Montessori Nuggets: geometry

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Geometry Cabinet Contents

Geometry Cabinet Drawers

Drawer I	Contains six circles varying in diameter from ten to five cm
Drawer II	Six variations of rectangles: 10x10 cm square, 10x9, 10x8, 10x7, 10x6, 10x5
Drawer III	Different types of triangles: equilateral, right-angled isosceles, obtuse-angled isosceles, acute-angled isosceles, right-angled scalene, obtuse angled scalene
Drawer IV	Different regular polygons: pentagon, hexagon, heptagon, octagon, nonagon, decagon
Drawer V	Four curvilinear figures: ellipse, quatrefoil, curvilinear triangle, oval
Drawer VI	Four quadrilaterals: rhombus, right trapezoid, isosceles trapezoid, parallelogram

Geometry Cabinet Definitions

Circle	a closed plane curve consisting of all points at a given distance from a point within it called the center
Rectangle	a parallelogram having four right angles
Square	a rectangle having all four sides of equal length
Triangle	closed plane figure having three sides and three angle
Equilateral	All sides are of equal length; an equilateral triangle is also equiangular
Right-angled	Having one right angle (90°)
Obtuse-angled	Having one angle of greater than 90°
Acute-angled	Having angle of less than 90°
Scalene	Having all sides unequal
Isosceles	Having 2 sides equal
Polygon	A figure, usually closed-plane, having three or more usually straight sides
Pentagon	A polygon having 5 angles and 5 sides
Hexagon	A polygon having 6 angles and 6 sides
Heptagon	A polygon having 7 angles and 7 sides
Octagon	A polygon having 8 angles and 8 sides
Nonagon	A polygon having 9 angles and 9 sides
Decagon	A polygon having 10 angles and 10 sides
Ellipse	a plane curve such that the sums of the distances of each point in its periphery from two fixed points, the foci, are equal; (x²/a²) + (y²/b²)=1; if a=b, the ellipse is a circle
Oval	having the general form, shape or outline of an egg, wider at one end than the other
Quatrefoil	leaf composed of four leaflets; a panel-like ornament composed of four lobes radiating from a common center
Curvilinear triangle	triangle consisting of or bound by curved lines
Rhombus	an oblique-angled equilateral parallelogram (any equilateral parallelogram except a square)
Parallelogram	quadrilateral having both pairs of opposite sides parallel to each another
Trapezoid	a quadrilateral plane figure having two parallel and two non-parallel sides
Right trapezoid	trapezoid with one right angle
Isosceles trapezoid	trapezoid with two sides of equal length

Geometry Cabinet Cards:

For each geometric shape in the Montessori geometry cabinet, there are 3 cards on which the shape/inset fits: a solidly filled in one, one with a thick outline and one with a thin outline.

Montessori Scope and Sequence: Elementary

The following links take you to a general scope and sequence for AMI elementary albums.

These Montessori scope and sequences -

follow ages 6-12
do not include remedial (bridge) work
are not set in stone - follow the children before you!
are general (they do not include every single exercise with an album page that might start in year 1 and extend to year 4 or 5) - so when you see a range of 1-3, figure that the album page probably starts in year 1 (or 2 at the latest) and should generally be done by year 3. It could be earlier, but it shouldn't be much later, because the age ranges are pretty generous already. If you want the detailed version that contains every exercise and follow-up, in both Word and Excel formats (easily adapted for personal use), you will find them at Garden of Francis (for elementary) and Garden of Francis (for primary)

The years you see noted in "start" and "end" refer to the year in Montessori elementary - year 1 is the equivalent of age 6, through year 6 which is the equivalent of age 11-12.

The last math work may or may not be completed in elementary and may extend into early adolescence. The other albums are typically easily completed in a 6 year elementary cycle.

Looking for detailed scope and sequence that includes every single exercise of the album page and more specific guidelines and ages? Check with your album provider. Or join us at Keys of the Universe.

Or looking for primary Montessori album scope and sequences? Those are trickier. Presentations that are introduced at age 3, might have extensions that go through age 4, 5, and 6 - so you'll want a very detailed one that includes all exercises for each album page. Ask your album provider for a scope and sequence to correspond with that album set.

Or some general ages are included here in various Montessori Nuggets.

Binomial Cube and Trinomial Cube

What do these represent?

First, let's consider the lid of the binomial:
Look at each side and label each section a or b - so the length of each side is a + b
To find the area of the whole thing, we would multiply the length times the height (a + b)*(a+b)
We can also write that as (a+b)squared

Ok, so let's multiply that out - we end up with these pieces:
a-squared + ab + ab + b-squared

And that is what we have: a red square, 2 rectangles representing a times b, and a blue square.

But we have a CUBE.
So we multiply length by height by depth - (a + b)*(a+b)*(a+b)
Multiplying this out, we end up with a component for each prism within the binomial cube box.

Each piece represents a mathematical principle.

The same concept applies for the trinomial cube, except there are 3 factors: a, b, c

As the children proceed through other mathematical experiences, they will begin seeing these same images in SO many other places. Because these principles surround us.

And most of you only remember those formulas from high school algebra, right!? ;)

Geometry - Protractors

Before a child uses regular protractors in the elementary Montessori environment, the Montessori protractor is introduced. It is used to show the 360 degrees of the circle and its parts (fractions). We show the children how to measure other angles that are not parts of circles, and this learning is extended into measuring any kind of angle with a regular protractor. Yes, there is an album page at elementary called "Regular Protractor." :)

Geometry - Table of Contents

Geometry is a very hands-on experience in Montessori. Nothing theoretical and everything practical here!

Primary:
There is no explicit geometry table of contents for primary. Geometric concepts are explored through the use of the following inexhaustible list of experiences:

Mathematics:
Golden beads (point, line, plane, 3D that becomes a point again)
Wooden Hierarchical Material
Bead Cabinet displayed and used

Language:
Environment Language Games
Card materials
Cutting and other art experiences
Metal Insets (design work - preparation for writing)

Exercises of Practical Life:
visual layout of materials
design work on materials and in art pieces
variety of shapes available in the environment

Sensorial:

Geometry Cabinet

Geometry Cards

Leaf Cabinet

Leaf Cards

Constructive Triangles

Rectangular Box A

Rectangular Box B

Triangular Box

Large Hexagonal Box

Small Hexagonal Box

12 Blue Triangles

Binomial Cube

Trinomial Cube

Graded Geometric Figures

Decanomial Square

Knobless Cylinders

Geometric Solids

Metal Insets (design work)

Elementary - 1st-6th grade:

I. Introduction to Geometry

Original Geometry Story

II. Congruency, Similarity, Equivalency I

Iron Material

Constructive Triangles/Blue Box

III. Polygons

Polygons

IV. Angles I

Angles

Parts of an Angle

How to Measure an Angle

Addition and Subtraction of Angles

Regular Protractor (including bisecting an angle)

V. Lines

Nomenclature

Position of Two Lines

Intersecting Lines

VI. Angles II

Relationships Between Angles

Rel Bet Angles Formed by Parallel Lines Cut by a Transversal

Size Relationships of These Angles

VII. Equivalency II

Pythagoras Plates I and II

Pythagoras with the Constructive Triangles

VIII. Polygons II

Nomenclature of Polygons

Sum of the Angles in Plane Figures

IX. Equivalency III

Equivalency with Iron Material

Euclid’s Plate

X. Area

Concept of Area

Deriving Formulae with the Yellow Material

Deriving Formulae with the Iron Material

XI. Circle I

The Circle Nomenclature

Relationship Between Lines and Circumferences

Relationships Between Two Circumferences

XII. Circle II

Area of a Circle:

Circle as a Special Polygon

Measuring the Circumference

Formula for Area of a Circle

Relationship Between the Apothem and Side of a Plane Figure

XIII. Solid Geometry

Concept of Volume

Equivalence as Related to Solid Figures

Three Important Dimensions

Equivalence Between Prisms with Various Bases

Derivation of the Formula

Solids of Rotation

Volume of the Pyramid

Volume of the Cylinder and Cone

Polyhedrons

Lateral and Total Surface Area of Solids

XIV. Addendum

Geometry Nomenclature Material

Geometry Commands

Geometry Summary

XV. Appendix

Geometry Charts

Geometry Stories

Pages