Geometry

Story of How Geometry Got Its Name

Fundamental Concepts in Geometry

Point - Line - Surface - Solid w/ Golden Beads (make/build; move)

Classified Nomenclature: From the Point to the Solid (drag & lift cards)

Folder & booklet of the Fundamental Concepts (pulling out: solid-surface-line-point; etymology)

Later Activities & Consideration: Point-Line-Surface-Solid: Infinite number of ... in the ...

Intro from the Solid (w/ box, can, ball) - Surface - Line - Point: limits

Dimensions - Abstract reasoning: 0,1,2,3-dimension; ideas of point, line, surface

Further Work w/ Geometry Insets

Congruence, Similarity, Equivalence

Solid Geometry

Polygons

Area

The Circle

Angles

Lines

Work w/ Divided Squares & Triangles

Work w/ Constructive Triangles

Intro to Symbols

Triangle Insets

Congruence, Similarity, Equivalence

Design work w/ Iron Material (design; trace, color)

Fractional Relationships

Equivalence F.E. - Two Key Triangles

Equivalence F.E. - Box of Blue Triangle

Equivalence F.E. - Addition

Equivalence - Reasoning

Equivalence F.E. Using Single Figure; Two Figures; Combining Boxes; Pictures

Constructive Triangles & Metal Insets

Further Exploration: Congruence, Similarity

Pythagoras & Euclid

Sensorial + Reasoning

Pentagon = Rectangle (half perimeter x apothem)

Decagon = broad Rect.

Pentagon = Trapezoid (sum of b = perimeter, height = apothem)

Decagon = narrow Rect.

Broad - Deca - Narrow

Rhombus = Rectangle (same base, same height)

Triangle = Rectangle (same base, half height)

Trapezoid = Rectangle (sum of bases, half the height)

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3 Triangles (Plate 5)

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Sensorial: Right Isosceles Triangle

Numeric & Algebraic: Right Scalene Triangle

Story of Pythagoras

Euclid Plate

Py. & Eu. Applied to Other Figures

Story: Pythagoras Triples

Story: Pythagoras & Regular Solids

w/ Equilateral Triangles

Theorem of Pythagoras

Euclid's Theorem

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w/ Geometry Insets - Algebraic

Yellow Material - Numeric & Algebraic

Obtuse-angled Triangle (3)

Intro to concept: Area of Rect.

Area of Parallelogram

Acute-angled Triangle (3 ways)

Right-angled Triangle(3)

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Trapezoid: A=(B+b) h/2

Decagon: A=P/2 a (broad); A=P a/2(double); A=Pa/2(double)

Rhombus: A = bh; A=d D/2; A=D d/2; A= Dd/2

Pentagon: A=P a/2 (trapezoid); A=P/2 a (rect.)

Triangle: A = b h/2

Parts of a Straight L.: Ray & Segment

Positions of a S.L.: Horizontal, Vertical, Oblique

Intro: Straight & Curved Line

Positions of a S.L. on a Plane (Extension: on earth & in space: zenith, nadir, antipode)

Positions of 2 S.L.: Parallel, Convergent, Divergent

Relationships of 2 S.L.: intersecting - oblique; perpendicular. Ext. Bisect a line

Measuring: Story of 360 & Montessori Protractor; Draw an angle; Bisecting (2w); +,-

Prep. for Angle Theorems

Relationship: Complementary, Supplementary, Vertical, Adjacent

Sum of Angles in Polygons

Intro: Concept; Parts; Variety

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Transversal; Interior, Exterior, Vertical, Supplementary, Alternate (int., ext.), Corresponding

2 Parallel, 1 Transversal: Alternate, corresponding: congruent; Interior (same side): supplementary

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Triangles

Types of Planar Closed Curves: circle, ellipse (tilt flashlight/water), oval

Quadrilateral

Regular & Irregular (rotate in frame)

Parts of a Polygon: vertex, side, perimeter, surface, center, radius, apothem, diagonal, interior angle, midpoint of side

Types by # of Sides (w/ blue polygons); Ext.: Naming Polygons by Sides

Concept of Plane Figures (polygon, closed curve)

7 Tri. of Reality (3x3 grid)

Classify by Sides; Classify by Angles

Parts (vertex, side, angle, perimeter, surface; base, height); Obtuse- & Right-angled Tri.

Parts: Rect., Parallelogram, Trapezoid

Family Tree

Types (7); Interrogation Game

Triangle as the Constructor (stable); Diagonals for Stability

More Classification of Polygon: Regular? Equilateral? Equiangular?

Relationship

Circle as a Special Polygon (with infinite # of sides): blue polygons in inset frame

Circumference: Draw line. Mark & roll blue circle. Measure with diameter. C=d𝝅=2𝝅r

Parts

Surface: surface, sector, segment

Linear: center, circumference, radius, diameter, semi-circumference, chord, arc

Circle & Line: external: d > r; tangent: d = r; secant: d < r

Circle & Circle: external; externally tangent; secant; internally tangent; internal; concentric; + d,r

Finding the Area: 1. Double the Area; 2. Half the Circumference

Intro: Story of Stake & Animal

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Inscribed & Circumscribed (w/ segment inset & blue polygons)

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Non-Rectangular Prisms & Polyhedrons

Solids of Rotation

Volume

Surface

Nomenclature: Classification; Polyhedrons; Height

Sensorial: Trace, cut, fold

Formula

Formula for Volume: Building in 3 Dimensions

V=lwh; A(B)=lw; -> V = A(B) h

Concept: Compare 2 Cubes; Unit Cube; Building Equivalent Prisms(w/ 24 unit cubes)

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© Huidan Xu, 2022