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1、旋转在数学解题中发挥的重要作用（The important role of rotation in solving mathematical problems）Data worth havingFrom the usual study, accumulation and summaryWhere there is a problem, there must be somePlease also criticize and correct me!The important role of rotation in solving mathematical problemsDawu County h
2、igh school Chen Jie Chen Rui shopAbstract: rotation is a kind of transformation of plane geometryIt is to circle a certain figure in the plane (the center of rotation)Rotate an angle (rotation angle) in a certain directionConstruct a new graphUsing the shape of a rotated figureThe trait of not chang
3、ing in sizeSo as to find a way to solve problemsKey words arouse interest, optimize integration, master method, difficult to easyFirst, create scenariosStimulate students interest in rotationMan lives on the earthThe earth always moves round the sun and rotates from west to EastMake spring, summer,
4、autumn and winter clear all year roundThe trains, planes, and ships that we travel are all tyres of wheelsThe rails and propellers kept turning us round to our destination; the clock hung on the blackboard in the classroomThe second hour and minute non-stop rotationBring us to a better tomorrowWaitL
5、ife is full of spinSpinning gives us courage to liveRotation adds wisdom to our problem solvingRotation gives us a great deal of power to solve difficult problemsDEG1 cases of delta ABC Delta DEF to rotate around the fixed point O positionFigure 1(1) what is the center of rotation? What is the angle
6、 of rotation? How many rotation angles do you have? What kind of relationship do they have?(2) where is the position of the rotating point A and the rotating point B respectively?(3) what is the relationship between the line segment OA and the length of OD? What about OB and OF, OC and OE?(4) the AB
7、C around the point O and rotated by 90 DEGIf you get a GHKConnect AG, BH, CKHow many isosceles right triangles are there in the picture?(5) the ABC O after the 180 degree rotation around a fixed pointIf you get a GHKThe position relationship between the ABC value and GHK is.S ABCS Delta Delta Delta
8、GHKcase 2 the following four figures are equilateral trianglesSquareFive-pointed starRegular hexagonAfter pointing out how many degrees they rotate at least, they can coincide with themselvesA B C DFigure 2Thus we obtain the properties of the graph before and after the rotation transformation:(1) th
9、e corresponding line segments are equalEqual angle(2) the corresponding positions are arranged in the same order(3) the angle of the straight line of any two corresponding line segments is equal to the rotation angle(4) the rotation center O is the fixed point under the rotation change(5) the rotati
10、on of a graph does not change the shape or size of a figureChange only the position of the figureThrough figure 1 and Figure 2, let us further understand the line segment, angle, area, graphics, the rotation of some of the links and lawsRotational transformation is widely used in plane geometryEspec
11、ially in solving (proof) about isosceles right triangle, equilateral triangle, squareExplore the angle, line, area and other issuesIs often used in the thinking methodsSo as to stimulate students interest in rotationTwo, optimize integrationTrain students consciousness of revolving innovationIn the
12、process of solving problemsWe will meet with all kinds of difficult problemsThe terms of the subject are more scatteredThe area of a line segment, an angle, a graph is not in the same figureRequired solutionJust when we feel helplessIncrease the awareness of rotation changes immediatelyThe idea of r
13、otational variation plays an irreplaceable role in solving geometric problemsThis mathematical thinking reflects the diversity of thinkingIt is also a rule that we can learn geometryexample 3 as shown in Figure 3P is a ABC in the equilateral DeltaPB=2PC=1Angle BPC=150 degreesFind the length of the P
14、AFigure 3(thought display) in graphics, the three lines of PA, PB, and PC are P around the point, and the outward is shaped like a projectionThe condition is very dispersed, and it is difficult to find the value of PA directlyHow do you focus PA, PB, and PC in a triangle?This is the key to solving t
15、he problemBecause ABC is an equilateral triangleSo will ACP around C counterclockwise 60 degrees to BCD positionThen PA=BD, connect PDThe PA, PB, PC on a delta BPDThe problem can be solvedSolution: Delta ACP around C counterclockwise 60 degrees to BPD positionConnect PD,Then CP=CDPA=DB / BCD= / ACPA
16、nd dreams / ACP+ / BCP=60 degreesL / BCP+ / BCD= / PCD=60.Star delta PCD is an equilateral triangleL PC=PD=1, angle CPD=60 degreesAnd dreams / BPD= / BPC- / CPD=150 -60 =90 Star delta BPD is a right trianglePB=2 and dreamsPC=1Please tick tick 22+12= BP2+PD2= * BD= 5R PA= 5 VExplore 1: as shown in Fi
17、gure 4In square ABCDPA=1PB=2PC=3The dot P is inside the square ABCDFind the angle APB degreesFigure 4Three, master the methodImprove students ability of revolving problem solvingWe know the nature of the rotation changesIn TeachingUsually pay more attention to train students to use revolving transfo
18、rmation thinking to solve problemsHow to rotate a graph and how many degrees it can be reached; how to master the method of rotationSome of the difficulties and stresses in geometry can be reducedHelp students gradually add auxiliary linesThese are rules to followThus, the speed and quality of solvi
19、ng problems are greatly improvedexample 4 as shown in Figure 5In a ABC=90 / ACB degreesAC=BCM and N are two points on ABAnd meet AM2+BN2=MN2For angle MCN degreesFigure 5(thought demonstration) because of the conditions given by the topic, AM, BN and MN are in the same line as the three linesAnd meet
20、 AM2+BN2=MN2Well, lets seeIn a triangle, the sum of squares of two sides equals the square of the third sidesOnly in right angled trianglesUsing the Pythagorean theorem can be obtainedBecause we have to move the line segments AM, BN, and MN to a right triangleTo explore the relationship between them
21、Because the subject conditions in =90 / ACB degreesAC=BCABC is an isosceles right triangleSo the BCN around the point C clockwise 90 degrees to ACD positionConnect DMUsing the inverse theorem of Pythagorean theoremCongruent knowledge of trianglesYou can find the angle MCN degreesSolution: Delta BCN
22、around C clockwise 90 degrees to ACD positionConnect DM* AD=BNCD=CNDreams angle ACB=90 degreesAC=ABStar delta ABC is an isosceles right triangleL / B= / DAC= / CAB=45.L / DAB=90 degreesStar delta ADM is a right triangle* AM2+ AD2=DM2 (1)AM2+BN2=MN2 and dreams* AM2+ AD2=MN2 (2)By (1) (2) getDM2=MN2*
23、DM=MNIn the delta CDM and delta MCNCD=CNDM=MN dreamsCM=CNStar delta CDM = MCNL / DCM= / MCN=90 / DCM+ / MCN degrees of imprisonmentL / MCN=45 degreesFour, bold conjectureIt is difficult to make use of rotationMathematical conjecture is one of the most active, active and active factors in the develop
24、ment of MathematicsAccording to the conditions in the subject, graphic featuresMake bold explorations and conjecturesMathematical conjectures have a certain regularityfor exampleWe use rotationDare to guess a maths problemcase 5 (2011 national junior high school mathematics contest questions) as sho
25、wn in Figure 6The square ABCD has a length of 1Point P and Q are two internal pointsAnd the angle PAQ= angle PCQ=45 degreesS Delta ABP+S Delta PCQ+S Delta QAD value(thinking display) according to subject to the conditions, side length of the square of ABCD is 1, and the angle PAQ= angle PCQ=45 degre
26、es. No congruent triangles in Fig.Except for square ABCDNor is there a parallelogramAnd point P, Q and the square each vertex line, divide square into six trianglesNow we must find S Delta ABP+S Delta PCQ+S Delta QAD valueWe are based on the fact that the three triangles are exactly interleaved in t
27、he square ABCDMake a bold guessS Delta ABP+S Delta PCQ+S Delta QAD=1/2S square ABCD=1/2According to this conjectureLets find a way to solve the problemBecause we do not know the height of the base and the bottom edge of any triangleMany lines in a graphThere are many trianglesComplex conditionThe di
28、fficulty is very greatIt is impossible not to solve by adding auxiliary linesAnd because quadrilateral ABCD is squareAngle PAQ= angle PCQ=45 degrees, we might put a triangle diagram or two triangles in a certain direction to rotate 90 degreesThe triangle is optimized and integratedTransform this tri
29、angle into a polygon AEFCQ equal to the square areaAnd use what we have learnedCan be solvedFigure 6Solution: Delta ADQ Delta ABE to A around the point position in a clockwise rotation of 90 DEGThe delta CDQ Delta BCF around the point C to the location by the counterclockwise rotation of 90 degreesC
30、onnect EQ, FQ,AE=AQ / FBC= / CDQ * CF=CQ / ABE= / ADQABCD is a quadrilateral of imprisonmentL / FBC+ / ABE= / CDQ+ / ADQ=90.Dreams angle ABC=90 degreesL / FBC+ / ABE+ / ABC=180.B, E, F * three collinear* BE=DQ=BFConnect EP and FP againR S Delta PBF=S Delta PBEDreams / PAQ= / PCQ=45 / 1= / 3 degrees.
31、6 / / 4=L / 2+ / 3= / 5+ / 6=45.For CFP = CPQ AEP = Delta Delta Delta APQS AEP= S APQ * S CFP= S CPQS square ABCD=S Pentagon AEFCQDreams of S Delta ABP+S Delta PCQ+S Delta QAD=1/2 * 2 (S Delta AEP+S Delta EBP+ Delta S CFP)=1/2 x 2 x 1/2=1/2R S Delta ABP+S Delta PCQ+S Delta QAD=1/2Guess and answer th
32、e questions aboveWe verify the correctness of the conjectureDare to guessIt is helpful to stimulate students interest in learning and enhance their learning motivationConducive to a more thorough understanding and mastery of mathematical knowledgeIt helps to find solutions to problems more quicklyBo
33、ld guessing is an important way to create mathematical thinkingBy guessWe have always revolved around the feature of rotation transformationChange the position of a graphOptimize graphic structureFurther integration of graphicsGradually develop students ability to analyze and solve problemsSmooth ou
34、t the more complicated problemsExplore 2: as shown in Figure 7P is a point within the equilateral triangle ABCPA=2PB=2 V 3. (1), and the length of ABC; (2), the area of delta ABCFigure 7Through the above examples of detailed explanation and explorationLet us further understand that the rotation of a
35、 graph is determined by the rotation center, rotation angle and rotation directionExplore the rotation before and after the two graphics corresponding points to the rotating center of the same distanceThe angles formed by the connection of the corresponding point and the center of rotation are equal
36、 to each otherThe figure before and after rotation is congruentThusWe derive general rules for solving problems of rotation change:OneWhen the graph of the subject is equilateral triangle, isosceles right triangle, squareThe conditions given are very scatteredWe need to calculate the edges, angles,
37、and areashereAnd its hard to solveRotation transformation should be considered to solve the problem2. angle of rotationThe title figures are equilateral triangles, isosceles right triangles and squaresGenerally rotate 60 degrees, 45 degrees, 90 degrees3. after rotationFurther integration of graphica
38、l optimizationUse what you have learned to solve itIn the future teaching workWe should focus on developing students ability to observe, think, guess, explore and innovateTrain students ability of thinking and transferring knowledgeImprove students ability to analyze and solve problemsAnd let studen
39、ts realize the importance of participation in the actual operationGive the students more room to succeedSo that students can really understand the vividness, flexibility, practicability and creativity of MathematicsStimulate students enthusiasm for learning mathematicsEstablish self-confidence to learn math well?One