Mary Koerber Geometry in two dimensional and three dimensional ...

Performance Standards:The NYS Learning Standards for mathematics that are addressed in this topic isMathematical Reasoning. This is used because we prove that a parallelogram exists andthen we use this conjecture to show the equation of a rhombus.The NCTM Principles and Standards for School Mathematics states that students mustestablish the validity of geometric conjectures using deduction, prove theorems, andcritique arguments made by others. In this lesson we prove the geometric figure of aparallelogram and make conjectures to make the equation of a rhombus.Opening Activity:In the opening activity the students will be guided through the proof that a parallelogramexists with the teacher. We are given that segment AC and segment BD bisect each othera E (view teacher’s notes). Then there are several other conjectures that we can say,such as angle AEB is congruent to angle CED, because vertical angles are congruent.By the same reasoning, we can say that angle BEC is congruent to angle DEA. We canalso see that segment AE is congruent to segment CE, because of the definition of asegment bisector. From here we can say that triangle AEB is congruent to triangle CEDand triangle BEC is congruent to triangle DEA; both by side- angle- side. Bycorresponding parts of congruent triangles are congruent, we have that angle BAE iscongruent to angle DCE and angle ECB is congruent to angle EAD. So if we havecongruent alterior angles then we have that segment AB is parallel to segment CD andsegment BC is parallel to segment AD. After all of these steps we can finally say thatABCD is a parallelogram.Developmental Activity:In the beginning of the developmental activity we ask students several questions about acertain rhombus. These questions expand on what the students know about prooftechniques. The students will be required to know how to answer questions about sideangle side, corresponding parts of congruent triangles are congruent, and anglemeasurement. In the second part of the developmental activity we use the proof thatsegment AC is perpendicular to segment BD to show that the area of a rhombus is equalto one-half the product of the diagonals. We are first given that ABCD is a rhombus,and by definition of a rhombus segment AB is congruent to segment AD. Since segmentAC is a diagonal of a rhombus it bisects angle BAD, and therefore angle BAE and DAEare congruent. By the Reflexsive Property of Congruence we have that segment AE iscongruent to segment AE; and by side – angle - side we see that triangle ABE iscongruent to triangle ADE. By corresponding parts of congruent triangles are congruentwe get that angle AEB is congruent to angle AED; and because these angles are bothcongruent and supplementary , they are right angles. We can now finish that by thedefinition of perpendicular we have that segment AC is perpendicular to segment BD.Once we have proven this we can now develop another proof that allows us to use aformula to calculate the area. The proof will be given as a homework assignment, so thatthe students can see where the formula comes from. It is essential that the teacher goesover the homework with the students so that anyone who gets the proof wrong knowswhere they went wrong.Closing Activity:In the closing activity we tell the students the formula to the area of a rhombus; and westate that a square is also a rhombus. Later in the homework the students will see whythe area is one-half the product of the diagonals. The teacher will also show the studentsthe formula to find the area of a trapezoid. The students will learn that in a trapezoid theparallel sides are the bases, and the non-parallel sides are the legs. Also, the height is theperpendicular distance between the two parallel bases.Koeber – Page 11