# Geometry & spatial reasoning

*Geometry *originates from Greek* ge or gaia *- meaning earth or land and *metria* meaning measuring, or measurement of earth or land.

*Analytic geometry* is the combination of geometry and algebra.

*Geometry* is the science of correct reasoning on incorrect figures.

## Concepts & Generalizations

### Generalizations

*Relative position*is used to locate objects in space (directions). Use vocabulary involving important prepositions-in, on, under, inside, between, upon, from, outside, above, and so on.*Spatial representations*can be shown with models, diagrams, ... (geoboards, visual representation).*Spatial relationship*s and*Geometric patterns*can be found in our everyday world.

### Concepts

*Area*is the surface space of a two-dimenional space measured in square measurements.is made where two lines intersect.*Angle**Circle*is a two dimensional closed figure with everypoint being equal distance from its center.- Circle properties include: circumference, diameter, radius, sector, chord, area, and infinite lines of symmetry

*Congruence*is when two figures have the same shape and size.*Coordinate system*is a spatial reference system that uses vertical and horizontal references from a point of origin.*Line*is- A line has different properties (line, line segments, ray)

*Patterns*can be described mathematically.*Parallel lines*are lines in the same plane that do not intersect.-
*Perimeter*is the distance around a shape measured in linear measurements. *Perpendicular lines*intersect so they form a right angles or 90 degree angle. Can be made on a piece of paper holding the points on either end of the line together and folding at the mid point.*Pi*is the relationship between the diameter of a circle and its circumference.*Plane*is a flat surface such that if a straight line is draw from from on point on the line to another, the line would be entirely on the surface.*Point*is a place in space that has a location, but no size. It is impossible to draw, because if it is, then it has size.*Polar coordinate system*is a spatial reference system that uses and angle and distance for reference from a point of origin.*Polygon*is a plane figure whose sides are line segments. They can be convex or non convex. Regular polygons have angles and sides all of equal measure.*Symmetry*is an identical relationship on two sides of a line drawn through a figure. A figure has line symmetry if it can be flipped about a line so that the resulting figure coincides with the original one. A mirror can be used to illustrate summetry by placing the mirror where the line of symmetry i thought to be and seeing if the reflection is the same as the figure in front of the mirror.*Tessellation*is a 2-dimensional pattern made from plane figures that fit together to fill the plane, leaving no spaces.*Triangle*is a two dimensional closed figure with three sides.- Perpendicular bisectors can be made by joining the vertices and folding between them.
- Midpoint can be found by folding all three perpendicular bisectors and finding their union, where they cross.

*Two-dimensional geometric figures*include: circle, square, triangle, rectangle, polygon, ...*Volume*is the amount of space an object occupies.

## Teacher tools to facilitate geometry & visual spatial reasoning

- Developmental sequence of geometric ability
- Topology worksheet (in, on, out, & draw the shape) preschool - first grade
- Volume, Surface Area, and Visualization Activities
- Visual and spatial reasoning notes
- Pierre and Dina van Hiele Geometry Model Outline Notes
- Longitudinal development of volume
- Assessment teacher directed for area and perimeter - hexaright
- Vocabulary
- Metatrons Cube and Platonic Solids
- Ruler and compass constructions
- Cross sections

## Puzzles:

- Make a square - Broken Square Puzzle 4 pieces
- Make a capital T - Broken Tee puzzle 1
- Try another capital T - Broken Tee puzzle 2
- Make the square - five piece green square
- How many Triangles in the shapes puzzle
- How many squares and triangles are there?
- Directions for making Tangrams from a square sheet of paper
- Tangram puzzles - Fifteen geometry puzzles to solve with Tangrams
- Tangram pieces in a square pattern
- Four name word puzzle
- What are the fractional parts?
- Pentominoes
- How many squares are there?
- Find what the Angles and Lengths of the Sides are in this star
- Surveilence camera puzzle
- Make a cube - pattern

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