible the rounded number couldhave been before rounding and the lower bound the opposite.Upper bound will be the number with 5 after the last decimal or as a unit depending on how it is rounded. The lower bound is the number 5 after one less than the last unit.ing coordinates - gradient: difference between y-coordinates/difference between x-coordinatesing coordinates - midpoints: Add end values and divide by two for each x and y separatelying coordinates -the distance between two points: Use pythagoras' thereom to find the distance by using thegradients as the sides of a right angled triangleing frequency tables to find the mean: To find the mean, the frequency must be multiplied by the by the class it is in. These final values should then be added up and divided by the total frequency. If the classes are grouped, use themiddle value for the multiplication.ing frequency tables to find the median: To find the median from a frequency table, add all the values to find the total number of cars surveyed then divide this by 2 to find the middle value. Now add the frequencies up cumulatively to find which group contains this middle number.ing rates: Temperature lowers 5 degrees every minute.After 6 minutes, temperature lowers (5x6=30) 30 degrees.Temperature lowers by 1 degree in (1/5=0.2) 0.2 minutes.Temperature lowers by 15 degrees in (15/5=3) 3 minutes 195.Vectors: Vectors have both a magnitude and a direction. They can be represented by the start and end points with an arrow over top or as a lower case letter printed in bold ORunderlined. On graphs they are represented by lines witharrows indicating direction196.Vectors on a coordinate grid: On a coordinate grid, vectors can be represented by two numbers in a bracket as a fraction without the line. The top number is the amount moved across (negative left) and the bottom number is the amountmoved vertically (negative down)197.Venn diagrams - A': Complement of A (all items not in A)198.Vulgar fraction: A fraction represented by a numerator and denominator and not by decimals.199.Vulgar fractions: All fractions excluding mixed fractions 200.Which ratio to use: SOHCAHTOA201.x to the power of -5: The reciprocal of x to the power of 5 (1/x to the 5)202.x to the power of ½: the square root of x203.x to the power of -¼: 1/the fourth root of x204.x to the power of ¾: (the fourth root of x)cubed205.x to the power of -¾: 1/(the fourth root of x)cubed 206.y=mx + c: m is the gradient of the line, c is where the line intercepts the y axis.207.Z angles on parallel lines:Called alternate angles. They are the same. 208.θ/360 x pi x diameter: Arc length209.θ/360 x pi x radius squared: Sector area210.θ (theta): the angle at the centre of a sector。
数学必修五笔记整理手写笔记
数学必修五笔记整理手写笔记Studying mathematics is often seen as a challenging and daunting task for many students. However, it is also a subject that provides a solid foundation for critical thinking, problem-solving, and analytical skills. Mathematics is not just about numbers and equations; it is about logic, reasoning, and the ability to think abstractly.学习数学常常被许多学生视为一项具有挑战性和令人望而却步的任务。
然而，数学也是一门能够为批判性思维、问题解决能力和分析技能提供坚实基础的学科。
数学不仅仅是关于数字和方程式，它还涉及逻辑、推理和抽象思维能力。
When it comes to mastering mathematics, it is essential to have a clear understanding of the basic concepts. Building a strong foundation in arithmetic, algebra, geometry, and calculus is crucialfor tackling more advanced topics. It is important for students to practice regularly and seek help from teachers or tutors when encountering difficulties.在掌握数学的过程中，对基本概念有清晰的理解是至关重要的。
七到九年级数学笔记
七到九年级数学笔记Mathematics is a subject that can both fascinate and challenge young students in grades seven through nine. From algebra and geometry to statistics and probability, there are a wide range of topics to explore. While some students may struggle with certain concepts, others may excel and develop a strong interest in math. It is important for teachers and parents to provide support and encouragement to help students build their confidence and skills in mathematics.数学是一个让七到九年级的学生既着迷又挑战的学科。
从代数和几何到统计学和概率论，有许多主题可以探索。
虽然一些学生可能在某些概念上挣扎，但其他人可能表现出色，并对数学产生浓厚的兴趣。
教师和家长需要提供支持和鼓励，帮助学生建立自信并掌握数学技能。
One way to make math more engaging for students is to incorporate real-world examples and applications into lessons. By showing how math is used in everyday life, students can better understand the relevance and importance of the concepts they are learning. For example, teachers can teach about fractions using recipes ordemonstrate the concept of slope using real-life examples of inclines. This not only makes math more interesting but also helps students see how it can be applied in practical situations.让数学对学生更有吸引力的一种方法是将真实世界的例子和应用融入到课程中。
四年级上册数学第七单元概念笔记
四年级上册数学第七单元概念笔记Math is a fundamental subject that lays the foundation for problem-solving and critical thinking skills. 数学是一个基础科目，它为问题解决和批判性思维能力奠定了基础。
Having a strong understanding of mathematical concepts is essential for success in many fields, including science, technology, engineering, and even everyday life. 对数学概念有深刻的理解对成功在许多领域都是至关重要的，包括科学、技术、工程，甚至日常生活。
In the seventh unit of the fourth-grade math textbook, students are introduced to a variety of new concepts, including fractions, decimals, and measurements. 在四年级数学教科书的第七单元中，学生将会接触到各种新的概念，包括分数、小数和测量。
These concepts provide a foundation for more advanced mathematical topics that will be covered in later grades. 这些概念为以后更高级的数学课题提供了基础。
One of the key concepts covered in the seventh unit is fractions. 分数是第七单元中涵盖的一个关键概念。
Students will learn how to identify and work with fractions, including adding, subtracting, multiplying, and dividing them. 学生将学会如何识别和处理分数，包括加法、减法、乘法和除法。
四年级上册数学运算律简单读书笔记
四年级上册数学运算律简单读书笔记Math is a fundamental subject that plays a crucial role in our daily lives. 数学是一个基础学科，在我们日常生活中扮演着至关重要的角色。
From solving simple addition and subtraction problems to tackling complex equations, math is an essential skill that everyone needs to master. 从解决简单的加减法问题到处理复杂的方程式，数学是每个人都需要掌握的重要技能。
In the fourth grade, students are introduced to various math concepts, including mathematical operation laws. 在四年级阶段，学生将接触到各种数学概念，其中包括数学运算律。
These operation laws are important as they help students understand and apply different arithmetic operations effectively. 这些运算律是重要的，因为它们有助于学生有效地理解和应用不同的算术运算。
One of the fundamental operation laws in math is the commutative property. 其中一个数学中的基础运算律是交换律。
This property states that the order of two numbers being added or multiplied does not change the sum or product. 这个运算律表明相加或相乘的两个数的顺序不会改变它们的和或积。
七年级下册数学书笔记
七年级下册数学书笔记Mathematics is a subject that many students find challenging, but with the right resources and support, it can become more manageable and even enjoyable. 数学是许多学生觉得具有挑战性的科目，但是通过正确的资源和支持，它可以变得更容易管理甚至更有趣。
One of the key resources for students studying math is their textbook. Textbooks provide a structured and organized way of learning mathematical concepts, as well as practice problems to reinforce understanding. 学生学习数学的关键资源之一是他们的教科书。
教科书提供了一个结构化和有组织的学习数学概念的方式，以及练习问题来加强理解。
In the seventh grade, students typically cover a wide range of math topics, including algebra, geometry, and statistics. These topics build on the foundational math skills students have developed in earlier grades and introduce them to more complex concepts and problem-solving strategies. 在七年级，学生通常涵盖了广泛的数学主题，包括代数、几何和统计学。
这些主题建立在学生在较早年级已经发展的基本数学技能之上，并向他们介绍了更复杂的概念和解决问题的策略。
七年级下册数学,第二单元 笔记
七年级下册数学,第二单元笔记Studying the second unit of math in the seventh grade is crucial for building a strong foundation in mathematical concepts. 这一单元主要涵盖了数学中的百分数、比例、和增长/减少百分数。
By understanding these concepts, students can apply them to real-life situations and solve problems more effectively. 通过理解这些概念，学生们可以将其运用到实际生活中的情境中，并更有效地解决问题。
One key topic covered in this unit is percentages. 百分数是一个非常重要的概念，它在日常生活中随处可见。
Whether it's calculating discounts while shopping or figuring out a tip at a restaurant, percentages play a crucial role in everyday transactions. 无论是在购物时计算折扣，还是在餐厅给小费，百分数都在日常交易中发挥着关键作用。
Understanding how to convert between fractions, decimals, and percentages is essential for mastering this concept. 理解如何在分数、小数和百分数之间进行转换对于掌握这一概念至关重要。
Another important aspect of this unit is understanding ratios and proportions. 另一个重要的内容是理解比例和比例。
数学课堂笔记五下
数学课堂笔记五下Math class has always been a challenging subject for many students. However, with the right approach and mindset, it can become an enjoyable and rewarding experience. 数学课堂一直是许多学生觉得具有挑战性的科目。
然而，通过正确的方法和心态，它可以成为一种愉快和有益的体验。
First and foremost, it’s important to have a positive attitude towards math. Many students develop a fear or dislike for math due to past struggles or negative experiences. However, approaching the subject with an open mind and determination can make a significant difference. 首先，对数学持有积极的态度至关重要。
许多学生由于过去的艰难或负面的经历，对数学产生了恐惧或厌恶。
然而，以开放的心态和决心来对待这个科目可以产生重大的影响。
Furthermore, seeking help and guidance from teachers or tutors can greatly improve understanding and performance in math. It's important to understand that it's okay to ask for help when needed. 而且，寻求老师或导师的帮助和指导可以极大提高对数学的理解和表现。
高一数学必修一笔记总结
高一数学必修一笔记总结Math is a subject that many students struggle with, but it is also an essential skill to have in life. In high school, the first year math curriculum often includes algebra, geometry, and basic statistics. These topics form the foundation for more advanced math coursesin the future.数学是许多学生苦苦挣扎的学科，但它也是生活中必不可少的技能。
在高中，高一数学课程通常包括代数、几何和基本统计学。
这些课题构成了将来更高级数学课程的基础。
Algebra is the study of mathematical symbols and the rules for manipulating these symbols. It involves solving equations, simplifying expressions, and understanding the relationships between variables. These skills are important for solving real-world problems and for progressing to more complex mathematical concepts.代数是研究数学符号和操作这些符号的规则。
它涉及解方程、简化表达式和理解变量之间的关系。
这些技能对于解决现实世界的问题以及进一步理解更复杂的数学概念至关重要。
Geometry is the branch of mathematics that deals with the properties and relationships of points, lines, angles, surfaces, and solids. It requires a strong visualization and spatial reasoning skills, as it involves concepts such as shapes, sizes, and properties of space.几何是研究点、线、角、面和立体的性质和关系的数学分支。
三年级下册第一课数学笔记
三年级下册第一课数学笔记Today, I want to share my thoughts on the first lesson of math in the second semester of the third grade. 今天，我想分享一下三年级下学期数学的第一课的感想。
Mathematics has always been a subject that challenges me, but also one that I find incredibly rewarding. 数学一直是一个挑战我的科目，但我也发现它是一个非常有益的学科。
In this lesson, we learned about addition and subtraction of numbers up to 1000, which is a significant step up from what we were learning in the first semester. 在这节课上，我们学习了1000以内的数字的加减法，这是跟上学期所学内容相比的一个重要进步。
It was fascinating to see how different strategies such as using place value and regrouping can help us solve complex math problems with ease. 看到不同的策略，如使用位值和换组等如何帮助我们轻松解决复杂的数学问题真是太迷人了。
I particularly enjoyed the hands-on activities that our teacher introduced to us, such as using manipulatives to visualize addition and subtraction problems. 我特别喜欢老师给我们介绍的实践活动，比如使用可视化工具来解决加减法问题。
三年级数学课堂笔记
三年级数学课堂笔记Math class in the third grade is an important subject that lays the foundation for future learning. 三年级的数学课是一个重要的科目，为未来的学习打下基础。
In math class, students learn various concepts such as addition, subtraction, multiplication, division, fractions, and geometry. 在数学课上，学生学习各种概念，如加法、减法、乘法、除法、分数和几何。
It is essential for students to grasp these concepts early on to build a solid mathematical foundation. 学生要早日掌握这些概念，才能建立起扎实的数学基础。
The teacher plays a crucial role in helping students understand these concepts through engaging lessons and activities. 老师在通过引人入胜的课程和活动帮助学生理解这些概念方面起着至关重要的作用。
Math class also teaches students problem-solving skills and critical thinking, which are valuable skills for their academic and futuresuccess. 数学课还教给学生解决问题的技能和批判性思维，这些是他们学术和未来成功所必备的宝贵技能。
Students may struggle with certain concepts in math class, but with patience and practice, they can overcome challenges and improve their skills. 学生在数学课上可能会遇到某些概念的困难，但通过耐心和实践，他们可以克服挑战，提高自己的技能。
完整版数学笔记知识点汇总
完整版数学笔记知识点汇总【完整版数学笔记知识点汇总】一、代数学代数学是数学的一个重要分支，它研究的是各种运算和数学结构的性质和规律。
1.1 线性代数线性代数是代数学的一个分支，主要研究向量空间和线性变换的性质和关系。
1.1.1 向量的基本概念向量是具有大小和方向的量，常用箭头表示。
向量的加法满足交换律和结合律。
1.1.2 向量的数量积向量的数量积又称为内积或点积，表示两个向量的乘积与夹角的余弦值的乘积。
1.1.3 向量的向量积向量的向量积又称为外积或叉积，表示两个向量构成的平行四边形的面积有向量的垂直分量。
1.2 多项式与方程多项式是由各级次幂的常数乘以各个系数之和所构成的一个函数表达式。
1.2.1 一元多项式的运算包括多项式的加法、减法、乘法和除法等基本运算规则。
1.2.2 一元多项式方程一元多项式方程是一个关于未知数的等式，多项式方程的解即为方程的根。
1.2.3 因式分解与根的性质因式分解是将一个多项式表示为若干个次数较低的因式相乘的形式。
二、几何学几何学是研究空间、形状、大小和其他属性的数学分支。
2.1 平面几何学平面几何学研究平面上的点、线和图形的性质和关系。
2.1.1 点、线和面的基本概念点是空间的基本要素，一维无宽度；线是由一系列相邻点构成，无宽度、无端点；面是由一系列相邻线构成，有宽度、有边界。
2.1.2 图形的基本性质图形包括直线、角、三角形、四边形、多边形等，具有不同的性质和特征。
2.1.3 圆和圆锥的性质圆是由平面内所有距离一个给定点的位置相等的点构成的图形；圆锥是由一条曲线和该曲线外一点重合的直线所围成的立体。
2.2 空间几何学空间几何学研究空间中点、线、面和立体的性质和关系。
2.2.1 空间中的直线和平面空间中的直线是由无限多个点构成，平面是由无限多条直线构成。
2.2.2 空间中的多面体多面体是由有限个平面围成的立体，包括三棱柱、四棱柱、四面体、正八面体等。
2.2.3 空间中的圆锥和圆柱圆锥是由一条曲线和该曲线外一点重合的直线所围成的立体；圆柱是由两条平行的曲线和两个平行于这两条曲线的曲面构成的立体。
数学七下笔记
数学七下笔记Mathematics is a subject that many students find challenging, especially when they reach the seventh grade. 数学是一个让许多学生感到挑战的学科，特别是当他们到了七年级。
As the complexity of mathematical concepts and problem-solving strategies increases, students may struggle to keep up with the pace of the curriculum. 随着数学概念的复杂性和解决问题的策略增加，学生可能会难以跟上课程的步伐。
This is why it’s crucial for students to take thorough and organized notes in their math class to ensure they have the necessary reference material when it comes to studying for exams and completing homework assignments. 这就是为什么学生在数学课上要做详细和有条不紊的笔记是至关重要的，以确保他们在备考考试和完成作业时有必要的参考资料。
When it comes to taking notes in mathematics, it is important for students to use clear and concise language. 在进行数学笔记时，学生重要的是要使用清晰简洁的语言。
Mathematical concepts can be complex and intricate, and using convoluted or ambiguous language in notes can make it difficult for students to review and understand the material later. 数学概念可能复杂而微妙，在笔记中使用复杂或模糊的语言可能会使学生难以复习和理解后续的内容。
Maths_速读笔记
速讀筆記這些速讀筆記只是希望能簡單地把我認為同學應該要識的內容、重點、技巧等概括列出。
至於詳細的內容、公式等同學可以參考其他數學書。
百分法、率及比、變數法百分法∙意義：某數的一部份 1 = 100%∙增加/減少的概念：新值＝舊值（１±改變百分率）∙買賣應用：如售價＝成本（１＋盈利率）；售價＝標價（１－折扣率）∙利息：單利息及複利息；留意複利息中的“期利率”與“年利率”的分別率及比∙率的概念：“每小時行3公里＝3km/h”∙比的概念：o利用“a:b”，“b:c”計“a:b:c”o分餅概念，e.g.$2000，按“2:3:5”分給三人變數法∙正變（x=ky）、反變（x=k / y）、聯變（x=kyz）、部份變（x=ky+Ky^2）估算∙須知道常用誤差（e.g.絕對誤差、相對誤差）的定義和公式。
多項式（包括公式）公式的主項變換∙二個多項式的加減乘除∙因式分解的技巧（例如抽共同因式、運用恆等式）∙分數形式之多項式的加減∙餘式定理∙因式定理（即餘式＝0）及其運用（因式分解一元三次多項式）指數定律指數定律公式∙根式、含根式的分數之有理化∙進制（進制轉換、位值）對數方程解方程技巧，包括∙一元一次方程∙一元二次方程∙二元一次聯立方程∙一元一次不等式∙一元二次方程∙根的特性（判別式的值）∙求最大／最小值的方法（配方法或利用對稱軸x=-b/2a）∙可變為一元二次方程的方程，e.g. y^4+5y^2 – 4 = 0坐標若有兩點A、B 的直角坐標。
同學須知道：∙A的極坐標∙AB的長度、斜率、中點坐標∙若P點把線段AB分成兩份，且AP：PB = m：n，求P點坐標∙利用直線的斜率及通過的一點來求其方程∙如兩條直線是平行，則它們的斜率( m1及m2 )相等∙如兩條直線是垂直，則m1 x m2 = -1∙把直線Ax + By + C = 0變成y = mx ＋c，從而看出s斜率及ｙ-軸截點∙圓形方程：∙標準式及一般式∙從已知方程求圓的圓心坐標及半徑函數的圖像睇“y = mx + c”的圖∙睇“y = ax2 + bx + c” 的圖∙應用：劃y = ax^2 + bx + c的草圖來求一元二次不等式(如xv2 +5x + 4＞0)的解的區域∙睇“二元一次不等式”的解的區域∙函數轉換對圖像的影響（e.g.平移）三角比∙sin, cos, tan的基本定義（即於直角三角形內）∙特別角的三角比值（e.g. sin60°）∙sin (180-x)=sinx 等公式∙三角比恆等式(tan x = sin x / cos x等)∙解三角方程，e.g. 2sinx = 1（留意角的取值範圍）∙正弦公式、餘弦公式之應用∙希羅公式（可用餘弦公式及1/2 absinC來求三角形面積）∙三角比應用（方位角，立體圖等）演繹推理幾何∙與直線及平行線有關的定理e.g. 直線上的鄰角、內錯角∙與三角形有關的定理，全等及相似的條件及應用∙畢氏定理及其逆定理∙與多邊形有關的定理（四邊形可略讀）∙與圓形有關的定理平面及立體圖形求面積及體積的公式、技巧∙留意會考試卷中首頁會提供幾條公式∙扇形的面積及弧長∙相似形狀的“邊長比”、“面積比”及“體積比”的關係等差、等比數列等差數列∙通項Tn = a + (n – 1)d∙n項和= (2a+(n-1)d) * n / 2或n項和= (頭項+尾項)*項數/ 2∙等比數列∙通項Tn = aR^(n – 1)∙n項和= a (1 - R^n) / (1 - R)∙如-1 < R < 1，可求得"無限項之和"=a / (1-R)統計閱讀各統計圖的技巧∙集中趨勢（平均值、加權平均數、中位數、眾數）∙離差之量度（標準差）∙箱形圖概率概率的基本定義及性質∙獨立、互補事件∙常用的公式，如：∙P(A事件發生) = 1 – P(A事件不發生)∙P(A或B)＝P(A) + P(B) – P(A及B)∙如A、B為互斥事件，o P(A及B)＝0o P(A或B)＝P(A) + P(B)∙如A、B為獨立事件，P(A及B)＝P(A) x P(B)∙考慮概率時常用的技巧，如列表法、數數目及考慮事件的發生經過等∙條件概率∙概念：P(B|A) = 已知事件A發生了，事件B發生的概率∙公式：P(B|A) = P(A及B) / P(A)∙期望值。
高中数学maths基础知识要点总结与归纳
高中数学基础知识要点总结与归纳集合I. 基础知识要点 1. 集合中元素具有确定性、无序性、互异性. 2. 集合的性质：①任何一个集合是它本身的子集，记为A A ⊆；②空集是任何集合的子集，记为A ⊆φ；③空集是任何非空集合的真子集； 如果B A ⊆，同时A B ⊆，那么A = B. 如果C A C B B A ⊆⊆⊆，那么，.[注]：①Z = {整数}（√） Z ={全体整数} （×）②已知集合S 中A 的补集是一个有限集，则集合A 也是有限集.（×）（例：S=N ； A=+N ，则C s A= {0}） ③ 空集的补集是全集.④若集合A =集合B ，则C B A = ∅， C A B = ∅ C S （C A B ）= D （ 注 ：C A B = ∅）. 3. ①{（x ，y ）|xy =0，x ∈R ，y ∈R }坐标轴上的点集. ②{（x ，y ）|xy ＜0，x ∈R ，y ∈R}二、四象限的点集.③{（x ，y ）|xy ＞0，x ∈R ，y ∈R } 一、三象限的点集. [注]：①对方程组解的集合应是点集. 例： ⎩⎨⎧=-=+1323y x y x 解的集合{(2，1)}.②点集与数集的交集是φ. （例：A ={(x ，y )| y =x +1} B={y |y =x 2+1} 则A ∩B =∅）4. ①n 个元素的子集有2n 个. ②n 个元素的真子集有2n －1个. ③n 个元素的非空真子集有2n －2个.5. ⑴①一个命题的否命题为真，它的逆命题一定为真. 否命题⇔逆命题. ②一个命题为真，则它的逆否命题一定为真. 原命题⇔逆否命题. 例：①若325≠≠≠+b a b a 或，则应是真命题.解：逆否：a = 2且 b = 3，则a+b = 5，成立，所以此命题为真. ②且21≠≠y x3≠+y x . 解：逆否：x + y =3x = 1或y = 2.21≠≠∴y x 且3≠+y x ,故3≠+y x 是21≠≠y x 且的既不是充分，又不是必要条件.⑵小范围推出大范围；大范围推不出小范围. 例：若255πφφx x x 或，⇒.II. 竞赛知识要点1. 集合的运算.De Morgan 公式 C u A ∩ C u B = C u （A ∪ B ） C u A ∪ C u B = C u （A ∩ B ）2. 容斥原理：对任意集合AB 有B A B A B A I Y -+=. ）（）（）（）（C B A C B A C B A C B A Y Y Y Y =⋂⋂=⋂⋂）（）（）（）（）（）（C A B A C B A C A B A C B A Y I Y I YI Y I Y I ==AB A A A B A A ==）（，）（I Y Y IC B A C B C A B A C B A C B A I I I I I Y Y +++-++=)(.函数I. 基础知识要点1. 函数的三要素：定义域，值域，对应法则.2. 函数的单调区间可以是整个定义域，也可以是定义域的一部分. 对于具体的函数来说可能有单调区间，也可能没有单调区间，如果函数在区间（0，1）上为减函数，在区间（1，2）上为减函数，就不能说函数在），（），（2110⋃上为减函数.3. 反函数定义：只有满足y x −−→←唯一，函数)(x f y =才有反函数. 例：2x y =无反函数.函数)(x f y =的反函数记为)(1y fx -=，习惯上记为)(1x fy -=. 在同一坐标系，函数)(x f y =与它的反函数)(1x fy -=的图象关于x y =对称.[注]：一般地，3)f(x 3)(x f 1+≠+-的反函数. 3)(x f 1+-是先f(x)的反函数，在左移三个单位.3)f(x +是先左移三个单位，在)f(x 的反函数.4. ⑴单调函数必有反函数，但并非反函数存在时一定是单调的.因此，所有偶函数不存在反函数. ⑵如果一个函数有反函数且为奇函数，那么它的反函数也为奇函数.⑶设函数y = f （x ）定义域，值域分别为X 、Y . 如果y = f （x ）在X 上是增（减）函数，那么反函数)(1x f y -=在Y 上一定是增（减）函数，即互为反函数的两个函数增减性相同.⑷一般地，如果函数)(x f y =有反函数，且b a f =)(，那么a b f =-)(1. 这就是说点（b a ,）在函数)(x f y =图象上，那么点（a b ,）在函数)(1x fy -=的图象上.5. 指数函数：x a y =（1,0≠a aφ），定义域R ，值域为（+∞,0）. ⑴①当1φa ，指数函数：x a y =在定义域上为增函数；②当10ππa ，指数函数：x a y =在定义域上为减函数.⑵当1φa 时，xa y =的a 值越大，越靠近y 轴；当10ππa 时，则相反.6. 对数函数：如果a （1,0≠a a φ）的b 次幂等于N ，就是N a b =，数b 就叫做以a 为底的N 的对数，记作b N a =log （1,0≠a a φ，负数和零没有对数）；其中a 叫底数，N 叫真数. ⑴对数运算：()na n a a a cb a b b a Na n a a n a a a aa a a a a a a a cb aN N Na Mn M M n M N M NMN M N M n a1121log log ...log log 1log log log log log log log 1log log log log log log log log )(log 32log )12)1(=⋅⋅⋅⇒=⋅⋅===±=-=+=⋅-推论：换底公式：（以上10且...a a ,a 1,c 0,c 1,b 0,b 1,a 0,a 0,N 0,M n 21≠≠≠≠φφφφφφ） 注⑴：当0,πb a 时，)log()log()log(b a b a -+-=⋅.⑵：当0φM 时，取“+”，当n 是偶数时且0πM 时，0φn M ，而0πM，故取“—”.例如：x x x a a a log 2(log 2log 2Θ≠中x ＞0而2log x a 中x ∈R ）.⑵x a y =（1,0≠a a φ）与x y a log =互为反函数.当1φa 时，x y a log =的a 值越大，越靠近x 轴；当10ππa 时，则相反. 7. 奇函数，偶函数： ⑴偶函数：)()(x f x f =-设（b a ,）为偶函数上一点，则（b a ,-）也是图象上一点. 偶函数的判定：两个条件同时满足①定义域一定要关于y 轴对称，例如：12+=x y 在)1,1[-上不是偶函数.②满足)()(x f x f =-，或0)()(=--x f x f ，若0)(≠x f 时，1)()(=-x f x f . ⑵奇函数：)()(x f x f -=-设（b a ,）为奇函数上一点，则（b a --,）也是图象上一点. 奇函数的判定：两个条件同时满足①定义域一定要关于原点对称，例如：3x y =在)1,1[-上不是奇函数.②满足)()(x f x f -=-，或0)()(=+-x f x f ，若0)(≠x f 时，1)()(-=-x f x f . 8. 对称变换：①y = f （x ））（轴对称x f y y -=−−−→− ②y =f （x ））（轴对称x f y x -=−−−→−③y =f （x ））（原点对称x f y --=−−−→−9. 判断函数单调性（定义）作差法：对带根号的一定要分子有理化，例如：在进行讨论.10. 外层函数的定义域是内层函数的值域. 例如：已知函数f （x ）= 1+xx-1的定义域为A ，函数f [f （x ）]的定义域是B ，则集合A 与集合B 之间的关系是 . 解：)(x f 的值域是))((x f f 的定义域B ，)(x f 的值域R ∈，故R B ∈，而A {}1|≠=x x ，故A B ⊃.11. 常用变换：①)()()()()()(y f x f y x f y f x f y x f =-⇔=+. 证：)()(])[()()()()(y f y x f y y x f x f x f y f y x f -=+-=⇔=- ②)()()()()()(y f x f y x f y f x f yx f +=⋅⇔-= 证：)()()()(y f yxf y y x f x f +=⋅= 12. ⑴熟悉常用函数图象：例：||2x y =→||x 关于y 轴对称. |2|21+⎪⎭⎫⎝⎛=x y →||21x y ⎪⎭⎫⎝⎛=→|2|21+⎪⎭⎫ ⎝⎛=x y22122212122222121)()()(b x b x x x x x b x b x x f x f x ++++-=+-+=-）（A B ⊃|122|2-+=xxy→||y关于x轴对称.⑵熟悉分式图象：例：372312-+=-+=xxxy⇒定义域},3|{Rxxx∈≠，值域},2|{Ryyy∈≠→值域≠x前的系数之比.数列知识要点1. ⑴等差、等比数列：⑵看数列是不是等差数列有以下三种方法：①),2(1为常数dndaann≥=--②211-++=nnnaaa(2≥n)③bknan+=(kn,为常数).⑶看数列是不是等比数列有以下四种方法：①,,2(1≠≥=-且为常数qnqaann②112-+⋅=nnnaaa(2≥n，11≠-+nnnaaa)①注①：i. acb=，是a、b、c成等比的双非条件，即acb=、b、c等比数列.ii. acb=（ac＞0）→为a、b、c等比数列的充分不必要.iii. acb±=→为a、b、c等比数列的必要不充分.iv. ac b ±=且0φac →为a 、b 、c 等比数列的充要.注意：任意两数a 、c 不一定有等比中项，除非有ac ＞0，则等比中项一定有两个.③nn cq a =(q c ,为非零常数).④正数列{n a }成等比的充要条件是数列{n x a log }（1φx ）成等比数列. ⑷数列{n a }的前n 项和n S 与通项n a 的关系：⎩⎨⎧≥-===-)2()1(111n s s n a s a n nn[注]： ①()()d a nd d n a a n -+=-+=111（d 可为零也可不为零→为等差数列充要条件（即常数列也是等差数列）→若d 不为0，则是等差数列充分条件）.②等差{n a }前n 项和n d a n d Bn An S n ⎪⎭⎫ ⎝⎛-+⎪⎭⎫⎝⎛=+=22122 →2d 可以为零也可不为零→为等差的充要条件→若d 为零，则是等差数列的充分条件；若d 不为零，则是等差数列的充分条件.③非零．．常数列既可为等比数列，也可为等差数列.（不是非零，即不可能有等比数列） 2. ①等差数列依次每k 项的和仍成等差数列，其公差为原公差的k 2倍...,,232k k k k k S S S S S --； ②若等差数列的项数为2()+∈Nn n ，则，奇偶nd S S=-1+=n n a a S S 偶奇；③若等差数列的项数为()+∈-Nn n 12，则()n n a n S1212-=-，且n a S S =-偶奇，1-=n n S S 偶奇得到所求项数到代入12-⇒n n . 3. 常用公式：①1+2+3 …+n =()21+n n ②()()61213212222++=+++n n n n Λ③()2213213333⎥⎦⎤⎢⎣⎡+=++n n n Λ[注]：熟悉常用通项：9，99，999，…110-=⇒nn a ； 5，55，555，…()11095-=⇒nn a . 4. 等比数列的前n 项和公式的常见应用题：⑴生产部门中有增长率的总产量问题. 例如，第一年产量为a ，年增长率为r ，则每年的产量成等比数列，公比为r +1. 其中第n 年产量为1)1(-+n r a ，且过n 年后总产量为：.)1(1])1([)1(...)1()1(12r r a a r a r a r a a n n +-+-=+++++++-⑵银行部门中按复利计算问题. 例如：一年中每月初到银行存a 元，利息为r ，每月利息按复利计算，则每月的a 元过n 个月后便成为n r a )1(+元. 因此，第二年年初可存款：)1(...)1()1()1(101112r a r a r a r a ++++++++=)1(1])1(1)[1(12r r r a +-+-+.⑶分期付款应用题：a 为分期付款方式贷款为a 元；m 为m 个月将款全部付清；r 为年利率.()()()()()()()()1111111 (1112)1-++=⇒-+=+⇒++++++=+--m m m mm m mr r ar x r r x r a x r x r x r x r a5. 数列常见的几种形式：⑴n n n qa pa a +=++12（p 、q 为二阶常数）→用特证根方法求解.具体步骤：①写出特征方程q Px x +=2（2x 对应2+n a ，x 对应1+n a ），并设二根21,x x ②若21x x ≠可设nn n xc x c a 2211.+=，若21x x =可设nn x n c c a 121)(+=；③由初始值21,a a 确定21,c c .⑵r Pa a n n +=-1（P 、r 为常数）→用①转化等差，等比数列；②逐项选代；③消去常数n 转化为n n n qa Pa a +=++12的形式，再用特征根方法求n a ；④121-+=n n Pc c a （公式法），21,c c 由21,a a 确定.①转化等差，等比：1)(11-=⇒-+=⇒+=+++P rx x Px Pa a x a P x a n n n n . ②选代法：=++=+=--r r Pa P r Pa a n n n )(21x P x a P r P P r a a n n n -+=---+=⇒--1111)(1)1(Λ r r P a P n n +++⋅+=--Pr 211Λ.③用特征方程求解：⇒⎭⎬⎫+=+=-+相减，r Pa a r Pa a n n n n 111+n a 1111-+--+=⇒-=-n n n n n n Pa a P a Pa Pa a ）（. ④由选代法推导结果：Pr P P r a c P c a P r a c P r c n n n -+-+=+=-+=-=--111111112121）（，，. 6. 几种常见的数列的思想方法：⑴等差数列的前n 项和为n S ，在0πd 时，有最大值. 如何确定使n S 取最大值时的n 值，有两种方法： 一是求使0,01π+≥n n a a ，成立的n 值；二是由n da n d S n )2(212-+=利用二次函数的性质求n 的值. ⑵如果数列可以看作是一个等差数列与一个等比数列的对应项乘积，求此数列前n 项和可依照等比数列前n 项和的推倒导方法：错位相减求和. 例如：, (2)1)12,...(413,211n n -⋅⑶两个等差数列的相同项亦组成一个新的等差数列，此等差数列的首项就是原两个数列的第一个相同项，公差是两个数列公差21d d ，的最小公倍数.三角函数I. 基础知识要点1. ①与α（0°≤α＜360°）终边相同的角的集合（角α与角β的终边重合）：{}Z k k ∈+⨯=,360|αββο②终边在x 轴上的角的集合： {}Z k k ∈⨯=,180|οββ ③终边在y 轴上的角的集合：{}Z k k ∈+⨯=,90180|οοββ ④终边在坐标轴上的角的集合：{}Z k k ∈⨯=,90|οββ ⑤终边在y =x 轴上的角的集合：{}Z k k ∈+⨯=,45180|οοββ ⑥终边在x y -=轴上的角的集合：{}Z k k ∈-⨯=,45180|οοββ⑦若角α与角β的终边关于x 轴对称，则角α与角β的关系：βα-=k ο360⑧若角α与角β的终边关于y 轴对称，则角α与角β的关系：βα-+=οο180360kSIN \COS 三角函数值大小关系图1、2、3、4表示第一、二、三、四象限一半所在区域⑨若角α与角β的终边在一条直线上，则角α与角β的关系：βα+=k ο180 ⑩角α与角β的终边互相垂直，则角α与角β的关系：οο90360±+=βαk 2. 角度与弧度的互换关系：360°=2π 180°=π 1°=0.01745 1=57.30°=57°18′ 注意：正角的弧度数为正数，负角的弧度数为负数，零角的弧度数为零. 3. 三角函数的定义域：4. 三角函数的公式： （一）基本关系公式组二 公式组三xx k x x k x x k x x k cot )2cot(tan )2tan(cos )2cos(sin )2sin(=+=+=+=+ππππ xx x x xx x x cot )cot(tan )tan(cos )cos(sin )sin(-=--=-=--=-公式组四 公式组五 公式组六xx x x x x x x cot )cot(tan )tan(cos )cos(sin )sin(=+=+-=+-=+ππππ xx x x x x x x cot )2cot(tan )2tan(cos )2cos(sin )2sin(-=--=-=--=-ππππ xx x x x x xx cot )cot(tan )tan(cos )cos(sin )sin(-=--=--=-=-ππππ（二）角与角之间的互换公式组一 公式组二βαβαβαsin sin cos cos )cos(-=+ αααcos sin 22sin =βαβαβαsin sin cos cos )cos(+=- ααααα2222sin 211cos 2sin cos 2cos -=-=-= βαβαβαsin cos cos sin )sin(+=+ ααα2tan 1tan 22tan -=βαβαβαsin cos cos sin )sin(-=- 2cos 12sinαα-±= βαβαβαtan tan 1tan tan )tan(-+=+ 2cos 12cos αα+±=βαβαβαtan tan 1tan tan )tan(+-=- 公式组三 公式组四 公式组五公式组一sin x ·csc x =1tan x =x xcos sin sin 2x +cos 2x =1cos x ·sec x x =xx sin cos 1+tan 2x =sec 2x tan x ·cot x =11+cot 2x =csc 2x=1()()[]βαβαβα-++=1sin sin 21cos sin αααααααsin cos 1cos 1sin cos 1cos 12tan -=+=+-±=2tan12tan2sin 2ααα+=2tan 12tan 1cos 22ααα+-=2tan 12tan2tan 2ααα-=42675cos 15sin -==οο,42615cos 75sin +==οο,3275cot 15tan -==οο,3215cot 75tan +==οο. 5. 正弦、余弦、正切、余切函数的图象的性质：注意：①与的单调性正好相反；与的单调性也同样相反.一般地，若)(x 在],[b a 上递增（减），则)(x f y -=在],[b a 上递减（增）.②x y sin =与x y cos =的周期是π.③)sin(ϕω+=x y 或)cos(ϕω+=x y （0≠ω）的周期ωπ2=T .2tan xy =的周期为2π（πωπ2=⇒=T T ，如图，翻折无效）.④)sin(ϕω+=x y 的对称轴方程是2ππ+=k x （Z k ∈），对称中心（0,πk ）；)cos(ϕω+=x y 的对称轴方程是πk x =（Z k ∈），对称2cos2sin2sin sin βαβαβα-+=+2sin2cos 2sin sin βαβαβα-+=-2cos 2cos 2cos cos βαβαβα-+=+2sin 2sin 2cos cos βαβαβα-+-=-ααπsin )21cos(-=+ααπcos )21sin(=+ααπcot )21tan(-=+ααπsin )21cos(=-ααπcos )21sin(=-ααπcot )21tan(=-中心（0,21ππ+k ）；)tan(ϕω+=x y 的对称中心（0,2πk ）. x x y x y 2cos )2cos(2cos -=--=−−−→−=原点对称 ⑤当αtan ·,1tan =β)(2Z k k ∈+=+ππβα；αtan ·,1tan -=β)(2Z k k ∈+=-ππβα.⑥x y cos =与⎪⎭⎫⎝⎛++=ππk x y 22sin 是同一函数,而)(ϕω+=x y 是偶函数，则 )cos()21sin()(x k x x y ωππωϕω±=++=+=.⑦函数x y tan =在R 上为增函数.（×） [只能在某个单调区间单调递增. 若在整个定义域，x y tan =为增函数，同样也是错误的].⑧定义域关于原点对称是)(x f 具有奇偶性的必要不充分条件.（奇偶性的两个条件：一是定义域关于原点对称（奇偶都要），二是满足奇偶性条件，偶函数：)()(x f x f =-，奇函数：)()(x f x f -=-） 奇偶性的单调性：奇同偶反. 例如：x y tan =是奇函数，)31tan(π+=x y 是非奇非偶.（定义域不关于原点对称）奇函数特有性质：若x ∈0的定义域，则)(x f 一定有0)0(=f .（x ∉0的定义域，则无此性质）⑨x y sin =不是周期函数；x y sin =为周期函数（π=T）； x y cos =是周期函数（如图）；x y cos =为周期函数（π=T ）； 212cos +=x y 的周期为π（如图），并非所有周期函数都有最小正周期，例如： R k k x f x f y ∈+===),(5)(.⑩abb a b a y =+++=+=ϕϕαβαcos )sin(sin cos 22 有y b a ≥+22. II. 竞赛知识要点一、反三角函数.1. 反三角函数：⑴反正弦函数x y arcsin =是奇函数，故x x arcsin )arcsin(-=-，[]1,1-∈x （一定要注明定义域，若()+∞∞-∈,x ，没有x 与y 一一对应，故x y sin =无反函数）注：x x =)sin(arcsin ，[]1,1-∈x ，⎥⎦⎤⎢⎣⎡-∈2,2arcsin ππx .⑵反余弦函数x y arccos =非奇非偶，但有ππk x x 2)arccos()arccos(+=+-，[]1,1-∈x .注：①x x =)cos(arccos ，[]1,1-∈x ，[]π,0arccos ∈x .②x y cos =是偶函数，x y arccos =非奇非偶，而x y sin =和x y arcsin =为奇函数.⑶反正切函数：x y arctan =，定义域),(+∞-∞，值域（2,2ππ-），x y arctan =是奇函数，x x arctan )arctan(-=-，∈x ),(+∞-∞.注：x x =)tan(arctan ，∈x ),(+∞-∞.⑷反余切函数：x arc y cot =，定义域),(+∞-∞，值域（2,2ππ-），x arc y cot =是非奇非偶.ππk x arc x arc 2)cot()cot(+=+-，∈x ),(+∞-∞.注：①x x arc =)cot cot(，∈x ),(+∞-∞.②x y arcsin =与)1arcsin(x y -=互为奇函数，x y arctan =同理为奇而x y arccos =与x arc y cot =非奇非偶但满足]1,1[,2)cot(cot ]1,1[,2arccos )arccos(-∈+=-+-∈+=+-x k x arc x arc x k x x ππππ.y=|cos2x +1/2|图象⑵ 正弦、余弦、正切、余切函数的解集：a 的取值范围 解集 a 的取值范围 解集①a x =sin 的解集 ②a x =cos 的解集a＞1 ∅ a ＞1 ∅ a=1 {}Z k a k x x ∈+=,arcsin 2|π a =1 {}Z k a k x x ∈+=,arccos 2|π a＜1 (){}Z k a k x x k∈-+=,arcsin 1|π a ＜1 {}Z k a k x x ∈±=,arccos |π③a x =tan 的解集：{}Z k a k x x ∈+=,arctan |π ③a x =cot 的解集：{}Z k a k x x ∈+=,cot arc |π 二、三角恒等式. 组一 组二∏===nk nn nk12sin2sin 2cos 8cos4cos2cos2cos αααααααΛ∑=++=+++++=+nk dnd x d n nd x d x x kd x 0sin )cos())1sin(()cos()cos(cos )cos(Λ∑=++=+++++=+nk dnd x d n nd x d x x kd x 0sin )sin())1sin(()sin()sin(sin )sin(Λαγγββαγβαγβαγβαtan tan tan tan tan tan 1tan tan tan tan tan tan )tan(----++=++组三 三角函数不等式x sin ＜x ＜)2,0(,tan π∈x x xxx f sin )(=在),0(π上是减函数 若π=++C B A ，则C xy B xz A yz z y x cos 2cos 2cos 2222++≥++平面向量1. 长度相等且方向相同的两个向量是相等的量.注意：①若b a ρρ,为单位向量，则b a ρρ=. （⨯） 单位向量只表示向量的模为1，并未指明向量的方向.②若b a ρρ=，则a ρ∥b ρ. （√）2. ①()a ρμλ=()a ρλμ ②()a a a ρρρμλμλ+=+ ③()b a b a ρρρρλλλ+=+④设()()R y x b y x a ∈==λ,,,,2211 ()2121,y y x x b a ++=+ρρ ()2121,y y x x b a --=-ρρ ()21,y x a λλλ=ρ 2121y y x x b a +=⋅ρρ 2121y x a +=ρ（向量的模，针对向量坐标求模） ⑤平面向量的数量积：θcos b a b a ⋅=⋅ρρ ⑥a b b a ρρρρ⋅=⋅ ⑦()()()b a b a b a ρρρρρρλλλ⋅=⋅=⋅⑧()c b c a c b a ρρρρρρρ⋅+⋅=⋅+注意：①()()c b a c b a ρρρρρρ⋅⋅=⋅⋅不一定成立；cb b a ρρρ⋅=⋅c a ρρ=.②向量无大小（“大于”、“小于”对向量无意义），向量的模有大小.ααααααcos 3cos 43cos sin 4sin 33sin 33-=-=()()αββαβαβα2222cos cos sin sin sin sin -=-+=-ααααααsin 22sin 2cos ...4cos 2cos cos 11++=n n n③长度为0的向量叫零向量，记0ρ，0ρ与任意向量平行，0ρ的方向是任意的，零向量与零向量相等，且00ρρ=-.④若有一个三角形ABC ，则0；此结论可推广到n 边形.⑤若a n a m ρρ=（R n m ∈,），则有n m =. （⨯） 当a ρ等于0ρ时，0ρρρ==a n a m ，而n m ,不一定相等.⑥a ρ·a ρ=2||a ρ，||a ρ=2a ρ（针对向量非坐标求模），||b a ρρ⋅≤||||b a ρρ⋅. ⑦当0ρρ≠a 时，由0=⋅b a ρρ不能推出0ρρ≠b ，这是因为任一与a ρ垂直的非零向量b ρ，都有a ρ·b ρ=0.⑧若a ∥b ，b ∥c ，则a ∥c （×）当b 等于0时，不成立.3. ①向量b ρ与非零向量．．．．a ρ共线的充要条件是有且只有一个实数λ，使得a b ρρλ=（平行向量或共线向量）. 当a ,0φλ与b 共线同向：当,0πλa 与b 共线反向；当b 则为0,0与任何向量共线.注意：若b a ,共线，则b a λ= （×）若c 是a 的投影，夹角为θ，则c a =⋅θcos ，c a =⋅θcos （√）②设a ρ=()11,y x ，()22,y x b =ρa ρ∥b ρ⇔=-⇔01221y x y x b a b a b a ⋅=⋅⇔=λ a ρ⊥b ρ001221=+⇔=⋅⇔y y x x b a③设()()()332211,,,,,y x C y x B y x A ，则A 、B 、C 三点共线⇔∥⇔=λ（0≠λ）⇔（1212,y y x x --）=λ（1313,y y x x --）（0≠λ） ⇔（12x x -）·（13y y -）=（13x x -）·（12y y -）④两个向量a ρ、b ρ的夹角公式：222221212121cos y x y x y y x x +⋅++=θ⑤线段的定比分点公式：（0≠λ和1-） 设 P 1P ρ=λPP 2ρ（或P 2P λ1P 1P ），且21,,P P P 的坐标分别是),(),,(,,2211y x y x y x ）（，则推广1：当1=λ时，得线段21P P 的中点公式：推广2λ=则λλ++=1PB PA PM （λ对应终点向量）.三角形重心坐标公式：△ABC 的顶点()()()332211,,,,,y x C y x B y x A ，重心坐标()y x G ,： 注意：在△ABC 中，若0为重心，则=++，这是充要条件.⑥平移公式：若点P ()y x ,按向量a ρ=()k h ,平移到P ‘()'',y x ，则⎪⎩⎪⎨⎧+=+=ky y h x x ''4. ⑴正弦定理：设△ABC 的三边为a 、b 、c ，所对的角为A 、B 、C ，则R CcB b A a 2sin sin sin ===. ⎪⎪⎩⎪⎪⎨⎧++=++=33321321y y y y x x x x ⎪⎪⎩⎪⎪⎨⎧+=+=222121x x x y y y ⎪⎪⎩⎪⎪⎨⎧++=++=λλλλ112121x x x y y y BPM⑵余弦定理：⎪⎪⎩⎪⎪⎨⎧-+=-+=-+=C ab a b c B ac c a b A bc c b a cos 2cos 2cos 2222222222⑶正切定理：2tan2tanB A BA ba b a -+=-+ ⑷三角形面积计算公式：设△ABC 的三边为a ，b ，c ，其高分别为h a ，h b ，h c ，半周长为P ，外接圆、内切圆的半径为R ，r . ①S △=1/2ah a =1/2bh b =1/2ch c ②S △=Pr ③S △=abc/4R④S △=1/2sin C ·ab=1/2ac ·sin B=1/2cb ·sin A ⑤S △=()()()c P b P a P P --- [海伦公式] ⑥S △=1/2（b+c-a ）r a [如下图]=1/2（b+a-c ）r c =1/2（a+c-b ）r b[注]：到三角形三边的距离相等的点有4个，一个是内心，其余3个是旁心.如图：图1中的I 为S △ABC 的内心， S △=PrI 为S △ABC 的一个旁心，S △=1/2（b+c-a ）r a图1 图2 图3 图4附：三角形的五个“心”； 重心：三角形三条中线交点.外心：三角形三边垂直平分线相交于一点. 内心：三角形三内角的平分线相交于一点. 垂心：三角形三边上的高相交于一点.旁心：三角形一内角的平分线与另两条内角的外角平分线相交一点.⑸已知⊙O 是△ABC 的内切圆，若BC =a ，AC =b ，AB =c [注：s 为△ABC 的半周长,即2cb a ++] 则：①AE=a s -=1/2（b+c-a ） ②BN=b s -=1/2（a+c-b ） ③FC=c s -=1/2（a+b-c ）综合上述：由已知得，一个角的邻边的切线长，等于半周长减去对边（如图4）. 特例：已知在Rt △ABC ，c 为斜边，则内切圆半径r =cb a abc b a ++=-+2（如图3）. ⑹在△ABC 中，有下列等式成立C B A C B A tan tan tan tan tan tan =++. 证明：因为,C B A -=+π所以()()C B A -=+πtan tan ，所以C BA BA tan tan tan 1tan tan -=-+，∴结论！⑺在△ABC 中，D 是BC 上任意一点，则DC BD BCBCAB BD AC AD ⋅-+=222.证明：在△ABCD 中，由余弦定理，有ΛB BD AB BD AB AD cos 2222⋅⋅-+=① 在△ABC 中，由余弦定理有ΛBC AB AC BC AB B ⋅-+=2cos 222②，②代入①，化简可得，DC BD BCBC AB BD AC AD ⋅-+=222（斯德瓦定理）B I A BCDEF I AB C DE F r ar ar abc a ab c CDACB图5①若AD 是BC 上的中线，2222221a cb m a -+=； ②若AD 是∠A 的平分线，()a p p bc cb t a -⋅+=2，其中p 为半周长； ③若AD 是BC 上的高，()()()c p b p a p p ah a ---=2，其中p 为半周长.⑻△ABC 的判定：⇔+=222b a c △ABC 为直角△⇔∠A + ∠B =2π2c ＜⇔+22b a △ABC 为钝角△⇔∠A + ∠B ＜2π 2c ＞⇔+22b a △ABC 为锐角△⇔∠A + ∠B ＞2π 附：证明：abc b a C 2cos 222-+=，得在钝角△ABC 中，222222,00cos c b a c b a C πππ+⇔-+⇔⑼平行四边形对角线定理：对角线的平方和等于四边的平方和.)2=不 等 式 知识要点1. ⑴平方平均≥算术平均≥几何平均≥调和平均（a 、b 为正数）：2112a b a b+≥+（当a = b 时取等）特别地，222()22a b a b ab ++≤≤（当a = b 时，222()22a b a b ab ++==）),,,(332222时取等c b a R c b a c b a c b a ==∈⎪⎭⎫ ⎝⎛+++≥++⇒幂平均不等式：22122221)...(1...n n a a a na a a +++≥+++ ⑵含立方的几个重要不等式（a 、b 、c 为正数）： ①3322a b a b ab +≥+ ②3332223()()ab c abc a b c a b c ab ac bc ++-=++++---⇒3333a b c abc ++≥（等式即可成立0φc b a ++，时取等或0=++==c b a c b a）； 3a b c ++≤⇒33a b c abc ++⎛⎫≤ ⎪⎝⎭3333a b c ++≤2)(31c b a ac ba ab +++≤++（时取等c b a ==）⑶绝对值不等式：123123(0)a a a a a a ab a b a b ab ++≤++-≤-≤+≥时,取等⑷算术平均≥几何平均（a 1、a 2…an 为正数）：12n a a a n+++≥L a 1=a 2…=a n 时取等）⑸柯西不等式：设),,,2,1(,n iR b a i i Λ=∈则))(()(222212222122211n n n n b b b a a a b a b a b a ++++++≤+++ΛΛΛ等号成立当且仅当nn b a b a b a ===Λ2211时成立.（约定0=i a 时，0=i b ）例如：22222()()()ac bd a b c d +≤++.⑹常用不等式的放缩法：①21111111(2)1(1)(1)1n n n n n n n n n n-==-≥++--p p1)n ==≥pp2. 常用不等式的解法举例（x 为正数）： ①231124(1)2(1)(1)()22327x x x x x -=⋅--≤=②2222232(1)(1)124(1)()22327x x x y x x y y --=-⇒=≤=⇒≤类似于22sin cos sin (1sin )y x x x x ==-③111||||||()2x x x xxx+=+≥与同号，故取等直线和圆的方程 知识要点一、直线方程.1. 直线的倾斜角：一条直线向上的方向与x 轴正方向所成的最小正角叫做这条直线的倾斜角，其中直线与x 轴平行或重合时，其倾斜角为0，故直线倾斜角的范围是)0(1800πααπποο≤≤.注：①当ο90=α或12x x =时，直线l 垂直于x 轴，它的斜率不存在.②每一条直线都存在惟一的倾斜角，除与x 轴垂直的直线不存在斜率外，其余每一条直线都有惟一的斜率，并且当直线的斜率一定时，其倾斜角也对应确定.2. 直线方程的几种形式：点斜式、截距式、两点式、斜切式.特别地，当直线经过两点),0(),0,(b a ，即直线在x 轴，y 轴上的截距分别为)0,0(,≠≠b a b a 时，直线方程是：1=+bya x . 注：若232--=x y 是一直线的方程，则这条直线的方程是232--=x y ，但若)0(232≥--=x x y 则不是这条线. 附：直线系：对于直线的斜截式方程b kx y +=，当b k ,均为确定的数值时，它表示一条确定的直线，如果b k ,变化时，对应的直线也会变化.①当b 为定植，k 变化时，它们表示过定点（0，b ）的直线束.②当k 为定值，b 变化时，它们表示一组平行直线. 3. ⑴两条直线平行：1l ∥212k k l =⇔两条直线平行的条件是：①1l 和2l 是两条不重合的直线. ②在1l 和2l 的斜率都存在的前提下得到的. 因此，应特别注意，抽掉或忽视其中任一个“前提”都会导致结论的错误.（一般的结论是：对于两条直线21,l l ，它们在y 轴上的纵截距是21,b b ，则1l ∥212k k l =⇔，且21b b ≠或21,l l 的斜率均不存在，即2121A B B A =是平行的必要不充分条件，且21C C ≠）推论：如果两条直线21,l l 的倾斜角为21,αα则1l ∥212αα=⇔l . ⑵两条直线垂直：两条直线垂直的条件：①设两条直线1l 和2l 的斜率分别为1k 和2k ，则有12121-=⇔⊥k k l l 这里的前提是21,l l 的斜率都存在. ②0121=⇔⊥k l l ，且2l 的斜率不存在或02=k ，且1l 的斜率不存在. （即01221=+B A B A 是垂直的充要条件） 4. 直线的交角：⑴直线1l 到2l 的角（方向角）；直线1l 到2l 的角，是指直线1l 绕交点依逆时针方向旋转到与2l 重合时所转动的角θ，它的范围是),0(π，当ο90≠θ时21121tan k k k k +-=θ.⑵两条相交直线1l 与2l 的夹角：两条相交直线1l 与2l 的夹角，是指由1l 与2l 相交所成的四个角中最小的正角θ，又称为1l 和2l 所成的角，它的取值范围是 ⎝⎛⎥⎦⎤2,0π，当ο90≠θ，则有21121tan k k k k +-=θ.5. 过两直线⎩⎨⎧=++=++0:0:22221111C y B x A l C y B x A l 的交点的直线系方程λλ(0)(222111=+++++C y B x A C y B x A 为参数，0222=++C y B x A 不包括在内）6. 点到直线的距离：⑴点到直线的距离公式：设点),(00y x P ，直线P C By Ax l ,0:=++到l 的距离为d ，则有2200BA C By Ax d +++=.⑵两条平行线间的距离公式：设两条平行直线)(0:,0:212211C C C By Ax l C By Ax l ≠=++=++，它们之间的距离为d ，则有2221BA C C d +-=.7. 关于点对称和关于某直线对称：⑴关于点对称的两条直线一定是平行直线，且这个点到两直线的距离相等.⑵关于某直线对称的两条直线性质：若两条直线平行，则对称直线也平行，且两直线到对称直线距离相等. 若两条直线不平行，则对称直线必过两条直线的交点，且对称直线为两直线夹角的角平分线.⑶点关于某一条直线对称，用中点表示两对称点，则中点在对称直线上（方程①），过两对称点的直线方程与对称直线方程垂直（方程②）①②可解得所求对称点.注：①曲线、直线关于一直线（b x y +±=）对称的解法：y 换x ，x 换y. 例：曲线f (x ,y )=0关于直线y =x –2对称曲线方程是f (y +2 ,x –2)=0. ②曲线C: f (x ,y )=0关于点(a ,b)的对称曲线方程是f (a – x , 2b – y )=0. 二、圆的方程.1. ⑴曲线与方程：在直角坐标系中，如果某曲线C 上的 与一个二元方程0),(=y x f 的实数建立了如下关系： ①曲线上的点的坐标都是这个方程的解. ②以这个方程的解为坐标的点都是曲线上的点.那么这个方程叫做曲线方程；这条曲线叫做方程的曲线（图形）.⑵曲线和方程的关系，实质上是曲线上任一点),(y x M 其坐标与方程0),(=y x f 的一种关系，曲线上任一点),(y x 是方程0),(=y x f 的解；反过来，满足方程0),(=y x f 的解所对应的点是曲线上的点.注：如果曲线C 的方程是f(x ,y)=0，那么点P 0(x 0 ,y)线C 上的充要条件是f(x 0 ,y 0)=02. 圆的标准方程：以点),(b a C 为圆心，r 为半径的圆的标准方程是222)()(r b y a x =-+-.特例：圆心在坐标原点，半径为r 的圆的方程是：222r y x =+.注：特殊圆的方程：①与x 轴相切的圆方程222)()(b b y a x =±+- )],(),(,[b a b a b r -=或圆心②与y 轴相切的圆方程222)()(a b y a x =-+± )],(),(,[b a b a a r -=或圆心③与x 轴y 轴都相切的圆方程222)()(a a y a x =±+± )],(,[a a a r ±±=圆心3. 圆的一般方程：022=++++F Ey Dx y x .当0422φF E D -+时，方程表示一个圆，其中圆心⎪⎭⎫⎝⎛--2,2E D C ，半径2422FE D r -+=.当0422=-+F E D 时，方程表示一个点⎪⎭⎫⎝⎛--2,2E D . 当0422πF E D -+时，方程无图形（称虚圆）. 注：①圆的参数方程：⎩⎨⎧+=+=θθsin cos r b y r a x （θ为参数）.②方程022=+++++F Ey Dx Cy Bxy Ax 表示圆的充要条件是：0=B 且0≠=C A 且0422φAF E D -+.③圆的直径或方程：已知0))(())((),(),(21212211=--+--⇒y y y y x x x x y x B y x A （用向量可征）.4. 点和圆的位置关系：给定点),(00y x M 及圆222)()(:r b y a x C =-+-.①M 在圆C 内22020)()(r b y a x π-+-⇔②M 在圆C 上22020)()r b y a x =-+-⇔（ ③M 在圆C 外22020)()(r b y a x φ-+-⇔5. 直线和圆的位置关系：设圆圆C ：)0()()(222φr r b y a x =-+-； 直线l ：)0(022≠+=++B A C By Ax ；圆心),(b a C 到直线l 的距离22BA C Bb Aa d +++=.①r d =时，l 与C 相切；附：若两圆相切，则⇒⎪⎩⎪⎨⎧=++++=++++002222211122F y E x D y x F y E x D y x 相减为公切线方程.②r d π时，l 与C 相交；附：公共弦方程：设有两个交点，则其公共弦方程为0)()()(212121=-+-+-F F y E E x D D . ③r d φ时，l 与C 相离.附：若两圆相离，则⇒⎪⎩⎪⎨⎧=++++=++++02222211122F y E x D y x F y E x D y x 相减为圆心21O O 的连线的中与线方程.由代数特征判断：方程组⎪⎩⎪⎨⎧=++=-+-0)()(222C Bx Ax r b y a x 用代入法，得关于x （或y ）的一元二次方程，其判别式为∆，则：l ⇔=∆0与C 相切； l ⇔∆0φ与C 相交； l ⇔∆0π与C 相离.注：若两圆为同心圆则011122=++++F y E x D y x ，022222=++++F y E x D y x 相减，不表示直线.:0:222222111221=++++=++++F y E x D y x C F y E x D y x C6. 圆的切线方程：圆222r y x =+的斜率为k 的切线方程是r k kx y 21+±=过圆022=++++F Ey Dx y x 上一点),(00y x P 的切线方程为：0220000=++++++F y y E x x Dy y x x . ①一般方程若点(x 0 ,y 0)在圆上，则(x – a)(x 0 – a)+(y – b)(y 0 – b)=R 2. 特别地，过圆222r y x =+上一点),(00y x P 的切线方程为200r y y x x =+.②若点(x 0 ,y 0)不在圆上，圆心为(a,b)则⎪⎩⎪⎨⎧+---=-=-1)()(2110101R x a k y b R x x k y y ，联立求出⇒k 切线方程. 7. 求切点弦方程：方法是构造图，则切点弦方程即转化为公共弦方程. 如四类共圆. 已知O Θ的方程022=++++F Ey Dx y x …① 又以ABCD 为圆为方程为))(())((k b x y y a x x x A A =--+--…②4)()(222b y a x R A A -+-=…③，所以BC 的方程即③代②，①②相切即为所求.圆锥曲线方程I. 基础知识要点一、椭圆方程.1. 椭圆方程的第一定义：为端点的线段以无轨迹方程为椭圆21212121212121,2,2,2F F F F a PF PF F F a PF PF F F a PF PF ==+=+=+πφ⑴①椭圆的标准方程：i. 中心在原点，焦点在x 轴上：)0(12222φφb a b y a x =+. ii. 中心在原点，焦点在y 轴上：)0(12222φφb a bx a y =+.②一般方程：)0,0(122φφB A By Ax =+.③椭圆的标准参数方程：12222=+b y a x 的参数方程为⎩⎨⎧==θθsin cos b y a x （一象限θ应是属于20πθππ）. ⑵①顶点：),0)(0,(b a ±±或)0,)(,0(b a ±±.②轴：对称轴：x 轴，y 轴；长轴长a 2，短轴长b 2.③焦点：)0,)(0,(c c -或),0)(,0(c c -.④焦距：2221,2b a c c F F -==.⑤准线：c a x 2±=或c a y 2±=.⑥离心率：)10(ππe ace =.⑦焦点半径： i. 设),(00y x P 为椭圆)0(12222φφb a by ax =+上的一点，21,F F 为左、右焦点，则 由椭圆方程的第二定义可以推出. ii.设),(00y x P 为椭圆)0(12222φφb a ay bx =+上的一点，21,F F 为上、下焦点，则 由椭圆方程的第二定义可以推出.由椭圆第二定义可知：)0()(),0()(0002200201φπx a ex x ca e pF x ex a c a x e pF -=-=+=+=归结起来为“左加右减”. 注意：椭圆参数方程的推导：得→)sin ,cos (θθb a N 方程的轨迹为椭圆. ⑧通径：垂直于x 轴且过焦点的弦叫做通经.坐标：),(2222a b c a b d -=和),(2ab c⑶共离心率的椭圆系的方程：椭圆)0(12222φφb a b y a x =+的离心率是(c a c e =t t by a x (2222=+是大于0的参数，)0φφb a 的离心率也是ace =我们称此方程为共离心率的椭圆系方程. B C)⇒-=+=0201,ex a PF ex a PF⇒-=+=0201,ey a PF ey a PF asin α,)α)⑸若P 是椭圆：12222=+b y a x 上的点.21,F F 为焦点，若θ=∠21PF F ，则21F PF ∆的面积为2tan 2θb （用余弦定理与a PF PF 221=+可得）. 若是双曲线，则面积为2cot2θ⋅b .二、双曲线方程. 1. 双曲线的第一定义：的一个端点的一条射线以无轨迹方程为双曲线21212121212121,222F F F F a PF PF F F a PF PF F F a PF PF ==-=-=-φπ⑴①双曲线标准方程：)0,(1),0,(122222222φφb a b x a y b a b y a x =-=-. 一般方程：)0(122πAC Cy Ax =+.⑵①i. 焦点在x 轴上：顶点：)0,(),0,(a a - 焦点：)0,(),0,(c c - 准线方程c a x 2±= 渐近线方程：0=±b ya x 或02222=-by a xii. 焦点在y 轴上：顶点：),0(),,0(a a -. 焦点：),0(),,0(c c -. 准线方程：c a y 2±=. 渐近线方程：0=±b x a y 或02222=-bx a y ，参数方程：⎩⎨⎧==θθtan sec b y a x 或⎩⎨⎧==θθsec tan a y b x .②轴y x ,为对称轴，实轴长为2a , 虚轴长为2b ，焦距2c. ③离心率ace =. ④准线距c a 22（两准线的距离）；通径a b 22. ⑤参数关系a ce b a c =+=,222. ⑥焦点半径公式：对于双曲线方程12222=-by a x （21,F F 分别为双曲线的左、右焦点或分别为双曲线的上下焦点）“长加短减”原则：aex MF a ex MF -=+=0201 构成满足a MF MF 221=-aex F M a ex F M +-='--='0201（与椭圆焦半径不同，椭圆焦半径要带符号计算，而双曲线不带符号）aey F M a ey F M aey MF a ey MF -'-='+'-='+=-=02010201 ⑶等轴双曲线：双曲线222a y x ±=-2=e .⑷共轭双曲线：以已知双曲线的虚轴为实轴，实轴为虚轴的双曲线，叫做已知双曲线的共轭双曲线.λ=-2222b y a x 与λ-=-2222by a x互为共轭双曲线，它们具有共同的渐近线：02222=-b y a x .⑸共渐近线的双曲线系方程：)0(2222≠=-λλb y a x 的渐近线方程为02222=-b y a x 如果双曲线的渐近线为0=±bya x 时，它的双曲线方程可设为)0(2222≠=-λλby ax .例如：若双曲线一条渐近线为x y 21=且过)21,3(-p ，求双曲线的方程？ 解：令双曲线的方程为：)0(422≠=-λλy x ，代入)21,3(-得12822=-y x . ⑹直线与双曲线的位置关系：区域①：无切线，2条与渐近线平行的直线，合计2条；区域②：即定点在双曲线上，1条切线，2条与渐近线平行的直线，合计3条； 区域③：2条切线，2条与渐近线平行的直线，合计4条；区域④：即定点在渐近线上且非原点，1条切线，1条与渐近线平行的直线，合计2条； 区域⑤：即过原点，无切线，无与渐近线平行的直线.小结：过定点作直线与双曲线有且仅有一个交点，可以作出的直线数目可能有0、2、3、4条.（2）若直线与双曲线一支有交点，交点为二个时，求确定直线的斜率可用代入”“∆法与渐近线求交和两根之和与两根之积同号. ⑺若P 在双曲线12222=-by ax ，则常用结论1：P 到焦点的距离为m = n ，则P 到两准线的距离比为m ︰n.简证：ePF e PF d d 2121= =nm. 常用结论2：从双曲线一个焦点到另一条渐近线的距离等于b. 三、抛物线方程.3. 设0φp ，抛物线的标准方程、类型及其几何性质：注：①x c by ay =++2顶点)244(2aba b ac --.②)0(22≠=p px y 则焦点半径2P x PF +=;)0(22≠=p py x 则焦点半径为2P y PF +=.③通径为2p ，这是过焦点的所有弦中最短的.④px y 22=（或py x 22=）的参数方程为⎩⎨⎧==pt y pt x 222（或⎩⎨⎧==222pt y ptx ）（t 为参数）. 四、圆锥曲线的统一定义..4. 圆锥曲线的统一定义：平面内到定点F 和定直线l 的距离之比为常数e 的点的轨迹. 当10ππe 时，轨迹为椭圆； 当1=e 时，轨迹为抛物线； 当1φe 时，轨迹为双曲线； 当0=e 时，轨迹为圆（ace =，当b a c ==,0时）.5. 圆锥曲线方程具有对称性. 例如：椭圆的标准方程对原点的一条直线与双曲线的交点是关于原点对称的. 因为具有对称性，所以欲证AB=CD, 即证AD 与BC 的中点重合即可.立体几何 复习范围：第九章 编写时间：2004-7修订时间：总计第三次 2005-4I. 基础知识要点 一、 平面.1. 经过不在同一条直线上的三点确定一个面.注：两两相交且不过同一点的四条直线必在同一平面内.2. 两个平面可将平面分成3或4部分.（①两个平面平行，②两个平面相交）3. 过三条互相平行的直线可以确定1或3个平面.（①三条直线在一个平面内平行，②三条直线不在一个平面内平行） [注]：三条直线可以确定三个平面，三条直线的公共点有0或1个.4. 三个平面最多可把空间分成 8 部分.（X 、Y 、Z 三个方向） 二、 空间直线.1. 空间直线位置分三种：相交、平行、异面. 相交直线—共面有反且有一个公共点；平行直线—共面没有公共点；异面直线—不同在任一平面内[注]：①两条异面直线在同一平面内射影一定是相交的两条直线.（×）（可能两条直线平行，也可能是点和直线等） ②直线在平面外，指的位置关系：平行或相交③若直线a 、b 异面，a 平行于平面α，b 与α的关系是相交、平行、在平面α内. ④两条平行线在同一平面内的射影图形是一条直线或两条平行线或两点.⑤在平面内射影是直线的图形一定是直线.（×）（射影不一定只有直线，也可以是其他图形）⑥在同一平面内的射影长相等，则斜线长相等.（×）（并非是从平面外一点．．向这个平面所引的垂线段和斜线段） ⑦b a ,是夹在两平行平面间的线段，若b a =，则b a ,的位置关系为相交或平行或异面.2. 异面直线判定定理：过平面外一点与平面内一点的直线和平面内不经过该点的直线是异面直线.（不在任何一个平面内的两条直线）3. 平行公理：平行于同一条直线的两条直线互相平行.4. 等角定理：如果一个角的两边和另一个角的两边分别平行并且方向相同，那么这两个角相等（如下图）.（二面角的取值范围[)οο180,0∈θ）（直线与直线所成角(]οο90,0∈θ） （斜线与平面成角()οο90,0∈θ）（直线与平面所成角[]οο90,0∈θ）（向量与向量所成角])180,0[οο∈θ推论：如果两条相交直线和另两条相交直线分别平行，那么这两组直线所成锐角（或直角）相等. 5. 两异面直线的距离：公垂线的长度.空间两条直线垂直的情况：相交（共面）垂直和异面垂直.21,l l 是异面直线，则过21,l l 外一点P ，过点P 且与21,l l 都平行平面有一个或没有，但与21,l l 距离相等的点在同一平面内. （1L 或2L 在这个做出的平面内不能叫1L 与2L 平行的平面） 三、 直线与平面平行、直线与平面垂直.1. 空间直线与平面位置分三种：相交、平行、在平面内.2. 直线与平面平行判定定理：如果平面外一条直线和这个平面内一条直线平行，那么这条直线和这个平面平行.（“线线平行，线面平行”） [注]：①直线a 与平面α内一条直线平行，则a ∥α. （×）（平面外一条直线） ②直线a 与平面α内一条直线相交，则a 与平面α相交. （×）（平面外一条直线）③若直线a 与平面α平行，则α内必存在无数条直线与a 平行. （√）（不是任意一条直线，可利用平行的传递性证之） ④两条平行线中一条平行于一个平面，那么另一条也平行于这个平面. （×）（可能在此平面内） ⑤平行于同一直线的两个平面平行.（×）（两个平面可能相交） ⑥平行于同一个平面的两直线平行.（×）（两直线可能相交或者异面） 12方向相同12方向不相同。
二年级数学学霸笔记和心得
二年级数学学霸笔记和心得Mathematics is an essential subject in elementary education, and as a second-grade student, I have worked hard to excel in this field. Throughout the year, I have taken meticulous notes and developed effective strategies to become a math whiz. In this article, I will share my study notes and personal insights, hoping to inspire other students to achieve academic success in mathematics.I. Numbers and OperationsMastering numbers and operations is fundamental in mathematics. Here are my key takeaways:1. Place Value: Each digit in a number has a specific value based on its position. Understanding place value is crucial for solving addition and subtraction problems accurately.2. Addition: When adding two numbers, start from the rightmost digit and carry over any extra values when necessary. Practice mental math by adding single-digit numbers in your head.3. Subtraction: Subtraction is the opposite of addition. Start from the leftmost digit and borrow when needed. Take your time to ensure accuracy, and double-check your work.4. Multiplication: Memorize the multiplication table up to 10x10. Apply this knowledge to solve larger multiplication problems. Learn different strategies like grouping, repeated addition, or using arrays.5. Division: Understand the concept of sharing equally. Memorize division facts and practice solving division problems using manipulatives or mental math.II. Geometric Shapes and MeasurementGeometry and measurement introduce students to the world of shapes and sizes. Here's what I've learned:1. 2D Shapes: Recognize and classify shapes such as squares, circles, triangles, rectangles, and hexagons. Understand their properties, including the number of sides and angles.2. 3D Shapes: Identify three-dimensional objects like cubes, cylinders, spheres, cones, and pyramids. Learn how to count faces, vertices, and edges.3. Length, Weight, and Capacity: Develop a sense of measurement by comparing the length of objects using non-standard units (e.g., paper clips, cubes). Understand weight using balance scales and capacity using containers.4. Time: Learn how to read an analog clock to the hour and half-hour. Practice telling time in different scenarios, including daily routines and scheduling activities.III. Patterns and AlgebraPatterns and algebraic thinking enhance logical reasoning and problem-solving abilities. Here are some essential concepts:1. Patterns: Recognize and create patterns using shapes, colors, numbers, and objects. Identify the core elements of a pattern, such as the rule or sequence.2. Number Lines: Understand how to use number lines for counting, addition, and subtraction. Learn the concept of forward and backward jumps on the number line.3. Simple Equations: Solve basic equations by finding the missing value. Practice using balance scales or manipulatives to model equations visually.IV. Data Analysis and ProbabilityData analysis helps students comprehend and interpret information. Here's what I've learned:1. Graphs: Read and interpret different types of graphs, including bar graphs, pictographs, and line plots. Understand the components of a graph, such as the title, labels, and scales.2. Probability: Explore the concept of chance and likelihood. Identify and describe events as certain, likely, unlikely, or impossible.Now that I have shared my study notes, here are some crucial insights I have gained throughout my journey as a math whiz:1. Practice Regularly: Dedicate a specific time each day to practice math problems. Consistent practice enhances your speed and accuracy.2. Seek Help: Don't hesitate to ask your teacher or classmates for help when you encounter difficulties. Collaboration and support from others can significantly improve your understanding.3. Make it Fun: Incorporate games, puzzles, and online resources into your math practice. Engaging activities make learning enjoyable and help reinforce important concepts.4. Celebrate Achievements: Whenever you solve a challenging problem or make progress in a specific area, celebrate your accomplishments. Positive reinforcement boosts confidence and motivates further success.Remember, becoming a math whiz takes time and effort. Embrace challenges, stay determined, and never lose sight of your goal. With consistent practice, you can become a second-grade math champion and develop essential skills for future academic endeavors.。
4年级数学读书笔记摘抄
4年级数学读书笔记摘抄在四年级的数学学习中，我们接触到了许多有趣的概念和解决问题的方法。
以下是我在阅读数学书籍时做的一些笔记摘抄，它们帮助我更好地理解和掌握了四年级数学的知识点。
1. 四则运算四则运算包括加法、减法、乘法和除法。
加法和减法是基础运算，乘法是加法的快捷方式，而除法则是乘法的逆运算。
在进行四则运算时，我们需要遵循运算顺序，即先乘除后加减，同级运算从左到右进行。
2. 分数分数是表示一个整体被等分后的部分。
分数由分子和分母组成，分子表示所取的部分数量，分母表示整体被分成的份数。
例如，1/2 表示一个整体被分成两份，我们取其中一份。
分数的加减法需要找到共同的分母，然后分子相加减。
3. 小数小数是一种表示数值的方法，它允许我们表示不是整数的数。
小数点左边的数字表示整数部分，右边的数字表示小数部分。
小数的加减法需要对齐小数点，然后逐位相加减。
小数的乘除法则需要按照乘除法的规则进行计算。
4. 几何图形几何图形是数学中研究形状和大小的分支。
在四年级，我们学习了基本的二维图形，如正方形、长方形、三角形和圆形。
我们学习了如何计算这些图形的周长和面积，以及它们的性质和特点。
5. 数据处理数据处理是数学中的一个重要部分，它涉及收集、整理和分析数据。
我们学习了如何制作条形图、折线图和饼图来表示数据。
这些图表帮助我们更直观地理解数据和进行比较。
6. 解决问题解决问题是数学的核心。
我们学习了如何使用数学知识来解决实际问题，比如购物时的计算、时间管理、距离和速度的计算等。
通过练习，我们提高了逻辑思维和问题解决的能力。
通过这些笔记摘抄，我不仅复习了四年级数学的知识点，还加深了对数学概念的理解。
数学是一门基础学科，它在我们的日常生活中有着广泛的应用。
通过不断的学习和实践，我们可以更好地掌握数学，为未来的学习和生活打下坚实的基础。
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速讀筆記
這些速讀筆記只是希望能簡單地把我認為同學應該要識的內容、重點、技巧等概括列出。
至於詳細的內容、公式等同學可以參考其他數學書。
百分法、率及比、變數法
百分法
∙意義：某數的一部份 1 = 100%
∙增加/減少的概念：新值＝舊值（１±改變百分率）
∙買賣應用：如售價＝成本（１＋盈利率）；售價＝標價（１－折扣率）∙利息：單利息及複利息；留意複利息中的“期利率”與“年利率”的分別率及比
∙率的概念：“每小時行3公里＝3km/h”
∙比的概念：
o利用“a:b”，“b:c”計“a:b:c”
o分餅概念，e.g.$2000，按“2:3:5”分給三人
變數法
∙正變（x=ky）、反變（x=k / y）、聯變（x=kyz）、部份變（x=ky+Ky^2）
估算
∙須知道常用誤差（e.g.絕對誤差、相對誤差）的定義和公式。
多項式（包括公式）
公式的主項變換
∙二個多項式的加減乘除
∙因式分解的技巧（例如抽共同因式、運用恆等式）
∙分數形式之多項式的加減
∙餘式定理
∙因式定理（即餘式＝0）及其運用（因式分解一元三次多項式）
指數定律
指數定律公式
∙根式、含根式的分數之有理化
∙進制（進制轉換、位值）
對數
方程
解方程技巧，包括
∙一元一次方程
∙一元二次方程
∙二元一次聯立方程
∙一元一次不等式
∙一元二次方程
∙根的特性（判別式的值）
∙求最大／最小值的方法（配方法或利用對稱軸x=-b/2a）
∙可變為一元二次方程的方程，e.g. y^4+5y^2 – 4 = 0
坐標
若有兩點A、B 的直角坐標。
同學須知道：
∙A的極坐標
∙AB的長度、斜率、中點坐標
∙若P點把線段AB分成兩份，且AP：PB = m：n，求P點坐標
∙利用直線的斜率及通過的一點來求其方程
∙如兩條直線是平行，則它們的斜率( m1及m2 )相等
∙如兩條直線是垂直，則m1 x m2 = -1
∙把直線Ax + By + C = 0變成y = mx ＋c，從而看出s斜率及ｙ-軸截點
∙圓形方程：
∙標準式及一般式
∙從已知方程求圓的圓心坐標及半徑
函數的圖像
睇“y = mx + c”的圖
∙睇“y = ax2 + bx + c” 的圖
∙應用：劃y = ax^2 + bx + c的草圖來求一元二次不等式(如xv2 +5x + 4＞0)的解的區域∙睇“二元一次不等式”的解的區域
∙函數轉換對圖像的影響（e.g.平移）
三角比
∙
sin, cos, tan的基本定義（即於直角三角形內）
∙特別角的三角比值（e.g. sin60°）
∙sin (180-x)=sinx 等公式
∙三角比恆等式(tan x = sin x / cos x等)
∙解三角方程，e.g. 2sinx = 1（留意角的取值範圍）
∙正弦公式、餘弦公式之應用
∙希羅公式（可用餘弦公式及1/2 absinC來求三角形面積）∙三角比應用（方位角，立體圖等）
演繹推理幾何
∙與直線及平行線有關的定理e.g. 直線上的鄰角、內錯角∙與三角形有關的定理，全等及相似的條件及應用
∙畢氏定理及其逆定理
∙與多邊形有關的定理（四邊形可略讀）
∙與圓形有關的定理
平面及立體圖形
求面積及體積的公式、技巧
∙留意會考試卷中首頁會提供幾條公式
∙扇形的面積及弧長
∙相似形狀的“邊長比”、“面積比”及“體積比”的關係
等差、等比數列
等差數列
∙通項Tn = a + (n – 1)d
∙n項和= (2a+(n-1)d) * n / 2
或
n項和= (頭項+尾項)*項數/ 2
∙等比數列
∙通項Tn = aR^(n – 1)
∙n項和= a (1 - R^n) / (1 - R)
∙如-1 < R < 1，可求得"無限項之和"=a / (1-R)
統計
閱讀各統計圖的技巧
∙集中趨勢（平均值、加權平均數、中位數、眾數）
∙離差之量度（標準差）
∙箱形圖
概率
概率的基本定義及性質
∙獨立、互補事件
∙常用的公式，如：
∙P(A事件發生) = 1 – P(A事件不發生)
∙P(A或B)＝P(A) + P(B) – P(A及B)
∙如A、B為互斥事件，
o P(A及B)＝0
o P(A或B)＝P(A) + P(B)
∙如A、B為獨立事件，P(A及B)＝P(A) x P(B)
∙考慮概率時常用的技巧，如列表法、數數目及考慮事件的發生經過等∙條件概率
∙概念：P(B|A) = 已知事件A發生了，事件B發生的概率
∙公式：P(B|A) = P(A及B) / P(A)
∙期望值。