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Geometry, Screen Shot 2022-07-25 at 4.57.20 PM, Screen Shot 2022-07-25 at…
Geometry
Polygons
Triangles
7 Tri. of Reality (3x3 grid)
Classify by Sides; Classify by Angles
Parts (vertex, side, angle, perimeter, surface; base, height); Obtuse- & Right-angled Tri.
Types of Planar Closed Curves: circle, ellipse (tilt flashlight/water), oval
Quadrilateral
Parts: Rect., Parallelogram, Trapezoid
Family Tree
Types (7); Interrogation Game
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)
Triangle as the Constructor (stable); Diagonals for Stability
More Classification of Polygon: Regular? Equilateral? Equiangular?
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
The Circle
Relationship
Circle & Line
: external: d > r; tangent: d = r; secant: d < r
Circle & Circle
: external; externally tangent; secant; internally tangent; internal; concentric; + d,r
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
Inscribed & Circumscribed (w/ segment inset & blue polygons)
Finding the
Area
: 1. Double the Area; 2. Half the Circumference
Intro: Story of Stake & Animal
Lines
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
Solid Geometry
Non-Rectangular Prisms & Polyhedrons
Solids of Rotation
Volume
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)
Surface
Sensorial: Trace, cut, fold
Formula
Nomenclature: Classification; Polyhedrons; Height
Angles
Measuring: Story of 360 & Montessori Protractor; Draw an angle; Bisecting (2w); +,-
Prep. for Angle Theorems
Transversal; Interior, Exterior, Vertical, Supplementary, Alternate (int., ext.), Corresponding
2 Parallel, 1 Transversal: Alternate, corresponding: congruent; Interior (same side): supplementary
Relationship: Complementary, Supplementary, Vertical, Adjacent
Sum of Angles in Polygons
Intro: Concept; Parts; Variety
Further Work w/ Geometry Insets
Pythagoras & Euclid
Sensorial: Right Isosceles Triangle
Numeric & Algebraic: Right Scalene Triangle
Story of Pythagoras
Euclid Plate
Theorem of Pythagoras
Euclid's Theorem
Py. & Eu. Applied to Other Figures
Story: Pythagoras Triples
Story: Pythagoras & Regular Solids
w/ Equilateral Triangles
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)
3 Triangles (Plate 5)
Congruence, Similarity, Equivalence
Work w/ Divided Squares & Triangles
Intro to Symbols
Triangle Insets
Congruence, Similarity, Equivalence
Design work w/ Iron Material (design; trace, color)
Fractional Relationships
Work w/ Constructive Triangles
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
Area
w/ Geometry Insets - Algebraic
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
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)
Story of How Geometry Got Its Name
© Huidan Xu, 2022
