Geometry AMS

transcript

Bridging Geometry Concepts with Art

by Joan A. Cotter, Ph.D.joancotter@rightstartmath.com

AMS ConferenceMarch 17, 2012

San Francisco CA

Presentations available: rightstartmath.com

2 © Joan A. Cotter, 2012

Drawing ToolsPaper

North American paper11 in. by 8.5 in.

Drawing ToolsPaper

International paper21.0cm by 29.7cm

Drawing ToolsDrawing board with paper

Affix tape only at the top.

Drawing ToolsT-square

T-square

Right-handed user.

Left-handed user.

T-square

Must hug the edge of the board.

Drawing ToolsPencil (mechanical) and eraser

Hold pencil 2-3cm (1 in.) from tip.

An eraser is an essential tool.

Parallel lines.

Drawing ToolsThe 30-60 triangle (a set square)

Drawing ToolsThe 30-60 triangle

Triangle must hug the t-square.

This doesn’t work!

More parallel lines.

Perpendicular lines.

Still more parallel lines.

Intersecting lines.

Drawing ToolsThe 45 triangle

45° 90°

Maintain a margin.

Equilateral TrianglesDraw the base line

Move the t-square down.

Equilateral Triangles

Draw the base line

Draw a side.

Draw the sides

Flip the triangle and repeat.

Draw the sides

A bit of erasing.

Draw the sides

Equilateral TrianglesDivide it into halves

The two halves are congruent.

Divide it in half another way.

Equilateral TrianglesDivide it into thirds

A bit of erasing.

Divide it into thirds

Can you figure out how to do it?

Equilateral TrianglesDivide it into fourths

Finding the center of the base line.

Divide it into fourths

Be sure the tick mark intersects the line.

tick mark

Flip the triangle over and repeat.

How do you draw the last line?

The smaller triangles are similar to the larger triangle.

Name the rhombuses:

Name the rhombuses: ABDF,

Name the rhombuses. ABDF, BCDF,

Name the rhombuses: ABDF, BCDF, BDEF.

Name the trapezoids:

Name the trapezoids: BCEF,

Name the trapezoids: BCEF, ACDF,

Name the trapezoids: BCEF, ACDF, ABDE.

Equilateral TrianglesTetrahedron

Cut it out….

and fold on the lines.

and fold on the dotted lines.

Equilateral TrianglesDivide it into eighths

How could you draw eighths?

Not very esthetic.

Equilateral TrianglesDivide it into sixths

Can you divide it into sixths?

Name an acute triangle:

Name an acute triangle: ACE.

Name a right triangle:

Name a right triangle: ADE,

Name a right triangle: ADE, ABE,

Name a right triangle: ADE, ABE, ABG.

Name an obtuse triangle:

Name an obtuse triangle: GCE.

Equilateral TrianglesDivide it into twelfths

Can you divide it into twelfths?

Do you see the cube?

Do you see the hexagon?

Ninths

Twenty-sevenths

Thirty-seconds

Stephanie, 6, a Montessori child, did 256.

Equilateral TrianglesSpiral

From Psychogeometry

HexagonsStart with an equilateral triangle

A bit of erasing.

HexagonsStart with a small equilateral triangle

HexagonsDraw all the diagonals

HexagonsFind the figures

Another hexagon.

2 non-congruent rectangles.

3 non-congruent right triangles.

2 non-congruent rhombuses.

3 similar equilateral triangles.

3 similar obtuse triangles.

3 non-congruent trapezoids.

2 similar kites.

HexagonsConstruct the small star

Apothem: perpendicular line from side to center.

Ratio of the areas of star and hexagon?

HexagonsConstruct the large star

The Clock112

The Clock

Squares

The 45 triangle.

Squares

The 45 triangle is half of a square.

SquaresConstruct a square

Draw the base line.

Squares

Construct a square

Squares

Construct a square

Squares

Draw the sides.

Construct a square

Squares

Draw the diagonal…

Construct a square

Squares

or just a tick mark.

Construct a square

Squares

Draw the top.

Construct a square

Squares

A bit of erasing.

Construct a square

SquaresConstruct a square

SquaresHalves

SquaresFourths

SquaresEighths

SquaresSixteenths

SquaresHalves

We need to find the center.

SquaresHalves

SquaresFourths

SquaresEighths

Find the center of a small square.

SquaresEighths

Repeat for the other line.

SquaresEighths

SquaresSixteenths

SquaresRight triangle spiral

Adapted from Psychogeometry

SquaresRight triangle design

SquaresCircumscribed squares

SquaresInscribed squares

Compare the area of the squares.

TangramsConstructing a tangram

3 green figures are equal in area.

TangramsConstruct a tangram arrangement

No tracing or measuring.

More examples to construct.

TangramsConstruct a tangram reflection

Use the left side of the paper.

Start at the left side of the paper.

Reflect the image across the line.

Pythagorean TheoremConstruct squares on all sides

First construct a right triangle.

Area of blue squares = yellow sq.

Draw a right triangle.

Turn the t-square upside down.

Do not move the t-square.

Triangle AreaConstruct the altitude

Draw the altitude.

Area is half of enclosing rectangle.

Another altitude. Find area.

Altitudes intersect at the orthocenter.

3 areas slightly different. Why?

Bridging Geometry Concepts with Art

by Joan A. Cotter, Ph.D.joancotter@rightstartmath.com

AMS ConferenceMarch 17, 2012

San Francisco CA

Presentations available: rightstartmath.com

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