Algebra and Geometry - sgml/signature GitHub Wiki

Notable Mathematicians Born South of the Equator Before 1500

Abu'l-Wafa al-Buzjani

Euclidean Distance

Geometric Mean

Imagine you're analyzing the annual returns of an investment portfolio. The geometric mean would provide a more accurate representation of the overall growth rate over multiple years, factoring in compounding effects.

---------------------------------------------------------------------------
SELECT EXP(SUM(LN(value)) / COUNT(value)) AS geometric_mean
FROM your_table;

Sure thing! Here is the last table as markdown source code:

### Geometry Postulates, Properties, and Theorems

| Category        | Name                           | Description                                                                                           |
|-----------------|--------------------------------|-------------------------------------------------------------------------------------------------------|
| Postulates      | Euclid's Postulates            | Fundamental postulates including straight lines, circle creation, right angles, and parallel lines.   |
|                 | Parallel Postulate             | Through a point not on a line, exactly one line can be drawn parallel to the given line.              |
| Properties      | Properties of Parallelograms   | Opposite sides and angles are equal, diagonals bisect each other.                                     |
|                 | Properties of Triangles        | Interior angles sum to 180 degrees, Triangle Inequality Theorem.                                       |
| Theorems        | Thales' Theorem                | Angle subtended by a diameter is a right angle.                                                       |
|                 | Congruence Theorems            | SAS, ASA, and SSS congruence criteria for triangles.                                                  |
|                 | Similar Triangles Theorem      | If triangles have equal corresponding angles, their sides are proportional.                           |
|                 | Circle Theorems                | Various properties of angles subtended by arcs and cyclic quadrilaterals.                             |

### Algebra and Geometry Tables

| Field      | Table Type                    | Description                                                                                           |
|------------|-------------------------------|-------------------------------------------------------------------------------------------------------|
| Algebra    | Addition Table                | Basic addition facts.                                                                                 |
|            | Subtraction Table             | Basic subtraction facts.                                                                              |
|            | Exponentiation Table          | Results of raising numbers to various powers.                                                         |
|            | Logarithm Table               | Finding logarithms of numbers.                                                                        |
|            | Factorial Table               | Factorials of numbers, useful in combinatorics and probability.                                       |
|            | Quadratic Formula Table       | Solutions to quadratic equations using the quadratic formula.                                         |
|            | Modulus Table                 | Results of numbers divided by a given modulus, used in modular arithmetic.                            |
| Geometry   | Trigonometric Table           | Values of sine, cosine, tangent, and their reciprocals for various angles.                            |
|            | Area and Volume Formulas Table| Formulas for calculating the area and volume of various shapes and solids.                            |
|            | Pythagorean Triples Table     | Sets of three integers satisfying the Pythagorean theorem.                                            |
|            | Polygon Properties Table      | Properties of different polygons, including sides and interior angles.                                |
|            | Coordinate Plane Quadrants Table | Properties and signs of coordinates in the four quadrants of the Cartesian plane.                    |

Applied Geometry in Paper Airplanes - Failure Modes and Recovery Logic

NASA Shape Your Flight and Klutz Book of Paper Airplanes

ISBN (Klutz Book): 9781570548306

Design Fault Theme Overlay Geometric Failure Recovery Affordance Notes
Uneven Wing Fold Human Error Asymmetric triangles, lift imbalance Re-fold using right triangle guides NASA recommends outlining wing triangles for area calculation
Nose-Heavy Design Anti-pattern Center of mass too far forward Redistribute weight, shorten nose folds Common in beginner designs; affects glide ratio
Overfolded Creases Mistake & Regret Warped surfaces, drag increase Use flat edge tools, reduce fold iterations Leads to unpredictable torque and yaw
Ignored Wing Area Disorganization No measurement of wing geometry Apply triangle area formulas, symmetry checks NASA uses Pythagorean theorem to calculate flight triangles
Crumpled Launch Disaster Scenario Deformed structure, no lift Rebuild with fresh sheet, document fold sequence Often caused by rushed handling or poor material choice
Overuse of One Design Golden Hammer Applying same geometry to all flight goals Encourage design variation, test multiple models Klutz book offers 10 distinct fold types for comparison
No Flight Line Calibration Oversight Inconsistent launch angles and distances Mark take-off points, measure perpendiculars NASA outlines precise launch geometry for repeatability

Elementary Geometry Activities - Tool-Based Calibration and Innovation

Tools: Ruler, Protractor, Pencil, Paper

Activity Name Grade Range Geometric Focus Calibration Logic Notes
Angle Hunt Drawing 2–5 Angle classification Use protractor to identify and label angles in a scene Ideal for reinforcing acute, right, and obtuse angles
Symmetry Creatures 1–4 Axis symmetry Use ruler to reflect half-drawn shapes across a line Builds spatial reasoning and visual balance
Polygon Zoo 2–5 Polygon construction Use ruler to draw animals made of triangles, squares, etc. Encourages shape recognition and creativity
Mandala Mapping 3–6 Radial symmetry Use protractor to divide circle into equal angles Supports pattern design and rotational geometry
Perspective Cityscape 4–6 1-point perspective Use ruler to draw buildings converging to vanishing point Introduces depth and scale through geometric layout
Tessellation Tiles 3–6 Repeating patterns Use ruler to replicate tile shapes across grid Teaches edge alignment and spatial repetition
Quilt Block Planning 2–5 Grid layout Use ruler to design square-based quilt blocks Connects geometry to textile design logic
Angle Constellations 3–6 Angle measurement Use protractor to measure angles between plotted stars Combines geometry with astronomy themes
Parallel Line Art 1–4 Line spacing Use ruler to draw evenly spaced parallel lines Builds precision and visual rhythm
Shape Collage Blueprint 2–5 Area estimation Use ruler to sketch layout of geometric shapes Prepares students for spatial budgeting and planning

Nintendo Switch Games with Geometry-Based Mechanics

  • Big Brain Academy: Brain vs. Brain – $29.99
    Visual-spatial puzzles including angle estimation, shape matching, and symmetry logic.

  • Game Builder Garage – $29.99
    Grid-based object placement and logic flow; teaches alignment, spacing, and design geometry.

  • Picross S Series – $9.99 each
    Grid logic and spatial deduction; simulates ruler-based counting and shape construction.

  • Snipperclips Plus: Cut it out, together! – $29.99
    Shape construction and angle logic through character cutting and spatial problem solving.

  • Pushmo World (via Nintendo Switch Online) – Included with NSO Expansion Pack
    Block-based spatial puzzles requiring geometric planning and movement logic.

  • Tetris 99 – Free (NSO required for online play)
    Rotation and spatial packing; builds foundational geometry skills like fit logic and angle awareness.

  • Dr. Kawashima's Brain Training for Nintendo Switch – $29.99
    Includes visual estimation and geometry-based mini-games for cognitive development.

References

Games

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