Bridging Geometry Concepts with Art                by Joan A. Cotter, Ph.D.             joancotter@rightstartmath.comAMS C...
Drawing Tools          Paper      North American paper        11 in. by 8.5 in.2                            © Joan A. Cott...
Drawing Tools          Paper       International paper       21.0cm by 29.7cm3                            © Joan A. Cotter...
Drawing Tools    Drawing board with paper4                              © Joan A. Cotter, 2012
Drawing Tools    Drawing board with paper    Affix tape only at the top.5                                 © Joan A. Cotter...
Drawing Tools          T-square            T-square    Right-handed user.6                        © Joan A. Cotter, 2012
Drawing Tools          T-square            T-square    Left-handed user.7                       © Joan A. Cotter, 2012
Drawing Tools            T-square    Must hug the edge of the board.8                                 © Joan A. Cotter, 2012
Drawing Tools            T-square    Must hug the edge of the board.9                                 © Joan A. Cotter, 2012
Drawing Tools             T-square     Must hug the edge of the board.10                                 © Joan A. Cotter,...
Drawing Tools             T-square     Must hug the edge of the board.11                                 © Joan A. Cotter,...
Drawing Tools             T-square     Must hug the edge of the board.12                                 © Joan A. Cotter,...
Drawing Tools             T-square     Must hug the edge of the board.13                                 © Joan A. Cotter,...
Drawing Tools             T-square     Must hug the edge of the board.14                                 © Joan A. Cotter,...
Drawing Tools             T-square     Must hug the edge of the board.15                                 © Joan A. Cotter,...
Drawing Tools             T-square     Must hug the edge of the board.16                                 © Joan A. Cotter,...
Drawing Tools     Pencil (mechanical) and eraser     Hold pencil 2-3cm (1 in.) from tip.17                                ...
Drawing Tools     Pencil (mechanical) and eraser     Hold pencil 2-3cm (1 in.) from tip.18                                ...
Drawing Tools     Pencil (mechanical) and eraser     Hold pencil 2-3cm (1 in.) from tip.19                                ...
Drawing Tools     Pencil (mechanical) and eraser     An eraser is an essential tool.20                                    ...
Drawing Tools     Pencil (mechanical) and eraser     Parallel lines.21                                    © Joan A. Cotter...
Drawing Tools     Pencil (mechanical) and eraser     Parallel lines.22                                    © Joan A. Cotter...
Drawing Tools     The 30-60 triangle (a set square)                       30°              60°      90°23                 ...
Drawing Tools          The 30-60 triangle     Triangle must hug the t-square.24                                 © Joan A. ...
Drawing Tools          The 30-60 triangle     This doesn’t work!25                             © Joan A. Cotter, 2012
Drawing Tools      The 30-60 triangle26                         © Joan A. Cotter, 2012
Drawing Tools          The 30-60 triangle     More parallel lines.27                             © Joan A. Cotter, 2012
Drawing Tools          The 30-60 triangle     More parallel lines.28                             © Joan A. Cotter, 2012
Drawing Tools         The 30-60 triangle     Perpendicular lines.29                            © Joan A. Cotter, 2012
Drawing Tools         The 30-60 triangle     Perpendicular lines.30                            © Joan A. Cotter, 2012
Drawing Tools      The 30-60 triangle31                         © Joan A. Cotter, 2012
Drawing Tools          The 30-60 triangle     Still more parallel lines.32                                © Joan A. Cotter...
Drawing Tools          The 30-60 triangle     Still more parallel lines.33                                © Joan A. Cotter...
Drawing Tools      The 30-60 triangle34                         © Joan A. Cotter, 2012
Drawing Tools          The 30-60 triangle     Intersecting lines.35                             © Joan A. Cotter, 2012
Drawing Tools       The 45 triangle                         45°            45°          90°36                             ...
Drawing Tools       The 45 triangle37                       © Joan A. Cotter, 2012
Drawing Tools       The 45 triangle38                       © Joan A. Cotter, 2012
Drawing Tools           The 45 triangle     Maintain a margin.39                           © Joan A. Cotter, 2012
Drawing Tools           The 45 triangle     Maintain a margin.40                           © Joan A. Cotter, 2012
Drawing Tools           The 45 triangle     Maintain a margin.41                           © Joan A. Cotter, 2012
42   © Joan A. Cotter, 2012
Equilateral Triangles          Draw the base line43                             © Joan A. Cotter, 2012
Equilateral Triangles          Draw the base line      Move the t-square down.44                              © Joan A. Co...
Equilateral Triangles          Draw the base line      Move the t-square down.45                              © Joan A. Co...
Equilateral Triangles            Draw the sides      Draw a side.46                           © Joan A. Cotter, 2012
Equilateral Triangles            Draw the sides      Flip the triangle and repeat.47                                    © ...
Equilateral Triangles            Draw the sides      Flip the triangle and repeat.48                                    © ...
Equilateral Triangles            Draw the sides      A bit of erasing.49                           © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into halves50                               © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into halves51                               © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into halves52                               © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into halves                  1      1                  2      2      The two halve...
Equilateral Triangles          Divide it into halves      Divide it in half another way.54                                ...
Equilateral Triangles          Divide it into halves      Divide it in half another way.55                                ...
Equilateral Triangles          Divide it into halves      Divide it in half another way.56                                ...
Equilateral Triangles          Divide it into halves      Divide it in half another way.57                                ...
Equilateral Triangles         Divide it into halves58                               © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into thirds59                               © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into thirds60                               © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into thirds61                               © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into thirds62                               © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into thirds      A bit of erasing.63                                © Joan A. Cot...
Equilateral Triangles          Divide it into thirds      Flip the triangle and repeat.64                                 ...
Equilateral Triangles         Divide it into thirds65                               © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into thirds                  1       1                  3       3                 ...
Equilateral Triangles          Divide it into thirds                       1                       3                  1   ...
Equilateral Triangles         Divide it into fourths68                                © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into fourths      Finding the center of the base line.69                          ...
Equilateral Triangles           Divide it into fourths                                     tick mark      Be sure the tick...
Equilateral Triangles         Divide it into fourths71                                © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into fourths      Flip the triangle over and repeat.72                           ...
Equilateral Triangles         Divide it into fourths      How do you draw the last line?73                                ...
Equilateral Triangles         Divide it into fourths      How do you draw the last line?74                                ...
Equilateral Triangles                      Divide it into fourths     The smaller triangles are similar to the larger tria...
Equilateral Triangles         Divide it into fourths                      1                      4                      1 ...
Equilateral Triangles               Divide it into fourths                           A                     B             F...
Equilateral Triangles               Divide it into fourths                           A                     B             F...
Equilateral Triangles               Divide it into fourths                           A                     B             F...
Equilateral Triangles               Divide it into fourths                           A                     B             F...
Equilateral Triangles                 Divide it into fourths                             A                       B        ...
Equilateral Triangles                 Divide it into fourths                             A                       B        ...
Equilateral Triangles                 Divide it into fourths                             A                       B        ...
Equilateral Triangles                 Divide it into fourths                             A                       B        ...
Equilateral Triangles             Tetrahedron      Cut it out….85                           © Joan A. Cotter, 2012
Equilateral Triangles             Tetrahedron      and fold on the lines.86                             © Joan A. Cotter, ...
Equilateral Triangles             Tetrahedron      and fold on the lines.87                             © Joan A. Cotter, ...
Equilateral Triangles             Tetrahedron      and fold on the dotted lines.88                                    © Jo...
Equilateral Triangles             Tetrahedron      and fold on the dotted lines.89                                    © Jo...
Equilateral Triangles             Tetrahedron      and fold on the dotted lines.90                                    © Jo...
Equilateral Triangles             Tetrahedron      and fold on the dotted lines.91                                    © Jo...
Equilateral Triangles             Tetrahedron      and fold on the dotted lines.92                                    © Jo...
Equilateral Triangles         Divide it into eighths      How could you draw eighths?93                                  ©...
Equilateral Triangles         Divide it into eighths94                                © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into eighths95                                © Joan A. Cotter, 2012
Equilateral Triangles         Divide it into eighths      Not very esthetic.96                                © Joan A. Co...
Equilateral Triangles         Divide it into eighths97                                © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into sixths      Can you divide it into sixths?98                                ...
Equilateral Triangles          Divide it into sixths      Can you divide it into sixths?99                                ...
Equilateral Triangles          Divide it into sixths100                               © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into sixths101                               © Joan A. Cotter, 2012
Equilateral Triangles                 Divide it into sixths                             A                       B         ...
Equilateral Triangles                 Divide it into sixths                             A                       B         ...
Equilateral Triangles                  Divide it into sixths                              A                        B      ...
Equilateral Triangles                 Divide it into sixths                             A                       B         ...
Equilateral Triangles                 Divide it into sixths                             A                       B         ...
Equilateral Triangles                 Divide it into sixths                             A                       B         ...
Equilateral Triangles                 Divide it into sixths                             A                       B         ...
Equilateral Triangles                Divide it into sixths                            A                      B            ...
Equilateral Triangles          Divide it into twelfths       Can you divide it into twelfths?110                          ...
Equilateral Triangles          Divide it into twelfths111                                 © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into twelfths112                                 © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into twelfths113                                 © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into twelfths114                                 © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into twelfths115                                 © Joan A. Cotter, 2012
Equilateral Triangles          Divide it into twelfths       Do you see the cube?116                                 © Joa...
Equilateral Triangles          Divide it into twelfths       Do you see the cube?117                                 © Joa...
Equilateral Triangles          Divide it into twelfths       Do you see the hexagon?118                                 © ...
Equilateral Triangles          Divide it into twelfths       Do you see the hexagon?119                                 © ...
Equilateral Triangles               Ninths120                           © Joan A. Cotter, 2012
Equilateral Triangles           Twenty-sevenths121                           © Joan A. Cotter, 2012
Equilateral Triangles           Twenty-sevenths122                           © Joan A. Cotter, 2012
Equilateral Triangles           Twenty-sevenths123                           © Joan A. Cotter, 2012
Equilateral Triangles           Twenty-sevenths124                           © Joan A. Cotter, 2012
Equilateral Triangles            Thirty-seconds125                           © Joan A. Cotter, 2012
Equilateral Triangles       Stephanie, 6, a Montessori child, did 256.126                                       © Joan A. ...
Equilateral Triangles       Stephanie, 6, a Montessori child, did 256.127                                       © Joan A. ...
Equilateral Triangles       Stephanie, 6, a Montessori child, did 256.128                                       © Joan A. ...
Equilateral Triangles       Stephanie, 6, a Montessori child, did 256.129                                       © Joan A. ...
Equilateral Triangles       Stephanie, 6, a Montessori child, did 256.130                                       © Joan A. ...
Equilateral Triangles              Spiral131                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral132                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral133                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral134                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral135                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral136                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral137                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral138                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral139                           © Joan A. Cotter, 2012
Equilateral Triangles                Spiral       From Psychogeometry140                           © Joan A. Cotter, 2012
Equilateral Triangles              Spiral141                           © Joan A. Cotter, 2012
142   © Joan A. Cotter, 2012
Hexagons      Start with an equilateral triangle143                                        © Joan A. Cotter, 2012
Hexagons      Start with an equilateral triangle144                                        © Joan A. Cotter, 2012
Hexagons      Start with an equilateral triangle145                                        © Joan A. Cotter, 2012
Hexagons      Start with an equilateral triangle146                                        © Joan A. Cotter, 2012
Hexagons      Start with an equilateral triangle147                                        © Joan A. Cotter, 2012
Hexagons      Start with an equilateral triangle148                                        © Joan A. Cotter, 2012
Hexagons      Start with an equilateral triangle       A bit of erasing.149                                        © Joan ...
Hexagons      Start with an equilateral triangle150                                        © Joan A. Cotter, 2012
Hexagons      Start with a small equilateral triangle151                                             © Joan A. Cotter, 2012
Hexagons      Start with a small equilateral triangle152                                             © Joan A. Cotter, 2012
Hexagons      Start with a small equilateral triangle153                                             © Joan A. Cotter, 2012
Hexagons      Start with a small equilateral triangle         Flip the triangle and repeat.154                            ...
Hexagons      Start with a small equilateral triangle155                                             © Joan A. Cotter, 2012
Hexagons      Start with a small equilateral triangle156                                             © Joan A. Cotter, 2012
Hexagons      Start with a small equilateral triangle157                                             © Joan A. Cotter, 2012
Hexagons      Start with a small equilateral triangle158                                             © Joan A. Cotter, 2012
Hexagons      Start with a small equilateral triangle159                                             © Joan A. Cotter, 2012
Hexagons      Draw all the diagonals160                            © Joan A. Cotter, 2012
Hexagons      Draw all the diagonals161                            © Joan A. Cotter, 2012
Hexagons      Draw all the diagonals162                            © Joan A. Cotter, 2012
Hexagons      Draw all the diagonals163                            © Joan A. Cotter, 2012
Hexagons      Draw all the diagonals164                            © Joan A. Cotter, 2012
Hexagons      Draw all the diagonals165                            © Joan A. Cotter, 2012
Hexagons           Find the figures      Another hexagon.166                           © Joan A. Cotter, 2012
Hexagons           Find the figures      Another hexagon.167                           © Joan A. Cotter, 2012
Hexagons           Find the figures      2 non-congruent rectangles.168                                 © Joan A. Cotter, ...
Hexagons           Find the figures      2 non-congruent rectangles.169                                 © Joan A. Cotter, ...
Hexagons           Find the figures      2 non-congruent rectangles.170                                 © Joan A. Cotter, ...
Hexagons            Find the figures      3 non-congruent right triangles.171                                  © Joan A. C...
Hexagons            Find the figures      3 non-congruent right triangles.172                                  © Joan A. C...
Hexagons            Find the figures      3 non-congruent right triangles.173                                  © Joan A. C...
Hexagons            Find the figures      3 non-congruent right triangles.174                                  © Joan A. C...
Hexagons           Find the figures      2 non-congruent rhombuses.175                                © Joan A. Cotter, 2012
Hexagons           Find the figures      2 non-congruent rhombuses.176                                © Joan A. Cotter, 2012
Hexagons           Find the figures      2 non-congruent rhombuses.177                                © Joan A. Cotter, 2012
Hexagons            Find the figures      3 similar equilateral triangles.178                                      © Joan ...
Hexagons            Find the figures      3 similar equilateral triangles.179                                      © Joan ...
Hexagons            Find the figures      3 similar equilateral triangles.180                                      © Joan ...
Hexagons            Find the figures      3 similar obtuse triangles.181                                 © Joan A. Cotter,...
Hexagons            Find the figures      3 similar obtuse triangles.182                                 © Joan A. Cotter,...
Hexagons            Find the figures      3 similar obtuse triangles.183                                 © Joan A. Cotter,...
Hexagons            Find the figures      3 similar obtuse triangles.184                                 © Joan A. Cotter,...
Hexagons           Find the figures      3 non-congruent trapezoids.185                                 © Joan A. Cotter, ...
Hexagons           Find the figures      3 non-congruent trapezoids.186                                 © Joan A. Cotter, ...
Hexagons           Find the figures      3 non-congruent trapezoids.187                                 © Joan A. Cotter, ...
Hexagons           Find the figures      3 non-congruent trapezoids.188                                 © Joan A. Cotter, ...
Hexagons            Find the figures      2 similar kites.189                            © Joan A. Cotter, 2012
Hexagons            Find the figures      2 similar kites.190                            © Joan A. Cotter, 2012
Hexagons            Find the figures      2 similar kites.191                            © Joan A. Cotter, 2012
Hexagons               Construct the small star Apothem: perpendicular line from side to center.192                       ...
Hexagons               Construct the small star Apothem: perpendicular line from side to center.193                       ...
Hexagons               Construct the small star Apothem: perpendicular line from side to center.194                       ...
Hexagons      Construct the small star195                              © Joan A. Cotter, 2012
Hexagons      Construct the small star196                              © Joan A. Cotter, 2012
Hexagons      Construct the small star197                              © Joan A. Cotter, 2012
Hexagons      Construct the small star198                              © Joan A. Cotter, 2012
Hexagons                Construct the small star      Ratio of the areas of star and hexagon?199                          ...
Hexagons                Construct the small star      Ratio of the areas of star and hexagon?200                          ...
Hexagons      Construct the large star201                              © Joan A. Cotter, 2012
Hexagons      Construct the large star202                              © Joan A. Cotter, 2012
Hexagons      Construct the large star203                              © Joan A. Cotter, 2012
Hexagons      Construct the large star204                              © Joan A. Cotter, 2012
Hexagons      Construct the large star205                              © Joan A. Cotter, 2012
Hexagons      Construct the large star206                              © Joan A. Cotter, 2012
Hexagons      Construct the large star207                              © Joan A. Cotter, 2012
Hexagons      Construct the large star208                              © Joan A. Cotter, 2012
209   © Joan A. Cotter, 2012
The Clock        1112 1      10       2      9         3      8        4        7 6 5210                             © Joa...
The Clock        1112 1      10       2      9         3      8        4        7 6 5211                             © Joa...
The Clock        1112 1      10       2      9         3      8        4        7 6 5212                             © Joa...
The Clock        1112 1      10       2      9         3      8        4        7 6 5213                             © Joa...
The Clock        1112 1      10       2      9         3      8        4        7 6 5214                             © Joa...
The Clock        1112 1      10       2      9         3      8        4        7 6 5                           11 12 1   ...
216   © Joan A. Cotter, 2012
Squares      The 45 triangle.217                      © Joan A. Cotter, 2012
Squares      The 45 triangle is half of a square.218                                  © Joan A. Cotter, 2012
Squares          Construct a square      Draw the base line.219                            © Joan A. Cotter, 2012
Squares         Construct a square      Move the t-square down.220                             © Joan A. Cotter, 2012
Squares         Construct a square      Move the t-square down.221                             © Joan A. Cotter, 2012
Squares         Construct a square      Draw the sides.222                           © Joan A. Cotter, 2012
Squares         Construct a square      Draw the diagonal…223                           © Joan A. Cotter, 2012
Squares          Construct a square      or just a tick mark.224                            © Joan A. Cotter, 2012
Squares          Construct a square      Draw the top.225                            © Joan A. Cotter, 2012
Squares          Construct a square      A bit of erasing.226                            © Joan A. Cotter, 2012
Squares      Construct a square227                        © Joan A. Cotter, 2012
Squares       Halves228             © Joan A. Cotter, 2012
Squares       Halves229             © Joan A. Cotter, 2012
Squares       Fourths230              © Joan A. Cotter, 2012
Squares       Eighths231              © Joan A. Cotter, 2012
Squares      Sixteenths232                © Joan A. Cotter, 2012
Squares      Sixteenths233                © Joan A. Cotter, 2012
Squares               Halves      We need to find the center.234                                 © Joan A. Cotter, 2012
Squares               Halves      We need to find the center.235                                 © Joan A. Cotter, 2012
Squares               Halves      We need to find the center.236                                 © Joan A. Cotter, 2012
Squares               Halves      We need to find the center.237                                 © Joan A. Cotter, 2012
Squares       Halves238             © Joan A. Cotter, 2012
Squares       Fourths239              © Joan A. Cotter, 2012
Squares       Fourths240              © Joan A. Cotter, 2012
Squares              Eighths      Find the center of a small square.241                                 © Joan A. Cotter, ...
Squares              Eighths      Find the center of a small square.242                                 © Joan A. Cotter, ...
Squares              Eighths      Find the center of a small square.243                                 © Joan A. Cotter, ...
Squares              Eighths      Find the center of a small square.244                                 © Joan A. Cotter, ...
Squares               Eighths      Repeat for the other line.245                                © Joan A. Cotter, 2012
Squares       Eighths246              © Joan A. Cotter, 2012
Squares      Sixteenths247                © Joan A. Cotter, 2012
Squares      Sixteenths248                © Joan A. Cotter, 2012
Squares      Sixteenths249                © Joan A. Cotter, 2012
Squares      Right triangle spiral250                           © Joan A. Cotter, 2012
Squares      Right triangle spiral251                           © Joan A. Cotter, 2012
Squares      Right triangle spiral252                           © Joan A. Cotter, 2012
Squares      Right triangle spiral253                           © Joan A. Cotter, 2012
Squares      Right triangle spiral254                           © Joan A. Cotter, 2012
Squares      Right triangle spiral255                           © Joan A. Cotter, 2012
Squares      Right triangle spiral256                           © Joan A. Cotter, 2012
Squares      Right triangle spiral257                           © Joan A. Cotter, 2012
Squares      Right triangle spiral258                           © Joan A. Cotter, 2012
Squares          Right triangle spiral      Adapted from Psychogeometry259                                 © Joan A. Cotte...
Squares      Right triangle design260                           © Joan A. Cotter, 2012
Squares      Right triangle design261                           © Joan A. Cotter, 2012
Squares      Right triangle design262                           © Joan A. Cotter, 2012
Squares      Right triangle design263                           © Joan A. Cotter, 2012
Squares      Right triangle design264                           © Joan A. Cotter, 2012
Squares      Right triangle design265                           © Joan A. Cotter, 2012
Squares      Right triangle design266                           © Joan A. Cotter, 2012
Squares      Right triangle design267                           © Joan A. Cotter, 2012
Squares      Right triangle design268                           © Joan A. Cotter, 2012
Squares      Right triangle design269                           © Joan A. Cotter, 2012
Squares      Circumscribed squares270                           © Joan A. Cotter, 2012
Squares      Circumscribed squares271                           © Joan A. Cotter, 2012
Squares      Circumscribed squares272                           © Joan A. Cotter, 2012
Squares      Circumscribed squares273                           © Joan A. Cotter, 2012
Squares      Circumscribed squares274                           © Joan A. Cotter, 2012
Squares      Circumscribed squares275                           © Joan A. Cotter, 2012
Squares      Circumscribed squares276                           © Joan A. Cotter, 2012
Squares      Circumscribed squares277                           © Joan A. Cotter, 2012
Squares      Circumscribed squares278                           © Joan A. Cotter, 2012
Squares      Circumscribed squares279                           © Joan A. Cotter, 2012
Squares      Circumscribed squares280                           © Joan A. Cotter, 2012
Squares      Circumscribed squares281                           © Joan A. Cotter, 2012
Squares      Inscribed squares282                       © Joan A. Cotter, 2012
Squares      Inscribed squares283                       © Joan A. Cotter, 2012
Squares      Inscribed squares284                       © Joan A. Cotter, 2012
Squares          Inscribed squares      Compare the area of the squares.285                               © Joan A. Cotter...
Squares          Inscribed squares      Compare the area of the squares.286                               © Joan A. Cotter...
Squares      Inscribed squares287                       © Joan A. Cotter, 2012
Squares      Inscribed squares288                       © Joan A. Cotter, 2012
Squares      Inscribed squares289                       © Joan A. Cotter, 2012
Squares      Inscribed squares290                       © Joan A. Cotter, 2012
Squares      Inscribed squares291                       © Joan A. Cotter, 2012
Squares      Inscribed squares292                       © Joan A. Cotter, 2012
Squares      Inscribed squares293                       © Joan A. Cotter, 2012
Squares      Inscribed squares294                       © Joan A. Cotter, 2012
295   © Joan A. Cotter, 2012
Tangrams      Constructing a tangram296                            © Joan A. Cotter, 2012
Tangrams      Constructing a tangram297                            © Joan A. Cotter, 2012
Tangrams      Constructing a tangram298                            © Joan A. Cotter, 2012
Tangrams      Constructing a tangram299                            © Joan A. Cotter, 2012
Tangrams      Constructing a tangram300                            © Joan A. Cotter, 2012
Tangrams      Constructing a tangram301                            © Joan A. Cotter, 2012
Tangrams      Constructing a tangram302                            © Joan A. Cotter, 2012
Tangrams        Constructing a tangram      3 green figures are equal in area.303                                  © Joan ...
Tangrams        Constructing a tangram      3 green figures are equal in area.304                                  © Joan ...
Tangrams      Construct a tangram arrangement       No tracing or measuring.305                                     © Joan...
Tangrams      Construct a tangram arrangement       No tracing or measuring.306                                     © Joan...
Tangrams      Construct a tangram arrangement       No tracing or measuring.307                                     © Joan...
Tangrams      Construct a tangram arrangement       No tracing or measuring.308                                     © Joan...
Tangrams      Construct a tangram arrangement       No tracing or measuring.309                                     © Joan...
Tangrams      Construct a tangram arrangement       No tracing or measuring.310                                     © Joan...
Tangrams      Construct a tangram arrangement       No tracing or measuring.311                                     © Joan...
Tangrams      Construct a tangram arrangement       No tracing or measuring.312                                     © Joan...
Tangrams      Construct a tangram arrangement       No tracing or measuring.313                                     © Joan...
Tangrams      Construct a tangram arrangement       More examples to construct.314                                     © J...
Tangrams      Construct a tangram reflection      Use the left side of the paper.315                                     ©...
Tangrams      Construct a tangram reflection      Start at the left side of the paper.316                                 ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.317                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.318                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.319                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.320                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.321                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.322                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.323                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.324                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.325                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.326                                   ...
Tangrams      Construct a tangram reflection      Reflect the image across the line.327                                   ...
Tangrams      Construct a tangram reflection328                                    © Joan A. Cotter, 2012
329   © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides        First construct a right triangle.330                     ...
Pythagorean Theorem        Construct squares on all sides331                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides332                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides333                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides334                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides335                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides336                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides337                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides338                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides        Area of blue squares = yellow sq.339                     ...
Pythagorean Theorem        Construct squares on all sides        Area of blue squares = yellow sq.340                     ...
Pythagorean Theorem        Construct squares on all sides        Draw a right triangle.341                                ...
Pythagorean Theorem        Construct squares on all sides342                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides343                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides344                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides345                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides346                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides347                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides        Turn the t-square upside down.348                        ...
Pythagorean Theorem        Construct squares on all sides349                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides350                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides351                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides        Do not move the t-square.352                             ...
Pythagorean Theorem        Construct squares on all sides        Do not move the t-square.353                             ...
Pythagorean Theorem        Construct squares on all sides        Do not move the t-square.354                             ...
Pythagorean Theorem        Construct squares on all sides        Do not move the t-square.355                             ...
Pythagorean Theorem        Construct squares on all sides356                                      © Joan A. Cotter, 2012
Pythagorean Theorem        Construct squares on all sides        Area of blue squares = yellow sq.357                     ...
358   © Joan A. Cotter, 2012
Triangle Area      Construct the altitude                       A      B                        C359                      ...
Triangle Area         Construct the altitude                          A         B                        C      Draw the a...
Triangle Area         Construct the altitude                          A          B                       C      Area is ha...
Triangle Area         Construct the altitude                          A          B                       C      Another al...
Triangle Area         Construct the altitude                          A          B                       C      Another al...
Triangle Area         Construct the altitude                          A          B                       C      Another al...
Triangle Area         Construct the altitude                          A          B                       C      Another al...
Triangle Area         Construct the altitude                          A          B                       C      Altitudes ...
Triangle Area         Construct the altitude                          A          B                       C      3 areas sl...
Geometric Approach        • Basis of CAD (computer          aided design).        • Integrates many concepts.        • Inc...
Geometric Approach                   Goals        • To use elementary math.        • To learn to read math texts.        •...
Bridging Geometry Concepts with Art                by Joan A. Cotter, Ph.D.             joancotter@rightstartmath.comAMS C...
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Geometry AMS

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Geometry AMS

  1. 1. Bridging Geometry Concepts with Art by Joan A. Cotter, Ph.D. joancotter@rightstartmath.comAMS Conference March 17, 2012San Francisco CA Presentations available: rightstartmath.com
  2. 2. Drawing Tools Paper North American paper 11 in. by 8.5 in.2 © Joan A. Cotter, 2012
  3. 3. Drawing Tools Paper International paper 21.0cm by 29.7cm3 © Joan A. Cotter, 2012
  4. 4. Drawing Tools Drawing board with paper4 © Joan A. Cotter, 2012
  5. 5. Drawing Tools Drawing board with paper Affix tape only at the top.5 © Joan A. Cotter, 2012
  6. 6. Drawing Tools T-square T-square Right-handed user.6 © Joan A. Cotter, 2012
  7. 7. Drawing Tools T-square T-square Left-handed user.7 © Joan A. Cotter, 2012
  8. 8. Drawing Tools T-square Must hug the edge of the board.8 © Joan A. Cotter, 2012
  9. 9. Drawing Tools T-square Must hug the edge of the board.9 © Joan A. Cotter, 2012
  10. 10. Drawing Tools T-square Must hug the edge of the board.10 © Joan A. Cotter, 2012
  11. 11. Drawing Tools T-square Must hug the edge of the board.11 © Joan A. Cotter, 2012
  12. 12. Drawing Tools T-square Must hug the edge of the board.12 © Joan A. Cotter, 2012
  13. 13. Drawing Tools T-square Must hug the edge of the board.13 © Joan A. Cotter, 2012
  14. 14. Drawing Tools T-square Must hug the edge of the board.14 © Joan A. Cotter, 2012
  15. 15. Drawing Tools T-square Must hug the edge of the board.15 © Joan A. Cotter, 2012
  16. 16. Drawing Tools T-square Must hug the edge of the board.16 © Joan A. Cotter, 2012
  17. 17. Drawing Tools Pencil (mechanical) and eraser Hold pencil 2-3cm (1 in.) from tip.17 © Joan A. Cotter, 2012
  18. 18. Drawing Tools Pencil (mechanical) and eraser Hold pencil 2-3cm (1 in.) from tip.18 © Joan A. Cotter, 2012
  19. 19. Drawing Tools Pencil (mechanical) and eraser Hold pencil 2-3cm (1 in.) from tip.19 © Joan A. Cotter, 2012
  20. 20. Drawing Tools Pencil (mechanical) and eraser An eraser is an essential tool.20 © Joan A. Cotter, 2012
  21. 21. Drawing Tools Pencil (mechanical) and eraser Parallel lines.21 © Joan A. Cotter, 2012
  22. 22. Drawing Tools Pencil (mechanical) and eraser Parallel lines.22 © Joan A. Cotter, 2012
  23. 23. Drawing Tools The 30-60 triangle (a set square) 30° 60° 90°23 © Joan A. Cotter, 2012
  24. 24. Drawing Tools The 30-60 triangle Triangle must hug the t-square.24 © Joan A. Cotter, 2012
  25. 25. Drawing Tools The 30-60 triangle This doesn’t work!25 © Joan A. Cotter, 2012
  26. 26. Drawing Tools The 30-60 triangle26 © Joan A. Cotter, 2012
  27. 27. Drawing Tools The 30-60 triangle More parallel lines.27 © Joan A. Cotter, 2012
  28. 28. Drawing Tools The 30-60 triangle More parallel lines.28 © Joan A. Cotter, 2012
  29. 29. Drawing Tools The 30-60 triangle Perpendicular lines.29 © Joan A. Cotter, 2012
  30. 30. Drawing Tools The 30-60 triangle Perpendicular lines.30 © Joan A. Cotter, 2012
  31. 31. Drawing Tools The 30-60 triangle31 © Joan A. Cotter, 2012
  32. 32. Drawing Tools The 30-60 triangle Still more parallel lines.32 © Joan A. Cotter, 2012
  33. 33. Drawing Tools The 30-60 triangle Still more parallel lines.33 © Joan A. Cotter, 2012
  34. 34. Drawing Tools The 30-60 triangle34 © Joan A. Cotter, 2012
  35. 35. Drawing Tools The 30-60 triangle Intersecting lines.35 © Joan A. Cotter, 2012
  36. 36. Drawing Tools The 45 triangle 45° 45° 90°36 © Joan A. Cotter, 2012
  37. 37. Drawing Tools The 45 triangle37 © Joan A. Cotter, 2012
  38. 38. Drawing Tools The 45 triangle38 © Joan A. Cotter, 2012
  39. 39. Drawing Tools The 45 triangle Maintain a margin.39 © Joan A. Cotter, 2012
  40. 40. Drawing Tools The 45 triangle Maintain a margin.40 © Joan A. Cotter, 2012
  41. 41. Drawing Tools The 45 triangle Maintain a margin.41 © Joan A. Cotter, 2012
  42. 42. 42 © Joan A. Cotter, 2012
  43. 43. Equilateral Triangles Draw the base line43 © Joan A. Cotter, 2012
  44. 44. Equilateral Triangles Draw the base line Move the t-square down.44 © Joan A. Cotter, 2012
  45. 45. Equilateral Triangles Draw the base line Move the t-square down.45 © Joan A. Cotter, 2012
  46. 46. Equilateral Triangles Draw the sides Draw a side.46 © Joan A. Cotter, 2012
  47. 47. Equilateral Triangles Draw the sides Flip the triangle and repeat.47 © Joan A. Cotter, 2012
  48. 48. Equilateral Triangles Draw the sides Flip the triangle and repeat.48 © Joan A. Cotter, 2012
  49. 49. Equilateral Triangles Draw the sides A bit of erasing.49 © Joan A. Cotter, 2012
  50. 50. Equilateral Triangles Divide it into halves50 © Joan A. Cotter, 2012
  51. 51. Equilateral Triangles Divide it into halves51 © Joan A. Cotter, 2012
  52. 52. Equilateral Triangles Divide it into halves52 © Joan A. Cotter, 2012
  53. 53. Equilateral Triangles Divide it into halves 1 1 2 2 The two halves are congruent.53 © Joan A. Cotter, 2012
  54. 54. Equilateral Triangles Divide it into halves Divide it in half another way.54 © Joan A. Cotter, 2012
  55. 55. Equilateral Triangles Divide it into halves Divide it in half another way.55 © Joan A. Cotter, 2012
  56. 56. Equilateral Triangles Divide it into halves Divide it in half another way.56 © Joan A. Cotter, 2012
  57. 57. Equilateral Triangles Divide it into halves Divide it in half another way.57 © Joan A. Cotter, 2012
  58. 58. Equilateral Triangles Divide it into halves58 © Joan A. Cotter, 2012
  59. 59. Equilateral Triangles Divide it into thirds59 © Joan A. Cotter, 2012
  60. 60. Equilateral Triangles Divide it into thirds60 © Joan A. Cotter, 2012
  61. 61. Equilateral Triangles Divide it into thirds61 © Joan A. Cotter, 2012
  62. 62. Equilateral Triangles Divide it into thirds62 © Joan A. Cotter, 2012
  63. 63. Equilateral Triangles Divide it into thirds A bit of erasing.63 © Joan A. Cotter, 2012
  64. 64. Equilateral Triangles Divide it into thirds Flip the triangle and repeat.64 © Joan A. Cotter, 2012
  65. 65. Equilateral Triangles Divide it into thirds65 © Joan A. Cotter, 2012
  66. 66. Equilateral Triangles Divide it into thirds 1 1 3 3 1 366 © Joan A. Cotter, 2012
  67. 67. Equilateral Triangles Divide it into thirds 1 3 1 1 3 3 Can you figure out how to do it?67 © Joan A. Cotter, 2012
  68. 68. Equilateral Triangles Divide it into fourths68 © Joan A. Cotter, 2012
  69. 69. Equilateral Triangles Divide it into fourths Finding the center of the base line.69 © Joan A. Cotter, 2012
  70. 70. Equilateral Triangles Divide it into fourths tick mark Be sure the tick mark intersects the line.70 © Joan A. Cotter, 2012
  71. 71. Equilateral Triangles Divide it into fourths71 © Joan A. Cotter, 2012
  72. 72. Equilateral Triangles Divide it into fourths Flip the triangle over and repeat.72 © Joan A. Cotter, 2012
  73. 73. Equilateral Triangles Divide it into fourths How do you draw the last line?73 © Joan A. Cotter, 2012
  74. 74. Equilateral Triangles Divide it into fourths How do you draw the last line?74 © Joan A. Cotter, 2012
  75. 75. Equilateral Triangles Divide it into fourths The smaller triangles are similar to the larger triangle.75 © Joan A. Cotter, 2012
  76. 76. Equilateral Triangles Divide it into fourths 1 4 1 1 4 1 4 476 © Joan A. Cotter, 2012
  77. 77. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses:77 © Joan A. Cotter, 2012
  78. 78. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF,78 © Joan A. Cotter, 2012
  79. 79. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses. ABDF, BCDF,79 © Joan A. Cotter, 2012
  80. 80. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF, BCDF, BDEF.80 © Joan A. Cotter, 2012
  81. 81. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids:81 © Joan A. Cotter, 2012
  82. 82. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF,82 © Joan A. Cotter, 2012
  83. 83. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF,83 © Joan A. Cotter, 2012
  84. 84. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF, ABDE.84 © Joan A. Cotter, 2012
  85. 85. Equilateral Triangles Tetrahedron Cut it out….85 © Joan A. Cotter, 2012
  86. 86. Equilateral Triangles Tetrahedron and fold on the lines.86 © Joan A. Cotter, 2012
  87. 87. Equilateral Triangles Tetrahedron and fold on the lines.87 © Joan A. Cotter, 2012
  88. 88. Equilateral Triangles Tetrahedron and fold on the dotted lines.88 © Joan A. Cotter, 2012
  89. 89. Equilateral Triangles Tetrahedron and fold on the dotted lines.89 © Joan A. Cotter, 2012
  90. 90. Equilateral Triangles Tetrahedron and fold on the dotted lines.90 © Joan A. Cotter, 2012
  91. 91. Equilateral Triangles Tetrahedron and fold on the dotted lines.91 © Joan A. Cotter, 2012
  92. 92. Equilateral Triangles Tetrahedron and fold on the dotted lines.92 © Joan A. Cotter, 2012
  93. 93. Equilateral Triangles Divide it into eighths How could you draw eighths?93 © Joan A. Cotter, 2012
  94. 94. Equilateral Triangles Divide it into eighths94 © Joan A. Cotter, 2012
  95. 95. Equilateral Triangles Divide it into eighths95 © Joan A. Cotter, 2012
  96. 96. Equilateral Triangles Divide it into eighths Not very esthetic.96 © Joan A. Cotter, 2012
  97. 97. Equilateral Triangles Divide it into eighths97 © Joan A. Cotter, 2012
  98. 98. Equilateral Triangles Divide it into sixths Can you divide it into sixths?98 © Joan A. Cotter, 2012
  99. 99. Equilateral Triangles Divide it into sixths Can you divide it into sixths?99 © Joan A. Cotter, 2012
  100. 100. Equilateral Triangles Divide it into sixths100 © Joan A. Cotter, 2012
  101. 101. Equilateral Triangles Divide it into sixths101 © Joan A. Cotter, 2012
  102. 102. Equilateral Triangles Divide it into sixths A B F G C E D Name an acute triangle:102 © Joan A. Cotter, 2012
  103. 103. Equilateral Triangles Divide it into sixths A B F G C E D Name an acute triangle: ACE.103 © Joan A. Cotter, 2012
  104. 104. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle:104 © Joan A. Cotter, 2012
  105. 105. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE,105 © Joan A. Cotter, 2012
  106. 106. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE, ABE,106 © Joan A. Cotter, 2012
  107. 107. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE, ABE, ABG.107 © Joan A. Cotter, 2012
  108. 108. Equilateral Triangles Divide it into sixths A B F G C E D Name an obtuse triangle:108 © Joan A. Cotter, 2012
  109. 109. Equilateral Triangles Divide it into sixths A B F G C E D Name an obtuse triangle: GCE.109 © Joan A. Cotter, 2012
  110. 110. Equilateral Triangles Divide it into twelfths Can you divide it into twelfths?110 © Joan A. Cotter, 2012
  111. 111. Equilateral Triangles Divide it into twelfths111 © Joan A. Cotter, 2012
  112. 112. Equilateral Triangles Divide it into twelfths112 © Joan A. Cotter, 2012
  113. 113. Equilateral Triangles Divide it into twelfths113 © Joan A. Cotter, 2012
  114. 114. Equilateral Triangles Divide it into twelfths114 © Joan A. Cotter, 2012
  115. 115. Equilateral Triangles Divide it into twelfths115 © Joan A. Cotter, 2012
  116. 116. Equilateral Triangles Divide it into twelfths Do you see the cube?116 © Joan A. Cotter, 2012
  117. 117. Equilateral Triangles Divide it into twelfths Do you see the cube?117 © Joan A. Cotter, 2012
  118. 118. Equilateral Triangles Divide it into twelfths Do you see the hexagon?118 © Joan A. Cotter, 2012
  119. 119. Equilateral Triangles Divide it into twelfths Do you see the hexagon?119 © Joan A. Cotter, 2012
  120. 120. Equilateral Triangles Ninths120 © Joan A. Cotter, 2012
  121. 121. Equilateral Triangles Twenty-sevenths121 © Joan A. Cotter, 2012
  122. 122. Equilateral Triangles Twenty-sevenths122 © Joan A. Cotter, 2012
  123. 123. Equilateral Triangles Twenty-sevenths123 © Joan A. Cotter, 2012
  124. 124. Equilateral Triangles Twenty-sevenths124 © Joan A. Cotter, 2012
  125. 125. Equilateral Triangles Thirty-seconds125 © Joan A. Cotter, 2012
  126. 126. Equilateral Triangles Stephanie, 6, a Montessori child, did 256.126 © Joan A. Cotter, 2012
  127. 127. Equilateral Triangles Stephanie, 6, a Montessori child, did 256.127 © Joan A. Cotter, 2012
  128. 128. Equilateral Triangles Stephanie, 6, a Montessori child, did 256.128 © Joan A. Cotter, 2012
  129. 129. Equilateral Triangles Stephanie, 6, a Montessori child, did 256.129 © Joan A. Cotter, 2012
  130. 130. Equilateral Triangles Stephanie, 6, a Montessori child, did 256.130 © Joan A. Cotter, 2012
  131. 131. Equilateral Triangles Spiral131 © Joan A. Cotter, 2012
  132. 132. Equilateral Triangles Spiral132 © Joan A. Cotter, 2012
  133. 133. Equilateral Triangles Spiral133 © Joan A. Cotter, 2012
  134. 134. Equilateral Triangles Spiral134 © Joan A. Cotter, 2012
  135. 135. Equilateral Triangles Spiral135 © Joan A. Cotter, 2012
  136. 136. Equilateral Triangles Spiral136 © Joan A. Cotter, 2012
  137. 137. Equilateral Triangles Spiral137 © Joan A. Cotter, 2012
  138. 138. Equilateral Triangles Spiral138 © Joan A. Cotter, 2012
  139. 139. Equilateral Triangles Spiral139 © Joan A. Cotter, 2012
  140. 140. Equilateral Triangles Spiral From Psychogeometry140 © Joan A. Cotter, 2012
  141. 141. Equilateral Triangles Spiral141 © Joan A. Cotter, 2012
  142. 142. 142 © Joan A. Cotter, 2012
  143. 143. Hexagons Start with an equilateral triangle143 © Joan A. Cotter, 2012
  144. 144. Hexagons Start with an equilateral triangle144 © Joan A. Cotter, 2012
  145. 145. Hexagons Start with an equilateral triangle145 © Joan A. Cotter, 2012
  146. 146. Hexagons Start with an equilateral triangle146 © Joan A. Cotter, 2012
  147. 147. Hexagons Start with an equilateral triangle147 © Joan A. Cotter, 2012
  148. 148. Hexagons Start with an equilateral triangle148 © Joan A. Cotter, 2012
  149. 149. Hexagons Start with an equilateral triangle A bit of erasing.149 © Joan A. Cotter, 2012
  150. 150. Hexagons Start with an equilateral triangle150 © Joan A. Cotter, 2012
  151. 151. Hexagons Start with a small equilateral triangle151 © Joan A. Cotter, 2012
  152. 152. Hexagons Start with a small equilateral triangle152 © Joan A. Cotter, 2012
  153. 153. Hexagons Start with a small equilateral triangle153 © Joan A. Cotter, 2012
  154. 154. Hexagons Start with a small equilateral triangle Flip the triangle and repeat.154 © Joan A. Cotter, 2012
  155. 155. Hexagons Start with a small equilateral triangle155 © Joan A. Cotter, 2012
  156. 156. Hexagons Start with a small equilateral triangle156 © Joan A. Cotter, 2012
  157. 157. Hexagons Start with a small equilateral triangle157 © Joan A. Cotter, 2012
  158. 158. Hexagons Start with a small equilateral triangle158 © Joan A. Cotter, 2012
  159. 159. Hexagons Start with a small equilateral triangle159 © Joan A. Cotter, 2012
  160. 160. Hexagons Draw all the diagonals160 © Joan A. Cotter, 2012
  161. 161. Hexagons Draw all the diagonals161 © Joan A. Cotter, 2012
  162. 162. Hexagons Draw all the diagonals162 © Joan A. Cotter, 2012
  163. 163. Hexagons Draw all the diagonals163 © Joan A. Cotter, 2012
  164. 164. Hexagons Draw all the diagonals164 © Joan A. Cotter, 2012
  165. 165. Hexagons Draw all the diagonals165 © Joan A. Cotter, 2012
  166. 166. Hexagons Find the figures Another hexagon.166 © Joan A. Cotter, 2012
  167. 167. Hexagons Find the figures Another hexagon.167 © Joan A. Cotter, 2012
  168. 168. Hexagons Find the figures 2 non-congruent rectangles.168 © Joan A. Cotter, 2012
  169. 169. Hexagons Find the figures 2 non-congruent rectangles.169 © Joan A. Cotter, 2012
  170. 170. Hexagons Find the figures 2 non-congruent rectangles.170 © Joan A. Cotter, 2012
  171. 171. Hexagons Find the figures 3 non-congruent right triangles.171 © Joan A. Cotter, 2012
  172. 172. Hexagons Find the figures 3 non-congruent right triangles.172 © Joan A. Cotter, 2012
  173. 173. Hexagons Find the figures 3 non-congruent right triangles.173 © Joan A. Cotter, 2012
  174. 174. Hexagons Find the figures 3 non-congruent right triangles.174 © Joan A. Cotter, 2012
  175. 175. Hexagons Find the figures 2 non-congruent rhombuses.175 © Joan A. Cotter, 2012
  176. 176. Hexagons Find the figures 2 non-congruent rhombuses.176 © Joan A. Cotter, 2012
  177. 177. Hexagons Find the figures 2 non-congruent rhombuses.177 © Joan A. Cotter, 2012
  178. 178. Hexagons Find the figures 3 similar equilateral triangles.178 © Joan A. Cotter, 2012
  179. 179. Hexagons Find the figures 3 similar equilateral triangles.179 © Joan A. Cotter, 2012
  180. 180. Hexagons Find the figures 3 similar equilateral triangles.180 © Joan A. Cotter, 2012
  181. 181. Hexagons Find the figures 3 similar obtuse triangles.181 © Joan A. Cotter, 2012
  182. 182. Hexagons Find the figures 3 similar obtuse triangles.182 © Joan A. Cotter, 2012
  183. 183. Hexagons Find the figures 3 similar obtuse triangles.183 © Joan A. Cotter, 2012
  184. 184. Hexagons Find the figures 3 similar obtuse triangles.184 © Joan A. Cotter, 2012
  185. 185. Hexagons Find the figures 3 non-congruent trapezoids.185 © Joan A. Cotter, 2012
  186. 186. Hexagons Find the figures 3 non-congruent trapezoids.186 © Joan A. Cotter, 2012
  187. 187. Hexagons Find the figures 3 non-congruent trapezoids.187 © Joan A. Cotter, 2012
  188. 188. Hexagons Find the figures 3 non-congruent trapezoids.188 © Joan A. Cotter, 2012
  189. 189. Hexagons Find the figures 2 similar kites.189 © Joan A. Cotter, 2012
  190. 190. Hexagons Find the figures 2 similar kites.190 © Joan A. Cotter, 2012
  191. 191. Hexagons Find the figures 2 similar kites.191 © Joan A. Cotter, 2012
  192. 192. Hexagons Construct the small star Apothem: perpendicular line from side to center.192 © Joan A. Cotter, 2012
  193. 193. Hexagons Construct the small star Apothem: perpendicular line from side to center.193 © Joan A. Cotter, 2012
  194. 194. Hexagons Construct the small star Apothem: perpendicular line from side to center.194 © Joan A. Cotter, 2012
  195. 195. Hexagons Construct the small star195 © Joan A. Cotter, 2012
  196. 196. Hexagons Construct the small star196 © Joan A. Cotter, 2012
  197. 197. Hexagons Construct the small star197 © Joan A. Cotter, 2012
  198. 198. Hexagons Construct the small star198 © Joan A. Cotter, 2012
  199. 199. Hexagons Construct the small star Ratio of the areas of star and hexagon?199 © Joan A. Cotter, 2012
  200. 200. Hexagons Construct the small star Ratio of the areas of star and hexagon?200 © Joan A. Cotter, 2012
  201. 201. Hexagons Construct the large star201 © Joan A. Cotter, 2012
  202. 202. Hexagons Construct the large star202 © Joan A. Cotter, 2012
  203. 203. Hexagons Construct the large star203 © Joan A. Cotter, 2012
  204. 204. Hexagons Construct the large star204 © Joan A. Cotter, 2012
  205. 205. Hexagons Construct the large star205 © Joan A. Cotter, 2012
  206. 206. Hexagons Construct the large star206 © Joan A. Cotter, 2012
  207. 207. Hexagons Construct the large star207 © Joan A. Cotter, 2012
  208. 208. Hexagons Construct the large star208 © Joan A. Cotter, 2012
  209. 209. 209 © Joan A. Cotter, 2012
  210. 210. The Clock 1112 1 10 2 9 3 8 4 7 6 5210 © Joan A. Cotter, 2012
  211. 211. The Clock 1112 1 10 2 9 3 8 4 7 6 5211 © Joan A. Cotter, 2012
  212. 212. The Clock 1112 1 10 2 9 3 8 4 7 6 5212 © Joan A. Cotter, 2012
  213. 213. The Clock 1112 1 10 2 9 3 8 4 7 6 5213 © Joan A. Cotter, 2012
  214. 214. The Clock 1112 1 10 2 9 3 8 4 7 6 5214 © Joan A. Cotter, 2012
  215. 215. The Clock 1112 1 10 2 9 3 8 4 7 6 5 11 12 1 10 2 9 3 8 4 7 6 5215 © Joan A. Cotter, 2012
  216. 216. 216 © Joan A. Cotter, 2012
  217. 217. Squares The 45 triangle.217 © Joan A. Cotter, 2012
  218. 218. Squares The 45 triangle is half of a square.218 © Joan A. Cotter, 2012
  219. 219. Squares Construct a square Draw the base line.219 © Joan A. Cotter, 2012
  220. 220. Squares Construct a square Move the t-square down.220 © Joan A. Cotter, 2012
  221. 221. Squares Construct a square Move the t-square down.221 © Joan A. Cotter, 2012
  222. 222. Squares Construct a square Draw the sides.222 © Joan A. Cotter, 2012
  223. 223. Squares Construct a square Draw the diagonal…223 © Joan A. Cotter, 2012
  224. 224. Squares Construct a square or just a tick mark.224 © Joan A. Cotter, 2012
  225. 225. Squares Construct a square Draw the top.225 © Joan A. Cotter, 2012
  226. 226. Squares Construct a square A bit of erasing.226 © Joan A. Cotter, 2012
  227. 227. Squares Construct a square227 © Joan A. Cotter, 2012
  228. 228. Squares Halves228 © Joan A. Cotter, 2012
  229. 229. Squares Halves229 © Joan A. Cotter, 2012
  230. 230. Squares Fourths230 © Joan A. Cotter, 2012
  231. 231. Squares Eighths231 © Joan A. Cotter, 2012
  232. 232. Squares Sixteenths232 © Joan A. Cotter, 2012
  233. 233. Squares Sixteenths233 © Joan A. Cotter, 2012
  234. 234. Squares Halves We need to find the center.234 © Joan A. Cotter, 2012
  235. 235. Squares Halves We need to find the center.235 © Joan A. Cotter, 2012
  236. 236. Squares Halves We need to find the center.236 © Joan A. Cotter, 2012
  237. 237. Squares Halves We need to find the center.237 © Joan A. Cotter, 2012
  238. 238. Squares Halves238 © Joan A. Cotter, 2012
  239. 239. Squares Fourths239 © Joan A. Cotter, 2012
  240. 240. Squares Fourths240 © Joan A. Cotter, 2012
  241. 241. Squares Eighths Find the center of a small square.241 © Joan A. Cotter, 2012
  242. 242. Squares Eighths Find the center of a small square.242 © Joan A. Cotter, 2012
  243. 243. Squares Eighths Find the center of a small square.243 © Joan A. Cotter, 2012
  244. 244. Squares Eighths Find the center of a small square.244 © Joan A. Cotter, 2012
  245. 245. Squares Eighths Repeat for the other line.245 © Joan A. Cotter, 2012
  246. 246. Squares Eighths246 © Joan A. Cotter, 2012
  247. 247. Squares Sixteenths247 © Joan A. Cotter, 2012
  248. 248. Squares Sixteenths248 © Joan A. Cotter, 2012
  249. 249. Squares Sixteenths249 © Joan A. Cotter, 2012
  250. 250. Squares Right triangle spiral250 © Joan A. Cotter, 2012
  251. 251. Squares Right triangle spiral251 © Joan A. Cotter, 2012
  252. 252. Squares Right triangle spiral252 © Joan A. Cotter, 2012
  253. 253. Squares Right triangle spiral253 © Joan A. Cotter, 2012
  254. 254. Squares Right triangle spiral254 © Joan A. Cotter, 2012
  255. 255. Squares Right triangle spiral255 © Joan A. Cotter, 2012
  256. 256. Squares Right triangle spiral256 © Joan A. Cotter, 2012
  257. 257. Squares Right triangle spiral257 © Joan A. Cotter, 2012
  258. 258. Squares Right triangle spiral258 © Joan A. Cotter, 2012
  259. 259. Squares Right triangle spiral Adapted from Psychogeometry259 © Joan A. Cotter, 2012
  260. 260. Squares Right triangle design260 © Joan A. Cotter, 2012
  261. 261. Squares Right triangle design261 © Joan A. Cotter, 2012
  262. 262. Squares Right triangle design262 © Joan A. Cotter, 2012
  263. 263. Squares Right triangle design263 © Joan A. Cotter, 2012
  264. 264. Squares Right triangle design264 © Joan A. Cotter, 2012
  265. 265. Squares Right triangle design265 © Joan A. Cotter, 2012
  266. 266. Squares Right triangle design266 © Joan A. Cotter, 2012
  267. 267. Squares Right triangle design267 © Joan A. Cotter, 2012
  268. 268. Squares Right triangle design268 © Joan A. Cotter, 2012
  269. 269. Squares Right triangle design269 © Joan A. Cotter, 2012
  270. 270. Squares Circumscribed squares270 © Joan A. Cotter, 2012
  271. 271. Squares Circumscribed squares271 © Joan A. Cotter, 2012
  272. 272. Squares Circumscribed squares272 © Joan A. Cotter, 2012
  273. 273. Squares Circumscribed squares273 © Joan A. Cotter, 2012
  274. 274. Squares Circumscribed squares274 © Joan A. Cotter, 2012
  275. 275. Squares Circumscribed squares275 © Joan A. Cotter, 2012
  276. 276. Squares Circumscribed squares276 © Joan A. Cotter, 2012
  277. 277. Squares Circumscribed squares277 © Joan A. Cotter, 2012
  278. 278. Squares Circumscribed squares278 © Joan A. Cotter, 2012
  279. 279. Squares Circumscribed squares279 © Joan A. Cotter, 2012
  280. 280. Squares Circumscribed squares280 © Joan A. Cotter, 2012
  281. 281. Squares Circumscribed squares281 © Joan A. Cotter, 2012
  282. 282. Squares Inscribed squares282 © Joan A. Cotter, 2012
  283. 283. Squares Inscribed squares283 © Joan A. Cotter, 2012
  284. 284. Squares Inscribed squares284 © Joan A. Cotter, 2012
  285. 285. Squares Inscribed squares Compare the area of the squares.285 © Joan A. Cotter, 2012
  286. 286. Squares Inscribed squares Compare the area of the squares.286 © Joan A. Cotter, 2012
  287. 287. Squares Inscribed squares287 © Joan A. Cotter, 2012
  288. 288. Squares Inscribed squares288 © Joan A. Cotter, 2012
  289. 289. Squares Inscribed squares289 © Joan A. Cotter, 2012
  290. 290. Squares Inscribed squares290 © Joan A. Cotter, 2012
  291. 291. Squares Inscribed squares291 © Joan A. Cotter, 2012
  292. 292. Squares Inscribed squares292 © Joan A. Cotter, 2012
  293. 293. Squares Inscribed squares293 © Joan A. Cotter, 2012
  294. 294. Squares Inscribed squares294 © Joan A. Cotter, 2012
  295. 295. 295 © Joan A. Cotter, 2012
  296. 296. Tangrams Constructing a tangram296 © Joan A. Cotter, 2012
  297. 297. Tangrams Constructing a tangram297 © Joan A. Cotter, 2012
  298. 298. Tangrams Constructing a tangram298 © Joan A. Cotter, 2012
  299. 299. Tangrams Constructing a tangram299 © Joan A. Cotter, 2012
  300. 300. Tangrams Constructing a tangram300 © Joan A. Cotter, 2012
  301. 301. Tangrams Constructing a tangram301 © Joan A. Cotter, 2012
  302. 302. Tangrams Constructing a tangram302 © Joan A. Cotter, 2012
  303. 303. Tangrams Constructing a tangram 3 green figures are equal in area.303 © Joan A. Cotter, 2012
  304. 304. Tangrams Constructing a tangram 3 green figures are equal in area.304 © Joan A. Cotter, 2012
  305. 305. Tangrams Construct a tangram arrangement No tracing or measuring.305 © Joan A. Cotter, 2012
  306. 306. Tangrams Construct a tangram arrangement No tracing or measuring.306 © Joan A. Cotter, 2012
  307. 307. Tangrams Construct a tangram arrangement No tracing or measuring.307 © Joan A. Cotter, 2012
  308. 308. Tangrams Construct a tangram arrangement No tracing or measuring.308 © Joan A. Cotter, 2012
  309. 309. Tangrams Construct a tangram arrangement No tracing or measuring.309 © Joan A. Cotter, 2012
  310. 310. Tangrams Construct a tangram arrangement No tracing or measuring.310 © Joan A. Cotter, 2012
  311. 311. Tangrams Construct a tangram arrangement No tracing or measuring.311 © Joan A. Cotter, 2012
  312. 312. Tangrams Construct a tangram arrangement No tracing or measuring.312 © Joan A. Cotter, 2012
  313. 313. Tangrams Construct a tangram arrangement No tracing or measuring.313 © Joan A. Cotter, 2012
  314. 314. Tangrams Construct a tangram arrangement More examples to construct.314 © Joan A. Cotter, 2012
  315. 315. Tangrams Construct a tangram reflection Use the left side of the paper.315 © Joan A. Cotter, 2012
  316. 316. Tangrams Construct a tangram reflection Start at the left side of the paper.316 © Joan A. Cotter, 2012
  317. 317. Tangrams Construct a tangram reflection Reflect the image across the line.317 © Joan A. Cotter, 2012
  318. 318. Tangrams Construct a tangram reflection Reflect the image across the line.318 © Joan A. Cotter, 2012
  319. 319. Tangrams Construct a tangram reflection Reflect the image across the line.319 © Joan A. Cotter, 2012
  320. 320. Tangrams Construct a tangram reflection Reflect the image across the line.320 © Joan A. Cotter, 2012
  321. 321. Tangrams Construct a tangram reflection Reflect the image across the line.321 © Joan A. Cotter, 2012
  322. 322. Tangrams Construct a tangram reflection Reflect the image across the line.322 © Joan A. Cotter, 2012
  323. 323. Tangrams Construct a tangram reflection Reflect the image across the line.323 © Joan A. Cotter, 2012
  324. 324. Tangrams Construct a tangram reflection Reflect the image across the line.324 © Joan A. Cotter, 2012
  325. 325. Tangrams Construct a tangram reflection Reflect the image across the line.325 © Joan A. Cotter, 2012
  326. 326. Tangrams Construct a tangram reflection Reflect the image across the line.326 © Joan A. Cotter, 2012
  327. 327. Tangrams Construct a tangram reflection Reflect the image across the line.327 © Joan A. Cotter, 2012
  328. 328. Tangrams Construct a tangram reflection328 © Joan A. Cotter, 2012
  329. 329. 329 © Joan A. Cotter, 2012
  330. 330. Pythagorean Theorem Construct squares on all sides First construct a right triangle.330 © Joan A. Cotter, 2012
  331. 331. Pythagorean Theorem Construct squares on all sides331 © Joan A. Cotter, 2012
  332. 332. Pythagorean Theorem Construct squares on all sides332 © Joan A. Cotter, 2012
  333. 333. Pythagorean Theorem Construct squares on all sides333 © Joan A. Cotter, 2012
  334. 334. Pythagorean Theorem Construct squares on all sides334 © Joan A. Cotter, 2012
  335. 335. Pythagorean Theorem Construct squares on all sides335 © Joan A. Cotter, 2012
  336. 336. Pythagorean Theorem Construct squares on all sides336 © Joan A. Cotter, 2012
  337. 337. Pythagorean Theorem Construct squares on all sides337 © Joan A. Cotter, 2012
  338. 338. Pythagorean Theorem Construct squares on all sides338 © Joan A. Cotter, 2012
  339. 339. Pythagorean Theorem Construct squares on all sides Area of blue squares = yellow sq.339 © Joan A. Cotter, 2012
  340. 340. Pythagorean Theorem Construct squares on all sides Area of blue squares = yellow sq.340 © Joan A. Cotter, 2012
  341. 341. Pythagorean Theorem Construct squares on all sides Draw a right triangle.341 © Joan A. Cotter, 2012
  342. 342. Pythagorean Theorem Construct squares on all sides342 © Joan A. Cotter, 2012
  343. 343. Pythagorean Theorem Construct squares on all sides343 © Joan A. Cotter, 2012
  344. 344. Pythagorean Theorem Construct squares on all sides344 © Joan A. Cotter, 2012
  345. 345. Pythagorean Theorem Construct squares on all sides345 © Joan A. Cotter, 2012
  346. 346. Pythagorean Theorem Construct squares on all sides346 © Joan A. Cotter, 2012
  347. 347. Pythagorean Theorem Construct squares on all sides347 © Joan A. Cotter, 2012
  348. 348. Pythagorean Theorem Construct squares on all sides Turn the t-square upside down.348 © Joan A. Cotter, 2012
  349. 349. Pythagorean Theorem Construct squares on all sides349 © Joan A. Cotter, 2012
  350. 350. Pythagorean Theorem Construct squares on all sides350 © Joan A. Cotter, 2012
  351. 351. Pythagorean Theorem Construct squares on all sides351 © Joan A. Cotter, 2012
  352. 352. Pythagorean Theorem Construct squares on all sides Do not move the t-square.352 © Joan A. Cotter, 2012
  353. 353. Pythagorean Theorem Construct squares on all sides Do not move the t-square.353 © Joan A. Cotter, 2012
  354. 354. Pythagorean Theorem Construct squares on all sides Do not move the t-square.354 © Joan A. Cotter, 2012
  355. 355. Pythagorean Theorem Construct squares on all sides Do not move the t-square.355 © Joan A. Cotter, 2012
  356. 356. Pythagorean Theorem Construct squares on all sides356 © Joan A. Cotter, 2012
  357. 357. Pythagorean Theorem Construct squares on all sides Area of blue squares = yellow sq.357 © Joan A. Cotter, 2012
  358. 358. 358 © Joan A. Cotter, 2012
  359. 359. Triangle Area Construct the altitude A B C359 © Joan A. Cotter, 2012
  360. 360. Triangle Area Construct the altitude A B C Draw the altitude.360 © Joan A. Cotter, 2012
  361. 361. Triangle Area Construct the altitude A B C Area is half of enclosing361 © Joan A. Cotter, 2012
  362. 362. Triangle Area Construct the altitude A B C Another altitude. Find area.362 © Joan A. Cotter, 2012
  363. 363. Triangle Area Construct the altitude A B C Another altitude. Find area.363 © Joan A. Cotter, 2012
  364. 364. Triangle Area Construct the altitude A B C Another altitude. Find area.364 © Joan A. Cotter, 2012
  365. 365. Triangle Area Construct the altitude A B C Another altitude. Find area.365 © Joan A. Cotter, 2012
  366. 366. Triangle Area Construct the altitude A B C Altitudes intersect at the366 © Joan A. Cotter, 2012
  367. 367. Triangle Area Construct the altitude A B C 3 areas slightly different. Why?367 © Joan A. Cotter, 2012
  368. 368. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. • Excellent preparation for high school mathematics.368 © Joan A. Cotter, 2012
  369. 369. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics. • To discover mathematics in the everyday world. • To enjoy mathematics.369 © Joan A. Cotter, 2012
  370. 370. Bridging Geometry Concepts with Art by Joan A. Cotter, Ph.D. joancotter@rightstartmath.comAMS Conference March 17, 2012San Francisco CA Presentations available: rightstartmath.com

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