# Engaging students: Area of parallelogram

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Perla Perez. Her topic, from Geometry: finding the area of a parallelogram.

How does this topic extend what your students should have learned in previous courses?
A parallelogram is a two dimensional shape in which the opposite sides of the shape are parallel to each other and the opposite angles are equal. To find the area of a parallelogram the height is multiplied by the base. Before being able to solve for the area of a parallelogram, a learner must have foundational knowledge of what defines a base and height of a shape; as well as be able to understand what it means for lines to be parallel and to intersect (which is taught in grade 4). There are many different types of parallelograms; to name a few: rectangles, rhombuses, and squares. A rectangle is a special parallelogram in which it not only fits the criteria to be considered a parallelogram but all angles are equal. Because of the fact that all angles are equal, students tend to learn how to find the area of a rectangle first, and later learn to apply it to other parallelograms. Although, during elementary education students learn how to measure an angle, define parallel lines, and can even define perpendicular lines these topics are also taught in their high school geometry classes, typically in the beginning of the year.
References:

http://ritter.tea.state.tx.us/rules/tac/chapter111/ch111c.html

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.
Khan Academy provides numerous amounts of resource such as videos, practice question, and even tools that can help illustrate certain topics. One tool available to students helps them understand that the method to find the area of any parallelogram is the same as that of a rectangle. This tool can be found here: https://www.khanacademy.org/math/cc-sixth-grade-math/cc-6th-geometry-topic/cc-6th-parallelogram-area/a/area-of-parallelogram
This tool allows students to translate a right triangle “cut” from the parallelogram to the opposite side to create a rectangle by moving the green dot above.

Educators can begin the lesson by starting out with a rectangle shape and having students find the area. Then, with the tool at hand, have either the teacher or student translate it to look different, and finally prompt the students to see if the area has changed or not. To solidify this concept, the website offers two problems they can solve and visually represents the formula of the area of a parallelogram. By using this tool students visualize the relationship
between a rectangle and any parallelogram and therefore the area as well.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)
Euclid is known as the father of geometry. “Euclid’s Elements form one of the most beautiful and influential works of science in the history of humankind. Its beauty lies in its logical development of geometry and other branches of mathematics.“ With that said, in this great book of knowledge, Euclid separates topics by smaller books. He proves what parallel lines are in book one as well as the theorem of an area of a parallelogram in proposition 34, “In
parallelogrammic areas the opposite sides and angles equal one another, and the diameter bisects the areas.” Euclid however does not necessarily defines the criteria to be considered a parallelogram. Throughout his books he comes back to the concept of this shape and continues to add more contextual understanding such as relations to parallel lines, triangles, and different bisections made. Although Euclid’s E lements was written in 300 BC, his work is still being taught in high school geometry classrooms today.
Resources:
http://aleph0.clarku.edu/~djoyce/elements/bookVI/bookVI.html
https://en.wikipedia.org/wiki/Euclid%27s_Elements

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