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# Geometry

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### Geometry

1. 1. Geometry Big ideas of Geometry Content Connections Levels of Geometric thought Types of Geometry activities By: Kelsie, Matt, Tera
2. 2. THE BIG IDEAS <ul><li>Shapes are alike and different based on geometric properties. </li></ul><ul><li>Shapes can be described in terms of where they are in a space. </li></ul><ul><li>Shapes can be moved in a plane or space using translations (slides, flips, etc.) </li></ul><ul><li>Shapes can be seen from different perspectives. </li></ul>Main Menu
3. 3. Content Connections <ul><li>Measurement: Measurement and geometry are related in terms of finding volume, area, perimeter of geometric shapes and solids. </li></ul><ul><li>Proportional Reasoning: Proportions can be seen through similar objects and the shapes provide a good visual for understanding proportions. </li></ul><ul><li>Algebra: Slope and other relationships relate geometry to algebra. The Pythagorean relationship is based on algebra, as is the distance formula often used to find distances on a coordinate plane. All formulas for finding area, perimeter, etc. are based on using an algebraic relationship and plugging in numbers or perhaps finding unknowns using these formulas. </li></ul><ul><li>Integers: The coordinate plane uses both negative and positive integers and offers a visual of this concept. </li></ul>Main Menu
4. 4. Van Hiele’s Levels of Geometric Thought <ul><li>Pierre and Dina van Hiele have provided some insight about geometric thinking and have made a hierarchy of understanding that describes the stages we go through in thinking about geometric concepts. </li></ul><ul><li>Characteristics of the van Hiele levels: </li></ul><ul><li>-levels are sequential </li></ul><ul><li>-the levels are not age-dependent </li></ul><ul><li>-Geometric experience is the greatest factor in moving from one level to the next. </li></ul><ul><li>-instruction and language needs to be at the level of the student </li></ul><ul><li>Click below to find out more information about each level of thought: </li></ul><ul><li>Level 0: Visualization </li></ul><ul><li>Level 1: Analysis </li></ul><ul><li>Level 2: Informal Deduction </li></ul><ul><li>Level 3: Deduction </li></ul><ul><li>Level 4: Rigor </li></ul>Main Menu
5. 5. Level 0: Visualization Moving from Shapes to classes of shapes <ul><li>Example Activities for level 0: </li></ul><ul><li>Shape sorts (student should decide how to sort, not the teacher) </li></ul><ul><li>“ What’s my shape” : Guess who type game in which you guess the shape of a partner by asking yes/no questions </li></ul><ul><li>Tessellations </li></ul><ul><li>Geoboard activities </li></ul><ul><li>Use shapes on coordinate plane to play a battleship style game in which one student directs another one how to put their shapes in the same spot by using only terms such as “over 3 and up 2.” </li></ul><ul><li>Constructing and dissecting shapes </li></ul>Back to levels Seeing how shapes are alike and dislike is the primary focus, involves lots of sorting and classifying: be sure to provide a sufficient variety of shapes so irrelevant features don’t become important. To help students move to level 1, ask questions like: “Let’s see if that is true for other rectangles?” and “Can you draw a triangle that does not have a right angle?” (have them make generalizations)
6. 6. Level 1: Analysis Moving from classes of shapes to properties of shapes <ul><li>“ A significant difference between level 1 and level 0 is the object of students’ thought. While students will continue to use models and drawings of shapes, they begin to see these as representatives of classes of shapes. </li></ul><ul><li>Classification is more complex: triangles are separated into subgroups such as equilateral, isosceles, etc. and quadrilaterals are separated into rhombuses, rectangles, parallelograms, etc. Note: students at level 1 do not easily understand sub-relationships such as: all parallelograms are trapezoids, but not all trapezoids are parallelograms. </li></ul><ul><li>Example Activities for level 1 </li></ul><ul><li>-Sorting two and three dimensional shapes according to special categories </li></ul><ul><li>- Mystery definitions (show shapes under statements such as: All of these have something in common, None of these has it, and then ask “which of these have it?” </li></ul><ul><li>- activities examining similar figures </li></ul><ul><li>- Working with relationships of a circle </li></ul><ul><li>-Discovering Pi (have students measure different circles and examine the ratio between the diameter and the circumference. </li></ul>Back to levels
7. 7. Level 2: Informal Deduction Moving from properties of shapes to relationships among properties <ul><li>In stage 2, students are moving from simply learning about shapes to engaging in simple proofs and ideas that connect to algebra, they ask questions like “why?” and “what if?” </li></ul><ul><li>Examples of types of activities: </li></ul><ul><li>Finding and applying the Pythagorean relationship </li></ul><ul><li>Using mid-segments of triangles </li></ul><ul><li>The distance formula </li></ul><ul><li>Finding the slope of an angle </li></ul>Back to levels
8. 8. Levels 3 and 4 Deduction and Rigor Back to levels Very few students will have even reached level two by grade 6 or 7, so the book spent few little time explaining levels 3 and 4 but here is a brief overview: Level 3: Moving from relationships among properties to deductive systems of properties. “The student at this level is able to work with abstract statements about geometric properties and make conclusions based more on logic than intuition. This is the typical level of the High school geometry class. Level 4: Moving from deductive systems of properties to analysis of deductive systems. The objects of attention are axiomatic systems themselves rather than the deductions within a system. (axiomatic: those things that are assumed as true: laws, principles, postulates) This is generally the level of a college math major studying geometry.
9. 9. Types of Geometry Activities <ul><li>Sorting and Classifying Shapes </li></ul><ul><li>Tessellations </li></ul><ul><li>Definitions and Proofs </li></ul><ul><li>Graphing </li></ul><ul><li>Equations and Formulas </li></ul>Main Page
10. 10. Sorting and Classifying Shapes <ul><li>Shape sorts vary in type according to level. As students learn more content, they will become more sophisticated in shape sorts. At level 0, students should begin classifying based on lines of symmetry, number of sides,etc. By level 1, they should be sorting based on special categories of shapes such as concave, convex shapes, types of triangles and quadrilaterals, etc. </li></ul>Geometry Activities Main Menu
11. 11. Tessellations Geometry Activities Main Menu One shape and two shape tessellations can vary in difficulty. They should be used for students in 1 st -8 th grade. When shapes can be put together in more than one pattern, the creativity and the problem solving level increase. Actual shape tiles can be used, construction paper, pencil and paper, dot grids, or on computer programs such as Tesselmania (click for download of demo) Note: when using computer programs, ensure that students understand what they are doing and not just playing around until they get an end product. Escher-type tessellations are ones that make an a shape of something. This are very popular for students fifth grade and up. You should be discussing the idea of slides and rotations and students should understand why it then tessellates. If students use color, limit the number of colors to about 3 for most tessellations.
12. 12. Definitions and Proofs <ul><li>Definitions and proofs are based on logic. They require students to use if type statements and to use more abstract thinking. </li></ul><ul><li>Examples </li></ul><ul><li>Minimal defining lists (lists that describe a shape and anything meeting the descriptions must be the shape you are defining.) </li></ul><ul><li>True or false: Have students determine whether or not the following types of statements are true or false: </li></ul>Geometry Activities Main Menu
13. 13. Graphing <ul><li>Most of the graphing used in geometry involves using a coordinate plane. Students should become familiar with using coordinates to describe the location of a vertex or point on a plane, as well as be able to flip, rotate and move objects on a coordinate plane based on instructions such as moving up 2 and over 3. </li></ul>Geometry Activities Main Menu
14. 14. Equations and Formulas <ul><li>Equations and formulas are a large part of geometry, but are best if discovered by students to ensure understanding of the formulas. There is a clear link between geometry and algebra through the many formulas. </li></ul>Geometry Activities Main Menu