How Do I Love Thee? Modeling and Dimensional Thinking

Module 6: Assignment #1

Geometry is a class full of drawing shapes, polygons, cubes, constructing 3-D shapes, 2-D nets, and calculating the volume/area of a figure.  Students are always drawing pictures to solve problems, talking about relationships, and calculating answers based on dimensions.  With that said, I pushed myself to find a way to use dimensional thinking in the study of the Pythagorean Theorem and triangles.  There are many different types of triangles and as I drew them out on my paper I started to think past 2-D.  I thought about adding a third dimension and thought about triangles creating 3-D figures.

Dimensional thinking “involves moving from 2-D to 3-D or vice versa” (Root-Bernstein, 1999).  It also involves scaling which is altering proportions within one set of dimensions.  Typically, we introduce the Pythagorean Theorem after proving triangles congruent.  Triangle similarity and dimensional analysis of areas/volumes of similar figures is not covered until second semester.  That is also when we go over different polygons and find the area/volume of them.  Even though these topics are found in different chapters, they are all intertwined together.

In order to incorporate more dimensional thinking, I began to explore through modeling.

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This had me thinking, why not let students play with various manipulatives and create shapes?  Through this exploration and construction of 3-D figures, they will discover a lot of the geometric concepts.  It may sound crazy, but allowing students to play with manipulatives would be very powerful.  Modeling allows them to look at the figure from all different angles and experiment with how changing lengths or angles affect the shape.  Students could use magnetic balls and plastic pieces, spaghetti noodles and marshmallows, or even toothpicks and pieces of clay.  After some play, the teacher could give directions to create different triangles, right triangles, equilateral, isosceles, etc.  From a triangle comes larger 2-D shapes and 3-D shapes.  You could even challenge them to create a shape with the largest volume, or largest surface area.  They would begin to see the concepts we cover at work and in real life.  This hands-on experience would be powerful for kinesthetic learners, students with special needs, and anyone who struggles with visualizing in Geometry.  They can create similar figures by measuring certain parts and comparing models.  This would lead right into dimensional analysis, area, and volume discussions.  I could also see this modeling and dimensional thinking being useful throughout the chapters when we cover these topics, or as a challenge activity towards the end of the chapter.

My geometry classes already do some activities with various spheres and cylinders to calculate surface area and volume.  I have found it powerful in building up their estimated skills and accuracy in measuring as well as deepening the concepts.  We also do a container water project where students calculate the capacity of different shaped containers before filling them with water to win a competition.  Not only are these ways of thinking helpful in making the ideas stick, but they bring math to life!

Root-Bernstein, R. S., & Root-Berstein, M. (1999). Sparks of Genius: The 13 Thinking Tools of the World’s Most Creative People. Boston, MA: Houghton Mifflin.