B can be made from 4 segments and two semi-circles, for example: The hardest ones are the ones composed of both segments and curves. None of the line segments are perpendicular or parallel. Of course, this isn't the smallest number of pieces needed to build the letter A, but it is a correct decomposition and analysis of it. For example, a student might see an angle greater than 180 degrees at the top of the letter A.Īlternatively, the a student might see the letter A as being composed of 5 line segments and see two straight-angles on the sides of the letter A where the horizontal segment meets them. Note that students might also count angles that are greater than 180 degrees, so it is important for students to explicitly identify the angles they see. Three of these angles are acute, and two are obtuse. The letter A is composed of 3 line segments which meet in three places and form 5 angles less than 180 degrees. It would be good to address this directly in a whole-class discussion. Often students only focus on the angle that is less than 180 degrees, but for example, the letter L can be thought of as defining both an angle that is 90 degrees and one that is 270 degrees. Note that whenever two lines segments meet at an endpoint, two angles are implicitly defined. This is what allows us to see that the letter F, for example, is composed of three line segments, two of which are parallel to each other and both of which are perpendicular to the third. Likewise, two line segments are perpendicular if and only if the lines that contain them are perpendicular. So two line segments are parallel if and only if the lines that contain them are parallel. Does this mean that these line segments are parallel?Įven though these segments do not meet, the answer is "no." The trick is to realize that every line segment is contained in an (infinite) line. For example, parallel lines are lines that never meet. Students are often confused about whether two line segments are parallel or perpendicular. Students should have access to a plastic or metal tracing ruler with different sized circles, a straight edge, and some colored pencils or markers. This task has students composing and decomposing figures, which is an important way of looking at geometric figures and a skill that students start working on in kindergarten and continue building throughout elementary school and beyond. The benefit of working with letters as opposed to made-up figures with these characteristics is that students have already accepted them as objects, and so it seems reasonable to study them from a geometrical perspective. So, for example, you can have a geometric figure that is not connected (like the lower case letters "i" and "j"), or you can have a geometric figure that is composed of a simple closed curve with a line segment sticking out of it (like the letter "P"). Most students recognize polygons, circles, ellipses, and simple closed curves with some kind of symmetry as "shapes," but a 2-D geometric figure in general is simply a specified collection of points in the plane. Letters provide a good opportunity for students to broaden their understanding of what constitutes a 2-dimensional geometric figure. The purpose of this task is for students to analyze the geometry of letters.
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