A Different Look at Trapezoid Area Prerequisite Knowledge


 August Walton
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1 Prerequisite Knowledge Conditional statement an ifthen statement (If A, then B) Converse the two parts of the conditional statement are reversed (If B, then A) Parallel lines are lines in the same plane that do not intersect. Skew lines are lines not in the same plane that do not intersect. A transversal is a line (or line segment) that cuts across or intersects two or more lines in the same plane (usually parallel but not necessarily). t c d ABOVE: c and d are parallel; t is the transversal Two parallel lines cut by a transversal form several types of angles (defined in pairs) that are congruent. Corresponding angles are pairs of angles that are on the same side of the transversal and occupy the same location at each intersection (ex: angles 3 and 7; angles 6 and 2). If the transversal intersects parallel lines, then the corresponding angles are congruent. Alternate interior angles lie inside the parallel lines and are on opposite sides of the transversal (ex: angles 3 and 6; angles 5 and 4). If the transversal intersects parallel lines, then the alternate interior angles are congruent. Alternate exterior angles lie outside the parallel lines and are on opposite sides of the transversal (ex: angles 1 and 8; angles 2 and 7). If the transversal intersects parallel lines, then the alternate exterior angles are congruent. The converses for the above are true. Ex: If two lines in the same plane are intersected so that the corresponding angles are equal, then the lines are parallel.
2 Prerequisite Knowledge (page 2) a b c d ABOVE: a and b are parallel; c and d are parallel; any of the 4 can be a transversal Two parallel lines cut by a transversal form several types of angles (defined in pairs) that are congruent. Corresponding angles are pairs of angles that are on the same side of the transversal and occupy the same location at each intersection (ex: angles 3 and 7; angles 9 and 13). If the transversal intersects parallel lines, then the corresponding angles are congruent. Alternate interior angles lie inside the parallel lines and are on opposite sides of the transversal (ex: angles 3 and 6; angles 12 and 13). If the transversal intersects parallel lines, then the alternate interior angles are congruent. Alternate exterior angles lie outside the parallel lines and are on opposite sides of the transversal (ex: angles 1 and 8; angles 2 and 7). If the transversal intersects parallel lines, then the alternate exterior angles are congruent. Note: There are other possibilities in the diagram. Ex: If lines a and b are the parallel lines and d is the transversal, then angles 5 and 13 are corresponding angles. Note: The names of angles (as a single angle or as a pair) are dependent on the context. For example, angle 12 is an interior angle if b is a transversal with c and d as the parallel lines, but angle 12 is an exterior angle if c is the transversal with a and b as the parallel lines. TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 2
3 Prerequisite Knowledge (page 3) A midpoint lies exactly halfway between the two endpoints of a line segment. A polygon is a set of line segments in the same plane, each connected endtoend to form a closed shape. The polygon is made up of the segments themselves. The inside of the polygon is the interior and the outside is the exterior. A quadrilateral is a 4sided polygon four line segments in the same plane linked endtoend to create a closed figure. A trapezoid is a special type of quadrilateral that has exactly one pair of parallel sides. **** (However, another definition states that a trapezoid is a quadrilateral that has at least one pair of parallel sides by this definition, a parallelogram may also be considered a trapezoid.) In a trapezoid, the median is the segment that connects the midpoints of the two nonparallel sides. The median also bisects the height. A parallelogram is a special type of quadrilateral that has exactly two pair of parallel sides. Parallelograms have several special properties: Both pairs of opposite sides of a parallelogram are congruent. The opposite angles of a parallelogram are congruent. Each pair of the opposite sides of a parallelogram is both congruent and parallel. Given a quadrilateral, if you can establish that any of the above are true, then the quadrilateral is a parallelogram. Any side of a parallelogram can be considered a base. The altitude (or height) of a parallelogram is the perpendicular distance from the base to the opposite side (which may have to be extended). Symbols used to show equal measures TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 3
4 Developing the Parallelogram Formula Work in groups at your respective tables. In the group resource packet, you will find parallelogram figures on vanillacolored stock. Each student is to take one of these and follow the instructions. TASK: a) Mark all the pairs of angles and/or sides that have equal measures. Discuss with your group why you know they are equal measures. b) Using the scissors provided, cut along each edge so that the figure is a parallelogram. c) Cut the parallelogram along the segment labeled as h (height) so that there are two new figures. d) Join the two figures so that they form a special type of quadrilateral (A trapezoid is possible, but that is not the solution here). What is the name of this quadrilateral? How do you know that it is this special quadrilateral? e) How do the area of the original parallelogram and that of the new quadrilateral compare? Why is that? f) What is the formula for the area of the special new quadrilateral? (This was something learned in previous math classes.) g) Use the area formula for the new quadrilateral to find the formula for the area of the original parallelogram. TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 4
5 Making Connections Work in groups at your respective tables. In the group resource packet, you will find trapezoid figures on lavender stock. The segments b 1 and b 2 are parallel. Each student is to take one of these and follow the instructions. TASK: 1) Observe the trapezoid. a) Notice (and remember) that there are transversals intersecting parallel lines. Also notice that although the lines are extended past the points of intersection that a trapezoid is formed. b) Find the midpoint of each of the two nonparallel sides. (Use the rulers found in the group resource packet.) Measure from the vertices, not from the end of the segments. c) Connect those two points with a line segment. This horizontal segment is a median, and it connects the midpoints of the two nonparallel sides and also bisects the height. d) Label all the matching pairs of segments and angles that you know are equal measures. (This is a very important step!!!) e) Using the scissors in the group resource packet, cut along each edge so that the extra lengths are removed and the remaining figure is a trapezoid. Next, use the scissors to cut the trapezoid into two pieces by cutting along the median. 2) Take the two pieces and join the nonparallel sides in such a way as to form a new 4 sided figure. 3) What type of figure is this new quadrilateral? How could you prove your assertion? TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 5
6 4) How does the area of the new figure compare to the original trapezoid? Why is that? 5) Remember that the original figure was a trapezoid. How does the height of the new figure compare to the original trapezoid? 6) Using this information and the labeling of the bases of the new figure, use the parallelogram area formula to determine a formula for the area of a trapezoid. Justify/prove your answer. TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 6
7 New Perspective The area of a trapezoid is given by the formula below: A = 1 2 h ( b 1 + b 2 ) 1) Review your previous work and perspective in deriving the trapezoid area formula, especially the transformation of the parallelogram formula to the trapezoid formula. 2) Edit/Reorganize the formula below to match the interpretation and write down an explanation or justification. A = 1 2 h ( b 1 + b 2 ) TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 7
8 Perspective Summary The activity began with a trapezoid. The height was cut into two pieces and the two resulting parts were joined to make a new figure that can be proven to be a parallelogram. Since there was nothing added or destroyed on the figures, the area is still the same. This perspective can then be expressed to show that we take the new height (which is one half the original trapezoid height) times the base of the new parallelogram which is represented by ( b 1 + b 2 ). A = b h A = h b Original transformed to A = [ 1 2 h ] ( b 1 + b 2 ) new A = 1 2 h ( b 1 + b 2 ) TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 8
9 b 1 b 2 b 1 b 2 TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 9
10 b 1 b 2 b 1 b 2 TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 10
11 b 1 b 2 b 1 b 2 TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 11
12 h b h b h b TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 12
13 Addressed TEKS TEKS Objective for the lesson: Geometry G.2 B make conjectures about angles, lines, polygons, circles, and threedimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. Other TEKS that focus on prerequisite knowledge: Mathematics, Grade B identify relationships involving angles in triangles and quadrilaterals Mathematics, Grade B use properties to classify triangles and quadrilaterals. 7.9 A estimate measurements and solve application problems involving length (including perimeter and circumference) and area of polygons and other shapes Geometry G.5 B use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles. G.7 B use slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. G.8 A find areas of regular polygons, circles, and composite figures. G.9 A formulate and test conjectures about the properties of parallel and perpendicular lines based on explorations and concrete models. G.9 B formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models. TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 13
14 TEKS Objective: Lesson Objectives Geometry G2.B make conjectures about angles, lines, polygons, circles, and threedimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. Content Objectives Students will derive and connect the formulas for the area of parallelograms and for trapezoids. Students will examine how their thinking and perspective can be expressed in mathematical statements such as formulas. Language Objectives Students will be able to use mathematical vocabulary to explain orally, through models, and in writing the attributes of trapezoids and parallelograms, in addition to foundational concepts such as parallel lines. Using solid models of trapezoids and parallelograms, students will work in cooperative groups to discuss and construct area formulas for those two polygons. Students will communicate their findings to the whole group and justify/prove why their discovery is valid. TEXAS COMPREHENSIVE CENTER at the Southwest Educational Development Laboratory 14
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