Course Guide - USAID Teacher Education Project

"For example, suppose a triangle has side lengths of 2 cm, 4 cm, and 7 cm.Multiplying these side lengths by 2 produces 4 cm, 8 cm, and 14 cm. Therefore, atriangle with these side lengths would be similar to the original. However, adding 2 toeach original side length will produce a triangle that is 4 cm, 6 cm, and 9 cm. It is notsimilar to the original, although students often mistakenly think that it is....Afterseveral semesters of attempting to explain, in abstract terms, why multiplication anddivision are necessary where similar figures are concerned, I realized that many preserviceteachers need visual and hands-on experiences." (Johnson, G. November,2010. " Mathematical Explorations: Similar Triangles." Mathematics Teaching in theMiddle School. Reston, Virginia, National Council of Teachers of Mathematics:16(4), pp. 248-254.)c) Even when children have had experiences that prove that the degrees in thecorresponding interior angles of two similar figures are equal, they often hold onto themistaken belief that a larger figure (with its longer sides) should also have moredegrees in its angle sum than the smaller figure.The fact that side lengths can vary (in proportion to the original) but that themeasurement in degrees of the corresponding angles remains the same is much likethe idea (to be discussed later) that the length of an angle's rays is irrelevant to theangle between those two rays.3. What is essential to know or do in class?a) Corresponding sides and corresponding angles are the key elements of similarity.Corresponding sides must be in proportion; corresponding angles must have the samemeasure in degrees.b) Scale factors to create similar shapes use the operations of multiplication anddivision, not addition and subtraction.c) Using a 90° benchmark angle allows for a relatively accurate estimation andcomparison of angles. This informal way of estimating angle measurements is aprecursor to greater accuracy when measuring and constructing angles using aprotractor.4. Class Activitiesa) Begin by reviewing any thoughts students voiced during the pre-assessment aboutsimilarity and angle measurement. Let students know that they will be exploring theconcept of similar figures in several drawing activities.b) Have students take a sheet of lined paper and draw a large triangle, with the vertexon the top line, and its base on the bottom line. Have students draw a horizontal lineparallel to the base from one side of the triangle to the other. What do they noticeabout the new triangle that they created inside the original one?