Thursday, April 2, 2015

Some Things to Keep In Mind Regarding Elementary Geometry...

Back in the old days, foundational geometry units were often reserved for the end of the year in many grade levels. Browsing through the preview materials of the new Everyday Math units for grades 3 through 6, I see geometry units and lessons occur at the beginning, middle and end of the year. I don't know why geometry wound up back-loaded into March, April, May and June, back in the day, but I remember it well.  For me, geometry was fun and not difficult, whereas all other forms of math gave me greater challenges.  For others, operations, algebra and statistics come easy but geometry poses problems.

Here are some things we all should keep in mind when we are teaching the basics of geometry to prevent confusion and future geometric mishaps.  Please let me know if you have others I should add to this list! These are in no particular order:

  • Many students have trouble remembering the relationship between a square and a rectangle.  They often mistakenly use the terms interchangeably.  It may take some time for your students to know that a square is a classification of rectangle, or a type of rectangle, and not the other way around.  The more you model the correct terminology for them, the sooner they will clear up their misconceptions.  A square is a closed figure with four ninety degree angles and four congruent sides.  A rectangle only has the four 90 degree angles.  When referring to rectangles, it is important that we use the word rectangle, even if it is somewhat close to being a square.
  • Students and adults get three-dimensional and two-dimensional figures confused all the time.  Like the rectangle-square issue, there is also a square-cube issue.  A cube is not a square, plain and simple.  But nevertheless, it happens, we catch ourselves saying to a child, "Can you put those wooden blocks away when you are finished? Yes, the triangle ones and the square ones..." There is no such thing as a "square block."  It is called a cube, even though cubes have squares on them.  Using three-dimensional geometric terms appropriately is super important to encourage correct use of geometric terminology.  Rectangular solid/prism, triangular prism, pyramid, cylinder, cone, cube and sphere, those are the words to use when describing, well, rectangular solids, triangular prisms, pyramids, cylinders, cones, cubes and spheres. 
  • Don't be afraid to use other geometric terms, even if they sound too advanced.  Words like congruent, vertex/vertices, figure, base, height, etc, should all be heard by students when discussing geometry.  For example, you can say "the bottom side of the triangle..," just say "the bottom side, or the base, of the triangle..," so students become familiar with the terminology.  The same is true for all categories of math, in fact.  Everyday Math likes to use kid-friendly terminology, but the rest of the world does not often recognize EM-talk.  When you say, "number model," or "number sentence," you can also say, "or an equation."  When you refer to "turn-around facts," you can say, "also known as the commutative property..."
  • Think carefully before making definitive statements about shapes, such as, "when you put two triangles together, you get a rectangle," or, "when you cut a rectangle in half, you get two triangles," as statements like these may be dependent upon certain conditions and could lead to significant misunderstandings.  Sometimes, two triangles put together make a parallelogram, or even an irregular quadrangle, and sometimes when you cut a rectangle in half, you get two smaller rectangles. 
  • Be careful when drawing/ modeling representations of shapes and angles for students.  Your "right angle" might look right from where you are sitting off to the side of your whiteboard or easel, but from the vantage point of your students it might not look like a right angle at all.  Sometimes I am horrified at what I have drawn or written for my students when I back away from the board and have a proper look!  
  • When students are drawing polygons and circles, have them use rulers, compasses and stencils whenever possible.   The more they are encouraged to represent such figures as accurately as possible, the less they will be inclined to think their distorted and/or deformed representations are acceptable.  Also, these are important mathematical tools they should be comfortable using.

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