Activity: Format: Ojectives: Related 009 SOL(s): Materials: Time Required: Directions: Discovering Area Formulas of Quadrilaterals y Using Composite Figures Small group or Large Group Participants will investigate te area formulas for a triangle, parallelogram, and trapezoid. 8. Te student will solve practical area and perimeter prolems involving composite plane figures. Scissors Tape Pencil or pen Nine cards made of one-inc grid paper printed on cardstock cut into 3 x5 (see appendix for template) per participant or nine 3x5 grid index cards One-inc square tiles (if desired) Document camera and projector (if desired) 60 minutes. Model te cutting of te cards for te participants. Use a document camera to sow ow to cut te cards or old up te cards as you cut tem.. Basic Assumption: area of a rectangle is ase times eigt. From were does tis premise come? One interpretation of multiplication is an array interpretation. We can divide our card into one-inc squares and count te numer of one-inc square to determine its area. Since te card is 3 inces y five inces, we can create an array y laying out tree rows of five tiles. Te result is 5 tiles, wic is te product of 3 and 5. 3. Have te participants lael te ase and eigt of a rectangle on one of teir cards. Does it matter wic side is laeled te ase and wic side is laeled te eigt? No. Base and eigt are perpendicular to one anoter. Te orientation of our card does not matter. 4. In te middle of te card, ave te participants write Area = ase * eigt and underneat tat write A =. We will compare all oter figures wit tis card. 5. See instructions for cutting te cards wit te diagrams on page 3. 6. As you cut tese various triangles and quadrilaterals, review wit te participants te similarities and differences etween triangles and quadrilaterals, te properties of quadrilaterals, and vocaulary words suc as ase, eigt/altitude, diagonals, congruent, otuse, rigt, acute, scalene, isosceles, etc. Have te participants lael ase and eigt and write te formula on teir cards.
7. Rigt Triangle and Non-rigt Triangle: A = 8. Parallelogram: A = For our example, we will use a non-rectangular parallelogram. Once we ave te formula for te area a parallelogram, we can find te area of oter quadrilaterals. Wic oter quadrilaterals area also parallelograms? Rectangle (wic we already knew and were using as our comparison), romus, and square. 9. Trapezoid: A = ( + ) As an extension, you can discuss te median of a trapezoid. 0. For an extension, sow te Romus: A = d d. Will tis formula work for oter quadrilaterals? Only te square.. Discussion ow would you use tis wit students? Closing and Deriefing: Possile questions to ask: Wat did you learn from tis session? How would you apply tis to your classroom? Wat is still unclear? Comments and/or concerns? Reflection for Presenter: (Please reflect on and complete te questions elow immediately after delivering te session) Wat specific examples of learning did you note? Wat specific errors and/or misconceptions still need to e corrected? Summarize te worksop evaluations.
Instructions/Diagrams for Cutting te Cards Area of a Rectangle A = 3 y 5 Index card We will compare all oter areas to tis rectangle. Area = ase * eigt A = Area of a Triangle A = Example : Rigt Triangle Area = ase * eigt A = Take a card and draw in one diagonal using a straigtedge. Cut along te diagonal to form congruent triangles. We ave cut te area in alf, ut te ase and eigt remain te same lengt. Since it takes two congruent triangle to form te rectangle, te area of te triangle is alf te area of te rectangle; terefore, te area of te triangle is. Example : A Non-Rigt Triangle Area = ase * eigt A = Take a card and mark a point anywere along one side. Ten use a straigtedge to draw two lines connecting te vertices on te opposite side to tat point. Cut along tose lines to form tree triangles. Is te area of te two smaller triangles te same as te larger triangle? Yes. We can tape togeter te two smaller triangles to form a triangle congruent to te larger triangle. Is te ase and eigt of te larger triangle te same as te original rectangle? Yes. We ave cut te area in alf, ut te ase and eigt remain te same lengt, so te area of te triangle is. 3
Area of a Non-Rectangular Parallelogram A = Area = ase * eigt A = Take a card and mark a point anywere along one side. Ten use a straigtedge to draw a line connecting one vertex on te opposite side to tat point. Cut along tat line to form a triangle and a trapezoid. Tape te triangle to te opposite side to form a parallelogram. Is te area te same as te original rectangle? Yes. Is te ase and eigt of te parallelogram te same as te original rectangle? Yes, so te area of te parallelogram is. Area of a Trapezoid A = ( + ) Example : Creating an isosceles trapezoid Area = *te sum of te ases * eigt A = ( + ) Take a card and mark a point anywere along one side. Ten use a straigtedge to draw a line connecting one vertex on te opposite side to tat point. Cut along tat line to form a triangle and a trapezoid. Flip te triangle and tape it to te opposite side of a trapezoid to form a larger trapezoid. Is te area te same as te original rectangle? Yes. Is te eigt of te trapezoid te same as te original rectangle? Yes. Wat aout te ases (te one pair of parallel sides)? Neiter is equal to te original ase, ut te average of tem is equal to te original ase ( + ) =. Ten te area of te trapezoid is ( + ). 4
Example : Area = *te sum of te ases * eigt A = ( + ) Take a card and create a trapezoid y cutting off two triangles. Lael te ases and te eigt on te card. Trace te trapezoid onto anoter card to create a congruent trapezoid. Tape te two trapezoids togeter to create a parallelogram. Te area of te parallelogram is ( + ). Since te parallelogram is formed from congruent trapezoids, te area of te trapezoid is alf tat of te parallelogram or ( + ). Extension: Te median of a trapezoid is te segment parallel to te ases wose endpoints are te midpoints of te non-parallel sides. Fold te trapezoid in alf, lining up te parallel sides. Tis fold creates te median of te trapezoid. Compare it wit te ase of your original rectangle and notice tat tey are te same lengt. Te median is te average of te two ases (parallel sides) in te trapezoid. So te area of a trapezoid can also e expressed as te product of te median and te ase: ( ) = ( median) ( ). A = + median = te average of te ases median = ( + ) 5
Extension: Area of a Romus: A = d d A romus is a parallelogram and one can find its area y using te parallelogram area formula, A =. Here is anoter way to find te area of a romus using its diagonals. Take a card and draw ot diagonals using a straigtedge. Cut along te diagonals to form pairs of congruent, isosceles triangles. How do you know te triangles are isosceles? Te diagonals of a rectangle are equal in lengt and isects eac oter. Tape te ases of two of te congruent isosceles triangles togeter to form a romus. (Note: you can make two romi from te index card.) Now tat we ave a romus, let s draw in its diagonals. Te diagonals in a romus are perpendicular to eac oter. Wy? Tey are te ase and eigt of te two isosceles triangles tat we taped togeter. If we add togeter te areas of te two congruent triangles we will ave te area of te romus. Since te triangles are congruent teir areas are equal. Te area of one triangle is = d d. Douling te area of te triangle gives us te area of te romus; terefore, te area of te romus is A = i d d = d d. d d d d 6
Appendix 7