vities for Learning, Inc., 2012 Personalized Learning Bridges Middle-School Math with a Geometric Approach Aplus+ Conference October 24, 2012 Presented by Tracy Mittleider, MSEd and Kathleen Lawler Based on work of Joan A. Cotter, Ph.D.
Personalized Learning Bridges Middle School Math with a Geometric Approach
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© Activities for Learning, Inc., 2012
Personalized LearningBridges Middle-School Math with a Geometric Approach
Aplus+ ConferenceOctober 24, 2012
Presented by Tracy Mittleider, MSEd and Kathleen Lawler
Based on work of Joan A. Cotter, Ph.D.
© Activities for Learning, Inc., 2012
Why a Geometric Approach?
© Activities for Learning, Inc., 2012
Most students in “middle school” are visual learners.
Why a Geometric Approach?
© Activities for Learning, Inc., 2012
Most students in “middle school” are visual learners.
90% of the math topics can be explored geometrically (visually).
Why a Geometric Approach?
© Activities for Learning, Inc., 2012
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.
Why a Geometric Approach?
© Activities for Learning, Inc., 2012
Drawing ToolsPaper
11 in. by 8.5 in.
© Activities for Learning, Inc., 2012
Drawing ToolsDrawing board with paper
© Activities for Learning, Inc., 2012
Drawing ToolsDrawing board with paper
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
T-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
T-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsT-square
© Activities for Learning, Inc., 2012
Drawing ToolsPencil (mechanical) and eraser
© Activities for Learning, Inc., 2012
Drawing ToolsPencil (mechanical) and eraser
© Activities for Learning, Inc., 2012
Drawing ToolsPencil (mechanical) and eraser
© Activities for Learning, Inc., 2012
Drawing ToolsPencil (mechanical) and eraser
© Activities for Learning, Inc., 2012
Drawing ToolsPencil (mechanical) and eraser
© Activities for Learning, Inc., 2012
Drawing ToolsPencil (mechanical) and eraser
© Activities for Learning, Inc., 2012
Drawing ToolsThe 45 triangle
45°
45° 90°
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle (a set square)
60°
30°
90°
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
Drawing ToolsThe 30-60 triangle
© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Equilateral TrianglesDraw the base line
© Activities for Learning, Inc., 2012
Equilateral TrianglesDraw the base line
© Activities for Learning, Inc., 2012
Equilateral TrianglesDraw the sides
© Activities for Learning, Inc., 2012
Equilateral TrianglesDraw the sides
© Activities for Learning, Inc., 2012
Equilateral TrianglesDraw the sides
© Activities for Learning, Inc., 2012
Equilateral TrianglesDraw the sides
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
12
12
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into halves
© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into thirds
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into thirds
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into thirds
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into thirds
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into thirds
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into thirds
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into thirds
© Activities for Learning, Inc., 2012
Equilateral Triangles
13
13
13
Divide it into thirds
© Activities for Learning, Inc., 2012
Equilateral Triangles
13
13
13
Divide it into thirds
© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
Find the center of the base line.
© Activities for Learning, Inc., 2012
Equilateral Triangles
tick mark
Divide it into fourths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
14
14
14
14
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the rhombuses:
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the rhombuses: ABDF
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the rhombuses: ABDF, BCDF
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the rhombuses: ABDF, BCDF, BDEF
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the trapezoids:
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the trapezoids: BCEF
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the trapezoids: BCEF, ACDF
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into fourths
A
B
CD
E
F
Name the trapezoids: BCEF, ACDF, ABDE
© Activities for Learning, Inc., 2012
Equilateral Triangles
What if we cut this shape out….
© Activities for Learning, Inc., 2012
Equilateral Triangles
…and folded it on the lines….
© 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
© Activities for Learning, Inc., 2012
Equilateral Triangles
It’s a tetrahedron!
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into eighths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into eighths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into eighths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into eighths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into eighths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
A
B
CD
E
FG
Name an acute triangle:
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
A
B
CD
E
FG
Name an acute triangle: ACE
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
A
B
CD
E
FG
Name a right triangle:
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
A
B
CD
E
FG
Name a right triangle: ADE
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
A
B
CD
E
FG
Name a right triangle: ADE, ABE
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
A
B
CD
E
FG
Name a right triangle: ADE, ABE, ABG
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
A
B
CD
E
FG
Name an obtuse triangle:
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into sixths
A
B
CD
E
FG
Name an obtuse triangle: GCE
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
Can you see the cube?
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
Can you see the cube?
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
Do you see the hexagon?
© Activities for Learning, Inc., 2012
Equilateral TrianglesDivide it into twelfths
Do you see the hexagon?
© Activities for Learning, Inc., 2012
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 TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Equilateral TrianglesSpiral
© Activities for Learning, Inc., 2012
Geometric Approach
© Activities for Learning, Inc., 2012
Geometric Approach
• Basis of CAD (computer aided design).
© Activities for Learning, Inc., 2012
Geometric Approach
• Basis of CAD (computer aided design).
• Integrates many concepts.
© Activities for Learning, Inc., 2012
Geometric Approach
• Basis of CAD (computer aided design).
• Integrates many concepts.• Incorporates measuring.
© Activities for Learning, Inc., 2012
Geometric Approach
• Basis of CAD (computer aided design).
• Integrates many concepts.• Incorporates measuring.• Allows independent work.
© Activities for Learning, Inc., 2012
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
Geometric Approach
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• To use elementary math.
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• To use elementary math.• To learn to read math texts.
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• To use elementary math.• To learn to read math texts.• To prepare for more advanced
mathematics.
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• To use elementary math.• To learn to read math texts.• To prepare for more advanced
mathematics.• To discover mathematics in
the everyday world.
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• 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.
Goals
© Activities for Learning, Inc., 2012
Why a Geometric Approach?
© Activities for Learning, Inc., 2012
Most students in “middle school” are visual learners.
Why a Geometric Approach?
© Activities for Learning, Inc., 2012
Most students in “middle school” are visual learners.
90% of the math topics can be explored geometrically (visually).
Why a Geometric Approach?
© Activities for Learning, Inc., 2012
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.
Why a Geometric Approach?
© Activities for Learning, Inc., 2012
© 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
SquaresHalves
We need to find the center.
© Activities for Learning, Inc., 2012
SquaresHalves
We need to find the center.
© Activities for Learning, Inc., 2012
SquaresHalves
We need to find the center.
© Activities for Learning, Inc., 2012
SquaresHalves
© Activities for Learning, Inc., 2012
SquaresHalves
© Activities for Learning, Inc., 2012
SquaresFourths
© Activities for Learning, Inc., 2012
SquaresFourths
© Activities for Learning, Inc., 2012
SquaresEighths
Find the center of a small square.
© Activities for Learning, Inc., 2012
SquaresEighths
Find the center of a small square.
© Activities for Learning, Inc., 2012
SquaresEighths
Find the center of a small square.
© Activities for Learning, Inc., 2012
SquaresEighths
Find the center of a small square.
© Activities for Learning, Inc., 2012
SquaresEighths
© Activities for Learning, Inc., 2012
SquaresEighths
© Activities for Learning, Inc., 2012
SquaresSixteenths
© Activities for Learning, Inc., 2012
SquaresSixteenths
© Activities for Learning, Inc., 2012
SquaresSixteenths
© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
SquaresCircumscribed squares
© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
Compare the area of the squares.
© Activities for Learning, Inc., 2012
SquaresInscribed squares
Compare the area of the squares.
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresInscribed squares
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
SquaresRight triangle spiral
© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
Hexagons
Start with a small equilateral triangle.
© 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
© 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
HexagonsFind the figures
Do you see another hexagon?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 non-congruent rectangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 non-congruent rectangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 non-congruent rectangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 non-congruent right triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 non-congruent right triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 non-congruent right triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 non-congruent right triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 non-congruent rhombuses?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 non-congruent rhombuses?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 non-congruent rhombuses?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 similar equilateral triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 similar equilateral triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 similar equilateral triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 similar obtuse triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 similar obtuse triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 similar obtuse triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 similar obtuse triangles?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 non-congruent trapezoids?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 non-congruent trapezoids?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 non-congruent trapezoids?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
3 non-congruent trapezoids?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 similar kites?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 similar kites?
© Activities for Learning, Inc., 2012
HexagonsFind the figures
2 similar kites?
© Activities for Learning, Inc., 2012
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
Apothem: perpendicular line from side to center.
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
Apothem: perpendicular line from side to center.
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
Apothem: perpendicular line from side to center.
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
Ratio of the areas of star and hexagon?
© Activities for Learning, Inc., 2012
HexagonsConstructing the small star
Ratio of the areas of star and hexagon?
© Activities for Learning, Inc., 2012
HexagonsConstructing the large star
© Activities for Learning, Inc., 2012
HexagonsConstructing the large star
© Activities for Learning, Inc., 2012
HexagonsConstructing the large star
© Activities for Learning, Inc., 2012
HexagonsConstructing the large star
© Activities for Learning, Inc., 2012
HexagonsConstructing the large star
© Activities for Learning, Inc., 2012
HexagonsConstructing the large star
© Activities for Learning, Inc., 2012
HexagonsConstructing the large star
© Activities for Learning, Inc., 2012
HexagonsConstructing the large star
© Activities for Learning, Inc., 2012
The Clock112
23
4567
8910
11
© Activities for Learning, Inc., 2012
The Clock112
23
4567
8910
11
© Activities for Learning, Inc., 2012
The Clock112
23
4567
8910
11
© Activities for Learning, Inc., 2012
The Clock112
23
4567
8910
11
© Activities for Learning, Inc., 2012
The Clock112
23
4567
8910
11
© Activities for Learning, Inc., 2012
The Clock
12 12
3
4567
8
9
1011
11223
4567
8910
11
© 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
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 area?
© 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 Learning, Inc., 2012
Tangram Design Symmetry
Start at the left side of the paper.
© Activities for Learning, Inc., 2012
Tangram Design Symmetry
Reflect the image vertically across the line.
© Activities for Learning, Inc., 2012
Tangram Design Symmetry
Reflect the image vertically.
© 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
Reflect the image horizontally.
© Activities for Learning, Inc., 2012
Tangram Design Symmetry
Reflect the image horizontally.
© Activities for Learning, Inc., 2012
Tangram Design Symmetry
The blue rotated 180° is the yellow.
© Activities for Learning, Inc., 2012
Tangram Design Symmetry
Reflect the image vertically.
© 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
© Activities for Learning, Inc., 2012
Two horizontal reflections is the original.
Tangram Design Symmetry
© Activities for Learning, Inc., 2012
• Such constructions require thinking ahead.
Tangram Design Symmetry
© Activities for Learning, Inc., 2012
• Such constructions require thinking ahead.
• Each step must be justified; no guessing.
Tangram Design Symmetry
© Activities for Learning, Inc., 2012
• Such constructions require thinking ahead.
• Each step must be justified; no guessing.
• Symmetry is in everyday life.
Tangram Design Symmetry
© Activities for Learning, Inc., 2012
• Such constructions require thinking ahead.
• Each step must be justified; no guessing.
• Symmetry is in everyday life.
• Symmetry is often on tests.
Tangram Design Symmetry
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© Activities for Learning, Inc., 2012
Pythagorean Theorem
First construct 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
© Activities for Learning, Inc., 2012
Pythagorean Theorem
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Pythagorean Theorem
Area of blue squares = yellow square.
© Activities for Learning, Inc., 2012
Pythagorean Theorem
Area of blue squares = yellow square.
© Activities for Learning, Inc., 2012
Pythagorean Theorem
Draw a right triangle.
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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.
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Pythagorean Theorem
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Pythagorean Theorem
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Pythagorean Theorem
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Pythagorean Theorem
Do not move the t-square.
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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.
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Triangle Area
A
CB
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Triangle Area
Draw the altitude.
A
CB
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Triangle Area
Area is half of enclosing rectangle.
CB
A
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Triangle Area
Another altitude.
CB
A
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Triangle Area
Find the area.
CB
A
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Triangle Area
Another altitude.
A
CB
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Triangle Area
Find the area.
A
CB
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Triangle Area
Altitudes intersect at the orthocenter.
A
CB
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Triangle Area
3 areas slightly different. Why?
A
CB
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Geometric Approach
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Geometric Approach
• Basis of CAD (computer aided design).
© Activities for Learning, Inc., 2012
Geometric Approach
• Basis of CAD (computer aided design).
• Integrates many concepts.
© Activities for Learning, Inc., 2012
Geometric Approach
• Basis of CAD (computer aided design).
• Integrates many concepts.• Incorporates measuring.
© Activities for Learning, Inc., 2012
Geometric Approach
• Basis of CAD (computer aided design).
• Integrates many concepts.• Incorporates measuring.• Allows independent work.
© Activities for Learning, Inc., 2012
Geometric Approach
• Basis of CAD (computer aided design).
• Integrates many concepts.• Incorporates measuring.• Allows independent work.• Excellent preparation for
high school mathematics.
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Geometric Approach
Goals
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Geometric Approach
• To use elementary math.
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• To use elementary math.• To learn to read math texts.
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• To use elementary math.• To learn to read math texts.• To prepare for more advanced
mathematics.
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• To use elementary math.• To learn to read math texts.• To prepare for more advanced
mathematics.• To discover mathematics in
the everyday world.
Goals
© Activities for Learning, Inc., 2012
Geometric Approach
• 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.
Goals
© Activities for Learning, Inc., 2012
Personalized LearningBridges Middle-School Math
with a Geometric Approach – Part 2Presented by
Tracy Mittleider, MSEd and Kathleen LawlerBased on work of Joan A. Cotter, Ph.D.
Aplus+ ConferenceOctober 24, 2012
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