geometry ams
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TRANSCRIPT
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.
3 © Joan A. Cotter, 2012
Drawing ToolsPaper
International paper21.0cm by 29.7cm
4 © Joan A. Cotter, 2012
Drawing ToolsDrawing board with paper
5 © Joan A. Cotter, 2012
Drawing ToolsDrawing board with paper
Affix tape only at the top.
6 © Joan A. Cotter, 2012
Drawing ToolsT-square
T-square
Right-handed user.
7 © Joan A. Cotter, 2012
Drawing ToolsT-square
Left-handed user.
T-square
8 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
9 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
10 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
11 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
12 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
13 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
14 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
15 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
16 © Joan A. Cotter, 2012
Drawing ToolsT-square
Must hug the edge of the board.
17 © Joan A. Cotter, 2012
Drawing ToolsPencil (mechanical) and eraser
Hold pencil 2-3cm (1 in.) from tip.
18 © Joan A. Cotter, 2012
Drawing ToolsPencil (mechanical) and eraser
Hold pencil 2-3cm (1 in.) from tip.
19 © Joan A. Cotter, 2012
Drawing ToolsPencil (mechanical) and eraser
Hold pencil 2-3cm (1 in.) from tip.
20 © Joan A. Cotter, 2012
Drawing ToolsPencil (mechanical) and eraser
An eraser is an essential tool.
21 © Joan A. Cotter, 2012
Drawing ToolsPencil (mechanical) and eraser
Parallel lines.
22 © Joan A. Cotter, 2012
Drawing ToolsPencil (mechanical) and eraser
Parallel lines.
23 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle (a set square)
60°
30°
90°
24 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
Triangle must hug the t-square.
25 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
This doesn’t work!
26 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
27 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
More parallel lines.
28 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
More parallel lines.
29 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
Perpendicular lines.
30 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
Perpendicular lines.
31 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
32 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
Still more parallel lines.
33 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
Still more parallel lines.
34 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
35 © Joan A. Cotter, 2012
Drawing ToolsThe 30-60 triangle
Intersecting lines.
36 © Joan A. Cotter, 2012
Drawing ToolsThe 45 triangle
45°
45° 90°
37 © Joan A. Cotter, 2012
Drawing ToolsThe 45 triangle
38 © Joan A. Cotter, 2012
Drawing ToolsThe 45 triangle
39 © Joan A. Cotter, 2012
Drawing ToolsThe 45 triangle
Maintain a margin.
40 © Joan A. Cotter, 2012
Drawing ToolsThe 45 triangle
Maintain a margin.
41 © Joan A. Cotter, 2012
Drawing ToolsThe 45 triangle
Maintain a margin.
42 © Joan A. Cotter, 2012
43 © Joan A. Cotter, 2012
Equilateral TrianglesDraw the base line
44 © Joan A. Cotter, 2012
Equilateral TrianglesDraw the base line
Move the t-square down.
45 © Joan A. Cotter, 2012
Equilateral Triangles
Move the t-square down.
Draw the base line
46 © Joan A. Cotter, 2012
Equilateral Triangles
Draw a side.
Draw the sides
47 © Joan A. Cotter, 2012
Equilateral Triangles
Flip the triangle and repeat.
Draw the sides
48 © Joan A. Cotter, 2012
Equilateral Triangles
Flip the triangle and repeat.
Draw the sides
49 © Joan A. Cotter, 2012
Equilateral Triangles
A bit of erasing.
Draw the sides
50 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
51 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
52 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
53 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
12
12
The two halves are congruent.
54 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
Divide it in half another way.
55 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
Divide it in half another way.
56 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
Divide it in half another way.
57 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
Divide it in half another way.
58 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into halves
59 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into thirds
60 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into thirds
61 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into thirds
62 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into thirds
63 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into thirds
A bit of erasing.
64 © Joan A. Cotter, 2012
Equilateral Triangles
Flip the triangle and repeat.
Divide it into thirds
65 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into thirds
66 © Joan A. Cotter, 2012
Equilateral Triangles
13
13
13
Divide it into thirds
67 © Joan A. Cotter, 2012
Equilateral Triangles
13
13
13
Can you figure out how to do it?
Divide it into thirds
68 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
69 © Joan A. Cotter, 2012
Equilateral Triangles
Finding the center of the base line.
Divide it into fourths
70 © Joan A. Cotter, 2012
Equilateral Triangles
Be sure the tick mark intersects the line.
tick mark
Divide it into fourths
71 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
72 © Joan A. Cotter, 2012
Equilateral Triangles
Flip the triangle over and repeat.
Divide it into fourths
73 © Joan A. Cotter, 2012
Equilateral Triangles
How do you draw the last line?
Divide it into fourths
74 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
How do you draw the last line?
75 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
The smaller triangles are similar to the larger triangle.
76 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
14
14
14
14
77 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the rhombuses:
78 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the rhombuses: ABDF,
79 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the rhombuses. ABDF, BCDF,
80 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the rhombuses: ABDF, BCDF, BDEF.
81 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the trapezoids:
82 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the trapezoids: BCEF,
83 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the trapezoids: BCEF, ACDF,
84 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the trapezoids: BCEF, ACDF, ABDE.
85 © Joan A. Cotter, 2012
Equilateral TrianglesTetrahedron
Cut it out….
86 © Joan A. Cotter, 2012
Equilateral TrianglesTetrahedron
and fold on the lines.
87 © Joan A. Cotter, 2012
Equilateral TrianglesTetrahedron
and fold on the lines.
88 © Joan A. Cotter, 2012
Equilateral TrianglesTetrahedron
and fold on the dotted lines.
89 © Joan A. Cotter, 2012
Equilateral TrianglesTetrahedron
and fold on the dotted lines.
90 © Joan A. Cotter, 2012
Equilateral TrianglesTetrahedron
and fold on the dotted lines.
91 © Joan A. Cotter, 2012
Equilateral TrianglesTetrahedron
and fold on the dotted lines.
92 © Joan A. Cotter, 2012
Equilateral TrianglesTetrahedron
and fold on the dotted lines.
93 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into eighths
How could you draw eighths?
94 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into eighths
95 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into eighths
96 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into eighths
Not very esthetic.
97 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into eighths
98 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Can you divide it into sixths?
99 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Can you divide it into sixths?
100 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
101 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
102 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Name an acute triangle:
A
B
CD
E
FG
103 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Name an acute triangle: ACE.
A
B
CD
E
FG
104 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Name a right triangle:
A
B
CD
E
FG
105 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Name a right triangle: ADE,
A
B
CD
E
FG
106 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Name a right triangle: ADE, ABE,
A
B
CD
E
FG
107 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Name a right triangle: ADE, ABE, ABG.
A
B
CD
E
FG
108 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Name an obtuse triangle:
A
B
CD
E
FG
109 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into sixths
Name an obtuse triangle: GCE.
A
B
CD
E
FG
110 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
Can you divide it into twelfths?
111 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
112 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
113 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
114 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
115 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
116 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
Do you see the cube?
117 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
Do you see the cube?
118 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
Do you see the hexagon?
119 © Joan A. Cotter, 2012
Equilateral TrianglesDivide it into twelfths
Do you see the hexagon?
120 © Joan A. Cotter, 2012
Equilateral Triangles
Ninths
121 © Joan A. Cotter, 2012
Equilateral Triangles
Twenty-sevenths
122 © Joan A. Cotter, 2012
Equilateral Triangles
Twenty-sevenths
123 © Joan A. Cotter, 2012
Equilateral Triangles
Twenty-sevenths
124 © Joan A. Cotter, 2012
Equilateral Triangles
Twenty-sevenths
125 © Joan A. Cotter, 2012
Equilateral Triangles
Thirty-seconds
126 © Joan A. Cotter, 2012
Equilateral Triangles
Stephanie, 6, a Montessori child, did 256.
127 © Joan A. Cotter, 2012
Equilateral Triangles
Stephanie, 6, a Montessori child, did 256.
128 © Joan A. Cotter, 2012
Equilateral Triangles
Stephanie, 6, a Montessori child, did 256.
129 © Joan A. Cotter, 2012
Equilateral Triangles
Stephanie, 6, a Montessori child, did 256.
130 © Joan A. Cotter, 2012
Equilateral Triangles
Stephanie, 6, a Montessori child, did 256.
131 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
132 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
133 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
134 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
135 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
136 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
137 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
138 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
139 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
140 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
From Psychogeometry
141 © Joan A. Cotter, 2012
Equilateral TrianglesSpiral
142 © Joan A. Cotter, 2012
143 © Joan A. Cotter, 2012
HexagonsStart with an equilateral triangle
144 © Joan A. Cotter, 2012
HexagonsStart with an equilateral triangle
145 © Joan A. Cotter, 2012
HexagonsStart with an equilateral triangle
146 © Joan A. Cotter, 2012
HexagonsStart with an equilateral triangle
147 © Joan A. Cotter, 2012
HexagonsStart with an equilateral triangle
148 © Joan A. Cotter, 2012
HexagonsStart with an equilateral triangle
149 © Joan A. Cotter, 2012
HexagonsStart with an equilateral triangle
A bit of erasing.
150 © Joan A. Cotter, 2012
HexagonsStart with an equilateral triangle
151 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
152 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
153 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
154 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
Flip the triangle and repeat.
155 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
156 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
157 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
158 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
159 © Joan A. Cotter, 2012
HexagonsStart with a small equilateral triangle
160 © Joan A. Cotter, 2012
HexagonsDraw all the diagonals
161 © Joan A. Cotter, 2012
HexagonsDraw all the diagonals
162 © Joan A. Cotter, 2012
HexagonsDraw all the diagonals
163 © Joan A. Cotter, 2012
HexagonsDraw all the diagonals
164 © Joan A. Cotter, 2012
HexagonsDraw all the diagonals
165 © Joan A. Cotter, 2012
HexagonsDraw all the diagonals
166 © Joan A. Cotter, 2012
HexagonsFind the figures
Another hexagon.
167 © Joan A. Cotter, 2012
HexagonsFind the figures
Another hexagon.
168 © Joan A. Cotter, 2012
HexagonsFind the figures
2 non-congruent rectangles.
169 © Joan A. Cotter, 2012
HexagonsFind the figures
2 non-congruent rectangles.
170 © Joan A. Cotter, 2012
HexagonsFind the figures
2 non-congruent rectangles.
171 © Joan A. Cotter, 2012
HexagonsFind the figures
3 non-congruent right triangles.
172 © Joan A. Cotter, 2012
HexagonsFind the figures
3 non-congruent right triangles.
173 © Joan A. Cotter, 2012
HexagonsFind the figures
3 non-congruent right triangles.
174 © Joan A. Cotter, 2012
HexagonsFind the figures
3 non-congruent right triangles.
175 © Joan A. Cotter, 2012
HexagonsFind the figures
2 non-congruent rhombuses.
176 © Joan A. Cotter, 2012
HexagonsFind the figures
2 non-congruent rhombuses.
177 © Joan A. Cotter, 2012
HexagonsFind the figures
2 non-congruent rhombuses.
178 © Joan A. Cotter, 2012
HexagonsFind the figures
3 similar equilateral triangles.
179 © Joan A. Cotter, 2012
HexagonsFind the figures
3 similar equilateral triangles.
180 © Joan A. Cotter, 2012
HexagonsFind the figures
3 similar equilateral triangles.
181 © Joan A. Cotter, 2012
HexagonsFind the figures
3 similar obtuse triangles.
182 © Joan A. Cotter, 2012
HexagonsFind the figures
3 similar obtuse triangles.
183 © Joan A. Cotter, 2012
HexagonsFind the figures
3 similar obtuse triangles.
184 © Joan A. Cotter, 2012
HexagonsFind the figures
3 similar obtuse triangles.
185 © Joan A. Cotter, 2012
HexagonsFind the figures
3 non-congruent trapezoids.
186 © Joan A. Cotter, 2012
HexagonsFind the figures
3 non-congruent trapezoids.
187 © Joan A. Cotter, 2012
HexagonsFind the figures
3 non-congruent trapezoids.
188 © Joan A. Cotter, 2012
HexagonsFind the figures
3 non-congruent trapezoids.
189 © Joan A. Cotter, 2012
HexagonsFind the figures
2 similar kites.
190 © Joan A. Cotter, 2012
HexagonsFind the figures
2 similar kites.
191 © Joan A. Cotter, 2012
HexagonsFind the figures
2 similar kites.
192 © Joan A. Cotter, 2012
HexagonsConstruct the small star
Apothem: perpendicular line from side to center.
193 © Joan A. Cotter, 2012
HexagonsConstruct the small star
Apothem: perpendicular line from side to center.
194 © Joan A. Cotter, 2012
HexagonsConstruct the small star
Apothem: perpendicular line from side to center.
195 © Joan A. Cotter, 2012
HexagonsConstruct the small star
196 © Joan A. Cotter, 2012
HexagonsConstruct the small star
197 © Joan A. Cotter, 2012
HexagonsConstruct the small star
198 © Joan A. Cotter, 2012
HexagonsConstruct the small star
199 © Joan A. Cotter, 2012
HexagonsConstruct the small star
Ratio of the areas of star and hexagon?
200 © Joan A. Cotter, 2012
HexagonsConstruct the small star
Ratio of the areas of star and hexagon?
201 © Joan A. Cotter, 2012
HexagonsConstruct the large star
202 © Joan A. Cotter, 2012
HexagonsConstruct the large star
203 © Joan A. Cotter, 2012
HexagonsConstruct the large star
204 © Joan A. Cotter, 2012
HexagonsConstruct the large star
205 © Joan A. Cotter, 2012
HexagonsConstruct the large star
206 © Joan A. Cotter, 2012
HexagonsConstruct the large star
207 © Joan A. Cotter, 2012
HexagonsConstruct the large star
208 © Joan A. Cotter, 2012
HexagonsConstruct the large star
209 © Joan A. Cotter, 2012
210 © Joan A. Cotter, 2012
The Clock112
23
4567
8910
11
211 © Joan A. Cotter, 2012
The Clock112
23
4567
8910
11
212 © Joan A. Cotter, 2012
The Clock112
23
4567
8910
11
213 © Joan A. Cotter, 2012
The Clock112
23
4567
8910
11
214 © Joan A. Cotter, 2012
The Clock112
23
4567
8910
11
215 © Joan A. Cotter, 2012
The Clock
12 12
3
4567
8
9
1011
11223
4567
8910
11
216 © Joan A. Cotter, 2012
217 © Joan A. Cotter, 2012
Squares
The 45 triangle.
218 © Joan A. Cotter, 2012
Squares
The 45 triangle is half of a square.
219 © Joan A. Cotter, 2012
SquaresConstruct a square
Draw the base line.
220 © Joan A. Cotter, 2012
Squares
Move the t-square down.
Construct a square
221 © Joan A. Cotter, 2012
Squares
Move the t-square down.
Construct a square
222 © Joan A. Cotter, 2012
Squares
Draw the sides.
Construct a square
223 © Joan A. Cotter, 2012
Squares
Draw the diagonal…
Construct a square
224 © Joan A. Cotter, 2012
Squares
or just a tick mark.
Construct a square
225 © Joan A. Cotter, 2012
Squares
Draw the top.
Construct a square
226 © Joan A. Cotter, 2012
Squares
A bit of erasing.
Construct a square
227 © Joan A. Cotter, 2012
SquaresConstruct a square
228 © Joan A. Cotter, 2012
SquaresHalves
229 © Joan A. Cotter, 2012
SquaresHalves
230 © Joan A. Cotter, 2012
SquaresFourths
231 © Joan A. Cotter, 2012
SquaresEighths
232 © Joan A. Cotter, 2012
SquaresSixteenths
233 © Joan A. Cotter, 2012
SquaresSixteenths
234 © Joan A. Cotter, 2012
SquaresHalves
We need to find the center.
235 © Joan A. Cotter, 2012
SquaresHalves
We need to find the center.
236 © Joan A. Cotter, 2012
SquaresHalves
We need to find the center.
237 © Joan A. Cotter, 2012
SquaresHalves
We need to find the center.
238 © Joan A. Cotter, 2012
SquaresHalves
239 © Joan A. Cotter, 2012
SquaresFourths
240 © Joan A. Cotter, 2012
SquaresFourths
241 © Joan A. Cotter, 2012
SquaresEighths
Find the center of a small square.
242 © Joan A. Cotter, 2012
SquaresEighths
Find the center of a small square.
243 © Joan A. Cotter, 2012
SquaresEighths
Find the center of a small square.
244 © Joan A. Cotter, 2012
SquaresEighths
Find the center of a small square.
245 © Joan A. Cotter, 2012
SquaresEighths
Repeat for the other line.
246 © Joan A. Cotter, 2012
SquaresEighths
247 © Joan A. Cotter, 2012
SquaresSixteenths
248 © Joan A. Cotter, 2012
SquaresSixteenths
249 © Joan A. Cotter, 2012
SquaresSixteenths
250 © Joan A. Cotter, 2012
SquaresRight triangle spiral
251 © Joan A. Cotter, 2012
SquaresRight triangle spiral
252 © Joan A. Cotter, 2012
SquaresRight triangle spiral
253 © Joan A. Cotter, 2012
SquaresRight triangle spiral
254 © Joan A. Cotter, 2012
SquaresRight triangle spiral
255 © Joan A. Cotter, 2012
SquaresRight triangle spiral
256 © Joan A. Cotter, 2012
SquaresRight triangle spiral
257 © Joan A. Cotter, 2012
SquaresRight triangle spiral
258 © Joan A. Cotter, 2012
SquaresRight triangle spiral
259 © Joan A. Cotter, 2012
SquaresRight triangle spiral
Adapted from Psychogeometry
260 © Joan A. Cotter, 2012
SquaresRight triangle design
261 © Joan A. Cotter, 2012
SquaresRight triangle design
262 © Joan A. Cotter, 2012
SquaresRight triangle design
263 © Joan A. Cotter, 2012
SquaresRight triangle design
264 © Joan A. Cotter, 2012
SquaresRight triangle design
265 © Joan A. Cotter, 2012
SquaresRight triangle design
266 © Joan A. Cotter, 2012
SquaresRight triangle design
267 © Joan A. Cotter, 2012
SquaresRight triangle design
268 © Joan A. Cotter, 2012
SquaresRight triangle design
269 © Joan A. Cotter, 2012
SquaresRight triangle design
270 © Joan A. Cotter, 2012
SquaresCircumscribed squares
271 © Joan A. Cotter, 2012
SquaresCircumscribed squares
272 © Joan A. Cotter, 2012
SquaresCircumscribed squares
273 © Joan A. Cotter, 2012
SquaresCircumscribed squares
274 © Joan A. Cotter, 2012
SquaresCircumscribed squares
275 © Joan A. Cotter, 2012
SquaresCircumscribed squares
276 © Joan A. Cotter, 2012
SquaresCircumscribed squares
277 © Joan A. Cotter, 2012
SquaresCircumscribed squares
278 © Joan A. Cotter, 2012
SquaresCircumscribed squares
279 © Joan A. Cotter, 2012
SquaresCircumscribed squares
280 © Joan A. Cotter, 2012
SquaresCircumscribed squares
281 © Joan A. Cotter, 2012
SquaresCircumscribed squares
282 © Joan A. Cotter, 2012
SquaresInscribed squares
283 © Joan A. Cotter, 2012
SquaresInscribed squares
284 © Joan A. Cotter, 2012
SquaresInscribed squares
285 © Joan A. Cotter, 2012
SquaresInscribed squares
Compare the area of the squares.
286 © Joan A. Cotter, 2012
SquaresInscribed squares
Compare the area of the squares.
287 © Joan A. Cotter, 2012
SquaresInscribed squares
288 © Joan A. Cotter, 2012
SquaresInscribed squares
289 © Joan A. Cotter, 2012
SquaresInscribed squares
290 © Joan A. Cotter, 2012
SquaresInscribed squares
291 © Joan A. Cotter, 2012
SquaresInscribed squares
292 © Joan A. Cotter, 2012
SquaresInscribed squares
293 © Joan A. Cotter, 2012
SquaresInscribed squares
294 © Joan A. Cotter, 2012
SquaresInscribed squares
295 © Joan A. Cotter, 2012
296 © Joan A. Cotter, 2012
TangramsConstructing a tangram
297 © Joan A. Cotter, 2012
TangramsConstructing a tangram
298 © Joan A. Cotter, 2012
TangramsConstructing a tangram
299 © Joan A. Cotter, 2012
TangramsConstructing a tangram
300 © Joan A. Cotter, 2012
TangramsConstructing a tangram
301 © Joan A. Cotter, 2012
TangramsConstructing a tangram
302 © Joan A. Cotter, 2012
TangramsConstructing a tangram
303 © Joan A. Cotter, 2012
TangramsConstructing a tangram
3 green figures are equal in area.
304 © Joan A. Cotter, 2012
TangramsConstructing a tangram
3 green figures are equal in area.
305 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
306 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
307 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
308 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
309 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
310 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
311 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
312 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
313 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
No tracing or measuring.
314 © Joan A. Cotter, 2012
TangramsConstruct a tangram arrangement
More examples to construct.
315 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Use the left side of the paper.
316 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Start at the left side of the paper.
317 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
318 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
319 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
320 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
321 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
322 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
323 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
324 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
325 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
326 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
327 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
Reflect the image across the line.
328 © Joan A. Cotter, 2012
TangramsConstruct a tangram reflection
329 © Joan A. Cotter, 2012
330 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
First construct a right triangle.
331 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
332 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
333 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
334 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
335 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
336 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
337 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
338 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
339 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Area of blue squares = yellow sq.
340 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Area of blue squares = yellow sq.
341 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Draw a right triangle.
342 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
343 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
344 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
345 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
346 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
347 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
348 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Turn the t-square upside down.
349 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
350 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
351 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
352 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Do not move the t-square.
353 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Do not move the t-square.
354 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Do not move the t-square.
355 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Do not move the t-square.
356 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
357 © Joan A. Cotter, 2012
Pythagorean TheoremConstruct squares on all sides
Area of blue squares = yellow sq.
358 © Joan A. Cotter, 2012
359 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
A
CB
360 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
Draw the altitude.
A
CB
361 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
Area is half of enclosing rectangle.
CB
A
362 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
Another altitude. Find area.
CB
A
363 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
Another altitude. Find area.
CB
A
364 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
Another altitude. Find area.
A
CB
365 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
Another altitude. Find area.
A
CB
366 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
Altitudes intersect at the orthocenter.
A
CB
367 © Joan A. Cotter, 2012
Triangle AreaConstruct the altitude
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