OK......the trapezoid challenge.
The confusing thing is that both the inclusive and exclusive definitions are used in mathematics. Generally the inclusive definition is
used at the college-level, and the exclusive has historically been taught at the high school level. Our current Geometry textbook
(Holt Geometry) uses the exclusive definition and the new Geometry book (Big Ideas Geometry), being presented to the Board of
Education in March for adoption, also uses the exclusive definition. Specifically, Big Ideas Geometry defines a trapezoid as, "a
quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases."
As for Common Core, the following was taken from the Common Core progression document for K-6 geometry:
Note that in the U.S., that the term “trapezoid” may have two different meanings. In their study The Classification of Quadrilaterals
(Information Age Publishing, 2008), Usiskin et al. call these the exclusive and inclusive definitions:
T(E): a trapezoid is a quadrilateral with exactly one pair of parallel sides
T(I): a trapezoid is a quadrilateral with at least one pair of parallel sides.
These different meanings result in different classifications at the analytic level. According to T(E), a parallelogram is not a trapezoid;
according to T(I), a parallelogram is a trapezoid.
Both definitions are legitimate. However, Usiskin et al. conclude, “The preponderance of advantages to the inclusive definition of
trapezoid has caused all the articles we could find on the subject, and most college-bound geometry books, to favor the inclusive
definition.”
In reviewing the chart below from the CA Math Framework it appears that the classification of parallelograms is included within the
context of a trapezoid, thus indicating that a parallelogram is a trapezoid, therefore utilizing the inclusive definition in the
Framework.
That being said, I think it is best to actually do an activity that has students differentiate between the definitions to have an
understanding of each. Below are two tasks/activities from Illustrative Mathematics that may facilitate this understanding with your
students.
What is a trapezoid activity (Part 1): https://www.illustrativemathematics.org/content-standards/tasks/1504
What is a trapezoid activity (Part 2): https://www.illustrativemathematics.org/content-standards/tasks/1505
Hope this is at least some help.