# 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*- Symmetry or line symmetry is an identical relationship on two sides of a line drawn through a figure or parts of a figure. A figure has line symmetry if a part can be flipped about a line so both parts coincides. A mirror can be used to illustrate symmetry if placing the mirror on the line of symmetry the reflection is the same as the figure in front of the mirror. Symmetric parts are unchanged when reflected, rotated, or scaled.

*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

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