Trapezoids

3 Key excerpts on "Trapezoids"

• 

  eBook - ePub

  GED® Math Test Tutor, 2nd Edition

  • Sandra Rush(Author)
  • 2017(Publication Date)
  • Research & Education Association

    (Publisher)

  trapezoid is a quadrilateral with only one special fact: two sides are parallel. They don’t have to be equal, and the other two sides don’t have to be parallel or equal either.

  However, if the other two sides are equal, just as for triangles, the trapezoid has a special name: isosceles. An isosceles trapezoid has two parallel sides (FG and EH in the figure above), and the other two sides (EF = HG) are equal. The base is usually chosen to be the longest parallel side, and the base angles (the angles on either side of it) are equal (∠E = ∠H), although it is also true that the other two angles are equal (∠F = ∠G). In addition, for an isosceles trapezoid, the diagonals (here they would be EG and FH) are of equal length.

  The perimeter of any trapezoid is just the sum of the sides (AB + BC + CD + DA); for the isosceles trapezoid, it is (EF + FG + GH + HE). So for a trapezoid with four sides of lengths, say, a, b, c, and d, the perimeter is

  The area of any quadrilateral is based on the simple formula of base × height (also called altitude), but for a trapezoid we have to consider the average of the two bases as the base in this calculation. Otherwise, the area would be two different numbers, depending on which of the parallel sides is considered to be the base. The height is defined as perpendicular to the base, the same as for triangles. Since the bases are parallel to each other, the height is perpendicular to each base and its size doesn’t vary. How do we determine the average of the two bases? Averages are covered in Chapter 7 , but basically the average of two quantities is their sum divided by 2. So the formula to use for a trapezoid is (b1 + b2 ), where b1 and b2 are the lengths of the two parallel sides (bases). The area of a trapezoid is given by

  This formula is the same as the one provided on the GED® test formula sheet: A = h(b1 + b2 ) since multiplication is commutative (xy is the same as yx; see Chapter 1 ).

  Parallelogram

  A parallelogram

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  eBook - ePub

  Making Sense of Mathematics for Teaching Grades 6-8

  (Unifying Topics for an Understanding of Functions, Statistics, and Probability)

  • Edward C. Nolan, Juli K. Dixon(Authors)
  • 2016(Publication Date)
  • Solution Tree Press

    (Publisher)

  You also engaged in Mathematical Practice 7, “Look for and make use of structure,” in that you used the structure and properties of shapes to make sense of finding the area of an unknown shape, in this case, a trapezoid. By exploring the geometric structure of the trapezoid, the areas of rectangles, triangles, and parallelograms were used to make sense of the area of a trapezoid. The structure of equations and expressions also supported your viable arguments when you found equivalent expressions. This use of other topics in mathematics helps reinforce the connections between ideas in mathematics and the progression of those ideas throughout the grades.

  An Alternate Method

  Consider one more method for determining the area of this trapezoid (see figure 5.8 ). In this case, the trapezoid is transformed into a rectangle. How is this rectangle created? How can you determine the base and height of this rectangle?

  Figure 5.8: Transforming a trapezoid into a rectangle.

  The rectangle is created by first decomposing the trapezoid into a rectangle and two triangles similar to the decomposition in figure 5.3 (page 106 ). Next, the base of each decomposed triangle is bisected and two new smaller triangles are formed. These triangles are rotated 180° about the midpoint of the hypotenuse for each original triangle. The rotated smaller triangles form rectangles with the part of the triangles that were not rotated. These rectangles are composed with the rectangle from the trapezoid to make a larger rectangle. How, then, is the length of the newly formed base of the rectangle determined? Since you have rotated part of the base of each triangle to the opposite base of the trapezoid, the length of the opposite sides of the newly formed rectangle is the mean of the two bases of the original trapezoid, or ½(b 1 + b 2 ). Using the formula for the area of a rectangle leads to the formula

  Atrap

  = ½(b 1 + b 2 )h . This strategy provides another way that you can make sense of the formula for the area of a trapezoid.

  The Progression

  The formal development of understanding of measurement and geometry begins in kindergarten with students making sense of shapes and spatial relationships. As students progress throughout the elementary grades, the complexity and type of shapes change. In the middle grades, students apply the properties learned in the elementary grades to a variety of more complex shapes and in new ways. Following is a progression within measurement and geometry in grades 6–8.

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  eBook - ePub

  Polygons Galore

  A Mathematics Unit for High-Ability Learners in Grades 3-5

  • Dana T. Johnson, Marguerite M. Mason, Jill Adelson(Authors)
  • 2021(Publication Date)
  • Routledge

    (Publisher)
• I am thinking of a closed shape with four sides where no two of the sides are the same length. What could the shape be? (Trapezoid or dart)
• Ask students to create a graphic organizer showing the relationships of the various types of quadrilaterals. You may suggest a Venn diagram or tree diagram format. This can be done as a whole-class discussion or done independently by groups. Two examples are included: Venn Diagram of Quadrilateral Relationships (Teacher Resource 2) and Flow Diagram of Quadrilateral Relationships (Teacher Resource 3).
• Have students store their pieces in their sandwich bags, then complete Lesson Summary (Handout 2F) and discuss.

epub:type="title"> Assessment

• Observations (class discussions, partner work); probe for understanding of properties of quadrilaterals
• Quadrilateral Sorting Table (Handout 2D)
• Lesson 2 Summary (Handout 2F)

epub:type="title"> Notes to Teacher

• The plural of rhombus is either rhombuses or rhombi.
• Some geometric terms such as trapezoid and kite have slightly different definitions in different parts of the world. The definitions given in Handout 2E are the traditional definitions used in textbooks in the United States.
• Sometimes students think that a diagonal can only be inside of a polygon. However, in a concave polygon such as a dart, it may fall outside the quadrilateral as shown in Figure 2.1 .
  Figure 2.1.

  External diagonal of a concave polygon.
• Many students have only seen Venn diagrams as two overlapping circles that are used in a “compare and contrast” activity. The use of Venn diagrams in this lesson where some sets of shapes are entirely contained within another set might be a new idea for students.