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Bridging Geometry Concepts with Art
                by Joan A. Cotter, Ph.D.
             joancotter@rightstartmath.com




AMS Conference
 March 17, 2012
San Francisco CA



        Presentations available: rightstartmath.com
Drawing Tools
          Paper




      North American paper
        11 in. by 8.5 in.




2                            © Joan A. Cotter, 2012
Drawing Tools
          Paper




       International paper
       21.0cm by 29.7cm




3                            © Joan A. Cotter, 2012
Drawing Tools
    Drawing board with paper




4                              © Joan A. Cotter, 2012
Drawing Tools
    Drawing board with paper




    Affix tape only at the top.
5                                 © Joan A. Cotter, 2012
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, 2012
Drawing Tools
             T-square




     Must hug the edge of the board.
11                                 © Joan A. Cotter, 2012
Drawing Tools
             T-square




     Must hug the edge of the board.
12                                 © Joan A. Cotter, 2012
Drawing Tools
             T-square




     Must hug the edge of the board.
13                                 © Joan A. Cotter, 2012
Drawing Tools
             T-square




     Must hug the edge of the board.
14                                 © Joan A. Cotter, 2012
Drawing Tools
             T-square




     Must hug the edge of the board.
15                                 © Joan A. Cotter, 2012
Drawing Tools
             T-square




     Must hug the edge of the board.
16                                 © Joan A. Cotter, 2012
Drawing Tools
     Pencil (mechanical) and eraser




     Hold pencil 2-3cm (1 in.) from tip.
17                                    © Joan A. Cotter, 2012
Drawing Tools
     Pencil (mechanical) and eraser




     Hold pencil 2-3cm (1 in.) from tip.
18                                    © Joan A. Cotter, 2012
Drawing Tools
     Pencil (mechanical) and eraser




     Hold pencil 2-3cm (1 in.) from tip.
19                                    © Joan A. Cotter, 2012
Drawing Tools
     Pencil (mechanical) and eraser




     An eraser is an essential tool.
20                                     © Joan A. Cotter, 2012
Drawing Tools
     Pencil (mechanical) and eraser




     Parallel lines.
21                                    © Joan A. Cotter, 2012
Drawing Tools
     Pencil (mechanical) and eraser




     Parallel lines.
22                                    © Joan A. Cotter, 2012
Drawing Tools
     The 30-60 triangle (a set square)



                       30°




              60°      90°




23                                       © Joan A. Cotter, 2012
Drawing Tools
          The 30-60 triangle




     Triangle must hug the t-square.
24                                 © Joan A. Cotter, 2012
Drawing Tools
          The 30-60 triangle




     This doesn’t work!
25                             © Joan A. Cotter, 2012
Drawing Tools
      The 30-60 triangle




26                         © 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 triangle




31                         © Joan A. Cotter, 2012
Drawing Tools
          The 30-60 triangle




     Still more parallel lines.
32                                © Joan A. Cotter, 2012
Drawing Tools
          The 30-60 triangle




     Still more parallel lines.
33                                © Joan A. Cotter, 2012
Drawing Tools
      The 30-60 triangle




34                         © 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                             © Joan A. Cotter, 2012
Drawing Tools
       The 45 triangle




37                       © Joan A. Cotter, 2012
Drawing Tools
       The 45 triangle




38                       © 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 line




43                             © Joan A. Cotter, 2012
Equilateral Triangles
          Draw the base line




      Move the t-square down.
44                              © Joan A. Cotter, 2012
Equilateral Triangles
          Draw the base line




      Move the t-square down.
45                              © Joan A. Cotter, 2012
Equilateral Triangles
            Draw the sides




      Draw a side.
46                           © Joan A. Cotter, 2012
Equilateral Triangles
            Draw the sides




      Flip the triangle and repeat.
47                                    © Joan A. Cotter, 2012
Equilateral Triangles
            Draw the sides




      Flip the triangle and repeat.
48                                    © Joan A. Cotter, 2012
Equilateral Triangles
            Draw the sides




      A bit of erasing.
49                           © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into halves




50                               © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into halves




51                               © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into halves




52                               © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into halves




                  1      1
                  2      2




      The two halves are congruent.
53                                © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into halves




      Divide it in half another way.
54                                     © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into halves




      Divide it in half another way.
55                                     © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into halves




      Divide it in half another way.
56                                     © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into halves




      Divide it in half another way.
57                                     © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into halves




58                               © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into thirds




59                               © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into thirds




60                               © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into thirds




61                               © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into thirds




62                               © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into thirds




      A bit of erasing.
63                                © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into thirds




      Flip the triangle and repeat.
64                                    © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into thirds




65                               © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into thirds




                  1       1
                  3       3
                      1
                      3




66                               © Joan A. Cotter, 2012
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
Equilateral Triangles
         Divide it into fourths




68                                © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into fourths




      Finding the center of the base line.
69                                   © Joan A. Cotter, 2012
Equilateral Triangles
           Divide it into fourths




                                     tick mark




      Be sure the tick mark intersects the line.
70                                                 © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into fourths




71                                © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into fourths




      Flip the triangle over and repeat.
72                                    © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into fourths




      How do you draw the last line?
73                                     © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into fourths




      How do you draw the last line?
74                                     © Joan A. Cotter, 2012
Equilateral Triangles
                      Divide it into fourths




     The smaller triangles are similar to the larger triangle.
75                                                       © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into fourths



                      1
                      4
                      1
                 1    4     1
                 4          4




76                                © Joan A. Cotter, 2012
Equilateral Triangles
               Divide it into fourths

                           A



                     B             F



                C                       E
                            D



     Name the rhombuses:
77                                          © Joan A. Cotter, 2012
Equilateral Triangles
               Divide it into fourths

                           A



                     B             F



                C                       E
                            D



     Name the rhombuses: ABDF,
78                                          © Joan A. Cotter, 2012
Equilateral Triangles
               Divide it into fourths

                           A



                     B             F



                C                       E
                            D



     Name the rhombuses. ABDF, BCDF,
79                                          © Joan A. Cotter, 2012
Equilateral Triangles
               Divide it into fourths

                           A



                     B             F



                C                       E
                            D



     Name the rhombuses: ABDF, BCDF, BDEF.
80                                          © Joan A. Cotter, 2012
Equilateral Triangles
                 Divide it into fourths

                             A



                       B             F



                  C                       E
                              D



     Name the trapezoids:
81                                            © Joan A. Cotter, 2012
Equilateral Triangles
                 Divide it into fourths

                             A



                       B             F



                  C                       E
                              D



     Name the trapezoids: BCEF,
82                                            © Joan A. Cotter, 2012
Equilateral Triangles
                 Divide it into fourths

                             A



                       B             F



                  C                       E
                              D



     Name the trapezoids: BCEF, ACDF,
83                                            © Joan A. Cotter, 2012
Equilateral Triangles
                 Divide it into fourths

                             A



                       B             F



                  C                       E
                              D



     Name the trapezoids: BCEF, ACDF, ABDE.
84                                            © Joan A. Cotter, 2012
Equilateral Triangles
             Tetrahedron




      Cut it out….
85                           © Joan A. Cotter, 2012
Equilateral Triangles
             Tetrahedron




      and fold on the lines.
86                             © Joan A. Cotter, 2012
Equilateral Triangles
             Tetrahedron




      and fold on the lines.
87                             © Joan A. Cotter, 2012
Equilateral Triangles
             Tetrahedron




      and fold on the dotted lines.
88                                    © Joan A. Cotter, 2012
Equilateral Triangles
             Tetrahedron




      and fold on the dotted lines.
89                                    © Joan A. Cotter, 2012
Equilateral Triangles
             Tetrahedron




      and fold on the dotted lines.
90                                    © Joan A. Cotter, 2012
Equilateral Triangles
             Tetrahedron




      and fold on the dotted lines.
91                                    © Joan A. Cotter, 2012
Equilateral Triangles
             Tetrahedron




      and fold on the dotted lines.
92                                    © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into eighths




      How could you draw eighths?
93                                  © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into eighths




94                                © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into eighths




95                                © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into eighths




      Not very esthetic.
96                                © Joan A. Cotter, 2012
Equilateral Triangles
         Divide it into eighths




97                                © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into sixths




      Can you divide it into sixths?
98                                     © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into sixths




      Can you divide it into sixths?
99                                     © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into sixths




100                               © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into sixths




101                               © Joan A. Cotter, 2012
Equilateral Triangles
                 Divide it into sixths

                             A



                       B             F
                                 G


                 C                       E
                              D



      Name an acute triangle:
102                                          © Joan A. Cotter, 2012
Equilateral Triangles
                 Divide it into sixths

                             A



                       B             F
                                 G


                 C                       E
                              D



      Name an acute triangle: ACE.
103                                          © Joan A. Cotter, 2012
Equilateral Triangles
                  Divide it into sixths

                              A



                        B             F
                                  G


                  C                       E
                               D



      Name a right triangle:
104                                           © Joan A. Cotter, 2012
Equilateral Triangles
                 Divide it into sixths

                             A



                       B             F
                                 G


                 C                       E
                              D



      Name a right triangle: ADE,
105                                          © Joan A. Cotter, 2012
Equilateral Triangles
                 Divide it into sixths

                             A



                       B             F
                                 G


                 C                       E
                              D



      Name a right triangle: ADE, ABE,
106                                          © Joan A. Cotter, 2012
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
Equilateral Triangles
                 Divide it into sixths

                             A



                       B             F
                                 G


                 C                       E
                              D



      Name an obtuse triangle:
108                                          © Joan A. Cotter, 2012
Equilateral Triangles
                Divide it into sixths

                            A



                      B             F
                                G


                 C                      E
                             D



      Name an obtuse triangle: GCE.
109                                         © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




       Can you divide it into twelfths?
110                                   © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




111                                 © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




112                                 © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




113                                 © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




114                                 © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




115                                 © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




       Do you see the cube?
116                                 © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




       Do you see the cube?
117                                 © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




       Do you see the hexagon?
118                                 © Joan A. Cotter, 2012
Equilateral Triangles
          Divide it into twelfths




       Do you see the hexagon?
119                                 © Joan A. Cotter, 2012
Equilateral Triangles




               Ninths
120                           © Joan A. Cotter, 2012
Equilateral Triangles




           Twenty-sevenths
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




            Thirty-seconds
125                           © Joan A. Cotter, 2012
Equilateral Triangles




       Stephanie, 6, a Montessori child, did 256.
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
              Spiral




131                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




132                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




133                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




134                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




135                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




136                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




137                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




138                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




139                           © Joan A. Cotter, 2012
Equilateral Triangles
                Spiral




       From Psychogeometry
140                           © Joan A. Cotter, 2012
Equilateral Triangles
              Spiral




141                           © Joan A. Cotter, 2012
142   © Joan A. Cotter, 2012
Hexagons
      Start with an equilateral triangle




143                                        © Joan A. Cotter, 2012
Hexagons
      Start with an equilateral triangle




144                                        © Joan A. Cotter, 2012
Hexagons
      Start with an equilateral triangle




145                                        © Joan A. Cotter, 2012
Hexagons
      Start with an equilateral triangle




146                                        © Joan A. Cotter, 2012
Hexagons
      Start with an equilateral triangle




147                                        © Joan A. Cotter, 2012
Hexagons
      Start with an equilateral triangle




148                                        © Joan A. Cotter, 2012
Hexagons
      Start with an equilateral triangle




       A bit of erasing.
149                                        © Joan A. Cotter, 2012
Hexagons
      Start with an equilateral triangle




150                                        © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




151                                             © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




152                                             © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




153                                             © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




         Flip the triangle and repeat.
154                                             © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




155                                             © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




156                                             © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




157                                             © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




158                                             © Joan A. Cotter, 2012
Hexagons
      Start with a small equilateral triangle




159                                             © Joan A. Cotter, 2012
Hexagons
      Draw all the diagonals




160                            © Joan A. Cotter, 2012
Hexagons
      Draw all the diagonals




161                            © Joan A. Cotter, 2012
Hexagons
      Draw all the diagonals




162                            © Joan A. Cotter, 2012
Hexagons
      Draw all the diagonals




163                            © Joan A. Cotter, 2012
Hexagons
      Draw all the diagonals




164                            © Joan A. Cotter, 2012
Hexagons
      Draw all the diagonals




165                            © 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, 2012
Hexagons
           Find the figures




      2 non-congruent rectangles.
169                                 © Joan A. Cotter, 2012
Hexagons
           Find the figures




      2 non-congruent rectangles.
170                                 © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 non-congruent right triangles.
171                                  © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 non-congruent right triangles.
172                                  © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 non-congruent right triangles.
173                                  © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 non-congruent right triangles.
174                                  © Joan A. Cotter, 2012
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 A. Cotter, 2012
Hexagons
            Find the figures




      3 similar equilateral triangles.
179                                      © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 similar equilateral triangles.
180                                      © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 similar obtuse triangles.
181                                 © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 similar obtuse triangles.
182                                 © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 similar obtuse triangles.
183                                 © Joan A. Cotter, 2012
Hexagons
            Find the figures




      3 similar obtuse triangles.
184                                 © Joan A. Cotter, 2012
Hexagons
           Find the figures




      3 non-congruent trapezoids.
185                                 © Joan A. Cotter, 2012
Hexagons
           Find the figures




      3 non-congruent trapezoids.
186                                 © Joan A. Cotter, 2012
Hexagons
           Find the figures




      3 non-congruent trapezoids.
187                                 © Joan A. Cotter, 2012
Hexagons
           Find the figures




      3 non-congruent trapezoids.
188                                 © Joan A. Cotter, 2012
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                                        © Joan A. Cotter, 2012
Hexagons
               Construct the small star




 Apothem: perpendicular line from side to center.
193                                        © Joan A. Cotter, 2012
Hexagons
               Construct the small star




 Apothem: perpendicular line from side to center.
194                                        © Joan A. Cotter, 2012
Hexagons
      Construct the small star




195                              © Joan A. Cotter, 2012
Hexagons
      Construct the small star




196                              © Joan A. Cotter, 2012
Hexagons
      Construct the small star




197                              © Joan A. Cotter, 2012
Hexagons
      Construct the small star




198                              © Joan A. Cotter, 2012
Hexagons
                Construct the small star




      Ratio of the areas of star and hexagon?
199                                        © Joan A. Cotter, 2012
Hexagons
                Construct the small star




      Ratio of the areas of star and hexagon?
200                                        © Joan A. Cotter, 2012
Hexagons
      Construct the large star




201                              © Joan A. Cotter, 2012
Hexagons
      Construct the large star




202                              © Joan A. Cotter, 2012
Hexagons
      Construct the large star




203                              © Joan A. Cotter, 2012
Hexagons
      Construct the large star




204                              © Joan A. Cotter, 2012
Hexagons
      Construct the large star




205                              © Joan A. Cotter, 2012
Hexagons
      Construct the large star




206                              © Joan A. Cotter, 2012
Hexagons
      Construct the large star




207                              © Joan A. Cotter, 2012
Hexagons
      Construct the large star




208                              © Joan A. Cotter, 2012
209   © Joan A. Cotter, 2012
The Clock
        1112 1
      10       2
      9         3
      8        4
        7 6 5




210                             © Joan A. Cotter, 2012
The Clock
        1112 1
      10       2
      9         3
      8        4
        7 6 5




211                             © Joan A. Cotter, 2012
The Clock
        1112 1
      10       2
      9         3
      8        4
        7 6 5




212                             © Joan A. Cotter, 2012
The Clock
        1112 1
      10       2
      9         3
      8        4
        7 6 5




213                             © Joan A. Cotter, 2012
The Clock
        1112 1
      10       2
      9         3
      8        4
        7 6 5




214                             © Joan A. Cotter, 2012
The Clock
        1112 1
      10       2
      9         3
      8        4
        7 6 5



                           11 12 1
                      10               2
                      9                3
                       8               4
                           7   6   5




215                                        © Joan A. Cotter, 2012
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 square




227                        © Joan A. Cotter, 2012
Squares
       Halves




228             © Joan A. Cotter, 2012
Squares
       Halves




229             © Joan A. Cotter, 2012
Squares
       Fourths




230              © Joan A. Cotter, 2012
Squares
       Eighths




231              © Joan A. Cotter, 2012
Squares
      Sixteenths




232                © Joan A. Cotter, 2012
Squares
      Sixteenths




233                © 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
       Halves




238             © Joan A. Cotter, 2012
Squares
       Fourths




239              © Joan A. Cotter, 2012
Squares
       Fourths




240              © Joan A. Cotter, 2012
Squares
              Eighths




      Find the center of a small square.
241                                 © Joan A. Cotter, 2012
Squares
              Eighths




      Find the center of a small square.
242                                 © Joan A. Cotter, 2012
Squares
              Eighths




      Find the center of a small square.
243                                 © Joan A. Cotter, 2012
Squares
              Eighths




      Find the center of a small square.
244                                 © Joan A. Cotter, 2012
Squares
               Eighths




      Repeat for the other line.
245                                © Joan A. Cotter, 2012
Squares
       Eighths




246              © Joan A. Cotter, 2012
Squares
      Sixteenths




247                © Joan A. Cotter, 2012
Squares
      Sixteenths




248                © Joan A. Cotter, 2012
Squares
      Sixteenths




249                © Joan A. Cotter, 2012
Squares
      Right triangle spiral




250                           © Joan A. Cotter, 2012
Squares
      Right triangle spiral




251                           © Joan A. Cotter, 2012
Squares
      Right triangle spiral




252                           © Joan A. Cotter, 2012
Squares
      Right triangle spiral




253                           © Joan A. Cotter, 2012
Squares
      Right triangle spiral




254                           © Joan A. Cotter, 2012
Squares
      Right triangle spiral




255                           © Joan A. Cotter, 2012
Squares
      Right triangle spiral




256                           © Joan A. Cotter, 2012
Squares
      Right triangle spiral




257                           © Joan A. Cotter, 2012
Squares
      Right triangle spiral




258                           © Joan A. Cotter, 2012
Squares
          Right triangle spiral




      Adapted from Psychogeometry
259                                 © Joan A. Cotter, 2012
Squares
      Right triangle design




260                           © Joan A. Cotter, 2012
Squares
      Right triangle design




261                           © Joan A. Cotter, 2012
Squares
      Right triangle design




262                           © Joan A. Cotter, 2012
Squares
      Right triangle design




263                           © Joan A. Cotter, 2012
Squares
      Right triangle design




264                           © Joan A. Cotter, 2012
Squares
      Right triangle design




265                           © Joan A. Cotter, 2012
Squares
      Right triangle design




266                           © Joan A. Cotter, 2012
Squares
      Right triangle design




267                           © Joan A. Cotter, 2012
Squares
      Right triangle design




268                           © Joan A. Cotter, 2012
Squares
      Right triangle design




269                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




270                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




271                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




272                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




273                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




274                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




275                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




276                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




277                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




278                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




279                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




280                           © Joan A. Cotter, 2012
Squares
      Circumscribed squares




281                           © Joan A. Cotter, 2012
Squares
      Inscribed squares




282                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




283                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




284                       © Joan A. Cotter, 2012
Squares
          Inscribed squares




      Compare the area of the squares.
285                               © Joan A. Cotter, 2012
Squares
          Inscribed squares




      Compare the area of the squares.
286                               © Joan A. Cotter, 2012
Squares
      Inscribed squares




287                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




288                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




289                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




290                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




291                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




292                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




293                       © Joan A. Cotter, 2012
Squares
      Inscribed squares




294                       © Joan A. Cotter, 2012
295   © Joan A. Cotter, 2012
Tangrams
      Constructing a tangram




296                            © Joan A. Cotter, 2012
Tangrams
      Constructing a tangram




297                            © Joan A. Cotter, 2012
Tangrams
      Constructing a tangram




298                            © Joan A. Cotter, 2012
Tangrams
      Constructing a tangram




299                            © Joan A. Cotter, 2012
Tangrams
      Constructing a tangram




300                            © Joan A. Cotter, 2012
Tangrams
      Constructing a tangram




301                            © Joan A. Cotter, 2012
Tangrams
      Constructing a tangram




302                            © Joan A. Cotter, 2012
Tangrams
        Constructing a tangram




      3 green figures are equal in area.
303                                  © Joan A. Cotter, 2012
Tangrams
        Constructing a tangram




      3 green figures are equal in area.
304                                  © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
305                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
306                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
307                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
308                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
309                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
310                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
311                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
312                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       No tracing or measuring.
313                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram arrangement




       More examples to construct.
314                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Use the left side of the paper.
315                                     © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Start at the left side of the paper.
316                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
317                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
318                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
319                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
320                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
321                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
322                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
323                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
324                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
325                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
326                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




      Reflect the image across the line.
327                                    © Joan A. Cotter, 2012
Tangrams
      Construct a tangram reflection




328                                    © Joan A. Cotter, 2012
329   © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        First construct a right triangle.
330                                         © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




331                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




332                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




333                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




334                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




335                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




336                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




337                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




338                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Area of blue squares = yellow sq.
339                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Area of blue squares = yellow sq.
340                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Draw a right triangle.
341                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




342                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




343                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




344                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




345                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




346                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




347                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Turn the t-square upside down.
348                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




349                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




350                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




351                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Do not move the t-square.
352                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Do not move the t-square.
353                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Do not move the t-square.
354                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Do not move the t-square.
355                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




356                                      © Joan A. Cotter, 2012
Pythagorean Theorem
        Construct squares on all sides




        Area of blue squares = yellow sq.
357                                      © Joan A. Cotter, 2012
358   © Joan A. Cotter, 2012
Triangle Area
      Construct the altitude


                       A




      B                        C




359                                © Joan A. Cotter, 2012
Triangle Area
         Construct the altitude


                          A




         B                        C




      Draw the altitude.
360                                   © Joan A. Cotter, 2012
Triangle Area
         Construct the altitude


                          A




          B                       C




      Area is half of enclosing
361                                   © Joan A. Cotter, 2012
Triangle Area
         Construct the altitude


                          A




          B                       C




      Another altitude. Find area.
362                                   © Joan A. Cotter, 2012
Triangle Area
         Construct the altitude


                          A




          B                       C




      Another altitude. Find area.
363                                   © Joan A. Cotter, 2012
Triangle Area
         Construct the altitude


                          A




          B                       C




      Another altitude. Find area.
364                                   © Joan A. Cotter, 2012
Triangle Area
         Construct the altitude


                          A




          B                       C




      Another altitude. Find area.
365                                   © Joan A. Cotter, 2012
Triangle Area
         Construct the altitude


                          A




          B                       C




      Altitudes intersect at the
366                                   © Joan A. Cotter, 2012
Triangle Area
         Construct the altitude


                          A




          B                       C




      3 areas slightly different. Why?
367                                   © Joan A. Cotter, 2012
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
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
Bridging Geometry Concepts with Art
                by Joan A. Cotter, Ph.D.
             joancotter@rightstartmath.com




AMS Conference
 March 17, 2012
San Francisco CA



        Presentations available: rightstartmath.com

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Geometry AMS

  • 1. Bridging Geometry Concepts with Art by Joan A. Cotter, Ph.D. joancotter@rightstartmath.com AMS Conference March 17, 2012 San Francisco CA Presentations available: rightstartmath.com
  • 2. Drawing Tools Paper North American paper 11 in. by 8.5 in. 2 © Joan A. Cotter, 2012
  • 3. Drawing Tools Paper International paper 21.0cm by 29.7cm 3 © Joan A. Cotter, 2012
  • 4. Drawing Tools Drawing board with paper 4 © Joan A. Cotter, 2012
  • 5. Drawing Tools Drawing board with paper Affix tape only at the top. 5 © Joan A. Cotter, 2012
  • 6. Drawing Tools T-square T-square Right-handed user. 6 © Joan A. Cotter, 2012
  • 7. Drawing Tools T-square T-square Left-handed user. 7 © Joan A. Cotter, 2012
  • 8. Drawing Tools T-square Must hug the edge of the board. 8 © Joan A. Cotter, 2012
  • 9. Drawing Tools T-square Must hug the edge of the board. 9 © Joan A. Cotter, 2012
  • 10. Drawing Tools T-square Must hug the edge of the board. 10 © Joan A. Cotter, 2012
  • 11. Drawing Tools T-square Must hug the edge of the board. 11 © Joan A. Cotter, 2012
  • 12. Drawing Tools T-square Must hug the edge of the board. 12 © Joan A. Cotter, 2012
  • 13. Drawing Tools T-square Must hug the edge of the board. 13 © Joan A. Cotter, 2012
  • 14. Drawing Tools T-square Must hug the edge of the board. 14 © Joan A. Cotter, 2012
  • 15. Drawing Tools T-square Must hug the edge of the board. 15 © Joan A. Cotter, 2012
  • 16. Drawing Tools T-square Must hug the edge of the board. 16 © Joan A. Cotter, 2012
  • 17. Drawing Tools Pencil (mechanical) and eraser Hold pencil 2-3cm (1 in.) from tip. 17 © Joan A. Cotter, 2012
  • 18. Drawing Tools Pencil (mechanical) and eraser Hold pencil 2-3cm (1 in.) from tip. 18 © Joan A. Cotter, 2012
  • 19. Drawing Tools Pencil (mechanical) and eraser Hold pencil 2-3cm (1 in.) from tip. 19 © Joan A. Cotter, 2012
  • 20. Drawing Tools Pencil (mechanical) and eraser An eraser is an essential tool. 20 © Joan A. Cotter, 2012
  • 21. Drawing Tools Pencil (mechanical) and eraser Parallel lines. 21 © Joan A. Cotter, 2012
  • 22. Drawing Tools Pencil (mechanical) and eraser Parallel lines. 22 © Joan A. Cotter, 2012
  • 23. Drawing Tools The 30-60 triangle (a set square) 30° 60° 90° 23 © Joan A. Cotter, 2012
  • 24. Drawing Tools The 30-60 triangle Triangle must hug the t-square. 24 © Joan A. Cotter, 2012
  • 25. Drawing Tools The 30-60 triangle This doesn’t work! 25 © Joan A. Cotter, 2012
  • 26. Drawing Tools The 30-60 triangle 26 © Joan A. Cotter, 2012
  • 27. Drawing Tools The 30-60 triangle More parallel lines. 27 © Joan A. Cotter, 2012
  • 28. Drawing Tools The 30-60 triangle More parallel lines. 28 © Joan A. Cotter, 2012
  • 29. Drawing Tools The 30-60 triangle Perpendicular lines. 29 © Joan A. Cotter, 2012
  • 30. Drawing Tools The 30-60 triangle Perpendicular lines. 30 © Joan A. Cotter, 2012
  • 31. Drawing Tools The 30-60 triangle 31 © Joan A. Cotter, 2012
  • 32. Drawing Tools The 30-60 triangle Still more parallel lines. 32 © Joan A. Cotter, 2012
  • 33. Drawing Tools The 30-60 triangle Still more parallel lines. 33 © Joan A. Cotter, 2012
  • 34. Drawing Tools The 30-60 triangle 34 © Joan A. Cotter, 2012
  • 35. Drawing Tools The 30-60 triangle Intersecting lines. 35 © Joan A. Cotter, 2012
  • 36. Drawing Tools The 45 triangle 45° 45° 90° 36 © Joan A. Cotter, 2012
  • 37. Drawing Tools The 45 triangle 37 © Joan A. Cotter, 2012
  • 38. Drawing Tools The 45 triangle 38 © Joan A. Cotter, 2012
  • 39. Drawing Tools The 45 triangle Maintain a margin. 39 © Joan A. Cotter, 2012
  • 40. Drawing Tools The 45 triangle Maintain a margin. 40 © Joan A. Cotter, 2012
  • 41. Drawing Tools The 45 triangle Maintain a margin. 41 © Joan A. Cotter, 2012
  • 42. 42 © Joan A. Cotter, 2012
  • 43. Equilateral Triangles Draw the base line 43 © Joan A. Cotter, 2012
  • 44. Equilateral Triangles Draw the base line Move the t-square down. 44 © Joan A. Cotter, 2012
  • 45. Equilateral Triangles Draw the base line Move the t-square down. 45 © Joan A. Cotter, 2012
  • 46. Equilateral Triangles Draw the sides Draw a side. 46 © Joan A. Cotter, 2012
  • 47. Equilateral Triangles Draw the sides Flip the triangle and repeat. 47 © Joan A. Cotter, 2012
  • 48. Equilateral Triangles Draw the sides Flip the triangle and repeat. 48 © Joan A. Cotter, 2012
  • 49. Equilateral Triangles Draw the sides A bit of erasing. 49 © Joan A. Cotter, 2012
  • 50. Equilateral Triangles Divide it into halves 50 © Joan A. Cotter, 2012
  • 51. Equilateral Triangles Divide it into halves 51 © Joan A. Cotter, 2012
  • 52. Equilateral Triangles Divide it into halves 52 © Joan A. Cotter, 2012
  • 53. Equilateral Triangles Divide it into halves 1 1 2 2 The two halves are congruent. 53 © Joan A. Cotter, 2012
  • 54. Equilateral Triangles Divide it into halves Divide it in half another way. 54 © Joan A. Cotter, 2012
  • 55. Equilateral Triangles Divide it into halves Divide it in half another way. 55 © Joan A. Cotter, 2012
  • 56. Equilateral Triangles Divide it into halves Divide it in half another way. 56 © Joan A. Cotter, 2012
  • 57. Equilateral Triangles Divide it into halves Divide it in half another way. 57 © Joan A. Cotter, 2012
  • 58. Equilateral Triangles Divide it into halves 58 © Joan A. Cotter, 2012
  • 59. Equilateral Triangles Divide it into thirds 59 © Joan A. Cotter, 2012
  • 60. Equilateral Triangles Divide it into thirds 60 © Joan A. Cotter, 2012
  • 61. Equilateral Triangles Divide it into thirds 61 © Joan A. Cotter, 2012
  • 62. Equilateral Triangles Divide it into thirds 62 © Joan A. Cotter, 2012
  • 63. Equilateral Triangles Divide it into thirds A bit of erasing. 63 © Joan A. Cotter, 2012
  • 64. Equilateral Triangles Divide it into thirds Flip the triangle and repeat. 64 © Joan A. Cotter, 2012
  • 65. Equilateral Triangles Divide it into thirds 65 © Joan A. Cotter, 2012
  • 66. Equilateral Triangles Divide it into thirds 1 1 3 3 1 3 66 © Joan A. Cotter, 2012
  • 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. Equilateral Triangles Divide it into fourths 68 © Joan A. Cotter, 2012
  • 69. Equilateral Triangles Divide it into fourths Finding the center of the base line. 69 © Joan A. Cotter, 2012
  • 70. Equilateral Triangles Divide it into fourths tick mark Be sure the tick mark intersects the line. 70 © Joan A. Cotter, 2012
  • 71. Equilateral Triangles Divide it into fourths 71 © Joan A. Cotter, 2012
  • 72. Equilateral Triangles Divide it into fourths Flip the triangle over and repeat. 72 © Joan A. Cotter, 2012
  • 73. Equilateral Triangles Divide it into fourths How do you draw the last line? 73 © Joan A. Cotter, 2012
  • 74. Equilateral Triangles Divide it into fourths How do you draw the last line? 74 © Joan A. Cotter, 2012
  • 75. Equilateral Triangles Divide it into fourths The smaller triangles are similar to the larger triangle. 75 © Joan A. Cotter, 2012
  • 76. Equilateral Triangles Divide it into fourths 1 4 1 1 4 1 4 4 76 © Joan A. Cotter, 2012
  • 77. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: 77 © Joan A. Cotter, 2012
  • 78. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF, 78 © Joan A. Cotter, 2012
  • 79. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses. ABDF, BCDF, 79 © Joan A. Cotter, 2012
  • 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. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: 81 © Joan A. Cotter, 2012
  • 82. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF, 82 © Joan A. Cotter, 2012
  • 83. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF, 83 © Joan A. Cotter, 2012
  • 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. Equilateral Triangles Tetrahedron Cut it out…. 85 © Joan A. Cotter, 2012
  • 86. Equilateral Triangles Tetrahedron and fold on the lines. 86 © Joan A. Cotter, 2012
  • 87. Equilateral Triangles Tetrahedron and fold on the lines. 87 © Joan A. Cotter, 2012
  • 88. Equilateral Triangles Tetrahedron and fold on the dotted lines. 88 © Joan A. Cotter, 2012
  • 89. Equilateral Triangles Tetrahedron and fold on the dotted lines. 89 © Joan A. Cotter, 2012
  • 90. Equilateral Triangles Tetrahedron and fold on the dotted lines. 90 © Joan A. Cotter, 2012
  • 91. Equilateral Triangles Tetrahedron and fold on the dotted lines. 91 © Joan A. Cotter, 2012
  • 92. Equilateral Triangles Tetrahedron and fold on the dotted lines. 92 © Joan A. Cotter, 2012
  • 93. Equilateral Triangles Divide it into eighths How could you draw eighths? 93 © Joan A. Cotter, 2012
  • 94. Equilateral Triangles Divide it into eighths 94 © Joan A. Cotter, 2012
  • 95. Equilateral Triangles Divide it into eighths 95 © Joan A. Cotter, 2012
  • 96. Equilateral Triangles Divide it into eighths Not very esthetic. 96 © Joan A. Cotter, 2012
  • 97. Equilateral Triangles Divide it into eighths 97 © Joan A. Cotter, 2012
  • 98. Equilateral Triangles Divide it into sixths Can you divide it into sixths? 98 © Joan A. Cotter, 2012
  • 99. Equilateral Triangles Divide it into sixths Can you divide it into sixths? 99 © Joan A. Cotter, 2012
  • 100. Equilateral Triangles Divide it into sixths 100 © Joan A. Cotter, 2012
  • 101. Equilateral Triangles Divide it into sixths 101 © Joan A. Cotter, 2012
  • 102. Equilateral Triangles Divide it into sixths A B F G C E D Name an acute triangle: 102 © Joan A. Cotter, 2012
  • 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. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: 104 © Joan A. Cotter, 2012
  • 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. 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. 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. Equilateral Triangles Divide it into sixths A B F G C E D Name an obtuse triangle: 108 © Joan A. Cotter, 2012
  • 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. Equilateral Triangles Divide it into twelfths Can you divide it into twelfths? 110 © Joan A. Cotter, 2012
  • 111. Equilateral Triangles Divide it into twelfths 111 © Joan A. Cotter, 2012
  • 112. Equilateral Triangles Divide it into twelfths 112 © Joan A. Cotter, 2012
  • 113. Equilateral Triangles Divide it into twelfths 113 © Joan A. Cotter, 2012
  • 114. Equilateral Triangles Divide it into twelfths 114 © Joan A. Cotter, 2012
  • 115. Equilateral Triangles Divide it into twelfths 115 © Joan A. Cotter, 2012
  • 116. Equilateral Triangles Divide it into twelfths Do you see the cube? 116 © Joan A. Cotter, 2012
  • 117. Equilateral Triangles Divide it into twelfths Do you see the cube? 117 © Joan A. Cotter, 2012
  • 118. Equilateral Triangles Divide it into twelfths Do you see the hexagon? 118 © Joan A. Cotter, 2012
  • 119. Equilateral Triangles Divide it into twelfths Do you see the hexagon? 119 © Joan A. Cotter, 2012
  • 120. Equilateral Triangles Ninths 120 © Joan A. Cotter, 2012
  • 121. Equilateral Triangles Twenty-sevenths 121 © Joan A. Cotter, 2012
  • 122. Equilateral Triangles Twenty-sevenths 122 © Joan A. Cotter, 2012
  • 123. Equilateral Triangles Twenty-sevenths 123 © Joan A. Cotter, 2012
  • 124. Equilateral Triangles Twenty-sevenths 124 © Joan A. Cotter, 2012
  • 125. Equilateral Triangles Thirty-seconds 125 © Joan A. Cotter, 2012
  • 126. Equilateral Triangles Stephanie, 6, a Montessori child, did 256. 126 © Joan A. Cotter, 2012
  • 127. Equilateral Triangles Stephanie, 6, a Montessori child, did 256. 127 © Joan A. Cotter, 2012
  • 128. Equilateral Triangles Stephanie, 6, a Montessori child, did 256. 128 © Joan A. Cotter, 2012
  • 129. Equilateral Triangles Stephanie, 6, a Montessori child, did 256. 129 © Joan A. Cotter, 2012
  • 130. Equilateral Triangles Stephanie, 6, a Montessori child, did 256. 130 © Joan A. Cotter, 2012
  • 131. Equilateral Triangles Spiral 131 © Joan A. Cotter, 2012
  • 132. Equilateral Triangles Spiral 132 © Joan A. Cotter, 2012
  • 133. Equilateral Triangles Spiral 133 © Joan A. Cotter, 2012
  • 134. Equilateral Triangles Spiral 134 © Joan A. Cotter, 2012
  • 135. Equilateral Triangles Spiral 135 © Joan A. Cotter, 2012
  • 136. Equilateral Triangles Spiral 136 © Joan A. Cotter, 2012
  • 137. Equilateral Triangles Spiral 137 © Joan A. Cotter, 2012
  • 138. Equilateral Triangles Spiral 138 © Joan A. Cotter, 2012
  • 139. Equilateral Triangles Spiral 139 © Joan A. Cotter, 2012
  • 140. Equilateral Triangles Spiral From Psychogeometry 140 © Joan A. Cotter, 2012
  • 141. Equilateral Triangles Spiral 141 © Joan A. Cotter, 2012
  • 142. 142 © Joan A. Cotter, 2012
  • 143. Hexagons Start with an equilateral triangle 143 © Joan A. Cotter, 2012
  • 144. Hexagons Start with an equilateral triangle 144 © Joan A. Cotter, 2012
  • 145. Hexagons Start with an equilateral triangle 145 © Joan A. Cotter, 2012
  • 146. Hexagons Start with an equilateral triangle 146 © Joan A. Cotter, 2012
  • 147. Hexagons Start with an equilateral triangle 147 © Joan A. Cotter, 2012
  • 148. Hexagons Start with an equilateral triangle 148 © Joan A. Cotter, 2012
  • 149. Hexagons Start with an equilateral triangle A bit of erasing. 149 © Joan A. Cotter, 2012
  • 150. Hexagons Start with an equilateral triangle 150 © Joan A. Cotter, 2012
  • 151. Hexagons Start with a small equilateral triangle 151 © Joan A. Cotter, 2012
  • 152. Hexagons Start with a small equilateral triangle 152 © Joan A. Cotter, 2012
  • 153. Hexagons Start with a small equilateral triangle 153 © Joan A. Cotter, 2012
  • 154. Hexagons Start with a small equilateral triangle Flip the triangle and repeat. 154 © Joan A. Cotter, 2012
  • 155. Hexagons Start with a small equilateral triangle 155 © Joan A. Cotter, 2012
  • 156. Hexagons Start with a small equilateral triangle 156 © Joan A. Cotter, 2012
  • 157. Hexagons Start with a small equilateral triangle 157 © Joan A. Cotter, 2012
  • 158. Hexagons Start with a small equilateral triangle 158 © Joan A. Cotter, 2012
  • 159. Hexagons Start with a small equilateral triangle 159 © Joan A. Cotter, 2012
  • 160. Hexagons Draw all the diagonals 160 © Joan A. Cotter, 2012
  • 161. Hexagons Draw all the diagonals 161 © Joan A. Cotter, 2012
  • 162. Hexagons Draw all the diagonals 162 © Joan A. Cotter, 2012
  • 163. Hexagons Draw all the diagonals 163 © Joan A. Cotter, 2012
  • 164. Hexagons Draw all the diagonals 164 © Joan A. Cotter, 2012
  • 165. Hexagons Draw all the diagonals 165 © Joan A. Cotter, 2012
  • 166. Hexagons Find the figures Another hexagon. 166 © Joan A. Cotter, 2012
  • 167. Hexagons Find the figures Another hexagon. 167 © Joan A. Cotter, 2012
  • 168. Hexagons Find the figures 2 non-congruent rectangles. 168 © Joan A. Cotter, 2012
  • 169. Hexagons Find the figures 2 non-congruent rectangles. 169 © Joan A. Cotter, 2012
  • 170. Hexagons Find the figures 2 non-congruent rectangles. 170 © Joan A. Cotter, 2012
  • 171. Hexagons Find the figures 3 non-congruent right triangles. 171 © Joan A. Cotter, 2012
  • 172. Hexagons Find the figures 3 non-congruent right triangles. 172 © Joan A. Cotter, 2012
  • 173. Hexagons Find the figures 3 non-congruent right triangles. 173 © Joan A. Cotter, 2012
  • 174. Hexagons Find the figures 3 non-congruent right triangles. 174 © Joan A. Cotter, 2012
  • 175. Hexagons Find the figures 2 non-congruent rhombuses. 175 © Joan A. Cotter, 2012
  • 176. Hexagons Find the figures 2 non-congruent rhombuses. 176 © Joan A. Cotter, 2012
  • 177. Hexagons Find the figures 2 non-congruent rhombuses. 177 © Joan A. Cotter, 2012
  • 178. Hexagons Find the figures 3 similar equilateral triangles. 178 © Joan A. Cotter, 2012
  • 179. Hexagons Find the figures 3 similar equilateral triangles. 179 © Joan A. Cotter, 2012
  • 180. Hexagons Find the figures 3 similar equilateral triangles. 180 © Joan A. Cotter, 2012
  • 181. Hexagons Find the figures 3 similar obtuse triangles. 181 © Joan A. Cotter, 2012
  • 182. Hexagons Find the figures 3 similar obtuse triangles. 182 © Joan A. Cotter, 2012
  • 183. Hexagons Find the figures 3 similar obtuse triangles. 183 © Joan A. Cotter, 2012
  • 184. Hexagons Find the figures 3 similar obtuse triangles. 184 © Joan A. Cotter, 2012
  • 185. Hexagons Find the figures 3 non-congruent trapezoids. 185 © Joan A. Cotter, 2012
  • 186. Hexagons Find the figures 3 non-congruent trapezoids. 186 © Joan A. Cotter, 2012
  • 187. Hexagons Find the figures 3 non-congruent trapezoids. 187 © Joan A. Cotter, 2012
  • 188. Hexagons Find the figures 3 non-congruent trapezoids. 188 © Joan A. Cotter, 2012
  • 189. Hexagons Find the figures 2 similar kites. 189 © Joan A. Cotter, 2012
  • 190. Hexagons Find the figures 2 similar kites. 190 © Joan A. Cotter, 2012
  • 191. Hexagons Find the figures 2 similar kites. 191 © Joan A. Cotter, 2012
  • 192. Hexagons Construct the small star Apothem: perpendicular line from side to center. 192 © Joan A. Cotter, 2012
  • 193. Hexagons Construct the small star Apothem: perpendicular line from side to center. 193 © Joan A. Cotter, 2012
  • 194. Hexagons Construct the small star Apothem: perpendicular line from side to center. 194 © Joan A. Cotter, 2012
  • 195. Hexagons Construct the small star 195 © Joan A. Cotter, 2012
  • 196. Hexagons Construct the small star 196 © Joan A. Cotter, 2012
  • 197. Hexagons Construct the small star 197 © Joan A. Cotter, 2012
  • 198. Hexagons Construct the small star 198 © Joan A. Cotter, 2012
  • 199. Hexagons Construct the small star Ratio of the areas of star and hexagon? 199 © Joan A. Cotter, 2012
  • 200. Hexagons Construct the small star Ratio of the areas of star and hexagon? 200 © Joan A. Cotter, 2012
  • 201. Hexagons Construct the large star 201 © Joan A. Cotter, 2012
  • 202. Hexagons Construct the large star 202 © Joan A. Cotter, 2012
  • 203. Hexagons Construct the large star 203 © Joan A. Cotter, 2012
  • 204. Hexagons Construct the large star 204 © Joan A. Cotter, 2012
  • 205. Hexagons Construct the large star 205 © Joan A. Cotter, 2012
  • 206. Hexagons Construct the large star 206 © Joan A. Cotter, 2012
  • 207. Hexagons Construct the large star 207 © Joan A. Cotter, 2012
  • 208. Hexagons Construct the large star 208 © Joan A. Cotter, 2012
  • 209. 209 © Joan A. Cotter, 2012
  • 210. The Clock 1112 1 10 2 9 3 8 4 7 6 5 210 © Joan A. Cotter, 2012
  • 211. The Clock 1112 1 10 2 9 3 8 4 7 6 5 211 © Joan A. Cotter, 2012
  • 212. The Clock 1112 1 10 2 9 3 8 4 7 6 5 212 © Joan A. Cotter, 2012
  • 213. The Clock 1112 1 10 2 9 3 8 4 7 6 5 213 © Joan A. Cotter, 2012
  • 214. The Clock 1112 1 10 2 9 3 8 4 7 6 5 214 © Joan A. Cotter, 2012
  • 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 5 215 © Joan A. Cotter, 2012
  • 216. 216 © Joan A. Cotter, 2012
  • 217. Squares The 45 triangle. 217 © Joan A. Cotter, 2012
  • 218. Squares The 45 triangle is half of a square. 218 © Joan A. Cotter, 2012
  • 219. Squares Construct a square Draw the base line. 219 © Joan A. Cotter, 2012
  • 220. Squares Construct a square Move the t-square down. 220 © Joan A. Cotter, 2012
  • 221. Squares Construct a square Move the t-square down. 221 © Joan A. Cotter, 2012
  • 222. Squares Construct a square Draw the sides. 222 © Joan A. Cotter, 2012
  • 223. Squares Construct a square Draw the diagonal… 223 © Joan A. Cotter, 2012
  • 224. Squares Construct a square or just a tick mark. 224 © Joan A. Cotter, 2012
  • 225. Squares Construct a square Draw the top. 225 © Joan A. Cotter, 2012
  • 226. Squares Construct a square A bit of erasing. 226 © Joan A. Cotter, 2012
  • 227. Squares Construct a square 227 © Joan A. Cotter, 2012
  • 228. Squares Halves 228 © Joan A. Cotter, 2012
  • 229. Squares Halves 229 © Joan A. Cotter, 2012
  • 230. Squares Fourths 230 © Joan A. Cotter, 2012
  • 231. Squares Eighths 231 © Joan A. Cotter, 2012
  • 232. Squares Sixteenths 232 © Joan A. Cotter, 2012
  • 233. Squares Sixteenths 233 © Joan A. Cotter, 2012
  • 234. Squares Halves We need to find the center. 234 © Joan A. Cotter, 2012
  • 235. Squares Halves We need to find the center. 235 © Joan A. Cotter, 2012
  • 236. Squares Halves We need to find the center. 236 © Joan A. Cotter, 2012
  • 237. Squares Halves We need to find the center. 237 © Joan A. Cotter, 2012
  • 238. Squares Halves 238 © Joan A. Cotter, 2012
  • 239. Squares Fourths 239 © Joan A. Cotter, 2012
  • 240. Squares Fourths 240 © Joan A. Cotter, 2012
  • 241. Squares Eighths Find the center of a small square. 241 © Joan A. Cotter, 2012
  • 242. Squares Eighths Find the center of a small square. 242 © Joan A. Cotter, 2012
  • 243. Squares Eighths Find the center of a small square. 243 © Joan A. Cotter, 2012
  • 244. Squares Eighths Find the center of a small square. 244 © Joan A. Cotter, 2012
  • 245. Squares Eighths Repeat for the other line. 245 © Joan A. Cotter, 2012
  • 246. Squares Eighths 246 © Joan A. Cotter, 2012
  • 247. Squares Sixteenths 247 © Joan A. Cotter, 2012
  • 248. Squares Sixteenths 248 © Joan A. Cotter, 2012
  • 249. Squares Sixteenths 249 © Joan A. Cotter, 2012
  • 250. Squares Right triangle spiral 250 © Joan A. Cotter, 2012
  • 251. Squares Right triangle spiral 251 © Joan A. Cotter, 2012
  • 252. Squares Right triangle spiral 252 © Joan A. Cotter, 2012
  • 253. Squares Right triangle spiral 253 © Joan A. Cotter, 2012
  • 254. Squares Right triangle spiral 254 © Joan A. Cotter, 2012
  • 255. Squares Right triangle spiral 255 © Joan A. Cotter, 2012
  • 256. Squares Right triangle spiral 256 © Joan A. Cotter, 2012
  • 257. Squares Right triangle spiral 257 © Joan A. Cotter, 2012
  • 258. Squares Right triangle spiral 258 © Joan A. Cotter, 2012
  • 259. Squares Right triangle spiral Adapted from Psychogeometry 259 © Joan A. Cotter, 2012
  • 260. Squares Right triangle design 260 © Joan A. Cotter, 2012
  • 261. Squares Right triangle design 261 © Joan A. Cotter, 2012
  • 262. Squares Right triangle design 262 © Joan A. Cotter, 2012
  • 263. Squares Right triangle design 263 © Joan A. Cotter, 2012
  • 264. Squares Right triangle design 264 © Joan A. Cotter, 2012
  • 265. Squares Right triangle design 265 © Joan A. Cotter, 2012
  • 266. Squares Right triangle design 266 © Joan A. Cotter, 2012
  • 267. Squares Right triangle design 267 © Joan A. Cotter, 2012
  • 268. Squares Right triangle design 268 © Joan A. Cotter, 2012
  • 269. Squares Right triangle design 269 © Joan A. Cotter, 2012
  • 270. Squares Circumscribed squares 270 © Joan A. Cotter, 2012
  • 271. Squares Circumscribed squares 271 © Joan A. Cotter, 2012
  • 272. Squares Circumscribed squares 272 © Joan A. Cotter, 2012
  • 273. Squares Circumscribed squares 273 © Joan A. Cotter, 2012
  • 274. Squares Circumscribed squares 274 © Joan A. Cotter, 2012
  • 275. Squares Circumscribed squares 275 © Joan A. Cotter, 2012
  • 276. Squares Circumscribed squares 276 © Joan A. Cotter, 2012
  • 277. Squares Circumscribed squares 277 © Joan A. Cotter, 2012
  • 278. Squares Circumscribed squares 278 © Joan A. Cotter, 2012
  • 279. Squares Circumscribed squares 279 © Joan A. Cotter, 2012
  • 280. Squares Circumscribed squares 280 © Joan A. Cotter, 2012
  • 281. Squares Circumscribed squares 281 © Joan A. Cotter, 2012
  • 282. Squares Inscribed squares 282 © Joan A. Cotter, 2012
  • 283. Squares Inscribed squares 283 © Joan A. Cotter, 2012
  • 284. Squares Inscribed squares 284 © Joan A. Cotter, 2012
  • 285. Squares Inscribed squares Compare the area of the squares. 285 © Joan A. Cotter, 2012
  • 286. Squares Inscribed squares Compare the area of the squares. 286 © Joan A. Cotter, 2012
  • 287. Squares Inscribed squares 287 © Joan A. Cotter, 2012
  • 288. Squares Inscribed squares 288 © Joan A. Cotter, 2012
  • 289. Squares Inscribed squares 289 © Joan A. Cotter, 2012
  • 290. Squares Inscribed squares 290 © Joan A. Cotter, 2012
  • 291. Squares Inscribed squares 291 © Joan A. Cotter, 2012
  • 292. Squares Inscribed squares 292 © Joan A. Cotter, 2012
  • 293. Squares Inscribed squares 293 © Joan A. Cotter, 2012
  • 294. Squares Inscribed squares 294 © Joan A. Cotter, 2012
  • 295. 295 © Joan A. Cotter, 2012
  • 296. Tangrams Constructing a tangram 296 © Joan A. Cotter, 2012
  • 297. Tangrams Constructing a tangram 297 © Joan A. Cotter, 2012
  • 298. Tangrams Constructing a tangram 298 © Joan A. Cotter, 2012
  • 299. Tangrams Constructing a tangram 299 © Joan A. Cotter, 2012
  • 300. Tangrams Constructing a tangram 300 © Joan A. Cotter, 2012
  • 301. Tangrams Constructing a tangram 301 © Joan A. Cotter, 2012
  • 302. Tangrams Constructing a tangram 302 © Joan A. Cotter, 2012
  • 303. Tangrams Constructing a tangram 3 green figures are equal in area. 303 © Joan A. Cotter, 2012
  • 304. Tangrams Constructing a tangram 3 green figures are equal in area. 304 © Joan A. Cotter, 2012
  • 305. Tangrams Construct a tangram arrangement No tracing or measuring. 305 © Joan A. Cotter, 2012
  • 306. Tangrams Construct a tangram arrangement No tracing or measuring. 306 © Joan A. Cotter, 2012
  • 307. Tangrams Construct a tangram arrangement No tracing or measuring. 307 © Joan A. Cotter, 2012
  • 308. Tangrams Construct a tangram arrangement No tracing or measuring. 308 © Joan A. Cotter, 2012
  • 309. Tangrams Construct a tangram arrangement No tracing or measuring. 309 © Joan A. Cotter, 2012
  • 310. Tangrams Construct a tangram arrangement No tracing or measuring. 310 © Joan A. Cotter, 2012
  • 311. Tangrams Construct a tangram arrangement No tracing or measuring. 311 © Joan A. Cotter, 2012
  • 312. Tangrams Construct a tangram arrangement No tracing or measuring. 312 © Joan A. Cotter, 2012
  • 313. Tangrams Construct a tangram arrangement No tracing or measuring. 313 © Joan A. Cotter, 2012
  • 314. Tangrams Construct a tangram arrangement More examples to construct. 314 © Joan A. Cotter, 2012
  • 315. Tangrams Construct a tangram reflection Use the left side of the paper. 315 © Joan A. Cotter, 2012
  • 316. Tangrams Construct a tangram reflection Start at the left side of the paper. 316 © Joan A. Cotter, 2012
  • 317. Tangrams Construct a tangram reflection Reflect the image across the line. 317 © Joan A. Cotter, 2012
  • 318. Tangrams Construct a tangram reflection Reflect the image across the line. 318 © Joan A. Cotter, 2012
  • 319. Tangrams Construct a tangram reflection Reflect the image across the line. 319 © Joan A. Cotter, 2012
  • 320. Tangrams Construct a tangram reflection Reflect the image across the line. 320 © Joan A. Cotter, 2012
  • 321. Tangrams Construct a tangram reflection Reflect the image across the line. 321 © Joan A. Cotter, 2012
  • 322. Tangrams Construct a tangram reflection Reflect the image across the line. 322 © Joan A. Cotter, 2012
  • 323. Tangrams Construct a tangram reflection Reflect the image across the line. 323 © Joan A. Cotter, 2012
  • 324. Tangrams Construct a tangram reflection Reflect the image across the line. 324 © Joan A. Cotter, 2012
  • 325. Tangrams Construct a tangram reflection Reflect the image across the line. 325 © Joan A. Cotter, 2012
  • 326. Tangrams Construct a tangram reflection Reflect the image across the line. 326 © Joan A. Cotter, 2012
  • 327. Tangrams Construct a tangram reflection Reflect the image across the line. 327 © Joan A. Cotter, 2012
  • 328. Tangrams Construct a tangram reflection 328 © Joan A. Cotter, 2012
  • 329. 329 © Joan A. Cotter, 2012
  • 330. Pythagorean Theorem Construct squares on all sides First construct a right triangle. 330 © Joan A. Cotter, 2012
  • 331. Pythagorean Theorem Construct squares on all sides 331 © Joan A. Cotter, 2012
  • 332. Pythagorean Theorem Construct squares on all sides 332 © Joan A. Cotter, 2012
  • 333. Pythagorean Theorem Construct squares on all sides 333 © Joan A. Cotter, 2012
  • 334. Pythagorean Theorem Construct squares on all sides 334 © Joan A. Cotter, 2012
  • 335. Pythagorean Theorem Construct squares on all sides 335 © Joan A. Cotter, 2012
  • 336. Pythagorean Theorem Construct squares on all sides 336 © Joan A. Cotter, 2012
  • 337. Pythagorean Theorem Construct squares on all sides 337 © Joan A. Cotter, 2012
  • 338. Pythagorean Theorem Construct squares on all sides 338 © Joan A. Cotter, 2012
  • 339. Pythagorean Theorem Construct squares on all sides Area of blue squares = yellow sq. 339 © Joan A. Cotter, 2012
  • 340. Pythagorean Theorem Construct squares on all sides Area of blue squares = yellow sq. 340 © Joan A. Cotter, 2012
  • 341. Pythagorean Theorem Construct squares on all sides Draw a right triangle. 341 © Joan A. Cotter, 2012
  • 342. Pythagorean Theorem Construct squares on all sides 342 © Joan A. Cotter, 2012
  • 343. Pythagorean Theorem Construct squares on all sides 343 © Joan A. Cotter, 2012
  • 344. Pythagorean Theorem Construct squares on all sides 344 © Joan A. Cotter, 2012
  • 345. Pythagorean Theorem Construct squares on all sides 345 © Joan A. Cotter, 2012
  • 346. Pythagorean Theorem Construct squares on all sides 346 © Joan A. Cotter, 2012
  • 347. Pythagorean Theorem Construct squares on all sides 347 © Joan A. Cotter, 2012
  • 348. Pythagorean Theorem Construct squares on all sides Turn the t-square upside down. 348 © Joan A. Cotter, 2012
  • 349. Pythagorean Theorem Construct squares on all sides 349 © Joan A. Cotter, 2012
  • 350. Pythagorean Theorem Construct squares on all sides 350 © Joan A. Cotter, 2012
  • 351. Pythagorean Theorem Construct squares on all sides 351 © Joan A. Cotter, 2012
  • 352. Pythagorean Theorem Construct squares on all sides Do not move the t-square. 352 © Joan A. Cotter, 2012
  • 353. Pythagorean Theorem Construct squares on all sides Do not move the t-square. 353 © Joan A. Cotter, 2012
  • 354. Pythagorean Theorem Construct squares on all sides Do not move the t-square. 354 © Joan A. Cotter, 2012
  • 355. Pythagorean Theorem Construct squares on all sides Do not move the t-square. 355 © Joan A. Cotter, 2012
  • 356. Pythagorean Theorem Construct squares on all sides 356 © Joan A. Cotter, 2012
  • 357. Pythagorean Theorem Construct squares on all sides Area of blue squares = yellow sq. 357 © Joan A. Cotter, 2012
  • 358. 358 © Joan A. Cotter, 2012
  • 359. Triangle Area Construct the altitude A B C 359 © Joan A. Cotter, 2012
  • 360. Triangle Area Construct the altitude A B C Draw the altitude. 360 © Joan A. Cotter, 2012
  • 361. Triangle Area Construct the altitude A B C Area is half of enclosing 361 © Joan A. Cotter, 2012
  • 362. Triangle Area Construct the altitude A B C Another altitude. Find area. 362 © Joan A. Cotter, 2012
  • 363. Triangle Area Construct the altitude A B C Another altitude. Find area. 363 © Joan A. Cotter, 2012
  • 364. Triangle Area Construct the altitude A B C Another altitude. Find area. 364 © Joan A. Cotter, 2012
  • 365. Triangle Area Construct the altitude A B C Another altitude. Find area. 365 © Joan A. Cotter, 2012
  • 366. Triangle Area Construct the altitude A B C Altitudes intersect at the 366 © Joan A. Cotter, 2012
  • 367. Triangle Area Construct the altitude A B C 3 areas slightly different. Why? 367 © Joan A. Cotter, 2012
  • 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. 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. Bridging Geometry Concepts with Art by Joan A. Cotter, Ph.D. joancotter@rightstartmath.com AMS Conference March 17, 2012 San Francisco CA Presentations available: rightstartmath.com