This unit bundles student expectations that address angle relationships and measurements of circles, composite figures, rectangular and triangular prisms, and pyramids. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology."
Prior to this Unit
In Grade 6, students determined solutions for problems involving the area of rectangles, parallelograms, trapezoids, triangles, and the volume of right rectangular prisms where the dimensions were positive rational numbers.
During this Unit
Students revisit and solidify essential understandings of geometry. Students use the formulas for circumference and area of a circle to solve problems. Students extend previous knowledge of the area of rectangles, parallelograms, trapezoids, and triangles along with the area of circles in determining the area of composite figures consisting of rectangles, triangles, parallelograms, squares, quarter circles, semicircles, and trapezoids. Students also solve problems involving the volume of rectangular and triangular prisms and pyramids. Students extend their algebraic understandings of writing equations to represent geometric concepts, including the sum of the angles in a triangle, and other angle relationships. Students describe angle relationships such as adjacent angles, vertical angles, complementary angles, supplementary angles, and straight angles.
After this Unit
In Grade 8, students students will use informal arguments to establish facts about the sum of angles and measures of exterior angles of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. Additionally, in Grade 8, students will extend their knowledge of circles to finding the volume and surface area of cylinders, cones, and spheres. Students will describe the volume formula of a cylinder, V = Bh, in terms of its base area and its height and will model the relationship between the volume of a cylinder and a cone having both congruent bases and heights as well as connect that relationship to their respective formulas. Students will solve problems involving the volume of cylinders, cones, and spheres and will use previous knowledge of surface area to make connections to the formulas for lateral and total surface area to determine solutions for problems involving rectangular prisms, triangular prisms, and cylinders.
In Grade 7, solving problems involving the volume of rectangular prisms, triangular prisms, rectangular pyramids, and triangular pyramids, determining the circumference or circles, area of circles, and the area of composite figures containing combinations of rectangles, squares, parallelograms, trapezoids, triangles, semicircles, and quarter circles are identified as STAAR Readiness Standard 7.9A, 7.9B and 7.9C. Writing and solving equations using geometry concepts is identified as STAAR Supporting Standard 7.11C. These four standards are listed under the Grade 7 STAAR Reporting Category: Geometry and Measurement. These standards are part of the Grade 7 Focal Point: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems (TxRCFP). This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, II. Algebraic Reasoning, III. Geometry Reasoning, IV. Measurement Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Representation, and X Connections.
According to Curriculum and Evaluation Standards for School Mathematics (1989) by the National Council of Teachers of Mathematics (NCTM), “Geometric models provide a perspective from which students can analyze and solve problems, and geometric interpretations can help make an abstract (symbolic) representation more easily understood. Many real-world objects can be viewed geometrically. For example, the use of area models provides the interpretation for much of the arithmetic of decimals, fractions, ratios, proportions, and percents. At the middle school level, geometry should focus on investigating and using geometric ideas and relationships rather than on memorizing definitions and formulas” (p. 112). As students develop concepts of length and area, it should be noted that “Students need to become fluent in the use of procedures or formulas to solve problems; however, they also need to learn those skills with understanding rather than just through memorization. Algorithms and formulas have the potential to simplify calculation and clarify topics, but without understanding they can become an impediment to further learning” (NCTM, 2009, p. 7). In regards to the algorithm for calculating volume, “students who understand where formulas come from, do not seem as mysterious, tend to remember them, and reinforce the idea that mathematics makes sense. Rote use of formulas from a book offers none of these advantages” (Van de Walle, Karp, & Bay-Williams, 2010, p. 391).
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2009). Focus in Grades 6 – 8: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.