Pre-kindergarten – Grades 2/3Elementary School – Grades 3-5Middle School – Grades 6-8High School – Grades 9-12Post Secondary Grade 13-16ConclusionReferences:James P. DildineC & I 499 Fall 20041/14/2019 Page 1 of 8According to the NCTM (2000) “Instructional programs from pre-kindergarten through grade 12 should enable all students to:- Analyze characteristics and properties of two and three dimensional geometric shapes and develop mathematical arguments about geometric relationships;- Specify locations and describe spatial relationships using coordinate geometry and other representational systems;- Apply transformations and use symmetry to analyze mathematical situations;- Use visualization, spatial reasoning, and geometric modeling to solve problems.”I chose the 1st item to pursue through the Pre-K – 16 curriculum. I spent some time looking at Glencoe’s series of textbooks which are used in the Urbana School system. I also looked at the Illinois learning standards (ILS) and benchmarks.Most notable of the Illinois Learning Standards (ISBE, 1997) are included here:Pages 24 – 26 of the ILS describe the following goal and standards:Goal 9: Use geometric methods to analyze, categorize, and draw conclusions about points, lines, planes, and spaces. This is a fairly vague statement that is clarified by the Learning Standard (B):Identify, describe, classify, and compare relationships using points, lines, planes, and solids. They then establish benchmarks across the grade levels which I will not repeat here.I will explore a concept through the 1st recommendation from NCTM: The curriculum should enable all students to analyze characteristics and properties of two and three dimensional geometric shapes and develop mathematical arguments about geometric relationships. Following this concept through the curriculum from Pre-K to Grade 16 deals largely with building on ideas learned from each grade level. Finally we deconstructand refit these concepts at the college level, while building a new knowledge set of Non-Euclidean figures and shapes. This process can be summarized by the following example of curricular topic areas typically pursued:Students are first introduced to figures and shapes, say a Triangle in the early grades. Students learn about and name this triangle, and identify what figures are triangles and which are not. Next students should learn some features about specific triangles, right, equilateral, area, and perimeter, among other features. Students should use triangles to solve problems with similarity, and congruence. Students should then build upon this knowledge and apply it to volume and surface area of triangular prisms, tetrahedral, triangular pyramids, etc...One of the most important things that I have noticed is the use of more manipulatives andtechnologies, yet there is still not major adoption of these tools across curricula. Pre-kindergarten – Grades 2/3James P. DildineC & I 499 Fall 20041/14/2019 Page 2 of 8This period of time is an immense time for students to explore and become familiar with shapes and figures through the use of blocks, patterns and cut-outs of shapes. In the Pre –K and the early elementary school years (up to around grade 2 or 3) students are encouraged and expected to manipulate, draw, identify, and describe two dimensional shapes. They are encouraged to build two and three dimensional figures and shapes as well as use other figures or shapes to build more complex structures. They should be ableto describe two dimensional figures or shapes and discriminate between them. Example: Triangles vs. Rectangles.Students should begin hearing and using the terminology for these two and three dimensional figures and shapes. Students should begin using terms like cube, pyramid, cone, and cylinder. They should use these terms appropriately in the correct context. Elementary School – Grades 3-5Students should continue classifying figures and determining their characteristics. At thislevel students should have a level of comfort with working with the two-dimensional figures and should now be working to determine features about them; area, perimeter, acute vs. obtuse triangles, trapezoids, the entire quadrilateral family. From here they should be moving and typically are moving toward solids and describing their features. I recently visited a local elementary school and saw that students are exploring volumes of simple cubes and cones and cylinders. Curricular trends with figures and shapes have seen more and more explorations in reflections of figures, translations, tessellations, and rotations, especially at this level. This should and can be used more and more with dynamic software and manipulatives like the Mira. Middle School – Grades 6-8By the middle school level (Grades 6-8) students are generally using geometric figures and relationships to solve problems. Similar triangles are often used to determine the heights of buildings as in the example below (when the height of the tree is known and the angle of the viewer’s eye): A version of this is used at the 6th grade in a local school. BASE OF TREE TOP OF BUILDING BASE OF BUILDING EYE TOP OF TREE Students are moving from a strictly 2-D world and enhancing their knowledge of 3-D figures (solids) and should be using terms like Regular Polyhedra and describing what properties those figures have. At the same time Pre-algebra and Algebra students shouldJames P. DildineC & I 499 Fall 20041/14/2019 Page 3 of 8and typically are combining their knowledge of Geometry with their Algebraic learning. A common example I really like is the Difference of Two Squares. I believe that this activity offers a wonderful bridge between what is often thought of as two distinct areas of mathematics. I’ve taught this in the classroom as well to Geometry students, Algebra students, as well as teachers and I believe all really learned something about this relationship. I would like to see this used more and actually included in the school’s curriculum. I am currently revising my lesson so that schools in the area can use it with physical as well as virtual manipulatives. A SQUARED -B SQUARED  = 24.90 cm2(A-B)(A+B) = 24.90 cm2A SQUARED = 31.46 cm2B SQUARED = 6.56 cm2CGB or FF or BEFEBADCAGHigh School – Grades 9-12As students enter high school the geometry curriculum typically involves the realms of proof and even coordinate geometry. Each of these