For more roust techer experience, plese visit Techer Plce t mthdshord.com/cmp3 Mthemtics Bckground Extending Understnding of Two-Dimensionl Geometry In Grde 6, re nd perimeter were introduced to develop the ides of mesurement round nd within polygons in Covering nd Surrounding. This Unit focuses on polygons eyond tringles nd qudrilterls, developing the reltionships etween sides nd ngles. These reltionships led to such ides s tesselltions (tilings) of figures nd reflection nd rottion symmetry. Attention is lso given to the conditions needed to construct tringles nd qudrilterls. This leds to criteri for congruence of tringles, which is explored in the Grde 8 Unit Butterflies, Pinwheels, nd Wllpper. Students lso strengthen their mesurement skills. Students mke nd defend conjectures tht relte the sides nd ngles of polygon. Students lso model reltionships using vriles, concept first seen in Vriles nd Ptterns. The development in Shpes nd Designs is sed on the vn Hiele theory of geometry lerning. We egin y uilding from students experiences with recognition of shpes nd clssifiction of shpes in elementry grdes. Then, we move on to nlyzing the properties of those shpes. The overll development progresses from tctile nd visul experiences to more generl nd strct resoning. We ssume students hve hd some prior exposure to the sic shpes nd their nmes. Polygons A simple polygon is plnr figure consisting of t lest three points p 1, p 2, p n, clled vertices, tht re connected in order y line segments. These line segments re clled sides (with point p n connected to point p 1 ) so tht no two sides intersect except s prescried y the connection of consecutive vertices. Figures with pieces tht re not line segments, figures tht cnnot e trced completely from ny vertex ck to tht vertex, figures tht do not lie in single flt surfce, nd figures tht hve sides crossing t points other thn vertices re not usully clled polygons. Those distinctions re illustrted in the disply of polygons nd nonpolygons t the strt of Investigtion 1 in the Unit. 12 Shpes nd Designs Unit Plnning CMP14_TG07_U1_UP.indd 12 12/07/13 7:07 PM
OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT Exmples of Polygons Non-Exmples Polygons re generlly nmed y the numer of sides nd descried with severl other specil djectives. Below is summry of the types of polygons. Term Definition Picture Convex All interior ngles mesure less thn. Concve At lest one ngle mesures greter thn Regulr All sides nd ngles equl. Irregulr Not ll sides or ngles re equl. Cyclic All vertices lie on single circle. In this Unit, we will focus on simple convex polygons. continued on next pge Mthemtics Bckground 13 CMP14_TG07_U1_UP.indd 13 12/07/13 7:07 PM
Look for these icons tht point to enhnced content in Techer Plce Video Interctive Content An importnt distinction to keep in mind in geometric units is tht polygon consists of only the line segments (or sides) tht mke up the polygon. These line segments enclose region of the flt surfce. This region is sometimes clled the interior of the polygon or polygonl region. The points in the interior re not prt of the polygon, nd the points on the sides of the polygon re not prt of the interior. We cn lso tlk out the exterior region of polygon this is, the set of points tht re neither on the polygon nor in the interior of the polygon. The distinctions tht hold for polygons nd polygonl regions lso hold for ny closed plne figure, including circles. In the Grde 6 Unit Covering nd Surrounding, the primry focus ws on perimeter of the polygon nd re of the polygonl region. Techniclly speking, when we tlk out re we should sy re of the rectngulr region or tringulr region, nd so on, ut it hs ecome common prctice to sy re of rectngle. It is understood tht this is the re of the interior of the rectngle or the re of the rectngulr region creted y the rectngle. The distinction etween polygon nd polygonl region is importnt to note so tht students do not tke wy unintentionl misconceptions from the work or discussion in clss. Tesselltions The first ig question presented in Shpes nd Designs, to motivte nlysis of polygons, is the prolem of tiling, or tessellting, flt surfce. The key is tht mong the regulr polygons (polygons with ll edges the sme length nd ll ngles the sme mesure), only equilterl tringles, squres, nd regulr hexgons will tile plne. Mny other figures, nd comintion of figures, cn e used to tile flt surfce. When one understnds the importnt properties of simple polygons, one cn crete n undnce of estheticlly ppeling tiling ptterns, complete with rtistic emellishments in the style of rtist M. C. Escher. However, it is the discovery of the importnt properties of the figures tht mke the tiling possile, not the tiling question itself, tht is one of the foci of the Unit. 14 Shpes nd Designs Unit Plnning CMP14_TG07_U1_UP.indd 14 12/07/13 7:07 PM
OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT For regulr polygons to tile flt surfce, the ngle mesure of n interior ngle must e fctor of 360. So, n equilterl tringle (60 ngles), squre (90 ngles), nd regulr hexgon (120 ngles) re the only three regulr polygons tht cn tile flt surfce. Copies of ech of these will fit exctly round point in flt surfce (or plne). There re eight comintions of regulr polygons tht will tile. The numers in prentheses refer to the polygon y side numer 8 mens regulr octgon, 6 mens regulr hexgon, etc. The sequence of numers represents the order they pper round vertex of the tiling. Visit Techer Plce t mthdshord.com/cmp3 to see the complete imge gllery for the exmple elow. 4 8 8 2 octgons nd 1 squre Note tht there re two rrngements with tringles nd squres, ut depending on the rrngement they produce different tile ptterns, so order is importnt. In ddition, ny tringle or qudrilterl will tile flt surfce s in the exmples elow: c c c c c c c c c c c continued on next pge Mthemtics Bckground 15 CMP14_TG07_U1_UP.indd 15 12/07/13 7:07 PM
Look for these icons tht point to enhnced content in Techer Plce Video Interctive Content d c c c d d d c Symmetries of Shpes Among the most importnt properties of polygons re reflection nd rottion symmetry. Reflection symmetry is lso clled mirror symmetry, since the hlf of the figure on one side of the line looks like it is eing reflected in mirror. A polygon with reflection symmetry hs two hlves tht re mirror imges of ech other. If the polygon is folded over the line of symmetry, the two hlves of the polygon mtch exctly. Reflection Symmetry Rottion symmetry is lso clled turn symmetry, ecuse you cn turn the figure round its center point nd produce the sme imge. All shpes hve trivil rottion symmetry in the sense tht they cn e rotted 360º nd look the sme s efore the rottion. When we determine whether or not shpe hs rottion symmetry, we check for rottion symmetry for ngles less thn 360º. Rottion Symmetry 16 Shpes nd Designs Unit Plnning CMP14_TG07_U1_UP.indd 16 12/07/13 7:08 PM
OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT However, the convention is tht once we determine tht shpe hs rottion symmetry, when counting the rottion symmetries, we include tht trivil rottion s well. For exmple, the shpe elow hs 2 rottion symmetries: 180º nd 360º. This convention works out nicely ecuse we cn sy tht squre hs four rottion symmetries, regulr pentgon hs five rottion symmetries, nd regulr hexgon hs six rottion symmetries. In generl, regulr polygon with n sides will hve n rottion symmetries. Angle Mesures The shpe of polygon is strongly linked to the mesures of ngles formed where its sides meet. One stndrd definition of ngle is the union of two rys with common endpoint. Any pir of djcent sides in polygon determines n ngle if one imgines those sides extended without ound wy from the common vertex. The concept of rottion symmetry leds to nother wy of thinking out ngles s descriptors of turning motions crrying one side of n ngle onto the other. Both of these conceptions of the term ngle re developed in this Unit. A third conception of ngle s region (like sector of circle or piece of pizz) is introduced in the ACE exercises of the Unit. In ll contexts for thinking out ngles, it is usully helpful to mesure the figure or motion eing studied. For students, it is importnt to hve oth informl ngle sense nd skill in use of stndrd ngle mesurement tools. Angle sense is developed in this Unit y strting from the intuitive notion tht n ngle of mesure one degree is 1 1 90 of right ngle or squre corner or 360 of complete turn. Then, students develop fmilirity with importnt enchmrk ngles (multiples of 30 nd 45 ) y plying the gme Four in Row on circulr grid. This fmilirity with common enchmrk ngles will py mny dividends in future work with ngles. The need for more precision in ngle mesurement leds to theory nd techniques for mesuring ngles y the introduction of two common mesuring tools. The goniometer (goh nee AHM uh tur), or ngle ruler, is tool used in the medicl field for mesuring ngle of motion or the flexiility in ody joints, such s knees. continued on next pge Mthemtics Bckground 17 CMP14_TG07_U1_UP.indd 17 12/07/13 7:08 PM
Look for these icons tht point to enhnced content in Techer Plce Video Interctive Content center line rivet center line The protrctor is nother tool commonly used in the clssroom to mesure ngles. B 120 90 60 150 30 C 180 0 V A The next digrm illustrtes why nother method for mesuring ngles with the ngle ruler, clled the gripping method, gives the sme results s plcing the rivet over the vertex of the ngle eing mesured. The overlp of the sides of the ruler forms rhomus s you seprte them. In rhomus, opposite ngles re equl. This mens tht the rhomus ngle t the rivet nd the opposite ngle re equl. The ngle opposite the rivet in the rhomus is lso equl to the ngle etween the sides, since they re verticl ngles (i.e. ngles formed y two intersecting lines). So, when you plce shpe etween the rms of the ruler, the ngle t the rivet hs the sme mesure s the ngle etween the rms. 18 Shpes nd Designs Unit Plnning CMP14_TG07_U1_UP.indd 18 12/07/13 7:08 PM
OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT One of the criticl understndings to develop out ngle mesurement is the fct tht the mesure of n ngle does not depend on the lengths of the sides in drwing. When we mesure ngles, we re mesuring the opening or turn etween the edges of the ngle. The lengths of the two edges in specific drwing do not ffect the mesure of the ngle. Angle Sums in Polygons One of the most importnt theorems in ll of Eucliden geometry sttes tht the sum of the ngles of ny tringle is lwys equl to stright ngle, or. This property of tringles nd the ppliction to ngle sums of other polygons is developed experimentlly, since, in most forml developments of geometry, its stndrd proof depends on sutle xiom out prllel lines tht is not developed in this Unit. Bsed on tht property of tringles, students cn then reson to more generl results out the ngle sum of qudrilterls nd other polygons y showing how those figures cn e decomposed into tringles. One wy to reson out the ngle sum in polygon is to tringulte the polygon. Strt t ny vertex nd drw ll possile digonls from tht vertex. Tringulting qudrilterl gives two tringles, tringulting pentgon gives three tringles, tringulting hexgon gives four tringles, nd so on. Ech time the numer of sides increses y one, the numer of tringles increses y one. The numer of tringles formed y drwing digonls from given vertex to ll other nondjcent vertices in polygon is equl to n 2. The totl interior ngle mesure of ny polygon is T = (n 2). We cn use symols to stte rule for this pttern. If we let n represent the numer of sides in polygon, then n - 2 represents the numer of tringles we get y tringulting the polygon. If we multiply y for ech tringle, we hve the formul: (n - 2) * = the ngle sum in n n-sided polygon. Note tht this is true for oth regulr nd irregulr polygons. continued on next pge Mthemtics Bckground 19 CMP14_TG07_U1_UP.indd 19 20/12/13 2:54 PM
Look for these icons tht point to enhnced content in Techer Plce Video Interctive Content Another method tht students my use is to drw ll the line segments from point within polygon to ech vertex. This method sudivides the polygon into n tringles. In qudrilterl, four tringles re formed. The numer of tringles is the sme s the numer of vertices or sides of the qudrilterl. In the pentgon five tringles re formed. Agin, the numer of tringles is equl to the numer of sides or vertices of the pentgon. The sum of the ngles of the four tringles in the qudrilterl is * 4. However, this sum includes 360 round the centrl point. Therefore, to find the sum of the interior ngles of the qudrilterl, 360 must e sutrcted from the sum of the ngles of the four tringles. The sum of the interior ngles of the qudrilterl is * 4-360 = 360. The sum of the ngles of the five tringles formed in pentgon is * 5. However, this sum lso includes 360 round the centrl point. So, to find the sum of the interior ngles of pentgon, 360 must e sutrcted from the sum of the ngles of the five tringles. The sum of the interior ngles of the pentgon is * 5-360 = 540. We notice tht the sum of the interior ngles of qudrilterl or pentgon is times the numer of sides minus two. For the qudrilterl, the sum is * (4-2) nd for pentgon, the sum is * (5-2). This method works for ny polygon. For polygon with n sides, the sum of its interior ngles is: * n - 360 = * (n - 2). Interior Angles of Regulr Polygons If polygon is regulr, we cn find the numer of degrees in one of the ngles y dividing the sum y the numer of ngles. The expression (n - 2) * n represents the mesure of ech ngle of regulr n-sided polygon. Students my notice tht s the numer of sides of regulr polygon increses, the mesure for ech interior ngle lso increses. This mesure ctully pproches, which occurs s the shpe of the polygon pproches circle. Exterior Angles of Regulr Polygons In regulr polygon of n sides, the sum of the interior ngles is (n - 2) *. The (n - 2) * mesure of ech ngle is n. So, the mesure of ech corresponding exterior ngle is - sum of n exterior ngles (n - 2) * = n [ - n ] = n - (n - 2) * = n - n + 360 = 360 (n - 2) * n. The 20 Shpes nd Designs Unit Plnning CMP14_TG07_U1_UP.indd 20 20/12/13 2:56 PM
OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT This forml resoning is proly less convincing to most Grde 7 students thn the ctivity descried in Prolem 2.4 where students imgine wlking round polygon nd thinking out how they complete one full turn of their direction, or rottion of 360. Exploring Side Lengths of Polygons While ngles re importnt determinnts of the shpe of ny polygon, side lengths ply criticl role s well. Some experiments with ctul polystrip pieces will mke severl key properties of tringles nd qudrilterls cler. First, for ny three sides to mke tringle, the sum of ech pir of side lengths must e greter thn the third. This side length result is clled the Tringle Inequlity Theorem. Angles nd Prllel Lines Mny importnt geometric structures mke use of prllel lines, so it is useful to know how to check whether two given lines re prllel nd how to construct prllel lines. The key principle in oth tsks is the reltionship etween prllel lines nd ny third line tht intersects them. Below is pir of prllel lines tht re intersected y third line. The line tht intersects the prllel lines is clled trnsversl. As the trnsversl intersects the prllel lines, it cretes severl ngles. continued on next pge Mthemtics Bckground 21 CMP14_TG07_U1_UP.indd 21 12/07/13 7:08 PM
Look for these icons tht point to enhnced content in Techer Plce Video Interctive Content Angles nd e, ngles nd f, ngles c nd g, nd ngles d nd h re clled corresponding ngles. Angles d nd e nd ngles c nd f re clled lternte interior ngles. Prllel lines cut y trnsversl crete congruent (equl mesure) corresponding ngles nd congruent lternte interior ngles. Also note tht if two lines intersect, they crete two pirs of congruent opposite ngles. In the digrm, ngles nd c re congruent nd so re ngles nd d, e nd h, nd f nd g. These pirs of ngles re commonly clled verticl ngles. Angles nd d re supplementry ngles. Their sum is. At this point nmes re not stressed only the reltionship mong the ngles. Prllelogrms re defined in the Unit s qudrilterls with opposite sides prllel. There re other equivlent definitions (e.g., one pir of prllel nd congruent opposite sides). However, the focus on prllel lines is pproprite to the nme prllelogrm. Congruence Conditions One centrl theme of Investigtion 3 focuses on how vrious comintions of side lengths nd ngle mesurements determine the shpe of polygon. A question to pose is, How much informtion out polygon do you need to specify its shpe exctly? Clerly, you cnnot replicte tringle if ll you know is the length of one side or the mesure of one ngle. It certinly ought to e possile to reconstruct tringle if you know ll the side lengths nd ngle mesures, ut how out two pieces of informtion? How out three? One of the fundmentl results of Eucliden Geometry is set of necessry nd sufficient conditions to gurntee tht two tringles re congruent. The most common three such criteri re: If two sides nd the included ngle of one tringle re congruent respectively to two sides nd the included ngle of the other, the tringles will e congruent (in ll prts). This condition is commonly known s the Side-Angle-Side or SAS Theorem. If two sides nd the included ngle of one tringle re congruent respectively to two sides nd the included ngle of nother tringle, the tringles will e congruent (in ll prts). C F 6 cm 6 cm A 50 7 cm B 50 7 cm This condition is commonly known s the Side-Angle-Side or SAS Postulte. In the digrm ove, AB DE, A D, nd AC DF. So ABC DEF y the SAS Postulte. D E 22 Shpes nd Designs Unit Plnning CMP14_TG07_U1_UP.indd 22 12/07/13 7:08 PM
OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND INTRODUCTION PROJECT If two ngles nd the included side of one tringle re congruent respectively to two ngles nd the included side of the other, the tringles will e congruent (in ll prts). This condition is commonly known s the Angle-Side-Angle or ASA Theorem. If two ngles nd the included side of one tringle re congruent respectively to two ngles nd the included side of nother tringle, the tringles will e congruent (in ll prts). C F A 50 40 10 in B D 50 40 10 in This condition is commonly known s the Angle-Side-Angle or ASA Postulte. In the digrm ove, A D, AB = DE, nd B E. So ABC DEF y the ASA Postulte. E If the three sides of one tringle re congruent to three corresponding sides of the other, the tringles will e congruent (in ll prts). This condition is commonly known s the Side-Side-Side or SSS Theorem. If the three sides of one tringle re congruent to three corresponding sides of nother tringle, the tringles will e congruent (in ll prts). B E 6 ft 4.75 ft 6 ft 4.75 ft A 8 ft C 8 ft This condition is commonly known s the Side-Side-Side or SSS Postulte. In the digrm ove, AB = DE, BC = EF, nd AC = DF. So ABC DEF y the SSS Postulte. D F There re some other specil congruence criteri, ut these three re the most common nd useful. If you know less out the two tringles, you cnnot e sure tht they re congruent. You might know other sets of three mesurements nd still not e sure tht the tringles re congruent. For exmple, if the three ngles of one tringle re congruent respectively to the three ngles of nother, the tringles will e similr ut not necessrily congruent. Also, there re some comintions of mesurements in the SSA pttern tht produce three congruent corresponding prts, ut not congruent tringles. The theme of criteri gurnteeing congruence for tringles will e revisited in much more detil in CMP Grde 8 Units. At this point wht we im for is n informl understnding of how certin kinds of knowledge out tringle re telling. For n interesting extension of the ide, you might sk students to see how much informtion they would need out qudrilterl to know its shpe precisely. The simplest wy to think out this question is to sk, How could I strt drwing qudrilterl tht will hve the sme shpe s nother qudrilterl? Mthemtics Bckground 23 CMP14_TG07_U1_UP.indd 23 12/07/13 7:08 PM