Trapezoid Diagrams

Definition: A quadrilateral with exactly one set of parallel sides.

Given this definition of a trapezoid, our Venn Diagram must show the trapezoid section separate from parallelograms. Parallelograms have two sets of of opposite parallel sides so a trapezoid, with this definition, could never be a parallelogram.

A rhombus, rectangle and square all have two sets of parallel opposite sides, resulting in them being considered parallelograms.

A rectangle is an equiangular quadrilateral with two sets of parellel sides with equal lengths. Given this defintion, we know a square can always be considered a rectangle because a square is an equiangular quadrilateral with 2 sets of parallel sides at equal lengths. Not all rectangles are squares however, since squares are equilateral and not all rectangles are equilateral. Therefore, squares are represented in a portion of the rectangles section, but not in the entire rectangle section since there are rectangles that are not squares. This is represented in the diagram with only a portion of the rectangular circle containing squares.

A rhombus is a parallelogram with four sides of equal length. Given this defintion, we know that a square then is always a rhombus. A square however is always equiangular as well while a rhombus is not. Therefore, like with the rectangle, not all rhombii are squares but all squares are rhombii. Since squares are also rectangles, we can conclude that if a rhombii is a square, it must also be a rectangle. This is displayed in the above diagram where a portion of rhombii's overlap the reactangular portion that accounts for squares.

A kite is one of the trickier polygons to place. If one was to look at an image and the definition of a kite, they would typically place kites outside of the parallelogram box like trapezoids. However, since a kite is defined as a quadrilateral having two sets of equal adjacent sides, we can claim that a rhombus is a kite since a rhombus is equilateral and will always have two sets of equal adacent sides. Knowing that a rhombus is a kite, we know that a square will also always be a kite because a square always has two sets of equal adjacent sides. Since a square is a kite, then a kite that is a square is also a rectangle and rhombii since a square is both a rectangle and a rhombii. Therefore, the only time that a kite can overlap into the parallelogram section is when the kite is a rhombus, or a square (in the case of a square, a rectangle too). This is represented above with the overlap of the kite in the rhombus and square (which also contains a portion of the rectangle) categories. A kite goes down below parallelograms as well though because there are many kites that are not parallelograms, the only ones that are are the ones that are equilateral (so rhombus's) and regular (squares).

Definition: A quadrilateral with at least one set of parallel sides.

This defintion results in a different diagram and placement for trapezoids since this definition defines a trapezoid as a quadrilateral with at least one set of parallel sides. Since a trapezoid must have at least one set of parallel sides and parallelograms always have two sets of parallel sides, a parallelogram will always meet the requirement of being a trapezoid, as shown in the diagram above.

The explanation in the first diagram about the placement of the rectangle, square and rhombus will still hold true for this diagram. In this diagram however, these three polygons are now also defined trapezoids since each of these polygons does indeed have at least one set of parallel sides.

The explanation for the kite in terms of its overlap with a parallelogram only in the case of it being equilateral or regular, so a rhombi or square, holds true in this diagram too. An aspect to note about the kites placement in the venn diagram on this diagram is its placement of only being a trapezoid when it is a rhombi or square. We know, based off our knowledge of kites, that the only time kites can have parallel sides is when both sets of sides are parallel, so when a kite is a square (a rhombus, rectangle, and parallelogram). Due to parallelograms being trapezoids with this defintion, we then are able to claim that a kite is a trapezoid as well, but only when the kite is a square/rhombus. When a kite is not a square/rhombus, then a kite will be outside of the trapezoid section because a kite can never have one set of parallel sides, only two when it is a square or a rhombus; this is represented in the venn diagram.

The Differences

The reason for the change in placement of trapezoids between these two diagrams is because of the change in defintion. With the first definition, we define a trapezoid as having only one set of parallel sides. Since a parallelogram must have two sets of parallel sides and a rectangle, square, rhombus (and kite when it is a rhombus or square), fall into the parallelogram category, we know that each of those polygons must have at least two sets of parallel sides so can never be trapezoids. A kite has two sets of equal adjacent sides, resulting in it never being able to have only one set of parallel sides, which is why a kite also never overlaps with the trapezoid section.

In the second diagram however, a trapezoid's defintion is changed to a quadrilateral having at least one set of parallel sides. As we saw in the diagrams, this is a game changer. Since parallelograms have two sets of parallel sides, we know a parallelogram will always be categorized as a trapezoid as well because a parallelogram will always have at least one set of parallel sides. Since a rectangle, rhombus, square (and kite when a rhombus or square), fall under the categorization of being parallelograms, these polygons also will always be classified as trapezoids as well with this defintion. As stated above, since a kite can never have just one pair of parallel sides, the only time a kite can be classified as a parallelogram, so then a trapezoid as well, is when the kite is a rhombus or square.

The difference between these two defintions comes from the words: exactly and at least. The first diagram accounts for trapezoids that have exactly one set of parallel sides, no more, no less, while the second diagram accounts for trapezoids that must have, at the minimum, one set of parallel sides but are not limited to only one set.

Connection to Supplemental Reading

In one of the supplemental videos for this portfolio, fifth grade teacher Madeline Noonan discusses the concept of student perserverence when it comes to difficult concepts and mathematical problems that the students are not quite used to. I feel that this video on perserverence and the importance of not getting frustrated when working through a difficult mathematical concept held a lot of prevelance to the above venn diagrams we have been working to create throughout this quarter. We have been working on perfecting the two venn diagrams above since either the very first day of class or the second day and it has led to frustration among many because there was never one concrete answer given. Professor Champney however continued to encourage us to not get frustrated and like the teacher in the video said, continued to point out to us that we were not the only students struggling with the diagram, that it was a hard concept to grasp. With perserverance and a lot of group work, like shown in the supplemental video, my group finally came to an agreement on what the two diagrams should look like, as illustrated above.

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