Similarly, if you prove something about the parallel sides of a trapezoid, then you automatically prove it for both pairs of parallel sides of a parallelogram. The use of relating figures by inclusion serves a mathematical purpose: If you prove an idea that holds for a rectangle, then it automatically holds for a square. (3) To make theorems and proofs more general. Since rectangles are also parallelograms by inclusion, then parallelograms should be trapezoids by inclusion as well. This method (and its explanation) only makes sense if rectangles are also trapezoids. For example, calculus uses “trapezoidal approximations” for calculating integrals. (2) To make the definitions consistent with how they are used in all future math courses. It means that learning which figures are also other figures always follows the same principle-students don’t have to learn a bunch of “special cases.” This makes categorizing figures enormously easier, more intuitive, and overall far more coherent than the old common understandings-students only have to remember one principle and apply it appropriately. This conscious choice to make figures always related by inclusion rather than some inclusion and some rule-outs is a very important one for students. Thus, a parallelogram is also a trapezoid, an equilateral triangle is also an isosceles triangle, a square is also a rhombus and a rectangle, and so on. In Eureka Math, it was decided that all relationships between figures should be defined by inclusion instead of relating some figures by inclusion while defining other figures by rule-out. (1) To make categorizing figures easy to learn for students. There are at least three very good reasons to include parallelograms and rectangles as trapezoids: Thus it is common to see some books include parallelograms as trapezoids (at least one pair) and others rule-out parallelograms as trapezoids (exactly one pair). Any thoughts on this?įirst, in general, when it comes to whether an equilateral triangle is also an isosceles triangle, a square is a rhombus, and so on, it is very important to realize that the decision to include or rule-out one type of figure can be a purely human convention-that is, it is often a human decision rather than one based upon the structure of mathematics. a previous common understanding (or misunderstanding) that many of us had, that a trapezoid is a quadrilateral with exactly one pair of parallel sides? I was just thinking about this today and couldn’t think of how being able to classify all parallelograms also as trapezoids is useful, whereas it seemed very clean and clear to identify and recognize quads with only one pair of parallel sides as trapezoids and to be able to say “this is what a trapezoid is” rather than “this is one type of trapezoid”. I know this is correct, but my question is, how useful is this vs. I was reviewing the Grade 4 Module 4 lessons and videos and saw the definition of a trapezoid as a quadrilateral with at least one pair of parallel sides. Here is an excellent question from a user of Eureka Math:
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