# Geometry & spatial reasoning

## Introduction

* **Geometry *originates from Greek* ge or gaia * meaning earth or land and *metria* meaning measuring, or measurement of earth or land. Today geometry’s big ideas include the study of spatial properties and relations of points, lines, surfaces, solids, and higher dimensional quantities. Their shapes, variables or invariables, definitions, & proofs.

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

*Relative position* is used to locate objects in space with distance and directions.

## Knowledge base & tools for mathematical literacy

### Knowledge base

Mathematical knowledge base with concepts necessary for learners to conceptualize to become mathemtically literate and misconceptions that must be over come. Particularly important for geometry and spatial reasoning are:

### Tools to facilitate literacy in geometry & spatial reasoning & tools to achieve it

Background information for learning and teaching.

- Spatial and visual representations, abilities, & literacy - article
- Stages of geometric thinking - Pierre and Dina van Hiele' levels of geometric thinking in a chart
- Developmental sequence of geometric ability
- Longitudinal development of volume at the middle level - short article
- Visual and spatial reasoning - article includes visual thinking, related careers, research, skills, perception, memory, creative, six modes of visual learning (developed by the Polaroid Education Program), & instructional activities to develop visual spatial abilities
- Geometry vocabulary - lists of vocabulary for primary, middle, & upper levels.

## Activities to facilitate geometry & visual spatial literacy

- Assorted activity suggestions - No Work sheet or detailed explanation: topics include: Volume, Area, Visualization ...
- Area & perimeter asseassment - Work sheet with seven direct assessment questions. It defines a hexaright shape, which is used to assess the depth of understanding of area, perimeter and their relationships.
- Area & perimeter - pools and sidewalks - Work sheet patterns, relationships, & functions with area and perimeter
- Cartography map making, angle measurement, compass, clinometer, measure height, distance, locating objects with distance & angle combinations, contour maps, using triangles to measure
- Measuring heights of trees & other objects - clinometer, shadows, scale (Biltmore) stick
- Earth's Circumference, Eratosthenes, Astronomy, Solstices, Equinoxes, Latitude, Science, and Math - Background information and work sheet with instructions to calculate circumference of the Earth.
- Metatrons Cube and Platonic Solids - Link to external site

### Puzzles & challenges, with work sheets

- Area of nine shapes in a 17 square unit rectangle - The Pharaoh's problem
- Biggest body of water challenge - Select two bodies of water, lakes, rivers, sea, gulf, ocean.. and use their measurements and geometry to document reasoning for why you believe one is bigger than another. Decide how to measure: one dimension, length, width, height, or two dimensions - area, three dimensions volume or four? Time? …
- Broken Square Puzzle 4 pieces - Work sheet to make a square & solution
- Broken Tee puzzle 1 - Work sheet to make a capital T & solution
- Broken Tee puzzle 2 - Work sheet to make another capital T & solution
- Five - five piece puzzles - Classic square and four other shapes, directions, hints, solutions, worksheets and puzzle pieces patterns
- Triangles in the shapes puzzle - Work sheet to find triangles & hints
- Squares and triangles - Work sheet to find squares & triangles
- Tangrams - Directions for folding & cutting a square sheet of paper or oak tag to make the Tangrams
- Tangram puzzles - Work sheet to record fifteen different geometric shapes made with Tangrams
- Tangrams - Work sheet pattern with Tangram pieces in a square
- Three dimensional puzzles - images and patterns
- Four name word puzzle - Work sheet & puzzle
- Fractional parts of a subdivided square Challenge - Work sheet to find & record halves, fourths, eights, & sixteenths of a square with inscribed shapes
- Pentominoes - Describes Pentominoes shapes, history, and puzzles
- How many squares - Cooperative learning work sheet with puzzle that has 40 squares. Also for team building.
- Pattern block star challenge - Work sheet to find the angles of a trapezoid, triangle, parallelogram, and see if they can combine to make a perfect circle. Find what the Angles and Lengths of the Sides are in this
- Surveilence camera puzzle - Work sheet determine the best placement of cameras to surveil eight streets.
- Printable cube pattern - Work sheet

- Pythagoras theorem - Explanation sheet to justify a
^{2}+ b^{2}= c with 1. Area model & 2. lengths of sides model - Relative position & topology - Work sheet to identify: in, on, out, for 3d & 2d pictures, & draw the 5 shapes. Age: preschool - first grade
- Tessellation - Work sheet: How to make a tessellation
- Triangles - properties, kinds, angles, sides, and relationships
- Ruler and compass constructions
- Cross sections

## Tangram puzzle sets and tables

*Page Overview*:

- Geometry & spatial reasoning definitions
- Knowledge base & tools for professional development to faciliate math literacy
- Activities & puzzles to facilitate learning

*Other math knowledge bade & activities pages*

*Process dimensions*:

*Content dimensions*:

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