Personalized Learning                               Bridges Middle-School Math                                with a Geome...
Why a Geometric Approach?© Activities for Learning, Inc., 2012
Why a Geometric Approach?                    Most students in “middle school” are visual                    learners.© Act...
Why a Geometric Approach?                    Most students in “middle school” are visual                    learners.     ...
Why a Geometric Approach?                    Most students in “middle school” are visual                    learners.     ...
Drawing Tools                                             Paper                                           11 in. by 8.5 in...
Drawing Tools                                        Drawing board with paper© Activities for Learning, Inc., 2012
Drawing Tools                                        Drawing board with paper© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square                                             T-square© Act...
Drawing Tools                                           T-square                                             T-square© Act...
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                           T-square© Activities for Learning, Inc., 2012
Drawing Tools                                        Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
Drawing Tools                                        Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
Drawing Tools                                        Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
Drawing Tools                                        Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
Drawing Tools                                        Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
Drawing Tools                                        Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
Drawing Tools                                          The 45 triangle                                                    ...
Drawing Tools                                        The 30-60 triangle (a set square)                                    ...
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
Drawing Tools                                         The 30-60 triangle© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Draw the base line© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Draw the base line© Activities for Learning, Inc., 2012
Equilateral Triangles                                             Draw the sides© Activities for Learning, Inc., 2012
Equilateral Triangles                                             Draw the sides© Activities for Learning, Inc., 2012
Equilateral Triangles                                             Draw the sides© Activities for Learning, Inc., 2012
Equilateral Triangles                                             Draw the sides© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into halves© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into halves© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into halves© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into halves                                    ...
Equilateral Triangles                                            Divide it into halves© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into halves© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into halves© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into halves© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into halves© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into thirds© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into thirds© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into thirds© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into thirds© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into thirds© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into thirds© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into thirds© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into thirds                                    ...
Equilateral Triangles                                            Divide it into thirds                                    ...
© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into fourths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into fourths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into fourths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into fourths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into fourths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                            Divide it into fourths                                   ...
Equilateral Triangles                                          What if we cut this shape out….© Activities for Learning, I...
Equilateral Triangles                                          …and folded it on the lines….© Activities for Learning, Inc...
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles                                          It’s a tetrahedron!© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into eighths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into eighths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into eighths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into eighths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into eighths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into sixths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into sixths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into sixths© Activities for Learning, Inc., 2012
Equilateral Triangles                                            Divide it into sixths                                    ...
Equilateral Triangles                                            Divide it into sixths                                    ...
Equilateral Triangles                                            Divide it into sixths                                    ...
Equilateral Triangles                                            Divide it into sixths                                    ...
Equilateral Triangles                                            Divide it into sixths                                    ...
Equilateral Triangles                                            Divide it into sixths                                    ...
Equilateral Triangles                                            Divide it into sixths                                    ...
Equilateral Triangles                                            Divide it into sixths                                    ...
Equilateral Triangles                                           Divide it into twelfths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into twelfths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into twelfths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into twelfths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into twelfths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into twelfths© Activities for Learning, Inc., 2012
Equilateral Triangles                                           Divide it into twelfths                                   ...
Equilateral Triangles                                           Divide it into twelfths                                   ...
Equilateral Triangles                                           Divide it into twelfths                                   ...
Equilateral Triangles                                           Divide it into twelfths                                   ...
Equilateral Triangles                                          Ninths© Activities for Learning, Inc., 2012
Equilateral Triangles                                          Twenty-sevenths© Activities for Learning, Inc., 2012
Equilateral Triangles                                          Twenty-sevenths© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles                                          Thirty-seconds© Activities for Learning, Inc., 2012
Equilateral Triangles                                          Stephanie, 6, did 256!© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
Equilateral Triangles© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Equilateral Triangles                                                Spiral© Activities for Learning, Inc., 2012
Geometric Approach© Activities for Learning, Inc., 2012
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                                Goals© Activities for Learning, Inc., 2012
Geometric Approach                                                    Goals                                          • To ...
Geometric Approach                                                    Goals                                          • To ...
Geometric Approach                                                    Goals                                          • To ...
Geometric Approach                                                    Goals                                          • To ...
Geometric Approach                                                    Goals                                          • To ...
Why a Geometric Approach?© Activities for Learning, Inc., 2012
Why a Geometric Approach?                    Most students in “middle school” are visual                    learners.© Act...
Why a Geometric Approach?                    Most students in “middle school” are visual                    learners.     ...
Why a Geometric Approach?                    Most students in “middle school” are visual                    learners.     ...
© Activities for Learning, Inc., 2012
Squares                                        The 45 triangle.© Activities for Learning, Inc., 2012
Squares                                        The 45 triangle is half of a square.© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares© Activities for Learning, Inc., 2012
Squares                                        Halves© Activities for Learning, Inc., 2012
Squares                                        Fourths© Activities for Learning, Inc., 2012
Squares                                        Eighths© Activities for Learning, Inc., 2012
Squares                                        Sixteenths© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Squares                                                Halves                                        We need to find the c...
Squares                                                Halves                                        We need to find the c...
Squares                                                Halves                                        We need to find the c...
Squares                                         Halves© Activities for Learning, Inc., 2012
Squares                                         Halves© Activities for Learning, Inc., 2012
Squares                                         Fourths© Activities for Learning, Inc., 2012
Squares                                         Fourths© Activities for Learning, Inc., 2012
Squares                                               Eighths                                        Find the center of a ...
Squares                                               Eighths                                        Find the center of a ...
Squares                                               Eighths                                        Find the center of a ...
Squares                                               Eighths                                        Find the center of a ...
Squares                                         Eighths© Activities for Learning, Inc., 2012
Squares                                         Eighths© Activities for Learning, Inc., 2012
Squares                                        Sixteenths© Activities for Learning, Inc., 2012
Squares                                        Sixteenths© Activities for Learning, Inc., 2012
Squares                                        Sixteenths© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
Squares                                        Circumscribed squares© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                          Inscribed squares                                        Compare the area...
Squares                                          Inscribed squares                                        Compare the area...
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Inscribed squares© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
Squares                                        Right triangle spiral© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Hexagons                                        Start with a small equilateral triangle.© Activities for Learning, Inc., 2...
Hexagons© Activities for Learning, Inc., 2012
Hexagons© Activities for Learning, Inc., 2012
Hexagons© Activities for Learning, Inc., 2012
Hexagons© Activities for Learning, Inc., 2012
Hexagons© Activities for Learning, Inc., 2012
Hexagons© Activities for Learning, Inc., 2012
Hexagons© Activities for Learning, Inc., 2012
Hexagons© Activities for Learning, Inc., 2012
Hexagons                                        Draw all the diagonals.© Activities for Learning, Inc., 2012
Hexagons                                        Draw all the diagonals.© Activities for Learning, Inc., 2012
Hexagons                                        Draw all the diagonals.© Activities for Learning, Inc., 2012
Hexagons                                        Draw all the diagonals.© Activities for Learning, Inc., 2012
Hexagons                                        Draw all the diagonals.© Activities for Learning, Inc., 2012
Hexagons                                        Draw all the diagonals.© Activities for Learning, Inc., 2012
Hexagons                                        Do you see another hexagon?© Activities for Learning, Inc., 2012
Hexagons                                           Find the figures                                        Do you see anot...
Hexagons                                           Find the figures                                        2 non-congruent...
Hexagons                                           Find the figures                                        2 non-congruent...
Hexagons                                           Find the figures                                        2 non-congruent...
Hexagons                                           Find the figures                                        3 non-congruent...
Hexagons                                           Find the figures                                        3 non-congruent...
Hexagons                                           Find the figures                                        3 non-congruent...
Hexagons                                           Find the figures                                        3 non-congruent...
Hexagons                                           Find the figures                                        2 non-congruent...
Hexagons                                           Find the figures                                        2 non-congruent...
Hexagons                                           Find the figures                                        2 non-congruent...
Hexagons                                            Find the figures                                        3 similar equi...
Hexagons                                            Find the figures                                        3 similar equi...
Hexagons                                            Find the figures                                        3 similar equi...
Hexagons                                           Find the figures                                        3 similar obtus...
Hexagons                                           Find the figures                                        3 similar obtus...
Hexagons                                           Find the figures                                        3 similar obtus...
Hexagons                                           Find the figures                                        3 similar obtus...
Hexagons                                           Find the figures                                        3 non-congruent...
Hexagons                                           Find the figures                                        3 non-congruent...
Hexagons                                           Find the figures                                        3 non-congruent...
Hexagons                                           Find the figures                                        3 non-congruent...
Hexagons                                            Find the figures                                        2 similar kite...
Hexagons                                            Find the figures                                        2 similar kite...
Hexagons                                            Find the figures                                        2 similar kite...
© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the small star                        Apothem: perpendicular ...
Hexagons                                        Constructing the small star                        Apothem: perpendicular ...
Hexagons                                        Constructing the small star                        Apothem: perpendicular ...
Hexagons                                        Constructing the small star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the small star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the small star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the small star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the small star                                        Ratio o...
Hexagons                                        Constructing the small star                                        Ratio o...
Hexagons                                        Constructing the large star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the large star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the large star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the large star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the large star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the large star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the large star© Activities for Learning, Inc., 2012
Hexagons                                        Constructing the large star© Activities for Learning, Inc., 2012
The Clock                11 12 1              10        2              9          3              8         4              ...
The Clock                11 12 1              10        2              9          3              8         4              ...
The Clock                11 12 1              10        2              9          3              8         4              ...
The Clock                11 12 1              10        2              9          3              8         4              ...
The Clock                11 12 1              10        2              9          3              8         4              ...
The Clock                11 12 1              10        2              9          3              8         4              ...
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
Right Triangle Design© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Tangrams© Activities for Learning, Inc., 2012
Tangrams© Activities for Learning, Inc., 2012
Tangrams© Activities for Learning, Inc., 2012
Tangrams© Activities for Learning, Inc., 2012
Tangrams© Activities for Learning, Inc., 2012
Tangrams© Activities for Learning, Inc., 2012
Tangrams© Activities for Learning, Inc., 2012
Tangrams                                        Are the 3 green figures are equal in© Activities for Learning, Inc., 2012 ...
Tangrams© Activities for Learning, Inc., 2012
Tangrams                                        The 3 green figures ARE equal in area.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams                                        Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
Tangrams© Activities for Learning, Inc., 2012
Tangram Design Symmetry                                        Start at the left side of the paper.© Activities for Learni...
Tangram Design Symmetry                                        Start at the left side of the paper.© Activities for Learni...
Tangram Design Symmetry                                        Reflect the image vertically across the© Activities for Lea...
Tangram Design Symmetry                                        Reflect the image vertically.© Activities for Learning, Inc...
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry                                        Reflect the image horizontally.© Activities for Learning, I...
Tangram Design Symmetry                                        Reflect the image horizontally.© Activities for Learning, I...
Tangram Design Symmetry                                        The blue rotated 180° is the yellow.© Activities for Learni...
Tangram Design Symmetry                                        Reflect the image vertically.© Activities for Learning, Inc...
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry© Activities for Learning, Inc., 2012
Tangram Design Symmetry                                        Two horizontal reflections is the original.© Activities for...
Tangram Design Symmetry                                        • Such constructions require                               ...
Tangram Design Symmetry                                        • Such constructions require                               ...
Tangram Design Symmetry                                        • Such constructions require                               ...
Tangram Design Symmetry                                        • Such constructions require                               ...
© Activities for Learning, Inc., 2012
Pythagorean Theorem                                        First construct a right triangle.© Activities for Learning, Inc...
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem                                        Area of blue squares = yellow square.© Activities for Learning,...
Pythagorean Theorem                                        Area of blue squares = yellow square.© Activities for Learning,...
Pythagorean Theorem                                        Draw a right triangle.© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem                                        Turn the t-square upside down.© Activities for Learning, Inc., ...
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem                                        Do not move the t-square.© Activities for Learning, Inc., 2012
Pythagorean Theorem                                        Do not move the t-square.© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem© Activities for Learning, Inc., 2012
Pythagorean Theorem                                        Area of blue squares = yellow square.© Activities for Learning,...
Triangle Area                                                  A                                        B               C©...
Triangle Area                                                         A                                          B        ...
Triangle Area                                                          A                                          B       ...
Triangle Area                                                            A                                          B     ...
Triangle Area                                                         A                                          B        ...
Triangle Area                                                            A                                          B     ...
Triangle Area                                                         A                                          B        ...
Triangle Area                                                          A                                          B       ...
Triangle Area                                                         A                                          B        ...
Geometric Approach© Activities for Learning, Inc., 2012
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                          • Basis of CAD (computer                                      ...
Geometric Approach                                                Goals© Activities for Learning, Inc., 2012
Geometric Approach                                                    Goals                                          • To ...
Geometric Approach                                                    Goals                                          • To ...
Geometric Approach                                                    Goals                                          • To ...
Geometric Approach                                                    Goals                                          • To ...
Geometric Approach                                                    Goals                                          • To ...
Personalized Learning                      Bridges Middle-School Math                  with a Geometric Approach – Part 2 ...
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Personalized Learning Bridges Middle School Math with a Geometric Approach

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  • Right-handed user.
  • Hold pencil 2-3cm (1 in.)from tip.
  • Parallel lines.
  • Triangle must hug the t-square.
  • This doesn’t work!
  • Perpendicular lines.
  • Perpendicular lines.
  • Still more parallel lines.
  • Intersecting lines.
  • Intersecting lines.
  • Maintain a margin.
  • Move the t-square down.
  • Draw a side.
  • Flip the triangle and repeat.
  • Flip the triangle and repeat.
  • A bit of erasing.
  • The two halves are congruent.
  • Divide it in half another way.
  • Divide it in half another way.
  • Divide it in half another way.
  • Divide it in half another way.
  • A bit of erasing.
  • Flip the triangle and repeat.
  • Can you figure out how to do it?
  • Finding the center of the base line.
  • Be sure the tick mark intersects the line.
  • Flip the triangle over and repeat.
  • How do you draw the last line?
  • The smaller triangles are similar to the larger triangle.
  • Name the rhombuses:
  • Name the rhombuses: ABDF,
  • Name the rhombuses. ABDF, BCDF,
  • Name the rhombuses: ABDF, BCDF, BDEF.
  • Name the trapezoids:
  • Name the trapezoids: BCEF,
  • Name the trapezoids: BCEF, ACDF,
  • Name the trapezoids: BCEF, ACDF, ABDE.
  • Cut it out….
  • and fold on the lines.
  • and fold on the lines.
  • and fold on the dotted lines.
  • and fold on the dotted lines.
  • and fold on the dotted lines.
  • How could you draw eighths?
  • Not very esthetic.
  • Can you divide it into sixths?
  • Name an acute triangle:
  • Name an acute triangle: ACE.
  • Name a right triangle: ADE, ABE, ABG.
  • Draw the base line.
  • Move the t-square down.
  • Draw the sides.
  • Draw the diagonal…
  • or just a tick mark.
  • Draw the top.
  • A bit of erasing.
  • Outside square is twice the size of the inscribed square.
  • No tracing or measuring.
  • No tracing or measuring.
  • No tracing or measuring.
  • No tracing or measuring.
  • No tracing or measuring.
  • No tracing or measuring.
  • No tracing or measuring.
  • No tracing or measuring.
  • No tracing or measuring.
  • More examples to construct.
  • Personalized Learning Bridges Middle School Math with a Geometric Approach

    1. 1. Personalized Learning Bridges Middle-School Math with a Geometric Approach Presented by Tracy Mittleider, MSEd and Kathleen Lawler Based on work of Joan A. Cotter, Ph.D. Aplus+ Conference October 24, 2012© Activities for Learning, Inc., 2012
    2. 2. Why a Geometric Approach?© Activities for Learning, Inc., 2012
    3. 3. Why a Geometric Approach? Most students in “middle school” are visual learners.© Activities for Learning, Inc., 2012
    4. 4. Why a Geometric Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually).© Activities for Learning, Inc., 2012
    5. 5. Why a Geometric Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually). Therefore, it makes sense to teach them geometrically.© Activities for Learning, Inc., 2012
    6. 6. Drawing Tools Paper 11 in. by 8.5 in.© Activities for Learning, Inc., 2012
    7. 7. Drawing Tools Drawing board with paper© Activities for Learning, Inc., 2012
    8. 8. Drawing Tools Drawing board with paper© Activities for Learning, Inc., 2012
    9. 9. Drawing Tools T-square T-square© Activities for Learning, Inc., 2012
    10. 10. Drawing Tools T-square T-square© Activities for Learning, Inc., 2012
    11. 11. Drawing Tools T-square© Activities for Learning, Inc., 2012
    12. 12. Drawing Tools T-square© Activities for Learning, Inc., 2012
    13. 13. Drawing Tools T-square© Activities for Learning, Inc., 2012
    14. 14. Drawing Tools T-square© Activities for Learning, Inc., 2012
    15. 15. Drawing Tools T-square© Activities for Learning, Inc., 2012
    16. 16. Drawing Tools T-square© Activities for Learning, Inc., 2012
    17. 17. Drawing Tools T-square© Activities for Learning, Inc., 2012
    18. 18. Drawing Tools T-square© Activities for Learning, Inc., 2012
    19. 19. Drawing Tools T-square© Activities for Learning, Inc., 2012
    20. 20. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
    21. 21. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
    22. 22. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
    23. 23. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
    24. 24. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
    25. 25. Drawing Tools Pencil (mechanical) and eraser© Activities for Learning, Inc., 2012
    26. 26. Drawing Tools The 45 triangle 45° 45° 90°© Activities for Learning, Inc., 2012
    27. 27. Drawing Tools The 30-60 triangle (a set square) 30° 60° 90°© Activities for Learning, Inc., 2012
    28. 28. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    29. 29. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    30. 30. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    31. 31. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    32. 32. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    33. 33. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    34. 34. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    35. 35. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    36. 36. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    37. 37. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    38. 38. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    39. 39. Drawing Tools The 30-60 triangle© Activities for Learning, Inc., 2012
    40. 40. © Activities for Learning, Inc., 2012
    41. 41. Equilateral Triangles Draw the base line© Activities for Learning, Inc., 2012
    42. 42. Equilateral Triangles Draw the base line© Activities for Learning, Inc., 2012
    43. 43. Equilateral Triangles Draw the sides© Activities for Learning, Inc., 2012
    44. 44. Equilateral Triangles Draw the sides© Activities for Learning, Inc., 2012
    45. 45. Equilateral Triangles Draw the sides© Activities for Learning, Inc., 2012
    46. 46. Equilateral Triangles Draw the sides© Activities for Learning, Inc., 2012
    47. 47. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
    48. 48. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
    49. 49. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
    50. 50. Equilateral Triangles Divide it into halves 1 1 2 2© Activities for Learning, Inc., 2012
    51. 51. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
    52. 52. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
    53. 53. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
    54. 54. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
    55. 55. Equilateral Triangles Divide it into halves© Activities for Learning, Inc., 2012
    56. 56. © Activities for Learning, Inc., 2012
    57. 57. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
    58. 58. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
    59. 59. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
    60. 60. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
    61. 61. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
    62. 62. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
    63. 63. Equilateral Triangles Divide it into thirds© Activities for Learning, Inc., 2012
    64. 64. Equilateral Triangles Divide it into thirds 1 1 3 3 1 3© Activities for Learning, Inc., 2012
    65. 65. Equilateral Triangles Divide it into thirds 1 3 1 1 3 3© Activities for Learning, Inc., 2012
    66. 66. © Activities for Learning, Inc., 2012
    67. 67. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
    68. 68. Equilateral Triangles Divide it into fourths Find the center of the base line.© Activities for Learning, Inc., 2012
    69. 69. Equilateral Triangles Divide it into fourths tick mark© Activities for Learning, Inc., 2012
    70. 70. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
    71. 71. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
    72. 72. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
    73. 73. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
    74. 74. Equilateral Triangles Divide it into fourths© Activities for Learning, Inc., 2012
    75. 75. Equilateral Triangles Divide it into fourths 1 4 1 1 4 1 4 4© Activities for Learning, Inc., 2012
    76. 76. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses:© Activities for Learning, Inc., 2012
    77. 77. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF© Activities for Learning, Inc., 2012
    78. 78. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF, BCDF© Activities for Learning, Inc., 2012
    79. 79. Equilateral Triangles Divide it into fourths A B F C E D Name the rhombuses: ABDF, BCDF,© Activities for Learning, Inc., 2012 BDEF
    80. 80. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids:© Activities for Learning, Inc., 2012
    81. 81. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF© Activities for Learning, Inc., 2012
    82. 82. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF© Activities for Learning, Inc., 2012
    83. 83. Equilateral Triangles Divide it into fourths A B F C E D Name the trapezoids: BCEF, ACDF,© Activities for Learning, Inc., 2012 ABDE
    84. 84. Equilateral Triangles What if we cut this shape out….© Activities for Learning, Inc., 2012
    85. 85. Equilateral Triangles …and folded it on the lines….© Activities for Learning, Inc., 2012
    86. 86. Equilateral Triangles© Activities for Learning, Inc., 2012
    87. 87. Equilateral Triangles© Activities for Learning, Inc., 2012
    88. 88. Equilateral Triangles© Activities for Learning, Inc., 2012
    89. 89. Equilateral Triangles© Activities for Learning, Inc., 2012
    90. 90. Equilateral Triangles© Activities for Learning, Inc., 2012
    91. 91. Equilateral Triangles It’s a tetrahedron!© Activities for Learning, Inc., 2012
    92. 92. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
    93. 93. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
    94. 94. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
    95. 95. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
    96. 96. Equilateral Triangles Divide it into eighths© Activities for Learning, Inc., 2012
    97. 97. Equilateral Triangles Divide it into sixths© Activities for Learning, Inc., 2012
    98. 98. Equilateral Triangles Divide it into sixths© Activities for Learning, Inc., 2012
    99. 99. Equilateral Triangles Divide it into sixths© Activities for Learning, Inc., 2012
    100. 100. Equilateral Triangles Divide it into sixths A B F G C E D Name an acute triangle:© Activities for Learning, Inc., 2012
    101. 101. Equilateral Triangles Divide it into sixths A B F G C E D Name an acute triangle: ACE© Activities for Learning, Inc., 2012
    102. 102. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle:© Activities for Learning, Inc., 2012
    103. 103. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE© Activities for Learning, Inc., 2012
    104. 104. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE, ABE© Activities for Learning, Inc., 2012
    105. 105. Equilateral Triangles Divide it into sixths A B F G C E D Name a right triangle: ADE, ABE, ABG© Activities for Learning, Inc., 2012
    106. 106. Equilateral Triangles Divide it into sixths A B F G C E D Name an obtuse triangle:© Activities for Learning, Inc., 2012
    107. 107. Equilateral Triangles Divide it into sixths A B F G C E D Name an obtuse triangle: GCE© Activities for Learning, Inc., 2012
    108. 108. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
    109. 109. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
    110. 110. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
    111. 111. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
    112. 112. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
    113. 113. Equilateral Triangles Divide it into twelfths© Activities for Learning, Inc., 2012
    114. 114. Equilateral Triangles Divide it into twelfths Can you see the cube?© Activities for Learning, Inc., 2012
    115. 115. Equilateral Triangles Divide it into twelfths Can you see the cube?© Activities for Learning, Inc., 2012
    116. 116. Equilateral Triangles Divide it into twelfths Do you see the hexagon?© Activities for Learning, Inc., 2012
    117. 117. Equilateral Triangles Divide it into twelfths Do you see the hexagon?© Activities for Learning, Inc., 2012
    118. 118. Equilateral Triangles Ninths© Activities for Learning, Inc., 2012
    119. 119. Equilateral Triangles Twenty-sevenths© Activities for Learning, Inc., 2012
    120. 120. Equilateral Triangles Twenty-sevenths© Activities for Learning, Inc., 2012
    121. 121. Equilateral Triangles© Activities for Learning, Inc., 2012
    122. 122. Equilateral Triangles© Activities for Learning, Inc., 2012
    123. 123. Equilateral Triangles Thirty-seconds© Activities for Learning, Inc., 2012
    124. 124. Equilateral Triangles Stephanie, 6, did 256!© Activities for Learning, Inc., 2012
    125. 125. Equilateral Triangles© Activities for Learning, Inc., 2012
    126. 126. Equilateral Triangles© Activities for Learning, Inc., 2012
    127. 127. Equilateral Triangles© Activities for Learning, Inc., 2012
    128. 128. Equilateral Triangles© Activities for Learning, Inc., 2012
    129. 129. © Activities for Learning, Inc., 2012
    130. 130. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    131. 131. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    132. 132. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    133. 133. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    134. 134. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    135. 135. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    136. 136. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    137. 137. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    138. 138. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    139. 139. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    140. 140. Equilateral Triangles Spiral© Activities for Learning, Inc., 2012
    141. 141. Geometric Approach© Activities for Learning, Inc., 2012
    142. 142. Geometric Approach • Basis of CAD (computer aided design).© Activities for Learning, Inc., 2012
    143. 143. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts.© Activities for Learning, Inc., 2012
    144. 144. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring.© Activities for Learning, Inc., 2012
    145. 145. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work.© Activities for Learning, Inc., 2012
    146. 146. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. • Excellent preparation for high school mathematics.© Activities for Learning, Inc., 2012
    147. 147. Geometric Approach Goals© Activities for Learning, Inc., 2012
    148. 148. Geometric Approach Goals • To use elementary math.© Activities for Learning, Inc., 2012
    149. 149. Geometric Approach Goals • To use elementary math. • To learn to read math texts.© Activities for Learning, Inc., 2012
    150. 150. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics.© Activities for Learning, Inc., 2012
    151. 151. 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.© Activities for Learning, Inc., 2012
    152. 152. 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.© Activities for Learning, Inc., 2012
    153. 153. Why a Geometric Approach?© Activities for Learning, Inc., 2012
    154. 154. Why a Geometric Approach? Most students in “middle school” are visual learners.© Activities for Learning, Inc., 2012
    155. 155. Why a Geometric Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually).© Activities for Learning, Inc., 2012
    156. 156. Why a Geometric Approach? Most students in “middle school” are visual learners. 90% of the math topics can be explored geometrically (visually). Therefore, it makes sense to teach them geometrically.© Activities for Learning, Inc., 2012
    157. 157. © Activities for Learning, Inc., 2012
    158. 158. Squares The 45 triangle.© Activities for Learning, Inc., 2012
    159. 159. Squares The 45 triangle is half of a square.© Activities for Learning, Inc., 2012
    160. 160. Squares© Activities for Learning, Inc., 2012
    161. 161. Squares© Activities for Learning, Inc., 2012
    162. 162. Squares© Activities for Learning, Inc., 2012
    163. 163. Squares© Activities for Learning, Inc., 2012
    164. 164. Squares© Activities for Learning, Inc., 2012
    165. 165. Squares© Activities for Learning, Inc., 2012
    166. 166. Squares© Activities for Learning, Inc., 2012
    167. 167. Squares© Activities for Learning, Inc., 2012
    168. 168. Squares© Activities for Learning, Inc., 2012
    169. 169. Squares Halves© Activities for Learning, Inc., 2012
    170. 170. Squares Fourths© Activities for Learning, Inc., 2012
    171. 171. Squares Eighths© Activities for Learning, Inc., 2012
    172. 172. Squares Sixteenths© Activities for Learning, Inc., 2012
    173. 173. © Activities for Learning, Inc., 2012
    174. 174. Squares Halves We need to find the center.© Activities for Learning, Inc., 2012
    175. 175. Squares Halves We need to find the center.© Activities for Learning, Inc., 2012
    176. 176. Squares Halves We need to find the center.© Activities for Learning, Inc., 2012
    177. 177. Squares Halves© Activities for Learning, Inc., 2012
    178. 178. Squares Halves© Activities for Learning, Inc., 2012
    179. 179. Squares Fourths© Activities for Learning, Inc., 2012
    180. 180. Squares Fourths© Activities for Learning, Inc., 2012
    181. 181. Squares Eighths Find the center of a small square.© Activities for Learning, Inc., 2012
    182. 182. Squares Eighths Find the center of a small square.© Activities for Learning, Inc., 2012
    183. 183. Squares Eighths Find the center of a small square.© Activities for Learning, Inc., 2012
    184. 184. Squares Eighths Find the center of a small square.© Activities for Learning, Inc., 2012
    185. 185. Squares Eighths© Activities for Learning, Inc., 2012
    186. 186. Squares Eighths© Activities for Learning, Inc., 2012
    187. 187. Squares Sixteenths© Activities for Learning, Inc., 2012
    188. 188. Squares Sixteenths© Activities for Learning, Inc., 2012
    189. 189. Squares Sixteenths© Activities for Learning, Inc., 2012
    190. 190. © Activities for Learning, Inc., 2012
    191. 191. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    192. 192. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    193. 193. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    194. 194. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    195. 195. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    196. 196. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    197. 197. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    198. 198. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    199. 199. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    200. 200. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    201. 201. Squares Circumscribed squares© Activities for Learning, Inc., 2012
    202. 202. © Activities for Learning, Inc., 2012
    203. 203. Squares Inscribed squares© Activities for Learning, Inc., 2012
    204. 204. Squares Inscribed squares© Activities for Learning, Inc., 2012
    205. 205. Squares Inscribed squares© Activities for Learning, Inc., 2012
    206. 206. Squares Inscribed squares Compare the area of the squares.© Activities for Learning, Inc., 2012
    207. 207. Squares Inscribed squares Compare the area of the squares.© Activities for Learning, Inc., 2012
    208. 208. Squares Inscribed squares© Activities for Learning, Inc., 2012
    209. 209. Squares Inscribed squares© Activities for Learning, Inc., 2012
    210. 210. Squares Inscribed squares© Activities for Learning, Inc., 2012
    211. 211. Squares Inscribed squares© Activities for Learning, Inc., 2012
    212. 212. Squares Inscribed squares© Activities for Learning, Inc., 2012
    213. 213. Squares Inscribed squares© Activities for Learning, Inc., 2012
    214. 214. Squares Inscribed squares© Activities for Learning, Inc., 2012
    215. 215. Squares Inscribed squares© Activities for Learning, Inc., 2012
    216. 216. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    217. 217. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    218. 218. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    219. 219. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    220. 220. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    221. 221. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    222. 222. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    223. 223. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    224. 224. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    225. 225. Squares Right triangle spiral© Activities for Learning, Inc., 2012
    226. 226. © Activities for Learning, Inc., 2012
    227. 227. Hexagons Start with a small equilateral triangle.© Activities for Learning, Inc., 2012
    228. 228. Hexagons© Activities for Learning, Inc., 2012
    229. 229. Hexagons© Activities for Learning, Inc., 2012
    230. 230. Hexagons© Activities for Learning, Inc., 2012
    231. 231. Hexagons© Activities for Learning, Inc., 2012
    232. 232. Hexagons© Activities for Learning, Inc., 2012
    233. 233. Hexagons© Activities for Learning, Inc., 2012
    234. 234. Hexagons© Activities for Learning, Inc., 2012
    235. 235. Hexagons© Activities for Learning, Inc., 2012
    236. 236. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
    237. 237. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
    238. 238. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
    239. 239. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
    240. 240. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
    241. 241. Hexagons Draw all the diagonals.© Activities for Learning, Inc., 2012
    242. 242. Hexagons Do you see another hexagon?© Activities for Learning, Inc., 2012
    243. 243. Hexagons Find the figures Do you see another hexagon?© Activities for Learning, Inc., 2012
    244. 244. Hexagons Find the figures 2 non-congruent rectangles?© Activities for Learning, Inc., 2012
    245. 245. Hexagons Find the figures 2 non-congruent rectangles?© Activities for Learning, Inc., 2012
    246. 246. Hexagons Find the figures 2 non-congruent rectangles?© Activities for Learning, Inc., 2012
    247. 247. Hexagons Find the figures 3 non-congruent right triangles?© Activities for Learning, Inc., 2012
    248. 248. Hexagons Find the figures 3 non-congruent right triangles?© Activities for Learning, Inc., 2012
    249. 249. Hexagons Find the figures 3 non-congruent right triangles?© Activities for Learning, Inc., 2012
    250. 250. Hexagons Find the figures 3 non-congruent right triangles?© Activities for Learning, Inc., 2012
    251. 251. Hexagons Find the figures 2 non-congruent rhombuses?© Activities for Learning, Inc., 2012
    252. 252. Hexagons Find the figures 2 non-congruent rhombuses?© Activities for Learning, Inc., 2012
    253. 253. Hexagons Find the figures 2 non-congruent rhombuses?© Activities for Learning, Inc., 2012
    254. 254. Hexagons Find the figures 3 similar equilateral triangles?© Activities for Learning, Inc., 2012
    255. 255. Hexagons Find the figures 3 similar equilateral triangles?© Activities for Learning, Inc., 2012
    256. 256. Hexagons Find the figures 3 similar equilateral triangles?© Activities for Learning, Inc., 2012
    257. 257. Hexagons Find the figures 3 similar obtuse triangles?© Activities for Learning, Inc., 2012
    258. 258. Hexagons Find the figures 3 similar obtuse triangles?© Activities for Learning, Inc., 2012
    259. 259. Hexagons Find the figures 3 similar obtuse triangles?© Activities for Learning, Inc., 2012
    260. 260. Hexagons Find the figures 3 similar obtuse triangles?© Activities for Learning, Inc., 2012
    261. 261. Hexagons Find the figures 3 non-congruent trapezoids?© Activities for Learning, Inc., 2012
    262. 262. Hexagons Find the figures 3 non-congruent trapezoids?© Activities for Learning, Inc., 2012
    263. 263. Hexagons Find the figures 3 non-congruent trapezoids?© Activities for Learning, Inc., 2012
    264. 264. Hexagons Find the figures 3 non-congruent trapezoids?© Activities for Learning, Inc., 2012
    265. 265. Hexagons Find the figures 2 similar kites?© Activities for Learning, Inc., 2012
    266. 266. Hexagons Find the figures 2 similar kites?© Activities for Learning, Inc., 2012
    267. 267. Hexagons Find the figures 2 similar kites?© Activities for Learning, Inc., 2012
    268. 268. © Activities for Learning, Inc., 2012
    269. 269. Hexagons Constructing the small star Apothem: perpendicular line from side to center.© Activities for Learning, Inc., 2012
    270. 270. Hexagons Constructing the small star Apothem: perpendicular line from side to center.© Activities for Learning, Inc., 2012
    271. 271. Hexagons Constructing the small star Apothem: perpendicular line from side to center.© Activities for Learning, Inc., 2012
    272. 272. Hexagons Constructing the small star© Activities for Learning, Inc., 2012
    273. 273. Hexagons Constructing the small star© Activities for Learning, Inc., 2012
    274. 274. Hexagons Constructing the small star© Activities for Learning, Inc., 2012
    275. 275. Hexagons Constructing the small star© Activities for Learning, Inc., 2012
    276. 276. Hexagons Constructing the small star Ratio of the areas of star and hexagon?© Activities for Learning, Inc., 2012
    277. 277. Hexagons Constructing the small star Ratio of the areas of star and hexagon?© Activities for Learning, Inc., 2012
    278. 278. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
    279. 279. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
    280. 280. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
    281. 281. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
    282. 282. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
    283. 283. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
    284. 284. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
    285. 285. Hexagons Constructing the large star© Activities for Learning, Inc., 2012
    286. 286. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
    287. 287. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
    288. 288. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
    289. 289. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
    290. 290. The Clock 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
    291. 291. The Clock 11 12 1 10 2 9 3 8 4 7 6 5 11 12 1 10 2 9 3 8 4 7 6 5© Activities for Learning, Inc., 2012
    292. 292. Right Triangle Design© Activities for Learning, Inc., 2012
    293. 293. Right Triangle Design© Activities for Learning, Inc., 2012
    294. 294. Right Triangle Design© Activities for Learning, Inc., 2012
    295. 295. Right Triangle Design© Activities for Learning, Inc., 2012
    296. 296. Right Triangle Design© Activities for Learning, Inc., 2012
    297. 297. Right Triangle Design© Activities for Learning, Inc., 2012
    298. 298. Right Triangle Design© Activities for Learning, Inc., 2012
    299. 299. Right Triangle Design© Activities for Learning, Inc., 2012
    300. 300. Right Triangle Design© Activities for Learning, Inc., 2012
    301. 301. Right Triangle Design© Activities for Learning, Inc., 2012
    302. 302. © Activities for Learning, Inc., 2012
    303. 303. Tangrams© Activities for Learning, Inc., 2012
    304. 304. Tangrams© Activities for Learning, Inc., 2012
    305. 305. Tangrams© Activities for Learning, Inc., 2012
    306. 306. Tangrams© Activities for Learning, Inc., 2012
    307. 307. Tangrams© Activities for Learning, Inc., 2012
    308. 308. Tangrams© Activities for Learning, Inc., 2012
    309. 309. Tangrams© Activities for Learning, Inc., 2012
    310. 310. Tangrams Are the 3 green figures are equal in© Activities for Learning, Inc., 2012 area?
    311. 311. Tangrams© Activities for Learning, Inc., 2012
    312. 312. Tangrams The 3 green figures ARE equal in area.© Activities for Learning, Inc., 2012
    313. 313. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    314. 314. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    315. 315. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    316. 316. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    317. 317. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    318. 318. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    319. 319. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    320. 320. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    321. 321. Tangrams Constructing a tangram arrangement.© Activities for Learning, Inc., 2012
    322. 322. Tangrams© Activities for Learning, Inc., 2012
    323. 323. Tangram Design Symmetry Start at the left side of the paper.© Activities for Learning, Inc., 2012
    324. 324. Tangram Design Symmetry Start at the left side of the paper.© Activities for Learning, Inc., 2012
    325. 325. Tangram Design Symmetry Reflect the image vertically across the© Activities for Learning, Inc., 2012 line.
    326. 326. Tangram Design Symmetry Reflect the image vertically.© Activities for Learning, Inc., 2012
    327. 327. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    328. 328. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    329. 329. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    330. 330. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    331. 331. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    332. 332. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    333. 333. Tangram Design Symmetry Reflect the image horizontally.© Activities for Learning, Inc., 2012
    334. 334. Tangram Design Symmetry Reflect the image horizontally.© Activities for Learning, Inc., 2012
    335. 335. Tangram Design Symmetry The blue rotated 180° is the yellow.© Activities for Learning, Inc., 2012
    336. 336. Tangram Design Symmetry Reflect the image vertically.© Activities for Learning, Inc., 2012
    337. 337. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    338. 338. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    339. 339. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    340. 340. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    341. 341. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    342. 342. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    343. 343. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    344. 344. Tangram Design Symmetry© Activities for Learning, Inc., 2012
    345. 345. Tangram Design Symmetry Two horizontal reflections is the original.© Activities for Learning, Inc., 2012
    346. 346. Tangram Design Symmetry • Such constructions require thinking ahead.© Activities for Learning, Inc., 2012
    347. 347. Tangram Design Symmetry • Such constructions require thinking ahead. • Each step must be justified; no guessing.© Activities for Learning, Inc., 2012
    348. 348. Tangram Design Symmetry • Such constructions require thinking ahead. • Each step must be justified; no guessing. • Symmetry is in everyday life.© Activities for Learning, Inc., 2012
    349. 349. Tangram Design Symmetry • Such constructions require thinking ahead. • Each step must be justified; no guessing. • Symmetry is in everyday life. • Symmetry is often on tests.© Activities for Learning, Inc., 2012
    350. 350. © Activities for Learning, Inc., 2012
    351. 351. Pythagorean Theorem First construct a right triangle.© Activities for Learning, Inc., 2012
    352. 352. Pythagorean Theorem© Activities for Learning, Inc., 2012
    353. 353. Pythagorean Theorem© Activities for Learning, Inc., 2012
    354. 354. Pythagorean Theorem© Activities for Learning, Inc., 2012
    355. 355. Pythagorean Theorem© Activities for Learning, Inc., 2012
    356. 356. Pythagorean Theorem© Activities for Learning, Inc., 2012
    357. 357. Pythagorean Theorem© Activities for Learning, Inc., 2012
    358. 358. Pythagorean Theorem© Activities for Learning, Inc., 2012
    359. 359. Pythagorean Theorem© Activities for Learning, Inc., 2012
    360. 360. Pythagorean Theorem Area of blue squares = yellow square.© Activities for Learning, Inc., 2012
    361. 361. Pythagorean Theorem Area of blue squares = yellow square.© Activities for Learning, Inc., 2012
    362. 362. Pythagorean Theorem Draw a right triangle.© Activities for Learning, Inc., 2012
    363. 363. Pythagorean Theorem© Activities for Learning, Inc., 2012
    364. 364. Pythagorean Theorem© Activities for Learning, Inc., 2012
    365. 365. Pythagorean Theorem© Activities for Learning, Inc., 2012
    366. 366. Pythagorean Theorem© Activities for Learning, Inc., 2012
    367. 367. Pythagorean Theorem© Activities for Learning, Inc., 2012
    368. 368. Pythagorean Theorem© Activities for Learning, Inc., 2012
    369. 369. Pythagorean Theorem Turn the t-square upside down.© Activities for Learning, Inc., 2012
    370. 370. Pythagorean Theorem© Activities for Learning, Inc., 2012
    371. 371. Pythagorean Theorem© Activities for Learning, Inc., 2012
    372. 372. Pythagorean Theorem© Activities for Learning, Inc., 2012
    373. 373. Pythagorean Theorem Do not move the t-square.© Activities for Learning, Inc., 2012
    374. 374. Pythagorean Theorem Do not move the t-square.© Activities for Learning, Inc., 2012
    375. 375. Pythagorean Theorem© Activities for Learning, Inc., 2012
    376. 376. Pythagorean Theorem© Activities for Learning, Inc., 2012
    377. 377. Pythagorean Theorem© Activities for Learning, Inc., 2012
    378. 378. Pythagorean Theorem Area of blue squares = yellow square.© Activities for Learning, Inc., 2012
    379. 379. Triangle Area A B C© Activities for Learning, Inc., 2012
    380. 380. Triangle Area A B C Draw the altitude.© Activities for Learning, Inc., 2012
    381. 381. Triangle Area A B C Area is half of enclosing rectangle.© Activities for Learning, Inc., 2012
    382. 382. Triangle Area A B C Another altitude.© Activities for Learning, Inc., 2012
    383. 383. Triangle Area A B C Find the area.© Activities for Learning, Inc., 2012
    384. 384. Triangle Area A B C Another altitude.© Activities for Learning, Inc., 2012
    385. 385. Triangle Area A B C Find the area.© Activities for Learning, Inc., 2012
    386. 386. Triangle Area A B C Altitudes intersect at the orthocenter.© Activities for Learning, Inc., 2012
    387. 387. Triangle Area A B C 3 areas slightly different. Why?© Activities for Learning, Inc., 2012
    388. 388. Geometric Approach© Activities for Learning, Inc., 2012
    389. 389. Geometric Approach • Basis of CAD (computer aided design).© Activities for Learning, Inc., 2012
    390. 390. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts.© Activities for Learning, Inc., 2012
    391. 391. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring.© Activities for Learning, Inc., 2012
    392. 392. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work.© Activities for Learning, Inc., 2012
    393. 393. Geometric Approach • Basis of CAD (computer aided design). • Integrates many concepts. • Incorporates measuring. • Allows independent work. • Excellent preparation for high school mathematics.© Activities for Learning, Inc., 2012
    394. 394. Geometric Approach Goals© Activities for Learning, Inc., 2012
    395. 395. Geometric Approach Goals • To use elementary math.© Activities for Learning, Inc., 2012
    396. 396. Geometric Approach Goals • To use elementary math. • To learn to read math texts.© Activities for Learning, Inc., 2012
    397. 397. Geometric Approach Goals • To use elementary math. • To learn to read math texts. • To prepare for more advanced mathematics.© Activities for Learning, Inc., 2012
    398. 398. 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.© Activities for Learning, Inc., 2012
    399. 399. 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.© Activities for Learning, Inc., 2012
    400. 400. Personalized Learning Bridges Middle-School Math with a Geometric Approach – Part 2 Presented by Tracy Mittleider, MSEd and Kathleen Lawler Based on work of Joan A. Cotter, Ph.D. Aplus+ Conference October 24, 2012© Activities for Learning, Inc., 2012
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