MAC-CPTM Situations Project

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1 raft o not use witout permission -P ituations Project ituation 20: rea of Plane Figures Prompt teacer in a geometry class introduces formulas for te areas of parallelograms, trapezoids, and romi. e removes te formulas from te overead and poses several area prolems to er students. One student volunteers te correct answers very quickly. noter student asks, How did you memorize te formulas so fast? e first student responds, I didn t memorize te formulas. I can just see wat te area sould e. ommentary e four foci for tis situation reflect relationsips among tree classes of figures: parallelograms, trapezoids, and romi. In Foci 1 and 2, strategies for a class of figures are applied to one of its suclasses. e two foci differ in terms of weter te strategy is te application of a known formula or te application of a metod tat develops te known formulas. Foci 3 and 4 involve decomposition of quadrilaterals, wit te former empasizing efficient calculation and te latter targeting te logical development of matematics from wat is known to wat is needed. atematical Foci atematical Focus 1 ecause romi, squares, and rectangles are parallelograms, te area for eac can e found y =, were represents te ase and represents te eigt. parallelogram is a quadrilateral wit two pairs of parallel sides; romi, squares and rectangles are special parallelograms. e Venn diagram in Figure 1 represents tese relationsips. One could conclude tat te areas of tese it 20 rea of Plane Figures Page 1 of

2 raft o not use witout permission parallelograms would e found in te same way, te product of te lengt of a ase and te corresponding eigt. Parallelogram Rectangle quares Romi Figure 1 atematical Focus 2 ny parallelogram, trapezoid or romus can e decomposed into two triangles, te sum of wose areas is te area of te original quadrilateral wic suggests a way to generate familiar area formulas. e metod of constructing a diagonal to decompose a parallelogram, romus, or trapezoid into two triangles elucidates possile derivations of te area formulas. For example, in Figure 2, diagonal divides parallelogram into two congruent triangles,. Using te ase and eigt of te parallelogram, te area of eac triangle is triangle = 1. ince te area of 2 parallelogram is equal to te sum of te areas of te two triangles, parallelogram = = = Figure 2 imilarly, diagonal of trapezoid PR in Figure 3 sudivides te trapezoid into two triangles, PR and P. In general, te two triangles are not congruent ut teir areas sum to te area of te trapezoid. Using one ase and te eigt of te trapezoid, te area of PR is, and te area of P is it 20 rea of Plane Figures Page 2 of

3 raft o not use witout permission. e area of trapezoid PR is equal to te sum of te areas of PR and P:. R a P Figure 3 In te case of te romus, diagonal sudivides romus WXYZ into two congruent triangles, WXZ and YZX, as sown in Figure 4. Let V e te intersection of and, wic are diagonals of a romus and tus perpendicular isectors of eac oter. y defining diagonal, wic as lengt, to e te ase of ot WXZ and YZX, ten segment wit lengt would e te altitude of WXZ. Given WY a diagonal of lengt d2, segment wit lengt would e te altitude of YZX. e area of eac triangle, YZX and WXZ, would e triangle = 1 2 d d 2 = 1 4 d d 1 2, and tus te area of romus WXYZ would e: romus = 1 4 d d d d = d d 1 2 = 1 2 d d 1 2. W X Z V Y Figure 4 it 20 rea of Plane Figures Page 3 of

4 raft o not use witout permission It sould also e noted tat, ecause a romus is a parallelogram (see discussion of atematical Focus 1), it follows from te discussion of parallelogram aove tat a romus can e decomposed into two triangles wose sum is equal to te area of te romus. atematical Focus 3 parallelogram, trapezoid, or romus can e decomposed into a comination of polygons, te sum of wose areas can e calculated efficiently, and tat coice depends ot on te original figure and te measures involved. How we decompose a quadrilateral to determine its area depends on te specific type of quadrilateral we ave. ome decompositions are not possile. For example, one way to decompose a stereotypical scalene trapezoid like tat in Figure 5 involves two triangles and a rectangle. In contrast, it makes no sense to use two triangles and a rectangle in a decomposition of a rigt trapezoid suc as in Figure 6. J K 6 L E Figure 5 Figure 6 In some cases, te general nature of te decomposition migt e te same ut te calculation can e done more efficiently ased on oservations aout te numers involved. For example, te decomposition of a stereotypical trapezoid into one rectangle and two triangles works wit an isosceles trapezoid. o determine te area of te isosceles trapezoid JKL in Figure, we can decompose te figure into rectangle J and congruent triangles JK and L. e calculation is sligtly easier if we recognize te equal areas of te two rigt triangles: rea trapezoid JKL is 2(1/2 1/2 (-))() + ()() = 1(2)() + () = 10() = 0 square units. J K L Figure For figures of te same type, we migt use sligtly different computations or decompositions depending on te numers involved, particularly in te asence it 20 rea of Plane Figures Page 4 of

5 raft o not use witout permission of convenient formulas. For example, to determine te areas of te romus in Figure, tinking in terms of two triangles or four triangles yields easy calculations. e sligt cange in one measure creates te romus in Figure 9, for wic only one of te tree options seems to produce a sligtly easier mental calculation. = 4 = wice 1/2 () (1/2 4) or 2 4 (2), 16 square units wice 1/2 (4) (1/2 ) or 2 2 (4), 16 square units Four times 1/2 (1/2 ) (1/2 4) or 2 (4)(2), 16 square units Figure = 3 = wice 1/2 () (1/2 3) or 2 4 (1.5), square units wice 1/2 (3) (1/2 ) or 3 (4), square units Figure 9 Four times 1/2 (1/2 ) (1/2 3) or 2 (4)(1.5), square units atematical Focus 4 reas of rectangles, parallelograms, and triangles and trapezoids can e logically developed in tat order. deductive approac to te development of area formulas is consistent wit te systematic nature of Euclidean geometry. overing a rectangular region wit an array of unit squares illustrates te multiplicative nature of te area formula, = lw. e formula for te area of a rectangle can e used to develop te formula for te area of a parallelogram. n altitude from vertex in parallelogram creates rigt E. ranslating E along a vector from it 20 rea of Plane Figures Page 5 of

6 raft o not use witout permission to creates rectangle FE wose area is congruent to te area of parallelogram, as in Figure 10. us, =. E F Figure 10 e area formula for a parallelogram can e used to find te formulas for te area of a triangle and te area of a trapezoid. ot cases involve rotating a polygon 10 aout te midpoint of one side. Figure 11 sows ow te 10 rotation of aout midpoint, wic creates quadrilateral wit opposite sides congruent. o, is a parallelogram wose area is twice te area of original triangle; tus = implies. ' Figure 11 imilarly, Figure illustrates ow rotating trapezoid 10 aout midpoint of a non-parallel side creates quadrilateral wit opposite angles congruent. is a parallelogram wose area is twice te area of te original trapezoid; tus = (a+) implies = 1/2 (a+). it 20 rea of Plane Figures Page 6 of

7 raft o not use witout permission a ' a ' Figure Post ommentary Foci 1, 2 and 4 involve te conservation of area under translations and rotations. ese transformations are not as apparent in Focus 3, toug one can interpret te calculation string for te area of te isosceles trapezoid as 2(1/2 1/2(-))() is te area of two congruent triangles and te corresponding term, 1 1/2(- ))(), is te area of te rectangle created y te pairing of one triangle and te image of te second triangle after a translation and reflection are consecutively applied, as illustrated in Figure 1. First J en Reflected K L First ranslated Figure 1 ll of te foci ave implications for interpreting area formulas and related expressions. For example, Focus 1 suggests we would see =lw as a product of ase lengt l and eigt w. In Focus 2, te expressions are expanded or simplified versions of te sum of te areas of two triangles. e numerical expressions in Focus 3 are manipulated for ease of calculation. Focus 4 involves seeing formula as derivations or sources of oter area formulas. it 20 rea of Plane Figures Page of