Developing and Assessing Geometric and Measurement Concepts By Michelle Flaming
Agenda <ul><li>Geometry: </li></ul><ul><ul><li>Defining and Rationale </li></ul></ul><ul><ul><li>Two and Three Dimensional...
Geometry <ul><li>Brainstorm </li></ul><ul><ul><li>If a parent asked you, “What is geometry at your grade level, what types...
Why Teach Geometry So Early? <ul><li>If a parent approached you and asked, “Why do you teach geometry at such an early age...
The Why <ul><li>Geometry enables us to describe, analyze, and understand the physical world in which we live. </li></ul><u...
The Why <ul><li>Different, yet connected to the concept of number. </li></ul><ul><li>Connected to other topics such as: ar...
What Is Geometry? <ul><li>National Council of Teachers of Mathematics (NCTM) </li></ul><ul><ul><li>Two and Three Dimension...
Two and Three Dimensional Shapes <ul><li>Relate shapes to their real-world. </li></ul><ul><ul><li>May call rectangles “doo...
Two and Three Dimensional Shapes <ul><li>- Analyze characteristics and properties of two- and three- dimensional geometric...
Research on Two and Three Dimensional Shapes <ul><li>Circles are the easiest for students to recognize from a variety of s...
Research on Two and Three Dimensional Shapes <ul><li>Triangles are the hardest for students to recognize from a variety of...
Activity:  Different Shape Greens <ul><li>Pigs On The Ball  - by Amy Axelrod </li></ul><ul><li>1. Sort the “greens” into c...
Two and Three Dimensional Shapes <ul><li>- Analyze characteristics and properties of two- and three-dimensional geometric ...
Activity: Shape Blackout <ul><li>Partner Game </li></ul><ul><li>Choose 6 shapes, place on card </li></ul><ul><li>Shuffle n...
Three Dimensional Shapes <ul><li>- Analyze characteristics and properties of two- and three- dimensional geometric shapes ...
Terminology <ul><li>Students should begin by using their own vocabulary, such as “pointedness” to represent the corners an...
Two and Three Dimensional Shapes <ul><li>- Analyze characteristics and properties of two- and three- dimensional geometric...
Activity:  Combining Shapes <ul><li>Each person needs 30 triangles on card stock. </li></ul><ul><li>Work with partner, fin...
Teachers’ Role <ul><li>Provide materials for Hands On explorations. </li></ul><ul><ul><li>Toothpicks, marshmallows </li></...
Coordinate Geometry and Other Representational Systems <ul><li>Specify locations and describe relationships using coordina...
Coordinate Geometry and Other Representational Systems <ul><li>Specify locations and describe relationships using coordina...
Coordinate Geometry and Other Representational Systems <ul><li>Specify locations and describe relationships using coordina...
Coordinate Geometry and Other Representational Systems <ul><li>Terminology: </li></ul><ul><ul><li>Over/Under </li></ul></u...
Activity: Poetry Geometry <ul><li>Read  Five Little Chickens </li></ul><ul><li>Students manipulate objects to match poem. ...
Coordinate Geometry and Other Representational Systems <ul><ul><li>Research shows that students as early as 3 years old ca...
Transformational Geometry <ul><li>Apply transformations and use symmetry to analyze mathematical situations. </li></ul><ul...
Transformational Geometry <ul><li>Apply transformations and use symmetry to analyze mathematical situations. </li></ul><ul...
Activity:  Transformational  B-I-N-G-O Geometry <ul><li>Each student plays on one transformational bingo card. </li></ul><...
Terminology <ul><li>Sl ide/Tran sl ation </li></ul><ul><ul><li>“ An image is formed by moving every point on a figure the ...
Terminology <ul><li>Fl ip/Re fl ection: </li></ul><ul><ul><li>“ a figure is flipped over a line called the line of reflect...
Terminology <ul><li>Ro t ation/ T urn : </li></ul><ul><ul><li>“ The image is formed by turning it’s preimage about a point...
Terminology <ul><li>Congruent - Changing the shape’s position or orientation but not it’s shape or size.  </li></ul><ul><l...
Transformational Geometry <ul><li>Solving puzzles </li></ul><ul><li>Technology - Illuminations Website </li></ul><ul><li>K...
Activity: 5 Squares <ul><li>Use 5 square tiles.  Place the squares side by side to find all the possible composite shapes....
Transformational Geometry <ul><li>Recognize and create shapes that have symmetry </li></ul><ul><ul><li>Research by Clement...
Activity:Easter Eggs <ul><li>Use watercolors, color 1/2 of the “Easter egg”.  </li></ul><ul><li>Fold over the line of symm...
Visualization and Spatial Reasoning  <ul><li>Create mental images of geometric shapes using spatial memory and spatial vis...
Visualization and Spatial Reasoning <ul><li>Visualization - the ability to construct one or more images and/or directions ...
Visualization and Spatial Reasoning <ul><li>Ties to other areas of mathematics: </li></ul><ul><ul><li>Quickly recognize a ...
Research <ul><li>Clements and Battista in 2000 states: </li></ul><ul><ul><li>“Positive correlation’s have been found betwe...
Ideas for Building Visualization Skills <ul><li>1st - Building and Manipulating Concrete Objects: </li></ul><ul><ul><li>Us...
Ideas for Building Visualization Skills <ul><li>Mentally think about a square, what would you get if you cut the square in...
Activity: Enhancing Visualization <ul><li>Place objects/pictures on overhead. </li></ul><ul><li>Turn overhead on and off q...
<ul><li>Recognize geometric shapes and structures in the environment and specify their location. </li></ul><ul><ul><li>Loo...
Activity: Face Cards <ul><li>Students copy the face of each 3-D shape onto a tag board. </li></ul><ul><li>Pairs share thei...
 
Review of  Geometric Concepts <ul><ul><li>Two and Three Dimensional Shapes </li></ul></ul><ul><ul><ul><li>Sort shapes into...
Student Interviews
For more information … <ul><li>www.lulu.com (search Michelle Flaming) </li></ul><ul><li>Contact me at  [email_address] </l...
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Hands on Geometry for K-2 Learners

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Activities and strategies to help teachers engage K-2 learners with geometric concepts

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  • Hands on Geometry for K-2 Learners

    1. 1. Developing and Assessing Geometric and Measurement Concepts By Michelle Flaming
    2. 2. Agenda <ul><li>Geometry: </li></ul><ul><ul><li>Defining and Rationale </li></ul></ul><ul><ul><li>Two and Three Dimensional Shapes </li></ul></ul><ul><ul><li>Coordinate Geometry and Other Representational Systems </li></ul></ul><ul><ul><li>Transformational Geometry </li></ul></ul><ul><ul><li>Visualization and Spatial Reasoning </li></ul></ul><ul><ul><li>Assessing Geometric Concepts through Student Interviews </li></ul></ul><ul><li>Student Interviews </li></ul>
    3. 3. Geometry <ul><li>Brainstorm </li></ul><ul><ul><li>If a parent asked you, “What is geometry at your grade level, what types of activities do you do?” How would you respond? </li></ul></ul><ul><ul><li>Sort your list into categories. </li></ul></ul>
    4. 4. Why Teach Geometry So Early? <ul><li>If a parent approached you and asked, “Why do you teach geometry at such an early age? I didn’t learn it until high school.” How would you respond? </li></ul>
    5. 5. The Why <ul><li>Geometry enables us to describe, analyze, and understand the physical world in which we live. </li></ul><ul><li>Clements and Battista in 1992 states “positive correlations have been found between spatial ability and mathematics achievement at all grade levels”. </li></ul><ul><li>Numerous mathematical concepts have an obvious visual dimension. </li></ul><ul><ul><li>For example: </li></ul></ul>
    6. 6. The Why <ul><li>Different, yet connected to the concept of number. </li></ul><ul><li>Connected to other topics such as: art, science and social studies. </li></ul><ul><li>Often students who are not strong with numbers, excel in geometric and spatial concepts. </li></ul>
    7. 7. What Is Geometry? <ul><li>National Council of Teachers of Mathematics (NCTM) </li></ul><ul><ul><li>Two and Three Dimensional Shapes </li></ul></ul><ul><ul><li>Coordinate Geometry and Other Representational Systems </li></ul></ul><ul><ul><li>Transformational Geometry </li></ul></ul><ul><ul><li>Visualization and Spatial Reasoning </li></ul></ul>
    8. 8. Two and Three Dimensional Shapes <ul><li>Relate shapes to their real-world. </li></ul><ul><ul><li>May call rectangles “doors”. </li></ul></ul><ul><ul><li>May call spheres “balls”. </li></ul></ul><ul><ul><li>May call cubes “boxes”. </li></ul></ul>
    9. 9. Two and Three Dimensional Shapes <ul><li>- Analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships. </li></ul><ul><ul><li>recognize, name, build, draw, compare, and sort two and three dimensional shapes. </li></ul></ul>
    10. 10. Research on Two and Three Dimensional Shapes <ul><li>Circles are the easiest for students to recognize from a variety of shapes. </li></ul><ul><li>Rectangles are the next easiest. Although students don’t recognize that squares are a “special” rectangle. </li></ul><ul><li>Triangles are the hardest for students to recognize from a variety of shapes. </li></ul><ul><ul><li>Clements Research indicates only 60% accuracy. </li></ul></ul>
    11. 11. Research on Two and Three Dimensional Shapes <ul><li>Triangles are the hardest for students to recognize from a variety of shapes. </li></ul><ul><ul><li>Clements Research indicates only 60% accuracy. </li></ul></ul>
    12. 12. Activity: Different Shape Greens <ul><li>Pigs On The Ball - by Amy Axelrod </li></ul><ul><li>1. Sort the “greens” into categories of their choice. </li></ul><ul><li>2. Describe categories, introduce vocabulary. </li></ul><ul><li>3. Find appropriate “green.” </li></ul>
    13. 13. Two and Three Dimensional Shapes <ul><li>- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships. </li></ul><ul><li>- describe attributes and parts of two and three dimensional shapes: </li></ul>
    14. 14. Activity: Shape Blackout <ul><li>Partner Game </li></ul><ul><li>Choose 6 shapes, place on card </li></ul><ul><li>Shuffle name/property cards. </li></ul><ul><li>Take turns, if card matches the shape on board, cover with a marker. </li></ul><ul><li>Game is over when all shapes are covered. </li></ul>
    15. 15. Three Dimensional Shapes <ul><li>- Analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships. </li></ul><ul><ul><li>recognize, name, build, draw, compare, and sort two and three dimensional shapes . </li></ul></ul><ul><li>Cube: </li></ul><ul><li>Sphere: </li></ul><ul><li>Cone: </li></ul><ul><li>Cylinder: </li></ul><ul><li>Rectangular Prism: </li></ul><ul><li>-- Number and Shape of faces </li></ul><ul><li>-- Number of edges and corners (angles, vertices) </li></ul>
    16. 16. Terminology <ul><li>Students should begin by using their own vocabulary, such as “pointedness” to represent the corners and/or vertices. </li></ul><ul><li>Look for similarities and differences between shapes. -- McRel #1 Instructional Strategy. </li></ul><ul><li>Terminology should not be the focus in the early grades. </li></ul>
    17. 17. Two and Three Dimensional Shapes <ul><li>- Analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical arguments about geometric relationships. </li></ul><ul><li>- investigate and predict the results of putting together and taking apart two and three-dimensional shapes. </li></ul>
    18. 18. Activity: Combining Shapes <ul><li>Each person needs 30 triangles on card stock. </li></ul><ul><li>Work with partner, find all the possible composite shapes from 4 triangles. </li></ul><ul><li>Describe the new shape </li></ul><ul><li>Share and discuss all the shapes. </li></ul>
    19. 19. Teachers’ Role <ul><li>Provide materials for Hands On explorations. </li></ul><ul><ul><li>Toothpicks, marshmallows </li></ul></ul><ul><ul><li>String </li></ul></ul><ul><ul><li>Manipulatives </li></ul></ul><ul><ul><li>Real-world Objects </li></ul></ul><ul><li>Structures the classroom which encourages students to explore and communicate. </li></ul><ul><li>Look at examples and nonexamples - Frayer Model </li></ul>
    20. 20. Coordinate Geometry and Other Representational Systems <ul><li>Specify locations and describe relationships using coordinate geometry and other representational systems. </li></ul><ul><ul><li>Describe, name, and interpret relative position and direction and distance in navigating space and applying ideas about direction and distance; </li></ul></ul><ul><ul><li>Find and name locations with simple relationships such as “near to” and in coordinate systems such as maps. </li></ul></ul>
    21. 21. Coordinate Geometry and Other Representational Systems <ul><li>Specify locations and describe relationships using coordinate geometry and other representational systems. </li></ul><ul><li>Navigational questions: </li></ul><ul><ul><li>1. Which way?(direction) </li></ul></ul><ul><ul><li>2. How far? (distance) </li></ul></ul><ul><ul><li>3. Where? (location) </li></ul></ul><ul><ul><li>4. What objects? (representation) </li></ul></ul>
    22. 22. Coordinate Geometry and Other Representational Systems <ul><li>Specify locations and describe relationships using coordinate geometry and other representational systems. </li></ul><ul><li>Conversations </li></ul><ul><li>Demonstrations </li></ul><ul><li>Stories (Acting out/Making Models) </li></ul><ul><li>Poems </li></ul><ul><li>Songs </li></ul>
    23. 23. Coordinate Geometry and Other Representational Systems <ul><li>Terminology: </li></ul><ul><ul><li>Over/Under </li></ul></ul><ul><ul><li>Far/Near </li></ul></ul><ul><ul><li>Next to </li></ul></ul><ul><ul><li>Inside/Outside - Closed Shapes Only </li></ul></ul><ul><ul><li>Above/Below - For 2D (Label paper “top”) </li></ul></ul><ul><ul><li>Right/Left - Direction </li></ul></ul><ul><ul><li>Up/Down </li></ul></ul><ul><ul><li>Forward/Backward </li></ul></ul>
    24. 24. Activity: Poetry Geometry <ul><li>Read Five Little Chickens </li></ul><ul><li>Students manipulate objects to match poem. </li></ul><ul><li>Read Monkey March </li></ul><ul><li>Students act out the poem. </li></ul>
    25. 25. Coordinate Geometry and Other Representational Systems <ul><ul><li>Research shows that students as early as 3 years old can build simple 3D maps (Blaut and Stea in 1974) </li></ul></ul><ul><ul><li>Technology can also support this concept: </li></ul></ul><ul><ul><ul><li>Can You Get the Turtle to the Pond? </li></ul></ul></ul><ul><ul><ul><ul><li>Illuminations.nctm.org </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Click I-Math investigation </li></ul></ul></ul></ul><ul><ul><ul><ul><li>Click turtle and pond icon. </li></ul></ul></ul></ul><ul><ul><ul><li>E-example 4.2 Navigating Paths and Mazes (Part 1) </li></ul></ul></ul><ul><ul><ul><ul><li>Students can learn orientation, directions, and measurement concept. </li></ul></ul></ul></ul>
    26. 26. Transformational Geometry <ul><li>Apply transformations and use symmetry to analyze mathematical situations. </li></ul><ul><ul><li>Recognize and apply slides, flips, and turns. </li></ul></ul><ul><ul><li>Recognize and create shapes that have symmetry. </li></ul></ul>
    27. 27. Transformational Geometry <ul><li>Apply transformations and use symmetry to analyze mathematical situations. </li></ul><ul><ul><li>Recognize and apply slides, flips, and turns. </li></ul></ul><ul><ul><li>4 and 5 year old students can move objects to determine whether they are the same or identical to other shapes (Congruent) </li></ul></ul>
    28. 28. Activity: Transformational B-I-N-G-O Geometry <ul><li>Each student plays on one transformational bingo card. </li></ul><ul><li>As the teacher places one shape on the overhead, if the student has that shape on their board, they cover it with a marker. </li></ul>
    29. 29. Terminology <ul><li>Sl ide/Tran sl ation </li></ul><ul><ul><li>“ An image is formed by moving every point on a figure the same distance the same direction.” </li></ul></ul><ul><ul><ul><li>Geometry To Go </li></ul></ul></ul>
    30. 30. Terminology <ul><li>Fl ip/Re fl ection: </li></ul><ul><ul><li>“ a figure is flipped over a line called the line of reflection. All corresponding points in the image and preimage are equidistance from the line of reflection.” </li></ul></ul><ul><ul><ul><li>Geometry To Go </li></ul></ul></ul>
    31. 31. Terminology <ul><li>Ro t ation/ T urn : </li></ul><ul><ul><li>“ The image is formed by turning it’s preimage about a point.” </li></ul></ul><ul><ul><ul><li>Geometry To Go </li></ul></ul></ul>
    32. 32. Terminology <ul><li>Congruent - Changing the shape’s position or orientation but not it’s shape or size. </li></ul><ul><li>Any of the basic transformations creates a congruent shape. </li></ul>
    33. 33. Transformational Geometry <ul><li>Solving puzzles </li></ul><ul><li>Technology - Illuminations Website </li></ul><ul><li>Kid Pix Deluxe 3 (The Learning Company 2000) </li></ul><ul><li>Geometer’s Sketch Pad </li></ul>
    34. 34. Activity: 5 Squares <ul><li>Use 5 square tiles. Place the squares side by side to find all the possible composite shapes. </li></ul><ul><li>How will you know if it is a different shape? </li></ul>
    35. 35. Transformational Geometry <ul><li>Recognize and create shapes that have symmetry </li></ul><ul><ul><li>Research by Clements (2000) shows that vertical symmetry develops earlier and continues to be easier for young students to see than horizontal. </li></ul></ul>
    36. 36. Activity:Easter Eggs <ul><li>Use watercolors, color 1/2 of the “Easter egg”. </li></ul><ul><li>Fold over the line of symmetry to create the other side. </li></ul>
    37. 37. Visualization and Spatial Reasoning <ul><li>Create mental images of geometric shapes using spatial memory and spatial visualizations; </li></ul><ul><ul><li>Recognize and represent shapes from different perspectives </li></ul></ul><ul><ul><li>Relate ideas in geometry to ideas in number and measurement </li></ul></ul><ul><ul><li>Recognize geometric shapes and structures in the environment and specify their location. </li></ul></ul>
    38. 38. Visualization and Spatial Reasoning <ul><li>Visualization - the ability to construct one or more images and/or directions within the mind. </li></ul><ul><li>First - Static </li></ul><ul><li>Eventually be able to manipulate in the mind. </li></ul>
    39. 39. Visualization and Spatial Reasoning <ul><li>Ties to other areas of mathematics: </li></ul><ul><ul><li>Quickly recognize a set </li></ul></ul><ul><ul><li>Commutative Property </li></ul></ul>
    40. 40. Research <ul><li>Clements and Battista in 2000 states: </li></ul><ul><ul><li>“Positive correlation’s have been found between spatial ability and mathematics achievement at ALL grade levels.” </li></ul></ul>
    41. 41. Ideas for Building Visualization Skills <ul><li>1st - Building and Manipulating Concrete Objects: </li></ul><ul><ul><li>Use of tangrams, pattern blocks, and other concrete materials. </li></ul></ul><ul><li>2nd - Mental Representations </li></ul><ul><ul><li>Quick Images </li></ul></ul>
    42. 42. Ideas for Building Visualization Skills <ul><li>Mentally think about a square, what would you get if you cut the square in half from diagonal to diagonal? </li></ul><ul><li>Think about a circle, what would the shape look like if you cut it in half? Draw what you see. </li></ul><ul><li>“Quick images” </li></ul>
    43. 43. Activity: Enhancing Visualization <ul><li>Place objects/pictures on overhead. </li></ul><ul><li>Turn overhead on and off quickly. </li></ul><ul><li>Ask students to draw what they saw. </li></ul><ul><li>Images can be modified after another quick glance. </li></ul><ul><li>Teacher Questions: </li></ul><ul><ul><li>“ What did you see?” </li></ul></ul><ul><ul><li>“ How did you decide what to draw?” </li></ul></ul>
    44. 44. <ul><li>Recognize geometric shapes and structures in the environment and specify their location. </li></ul><ul><ul><li>Look for geometric shapes in nature and in the world in which we live. </li></ul></ul>
    45. 45. Activity: Face Cards <ul><li>Students copy the face of each 3-D shape onto a tag board. </li></ul><ul><li>Pairs share their tag board and shapes with other students to guess which shape goes with each tag board. </li></ul>
    46. 47. Review of Geometric Concepts <ul><ul><li>Two and Three Dimensional Shapes </li></ul></ul><ul><ul><ul><li>Sort shapes into: </li></ul></ul></ul><ul><ul><ul><ul><li>Circles, Rectangles (excluding the square), and Triangles </li></ul></ul></ul></ul><ul><ul><ul><li>Describe the properties (# of sides, angles, congruent length) of the basic shapes using their own vocabulary. </li></ul></ul></ul><ul><ul><ul><li>Take apart a shape and describe the shapes that made up the whole. </li></ul></ul></ul><ul><ul><li>Coordinate Geometry and Other Representational Systems </li></ul></ul><ul><ul><ul><li>Specify locations and describe relationships using terminology such as: under/below, right/left, etc. </li></ul></ul></ul><ul><ul><li>Transformational Geometry </li></ul></ul><ul><ul><ul><li>Recognize and apply slides, flips and turns. </li></ul></ul></ul><ul><ul><ul><li>Recognize and create shapes that have symmetry. </li></ul></ul></ul><ul><ul><li>Visualization and Spatial Reasoning </li></ul></ul><ul><ul><ul><li>Recognize and represent shapes mentally. </li></ul></ul></ul><ul><ul><ul><li>Recognize geometric shapes from a variety of perspectives. </li></ul></ul></ul>
    47. 48. Student Interviews
    48. 49. For more information … <ul><li>www.lulu.com (search Michelle Flaming) </li></ul><ul><li>Contact me at [email_address] </li></ul><ul><li>Thank you for joining me in PreK-2 Math Foundations </li></ul>

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