Geometry AMS

Bridging Geometry Concepts with Art

by Joan A. Cotter, [email protected]

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


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.



Parallel lines.


Drawing ToolsThe 30-60 triangle (a set square)

60°

30°

90°


Drawing ToolsThe 30-60 triangle

Triangle must hug the t-square.



This doesn’t work!





More parallel lines.



More parallel lines.



Perpendicular lines.



Perpendicular lines.





Still more parallel lines.



Still more parallel lines.





Intersecting lines.


Drawing ToolsThe 45 triangle

45°

45° 90°







Maintain a margin.



Maintain a margin.



Maintain a margin.



Equilateral TrianglesDraw the base line


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




Draw the sides



A bit of erasing.

Draw the sides


Equilateral TrianglesDivide it into halves







12

12

The two halves are congruent.



Divide it in half another way.













Equilateral TrianglesDivide it into thirds









A bit of erasing.




Divide it into thirds





13

13

13




13

13

13

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?




How do you draw the last line?



The smaller triangles are similar to the larger triangle.



14

14

14

14



A

B

CD

E

F

Name the rhombuses:



A

B

CD

E

F

Name the rhombuses: ABDF,



A

B

CD

E

F

Name the rhombuses. ABDF, BCDF,



A

B

CD

E

F

Name the rhombuses: ABDF, BCDF, BDEF.



A

B

CD

E

F

Name the trapezoids:



A

B

CD

E

F

Name the trapezoids: BCEF,



A

B

CD

E

F

Name the trapezoids: BCEF, ACDF,



A

B

CD

E

F

Name the trapezoids: BCEF, ACDF, ABDE.


Equilateral TrianglesTetrahedron

Cut it out….



and fold on the lines.



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?



Can you divide it into sixths?







Name an acute triangle:

A

B

CD

E

FG



Name an acute triangle: ACE.

A

B

CD

E

FG



Name a right triangle:

A

B

CD

E

FG



Name a right triangle: ADE,

A

B

CD

E

FG



Name a right triangle: ADE, ABE,

A

B

CD

E

FG



Name a right triangle: ADE, ABE, ABG.

A

B

CD

E

FG



Name an obtuse triangle:

A

B

CD

E

FG



Name an obtuse triangle: GCE.

A

B

CD

E

FG


Equilateral TrianglesDivide it into twelfths

Can you divide it into twelfths?













Do you see the cube?



Do you see the cube?



Do you see the hexagon?



Do you see the hexagon?



Ninths



Twenty-sevenths



Twenty-sevenths



Twenty-sevenths



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.



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.



2 similar kites.



2 similar kites.


HexagonsConstruct the small star

Apothem: perpendicular line from side to center.

















Ratio of the areas of star and hexagon?



Ratio of the areas of star and hexagon?


HexagonsConstruct the large star

















The Clock112

23

4567

8910

11


The Clock112

23

4567

8910

11


The Clock112

23

4567

8910

11


The Clock112

23

4567

8910

11


The Clock112

23

4567

8910

11


The Clock

12 12

3

4567

8

9

1011

11223

4567

8910

11



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


SquaresHalves


SquaresFourths


SquaresEighths


SquaresSixteenths


SquaresSixteenths


SquaresHalves

We need to find the center.


SquaresHalves



SquaresHalves



SquaresHalves



SquaresHalves


SquaresFourths


SquaresFourths


SquaresEighths

Find the center of a small square.


SquaresEighths



SquaresEighths



SquaresEighths



SquaresEighths

Repeat for the other line.


SquaresEighths


SquaresSixteenths


SquaresSixteenths


SquaresSixteenths


SquaresRight triangle spiral



















Adapted from Psychogeometry


SquaresRight triangle design




















SquaresCircumscribed squares
























SquaresInscribed squares







Compare the area of the squares.



Compare the area of the squares.



















TangramsConstructing a tangram















3 green figures are equal in area.



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

A

CB



Draw the altitude.

A

CB



Area is half of enclosing rectangle.

CB

A



Another altitude. Find area.

CB

A




CB

A




A

CB




A

CB



Altitudes intersect at the orthocenter.

A

CB



3 areas slightly different. Why?

A

CB

Bridging Geometry Concepts with Art

by Joan A. Cotter, [email protected]

AMS ConferenceMarch 17, 2012

San Francisco CA

Presentations available: rightstartmath.com

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Geometry AMS

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