Thank you for participating in Teach It First!


 Kristina Stanley
 1 years ago
 Views:
Transcription
1 Thnk you for prtiipting in Teh It First! This Teh It First Kit ontins Common Core Coh, Mthemtis teher lesson followed y the orresponding student lesson. We re onfident tht using this lesson will help you hieve your ssessment preprtion gols for your entire lss. The Common Core Coh, Mthemtis progrm is sed on the philosophy tht mthemtil skills re uilt on onepts. Mth, mye more thn ny other shool sujet, uilds from onept to onept, one on top of the other, over severl yers. When students understnd onepts nd how they onnet to skills, they re etter equipped to solve prolems tht they enounter in the rel world. This progrm is 100% ligned to the Common Core Stte Stndrds nd provides set of lessons for eh of the five CCSS domins, with eh lesson ligning to one or more stndrd together, lessons over ll the domin s stndrds. Conept Lessons egin with n underlying onept tht onnets diretly to the skill or skills tught in tht lesson. Students will use fourstep prolemsolving proess Red, Pln, Solve, Chek to pproh ny mthemtil prolem. Intertive questions follow emples nd sk students to disuss topi, model sitution, try to solve prolem on their own, or hek their work. With this instrutionl nhor, you n implement the Common Core Stte Stndrds with onfidene. We re hppy to provide you this omplimentry smple nd would love to know wht you think. One you hve red through this lesson, do wht you do est present it to your students. Then, don t forget to omplete quik survey y going to Regrds, Triumph Lerning Join the onverstion out Common Core tody y visiting ommonore.om, the ple where tehers, prents, nd eperts ome together to shre est prties nd prtil informtion for suessfully implementing Common Core stndrds in the lssroom. 136 Mdison Avenue New York, NY p: f :
2 25 Eplining the Pythgoren Theorem LESSON Lerning Ojetives Students will understnd proof of the Pythgoren theorem nd its onverse. Students will use the Pythgoren theorem to find the length of missing side of right tringle nd the onverse to hek if tringle is right tringle. Common Core Stte Stndrd 8.G.6 Eplin proof of the Pythgoren Theorem nd its onverse. Voulry hypotenuse in right tringle, the side opposite the right ngle leg in right tringle, one of the two shorter sides; it is opposite one of the two ute ngles Pythgoren theorem sttes tht the sum of the squres of the lengths of the legs in right tringle is equl to the squre of the length of the hypotenuse Before the Lesson Review the definition of right tringle nd the speil voulry ssoited with right tringles. Remind students tht the longest side of tringle is opposite the ngle with the gretest mesure. Point out tht the ngle with the gretest mesure in right tringle is the right ngle. Disuss why. Emphsize tht the side opposite the right ngle in right tringle is lled the hypotenuse. The two sides tht re djent to the right ngle re the legs. Understnd Connet To help develop oneptul understnding, disuss the proof in detil. Emphsize tht euse the figures re ongruent, their res re ongruent. The proof relies on this ft. The ongruent res re lgerilly simplified to produe the Pythgoren theorem. Chek tht students understnd the formul for the re of eh figure. Work through the steps tht re used to simplify eh form of the re. Cll ttention to the epressions for eh re. Point out tht students n lso find the res of the 4 qudrilterls of Figure 1 to determine its re: the smll squre hs n re of 2, the two retngles hve res of, nd the lrge squre hs n re of 2. The re of Figure 1 is the sum of the res of the qudrilterls, Hve students eplin in their own words why the epressions re equted to find the Pythgoren theorem nd how it simplifies to the theorem. Point out tht the onverse of the Pythgoren theorem strts with the speil reltionship etween the sum of the squres of tringle nd sys tht if this reltionship is met, then the tringle is right tringle. To onnet the onept to proedurl understnding, sk, Why n the Pythgoren theorem e used to find the missing side length? (The tringle is right tringle.) Emphsize tht the Pythgoren theorem is only for right tringles. Disuss the importne of distinguishing etween the legs nd the hypotenuse of right tringle to pply the theorem. Ask: Whih side is lwys the Dupliting ny prt of this ook is prohiited y lw. 72
3 hypotenuse in right tringle? (The side opposite the right ngle.) How does the length of the hypotenuse ompre to the length of the legs? (It is lwys the longest side.) Disuss how to use this ft to hek tht the Pythgoren theorem hs een orretly nd urtely used. Suggest tht students lwys write the Pythgoren theorem s the first step when using it to find missing length. Then sustitute nd solve. Review how to use the order of opertions to solve for. DISCUSS MP2 MP4 Disuss how to reognize tht given tringle is right tringle. No, the Pythgoren theorem nnot e used. Eplntions my vry. Possile eplntion: The Pythgoren theorem only reltes the lengths of the sides in right tringle. Tringle MNP is n otuse tringle, so the Pythgoren theorem nnot e used. Emples EXAMPLE A This emple introdues students to rellife prolem tht is solved y using the Pythgoren theorem. Disuss how the digrm illustrtes the informtion in the prolem. DISCUSS MP1 MP5 Disuss how to evlute the squre root of greter numer. Answers my vry. Possile nswer. One wy is to use guess nd hek. For emple, I know tht , so I would guess numer suh s , whih is still too low. I would guess higher numer, suh s , whih is too high. However, now I know tht 576 is etween 20 nd 25. If I guessed 24 net, I would find out tht , so Prtie As students re working, py speil ttention to prolems 4 9, whih provide n opportunity for students to use the onverse of the Pythgoren theorem to determine if the given sides form right tringle. For nswers, see pges 107 nd 108. Common Errors Students my inorretly pply the Pythgoren theorem y not following the order of opertions in their omputtions. Review the order of opertions nd how they pply in prtiulr to prolems tht use the Pythgoren theorem to find the length of missing side of right tringle. Domin 4 Dupliting ny prt of this ook is prohiited y lw. EXAMPLE B In this emple students use the onverse of the Pythgoren theorem to determine if tringle is right tringle. Review how to hoose the lengths of the sides to test the theorem. TRY MP3 MP6 Point out tht Ros only hnges the length of the longest penil. The other two penils sty the sme length. Disuss why this mens the Pythgoren theorem n e used to find the desired length of the longest penil. 17 entimeters
4 25 LESSON Eplining the Pythgoren Theorem UNDERSTAND The Pythgoren theorem sttes tht, in ny right tringle, the sum of the squres of the lengths of the legs is equl to the squre of the length of the hypotenuse. The two ongruent squres shown elow were uilt using ongruent right tringles nd squres with lengths,, nd. Use Figures 1 nd 2 to prove the Pythgoren theorem. leg hypotenuse leg Figure 1 Figure 2 Let ( 1 ) 2, or ( 1 )( 1 ), represent the re of Figure 1. Simplify. A of Figure 15 ( 1 )( 1 ) Use the distriutive property to simplify. 5 ()( 1 ) 1 ( 1 ) Comine like terms Figure 2 ws uilt using 4 ongruent right tringles with se nd height, nd squre with sides, so: A of Figure 2 5 [4 3 (A of eh tringle)] 1 (A of squre) Sustitute Multiply Sine the figures re ongruent, set the epressions for their res equl. Simplify. (A of Figure 1) 5 (A of Figure 2) Sustitute Sutrt 2 from oth sides The onverse of the Pythgoren theorem sttes tht if tringle hs sides of length,, nd suh tht , then the tringle is right tringle with right ngle opposite. Dupliting ny prt of this ook is prohiited y lw. 136 Domin 4: Geometry
5 Connet Determine the length of KL in njkl. K 6 J 8 L 1 Wht kind of tringle is njkl? The rightngle symol shows tht /J is right ngle. n JKL is right tringle. 2 Identify the lengths of the legs nd the hypotenuse. 3 Use the formul for the Pythgoren theorem. The legs hve lengths of 6 units nd 8 units. The hypotenuse, KL, is units long. Dupliting ny prt of this ook is prohiited y lw. Sustitute 6 for nd 8 for. Solve for So, equls 10. The length of KL is 10 units. DISCUSS Could you use the Pythgoren theorem to find the missing length of NP in this tringle? Eplin. N 6 M 8 P Lesson 25: Eplining the Pythgoren Theorem 137
6 EXAMPLE A A guy wire tht is 26 feet long is tthed to the top of pole. The wire is tthed to the ground t point tht is 10 feet from the se of the pole. Determine h, the height of the pole. 26 ft h 10 ft 1 How will you solve this prolem? The rightngle symol shows tht the wire, the pole, nd the distne from the se of the pole to the ple where the wire is tthed form right tringle. Use the formul for the Pythgoren theorem. 2 Identify the legs nd the hypotenuse. 3 Sustitute those lengths into the formul nd solve for h. The legs hve lengths of 10 feet nd h feet. The hypotenuse is 26 feet long h Sustitute h Sutrt 100 from oth sides. h h h 5 24 The height of the pole is 24 feet. DISCUSS If you hve lultor, finding the integer vlue of 576 is simple. How ould you evlute 576 if you do not hve lultor? Dupliting ny prt of this ook is prohiited y lw. 138 Domin 4: Geometry
7 EXAMPLE B Ros hs three penils, eh different length. The lengths re 18 entimeters, 15 entimeters, nd 8 entimeters. Could she form right tringle using these penils s the sides? 1 Wht does the onverse of the Pythgoren theorem stte? It sttes tht if tringle hs side lengths,, nd suh tht , then the tringle is right tringle. 2 Identify the shorter penil lengths nd the longest penil length. 3 Test the vlues in the formul entimeters, 15 entimeters, nd 18 entimeters re not the side lengths of right tringle. It is not possile for Ros to use those penils to mke right tringle. The shorter lengths re 8 entimeters nd 15 entimeters. Those would represent the shorter sides of the tringle. Let 5 8 nd The longest length is 18 entimeters. Let Dupliting ny prt of this ook is prohiited y lw. TrY Ros finds tht if she shrpens her longest penil nd shortens its length, she n use it to form right tringle with the other two penils. Wht length would she need to mke the longest penil? Lesson 25: Eplining the Pythgoren Theorem 139
8 Prtie Write n eqution tht shows the reltionship etween the given side lengths of these right tringles. Simplify if possile. 1. d 2. m p n 3. y REMEMBER The sum of the squres of the leg lengths equls the squre of the hypotenuse length. Use the onverse of the Pythgoren theorem to determine whether or not tringle with the given side lengths is right tringle. Show your work in., 4 in., 5 in yd, 7 yd, 11 yd 6. 5 m, 10 m, 15 m 7. 9 m, 39 m, 41 m mm, 99 mm, 101 mm 9. 4 m, 7.5 m, 8.5 m Choose the est nswer. 10. A right tringle hs legs mesuring 15 meters nd 20 meters. Wht is the length of the hypotenuse? A. 13 meters B. 17 meters C. 25 meters D. 35 meters 11. A right tringle hs leg mesuring 24 units nd hypotenuse mesuring 25 units. Wht is the length of the other leg? A. 7 units B. 9 units C. 35 units D. 49 units Dupliting ny prt of this ook is prohiited y lw. 140 Domin 4: Geometry
9 Find, the missing side length in eh right tringle. Show your work Prove. Use entimeter ruler nd protrtor, s needed. 18. PROVE Given: nfgh hs sides,, nd units long, nd it is true tht To prove tht the onverse of the Pythgoren theorem is true, show tht nfgh is right tringle. F Dupliting ny prt of this ook is prohiited y lw. H G To do this, mesure lengths nd, in entimeters. Then drw right tringle JKL, with legs nd units long, nd hypotenuse leled d. Mke /J the right ngle. Eplin how you know tht d 2. Then use sequene of rigid motions to prove tht nfgh is right tringle. How does this show tht nfgh is right tringle? Lesson 25: Eplining the Pythgoren Theorem 141
The remaining two sides of the right triangle are called the legs of the right triangle.
10 MODULE 6. RADICAL EXPRESSIONS 6 Pythgoren Theorem The Pythgoren Theorem An ngle tht mesures 90 degrees is lled right ngle. If one of the ngles of tringle is right ngle, then the tringle is lled right
More information4.5 The Converse of the
Pge 1 of. The onverse of the Pythgoren Theorem Gol Use the onverse of Pythgoren Theorem. Use side lengths to lssify tringles. Key Words onverse p. 13 grdener n use the onverse of the Pythgoren Theorem
More informationThe area of the larger square is: IF it s a right triangle, THEN + =
8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht
More information11. PYTHAGORAS THEOREM
11. PYTHAGORAS THEOREM 111 Along the Nile 2 112 Proofs of Pythgors theorem 3 113 Finding sides nd ngles 5 114 Semiirles 7 115 Surds 8 116 Chlking hndll ourt 9 117 Pythgors prolems 10 118 Designing
More informationProving the Pythagorean Theorem
Proving the Pythgoren Theorem Proposition 47 of Book I of Eulid s Elements is the most fmous of ll Eulid s propositions. Disovered long efore Eulid, the Pythgoren Theorem is known y every high shool geometry
More informationEssential Question What are the Law of Sines and the Law of Cosines?
9.7 TEXS ESSENTIL KNOWLEDGE ND SKILLS G.6.D Lw of Sines nd Lw of osines Essentil Question Wht re the Lw of Sines nd the Lw of osines? Disovering the Lw of Sines Work with prtner.. opy nd omplete the tle
More informationThree squares with sides 3, 4, and 5 units are used to form the right triangle shown. In a right triangle, the sides have special names.
1 The Pythgoren Theorem MAIN IDEA Find length using the Pythgoren Theorem. New Voulry leg hypotenuse Pythgoren Theorem Mth Online glenoe.om Extr Exmples Personl Tutor SelfChek Quiz Three squres with
More informationexcenters and excircles
21 onurrene IIi 2 lesson 21 exenters nd exirles In the first lesson on onurrene, we sw tht the isetors of the interior ngles of tringle onur t the inenter. If you did the exerise in the lst lesson deling
More information81. The Pythagorean Theorem and Its Converse. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary
81 The Pythgoren Theorem nd Its Converse Voulry Review 1. Write the squre nd the positive squre root of eh numer. Numer Squre Positive Squre Root 9 81 3 1 4 1 16 1 2 Voulry Builder leg (noun) leg Relted
More informationThe Pythagorean Theorem Tile Set
The Pythgoren Theorem Tile Set Guide & Ativities Creted y Drin Beigie Didx Edution 395 Min Street Rowley, MA 01969 www.didx.om DIDAX 201 #211503 1. Introdution The Pythgoren Theorem sttes tht in right
More informationState the size of angle x. Sometimes the fact that the angle sum of a triangle is 180 and other angle facts are needed. b y 127
ngles 2 CHTER 2.1 Tringles Drw tringle on pper nd lel its ngles, nd. Ter off its orners. Fit ngles, nd together. They mke stright line. This shows tht the ngles in this tringle dd up to 180 ut it is not
More informationTHE PYTHAGOREAN THEOREM
THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most wellknown nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this
More informationKnow the sum of angles at a point, on a straight line and in a triangle
2.1 ngle sums Know the sum of ngles t point, on stright line n in tringle Key wors ngle egree ngle sum n ngle is mesure of turn. ngles re usully mesure in egrees, or for short. ngles tht meet t point mke
More informationLesson 2.1 Inductive Reasoning
Lesson.1 Inutive Resoning Nme Perio Dte For Eerises 1 7, use inutive resoning to fin the net two terms in eh sequene. 1. 4, 8, 1, 16,,. 400, 00, 100, 0,,,. 1 8, 7, 1, 4,, 4.,,, 1, 1, 0,,. 60, 180, 10,
More informationActivity I: Proving the Pythagorean Theorem (Grade Levels: 69)
tivity I: Proving the Pythgoren Theorem (Grde Levels: 69) Stndrds: Stndrd 7: Resoning nd Proof Ojetives: The Pythgoren theorem n e proven using severl different si figures. This tivity introdues student
More informationFinal Exam covers: Homework 0 9, Activities 1 20 and GSP 1 6 with an emphasis on the material covered after the midterm exam.
MTH 494.594 / FINL EXM REVIEW Finl Exm overs: Homework 0 9, tivities 1 0 nd GSP 1 6 with n emphsis on the mteril overed fter the midterm exm. You my use oth sides of one 3 5 rd of notes on the exm onepts
More informationSimple Electric Circuits
Simple Eletri Ciruits Gol: To uild nd oserve the opertion of simple eletri iruits nd to lern mesurement methods for eletri urrent nd voltge using mmeters nd voltmeters. L Preprtion Eletri hrges move through
More informationAngles 2.1. Exercise 2.1... Find the size of the lettered angles. Give reasons for your answers. a) b) c) Example
2.1 Angles Reognise lternte n orresponing ngles Key wors prllel lternte orresponing vertilly opposite Rememer, prllel lines re stright lines whih never meet or ross. The rrows show tht the lines re prllel
More informationLesson 18.2: Right Triangle Trigonometry
Lesson 8.: Right Tringle Trigonometry lthough Trigonometry is used to solve mny prolems, historilly it ws first pplied to prolems tht involve right tringle. This n e extended to nonright tringles (hpter
More informationRight Triangle Trigonometry 8.7
304470_Bello_h08_se7_we 11/8/06 7:08 PM Pge R1 8.7 Right Tringle Trigonometry R1 8.7 Right Tringle Trigonometry T E G T I N G S T R T E D The origins of trigonometry, from the Greek trigonon (ngle) nd
More informationRight Triangle Trigonometry
CONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use
More informationProving the Pythagorean Theorem
CONCEPT DEVELOPMENT Mthemtis Assessment Projet CLASSROOM CHALLENGES A Formtive Assessment Lesson Proving the Pythgoren Theorem Mthemtis Assessment Resoure Servie University of Nottinghm & UC Berkeley For
More information8.2 Trigonometric Ratios
8.2 Trigonometri Rtios Ojetives: G.SRT.6: Understnd tht y similrity, side rtios in right tringles re properties of the ngles in the tringle, leding to definitions of trigonometri rtios for ute ngles. For
More information1. Definition, Basic concepts, Types 2. Addition and Subtraction of Matrices 3. Scalar Multiplication 4. Assignment and answer key 5.
. Definition, Bsi onepts, Types. Addition nd Sutrtion of Mtries. Slr Multiplition. Assignment nd nswer key. Mtrix Multiplition. Assignment nd nswer key. Determinnt x x (digonl, minors, properties) summry
More informationSECTION 72 Law of Cosines
516 7 Additionl Topis in Trigonometry h d sin s () tn h h d 50. Surveying. The lyout in the figure t right is used to determine n inessile height h when seline d in plne perpendiulr to h n e estlished
More informationD e c i m a l s DECIMALS.
D e i m l s DECIMALS www.mthletis.om.u Deimls DECIMALS A deiml numer is sed on ple vlue. 214.84 hs 2 hundreds, 1 ten, 4 units, 8 tenths nd 4 hundredths. Sometimes different 'levels' of ple vlue re needed
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationISTM206: Lecture 3 Class Notes
IST06: Leture 3 Clss otes ikhil Bo nd John Frik 9905 Simple ethod. Outline Liner Progrmming so fr Stndrd Form Equlity Constrints Solutions, Etreme Points, nd Bses The Representtion Theorem Proof of the
More informationSine and Cosine Ratios. For each triangle, find (a) the length of the leg opposite lb and (b) the length of the leg adjacent to lb.
 Wht You ll ern o use sine nd osine to determine side lengths in tringles... nd Wh o use the sine rtio to estimte stronomil distnes indiretl, s in Emple Sine nd osine tios hek Skills You ll Need for Help
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationThe Math Learning Center PO Box 12929, Salem, Oregon 97309 0929 Math Learning Center
Resource Overview Quntile Mesure: Skill or Concept: 1010Q Determine perimeter using concrete models, nonstndrd units, nd stndrd units. (QT M 146) Use models to develop formuls for finding res of tringles,
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationRatio and Proportion
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
More informationHow to Graphically Interpret the Complex Roots of a Quadratic Equation
Universit of Nersk  Linoln DigitlCommons@Universit of Nersk  Linoln MAT Em Epositor Ppers Mth in the Middle Institute Prtnership 7007 How to Grphill Interpret the Comple Roots of Qudrti Eqution Crmen
More informationa 2 + b 2 = c 2. There are many proofs of this theorem. An elegant one only requires that we know that the area of a square of side L is L 2
Pythgors Pythgors A right tringle, suh s shown in the figure elow, hs one 90 ngle. The long side of length is the hypotenuse. The short leg (or thetus) hs length, nd the long leg hs length. The theorem
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More information8. Hyperbolic triangles
8. Hyperoli tringles Note: This yer, I m not doing this mteril, prt from Pythgors theorem, in the letures (nd, s suh, the reminder isn t exminle). I ve left the mteril s Leture 8 so tht (i) nyody interested
More informationChess and Mathematics
Chess nd Mthemtis in UK Seondry Shools Dr Neill Cooper Hed of Further Mthemtis t Wilson s Shool Mnger of Shool Chess for the English Chess Federtion Mths in UK Shools KS (up to 7 yers) Numers: 5 + 7; x
More informationTwo special Righttriangles 1. The
Mth Right Tringle Trigonometry Hndout B (length of )  c  (length of side ) (Length of side to ) Pythgoren s Theorem: for tringles with right ngle ( side + side = ) + = c Two specil Righttringles. The
More informationOVERVIEW Prove & Use the Laws of Sines & Cosines G.SRT.10HONORS
OVERVIEW Prove & Use te Lws of Sines & osines G.SRT.10HONORS G.SRT.10 (HONORS ONLY) Prove te Lws of Sines nd osines nd use tem to solve prolems. No interprettion needed  prove te Lw of Sines nd te Lw
More informationMATH PLACEMENT REVIEW GUIDE
MATH PLACEMENT REVIEW GUIDE This guie is intene s fous for your review efore tking the plement test. The questions presente here my not e on the plement test. Although si skills lultor is provie for your
More information10.3 Systems of Linear Equations: Determinants
758 CHAPTER 10 Systems of Equtions nd Inequlities 10.3 Systems of Liner Equtions: Determinnts OBJECTIVES 1 Evlute 2 y 2 Determinnts 2 Use Crmer s Rule to Solve System of Two Equtions Contining Two Vriles
More informationFunctions A B C D E F G H I J K L. Contents:
Funtions Contents: A reltion is n set of points whih onnet two vriles. A funtion, sometimes lled mpping, is reltion in whih no two different ordered pirs hve the sme oordinte or first omponent. Algeri
More informationRight Triangle Trigonometry for College Algebra
Right Tringle Trigonometry for ollege Alger B A sin os A = = djent A = = tn A = = djent sin B = = djent os B = = tn B = = djent ontents I. Bkground nd Definitions (exerises on pges 34) II. The Trigonometri
More informationRightangled triangles
13 13A Pythgors theorem 13B Clulting trigonometri rtios 13C Finding n unknown side 13D Finding ngles 13E Angles of elevtion nd depression Rightngled tringles Syllus referene Mesurement 4 Rightngled tringles
More informationThe AVL Tree Rotations Tutorial
The AVL Tree Rottions Tutoril By John Hrgrove Version 1.0.1, Updted Mr222007 Astrt I wrote this doument in n effort to over wht I onsider to e drk re of the AVL Tree onept. When presented with the tsk
More informationEXAMPLE EXAMPLE. Quick Check EXAMPLE EXAMPLE. Quick Check. EXAMPLE RealWorld Connection EXAMPLE
 Wht You ll Lern To use the Pthgoren Theorem To use the onverse of the Pthgoren Theorem... nd Wh To find the distne etween two doks on lke, s in Emple The Pthgoren Theorem nd Its onverse hek Skills You
More informationIt may be helpful to review some right triangle trigonometry. Given the right triangle: C = 90º
Ryn Lenet Pge 1 Chemistry 511 Experiment: The Hydrogen Emission Spetrum Introdution When we view white light through diffrtion grting, we n see ll of the omponents of the visible spetr. (ROYGBIV) The diffrtion
More informationPYTHAGORAS THEOREM. Answers. Edexcel GCSE Mathematics (Linear) 1MA0
Edexel GSE Mthemtis (Liner) 1M0 nswers PYTHGORS THEOREM Mterils required for exmintion Ruler grduted in entimetres nd millimetres, protrtor, ompsses, pen, H penil, erser. Tring pper my e used. Items inluded
More informationContent Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem
Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationThe Cat in the Hat. by Dr. Seuss. A a. B b. A a. Rich Vocabulary. Learning Ab Rhyming
MINILESSON IN TION The t in the Ht y Dr. Seuss Rih Voulry tme dj. esy to hndle (not wild) LERNING Lerning Rhyming OUT Words I know it is wet nd the sun is not sunny. ut we n hve Lots of good fun tht is
More informationFractions: Arithmetic Review
Frtions: Arithmeti Review Frtions n e interprete s rtios omprisons of two quntities. For given numer expresse in frtion nottion suh s we ll the numertor n the enomintor n it is helpful to interpret this
More informationPYTHAGORAS THEOREM 8YEARS. A guide for teachers  Years 8 9. The Improving Mathematics Education in Schools (TIMES) Project
The Improving Mthemtis Edution in Shools (TIMES) Projet PYTHGORS THEOREM guide for tehers  Yers 8 9 MESUREMENT ND GEOMETRY Module 15 June 2011 8YERS 9 Pythgors theorem (Mesurement nd Geometry: Module
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationCHAPTER 4: POLYGONS AND SOLIDS. 3 Which of the following are regular polygons? 4 Draw a pentagon with equal sides but with unequal angles.
Mthemtis for Austrli Yer 6  Homework POLYGONS AND SOLIDS (Chpter 4) CHAPTER 4: POLYGONS AND SOLIDS 4A POLYGONS 3 Whih of the following re regulr polygons? A polygon is lose figure whih hs only stright
More informationAngles and Triangles
nges nd Tringes n nge is formed when two rys hve ommon strting point or vertex. The mesure of n nge is given in degrees, with ompete revoution representing 360 degrees. Some fmiir nges inude nother fmiir
More informationMaximum area of polygon
Mimum re of polygon Suppose I give you n stiks. They might e of ifferent lengths, or the sme length, or some the sme s others, et. Now there re lots of polygons you n form with those stiks. Your jo is
More informationLesson 32: Using Trigonometry to Find Side Lengths of an Acute Triangle
: Using Trigonometry to Find Side Lengths of n Aute Tringle Clsswork Opening Exerise. Find the lengths of d nd e.. Find the lengths of x nd y. How is this different from prt ()? Exmple 1 A surveyor needs
More informationP.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
More informationPythagoras theorem is one of the most popular theorems. Paper Folding And The Theorem of Pythagoras. Visual Connect in Teaching.
in the lssroom Visul Connet in Tehing Pper Folding And The Theorem of Pythgors Cn unfolding pper ot revel proof of Pythgors theorem? Does mking squre within squre e nything more thn n exerise in geometry
More information11.1 Conic sections (conics)
. Coni setions onis Coni setions re formed the intersetion of plne with right irulr one. The tpe of the urve depends on the ngle t whih the plne intersets the surfe A irle ws studied in lger in se.. We
More informationSOLVING EQUATIONS BY FACTORING
316 (560) Chpter 5 Exponents nd Polynomils 5.9 SOLVING EQUATIONS BY FACTORING In this setion The Zero Ftor Property Applitions helpful hint Note tht the zero ftor property is our seond exmple of getting
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationChapter. Radicals (Surds) Contents: A Radicals on a number line. B Operations with radicals C Expansions with radicals D Division by radicals
Chter 4 Rdils (Surds) Contents: A Rdils on numer line B Oertions with rdils C Exnsions with rdils D Division y rdils 88 RADICALS (SURDS) (Chter 4) INTRODUCTION In revious yers we used the Theorem of Pythgors
More informationPrealgebra 7* In your group consider the following problems:
Prelger * Group Activit # Group Memers: In our group consider the following prolems: 1) If ever person in the room, including the techer, were to shke hnds with ever other person ectl one time, how mn
More informationUnit 6: Exponents and Radicals
Eponents nd Rdicls : The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N):  counting numers. {,,,,, } Whole Numers (W):  counting numers with 0. {0,,,,,, } Integers (I): 
More informationLesson 4.1 Triangle Sum Conjecture
Lesson 4.1 ringle um onjecture Nme eriod te n ercises 1 9, determine the ngle mesures. 1. p, q 2., y 3., b 31 82 p 98 q 28 53 y 17 79 23 50 b 4. r, s, 5., y 6. y t t s r 100 85 100 y 30 4 7 y 31 7. s 8.
More informationIntroduction. Law of Cosines. a 2 b2 c 2 2bc cos A. b2 a 2 c 2 2ac cos B. c 2 a 2 b2 2ab cos C. Example 1
3330_060.qxd 1/5/05 10:41 M Pge 439 Setion 6. 6. Lw of osines 439 Lw of osines Wht you should lern Use the Lw of osines to solve olique tringles (SSS or SS). Use the Lw of osines to model nd solve rellife
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationChapter15 SAMPLE. Simultaneous equations. Contents: A B C D. Graphical solution Solution by substitution Solution by elimination Problem solving
Chpter15 Simultneous equtions Contents: A B C D Grphil solution Solution y sustitution Solution y elimintion Prolem solving 308 SIMULTANEOUS EQUATIONS (Chpter 15) Opening prolem Ewen wnts to uy pie, ut
More informationThe Parallelogram Law. Objective: To take students through the process of discovery, making a conjecture, further exploration, and finally proof.
The Prllelogrm Lw Objective: To tke students through the process of discovery, mking conjecture, further explortion, nd finlly proof. I. Introduction: Use one of the following Geometer s Sketchpd demonstrtion
More informationThe theorem of. Pythagoras. Opening problem
The theorem of 8 Pythgors ontents: Pythgors theorem [4.6] The onverse of Pythgors theorem [4.6] Prolem solving [4.6] D irle prolems [4.6, 4.7] E Threedimensionl prolems [4.6] Opening prolem The Louvre
More informationChapter. Contents: A Constructing decimal numbers
Chpter 9 Deimls Contents: A Construting deiml numers B Representing deiml numers C Deiml urreny D Using numer line E Ordering deimls F Rounding deiml numers G Converting deimls to frtions H Converting
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationConsumption and investment spending. Cambridge University Press 2012 Economics for the IB Diploma 1
Supplementry mteril for Chpter 9 9.8 Understnding ggregte demnd nd the multiplier in terms of the Keynesin ross model (supplementry mteril, reommended for higher level) This mteril is inluded for the interested
More informationFor the Final Exam, you will need to be able to:
Mth B Elementry Algebr Spring 0 Finl Em Study Guide The em is on Wednesdy, My 0 th from 7:00pm 9:0pm. You re lloed scientific clcultor nd " by 6" inde crd for notes. On your inde crd be sure to rite ny
More informationHeron, Brahmagupta, Pythagoras, and the Law of Cosines
University of Nersk  Linoln DigitlCommons@University of Nersk  Linoln MAT Exm Expository Ppers Mth in the Middle Institute Prtnership 71006 Heron, Brhmgupt, Pythgors, nd the Lw of Cosines Kristin K.
More informationSection 55 Solving Right Triangles*
55 Solving Right Tringles 379 79. Geometry. The re of retngulr nsided polygon irumsried out irle of rdius is given y A n tn 80 n (A) Find A for n 8, n 00, n,000, nd n 0,000. Compute eh to five deiml
More informationQuick Guide to Lisp Implementation
isp Implementtion Hndout Pge 1 o 10 Quik Guide to isp Implementtion Representtion o si dt strutures isp dt strutures re lled Sepressions. The representtion o n Sepression n e roken into two piees, the
More information5.6 The Law of Cosines
44 HPTER 5 nlyti Trigonometry 5.6 The Lw of osines Wht you ll lern out Deriving the Lw of osines Solving Tringles (SS, SSS) Tringle re nd Heron s Formul pplitions... nd why The Lw of osines is n importnt
More informationTallahassee Community College. Simplifying Radicals
Tllhssee Communit College Simplifing Rdils The squre root of n positive numer is the numer tht n e squred to get the numer whose squre root we re seeking. For emple, 1 euse if we squre we get 1, whih is
More informationEnd of term: TEST A. Year 4. Name Class Date. Complete the missing numbers in the sequences below.
End of term: TEST A You will need penil nd ruler. Yer Nme Clss Dte Complete the missing numers in the sequenes elow. 8 30 3 28 2 9 25 00 75 25 2 Put irle round ll of the following shpes whih hve 3 shded.
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationNapoleon and Pythagoras with Geometry Expressions
Npoleon nd Pythgors with eometry xpressions NPOLON N PYTORS WIT OMTRY XPRSSIONS... 1 INTROUTION... xmple 1: Npoleon s Theorem... 3 xmple : n unexpeted tringle from Pythgorslike digrm... 5 xmple 3: Penequilterl
More information2.1 ANGLES AND THEIR MEASURE. y I
.1 ANGLES AND THEIR MEASURE Given two interseting lines or line segments, the mount of rottion out the point of intersetion (the vertex) required to ring one into orrespondene with the other is lled the
More information1. Area under a curve region bounded by the given function, vertical lines and the x axis.
Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given
More informationProblem Set 2 Solutions
University of Cliforni, Berkeley Spring 2012 EE 42/100 Prof. A. Niknej Prolem Set 2 Solutions Plese note tht these re merely suggeste solutions. Mny of these prolems n e pprohe in ifferent wys. 1. In prolems
More informationPROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * challenge questions
PROJECTILE MOTION PRACTICE QUESTIONS (WITH ANSWERS) * hllenge questions e The ll will strike the ground 1.0 s fter it is struk. Then v x = 20 m s 1 nd v y = 0 + (9.8 m s 2 )(1.0 s) = 9.8 m s 1 The speed
More informationPractice Test 2. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn
Prtie Test 2 1. A highwy urve hs rdius of 0.14 km nd is unnked. A r weighing 12 kn goes round the urve t speed of 24 m/s without slipping. Wht is the mgnitude of the horizontl fore of the rod on the r?
More informationChapter 10 Geometry: Angles, Triangles and Distance
hpter 10 Geometry: ngles, Tringles nd Distne In setion 1 we egin y gthering together fts out ngles nd tringles tht hve lredy een disussed in previous grdes. This time the ide is to se student understnding
More informationThe Quadratic Formula and the Discriminant
99 The Qudrtic Formul nd the Discriminnt Objectives Solve qudrtic equtions by using the Qudrtic Formul. Determine the number of solutions of qudrtic eqution by using the discriminnt. Vocbulry discriminnt
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationInterior and exterior angles add up to 180. Level 5 exterior angle
22 ngles n proof Ientify interior n exterior ngles in tringles n qurilterls lulte interior n exterior ngles of tringles n qurilterls Unerstn the ie of proof Reognise the ifferene etween onventions, eﬁnitions
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationMBF 3C Unit 2 Trigonometry Outline
Dy MF 3 Unit 2 Trigonometry Outline Lesson Title Speifi Expettions 1 Review Trigonometry Solving for Sides Review Gr. 10 2 Review Trigonometry Solving for ngles Review Gr. 10 3 Trigonometry in the Rel
More informationThis unit will help you to calculate perimeters and areas of circles and sectors, and to find the radius given the circumference or area.
Get strte 1 Cirles This unit will help you to lulte perimeters n res of irles n setors, n to fin the rius given the irumferene or re. AO1 Flueny hek 1 Roun 4.635 to 2 eiml ples (.p.) 2 Roun 5.849 to 1.p.
More information