Developers

M-GLAnCE Project Directors
Debbie Ferry
Macomb ISD
Mathematics Consultant / Carol Nowakowski
Retired
Mathematics Consultant
K-4 Project Coordinator / Marie Copeland
Warren Consolidated
Macomb MSTC
5-8 Project Coordinator
2004 Project Contributors
David Andrews
Chippewa Valley Schools / William Ashton
FraserPublic Schools / Lynn Bieszki
Chippewa Valley Schools
Sharon Chriss
Romeo Schools / Kimberly DeShon
AnchorBaySchool District / Barbara Diliegghio
Retired, Math Consultant
Kimberly Dolan
AnchorBaySchool District / Jodi Giraud
L’Anse Creuse Schools / Julie Hessell
Romeo Schools
Amy Holloway
Clintondale Schools / Barbara Lipinski
AnchorBaySchool District / Linda Mayle
Romeo Schools
Therese Miekstyn
Chippewa Valley Schools / James Navetta
Chippewa Valley Schools / Gene Ogden
AnchorBaySchool District
Rebecca Phillion
Richmond Comm. Schools / Charlene Pitrucelle
AnchorBaySchool District / Shirley Starman
Van Dyke Public Schools
Ronald Studley
AnchorBaySchool District
2005 and 2006 Session/Module Developers
Carol Nowakowski
Retired, Math Consultant / Deb Barnett
Lake Shore Public Schools / Luann Murray
Genesee ISD
Kathy Albrecht
Retired, Math Consultant / Jo-Anne Schimmelpfenneg
Retired, Math Consultant / Marie Copeland
Warren Consolidated
Terri Faitel
Trenton Public Schools / Debbie Ferry
Macomb ISD

Gr. 2 Participant Packet – Session 5

Focus on: Geometry

Name of Activity / Description of Activity / Materials/Handouts / Key Tips for Teacher
I. Warm-up activity
Geometry Matching Exercise
Time: 10 minutes /
  • As teachers walk-in have the handout, Geometry Matching Exercise, for them to complete at their table.
/
  • Geometry Matching Exercise (Pg. 9)
/
  • May wish to give participants a clue on how to get started in their thinking.
  • Most of these entries are a real “play on words”.

II. Van Hiele Theory of Geometric Thought
Time: 15 minutes /
  • Walk participants through the theory of geometric thought by Van Hiele.
  • Discuss the first two levels (level 0 and level 1) that apply to K-3 students.
  • Use the research regarding common misconceptions to show teachers how the Frayer’s Model will help their students classify examples and non-examples of 2- and 3-dimensional shapes.
/
  • Van Hiele Levels of Instruction (Pg. 10, 11, 12, 13)
  • Misconceptions in Geometry (Pg. 14)
  • Frayer’s Model (Pg. 15)
/
  • Frayer’s Model is a generic model that can be used for any curriculum area as a way to help students identify properties of a given concept.

III. Frayer’s Model Activity
Time: 10 minutes /
  • Model for participants how they would develop the definition and characteristics of a given concept together as a group.
  • Use the concept of the GLCE’s to model using Frayer’s.
  • Teachers will then complete the bottom two sections (examples and non-examples) of the Frayer’s Model graphic organizer thinking about the GLCEs.
/
  • Frayer’s model (Pg. 15)
  • Markers
/
  • Use the graphic organizer to complete what it is, what it isn’t, examples and non-examples.
  • For young children, instead of writing – students can draw pictures to describe the examples and non-examples of the given concept.

IV. Instructional sequence for the GLCE’s
Time: 5 minutes /
  • Discuss with teachers all of the grade level content expectations that will be integrated within the development of geometric skills.
/
  • Geometry/GLCEs Connections sheet (Pg. 16)
/
  • Participants need to see connections and the rationale for how these expectations are woven into Geometric Skills.

V. The Greedy Triangle
By Marilyn Burns
Time: 10 minutes /
  • Read the story of The Greedy Triangle by Marilyn Burns.
  • Have teachers stand up and put their hands on their hips.
  • Discuss how they could have a student come up and trace the triangle inside their arms.
  • Have students explain how they are sure when a shape is a triangle.
  • Talk about how they could have a large piece of yarn tied end to end. Students help determine how many people will be needed to make a triangle…a square…a rectangle, etc. How many corners will your shape have? How many sides? Will the sides be different sizes? Will the edges be straight or curved?
/
  • The Greedy Triangle by Marilyn Burns
  • Yarn
/ “One of the benefits of The Greedy Triangle is that it shows shapes in many different forms and positions and places them in contexts that make them accessible to children. The illustrations give children a chance to think more flexibly about shapes and how they are used.”
Marilyn Burns
When introducing triangles – DO NOT SAY:
  • Have 2 points at the bottom and 1 at the top.
  • Have a point in the middle.
  • Have a flat bottom.
  • Are pointy.
  • Are like the open triangle used in music class or a cone-shaped clown hat.
  • Can be made from any three line segments.

VI. Geometry Walk
Time: 15 minutes – this includes their break time /
  • Have teachers complete a booklet titled “Geometry in My World.”
  • Have teachers take a walk around the building site.
  • While on the walk, they illustrate objects that contain geometric shapes using the Geometry Walk booklet. Participants (or students) draw a real world example thatillustrates the shape. Example: for a circle participants might draw a face of a clock. They describe their real world example in words, where it is located, etc.
/
  • Geometry Walk booklet
  • Pencils
/
  • Share Shapes, Shapes, Shapes by Tana Hoban as a segway to shapes in the real world.

VII. Shape Up by David Adler
Time: 15 minutes
Cheesy Triangles
Time: 15 minutes /
  • Read each page in the book and have participants follow the instructions.
  • Participants need a partner and will take turns in this activity.
  • Each player needs a set of Tangrams that is of a different color than their partner.
  • The object of this activity is to be the first player to use all of their Tangram pieces before their partner does. Players work to block their partner.
  • Choose one Tangram piece to place on the board at a time. Be sure the edges of the piece fit on the lines of the game board.
/
  • Shape Up, David Adler
  • Bread slice sheet (Pg. 17, 18)
  • 2 slices of American cheese
  • Toothpicks
  • Paper plates
  • Pretzel sticks
  • Plain paper
  • Graph paper
  • Pencils
  • Scissors
  • Large circle die cut – 15 copies
  • Tangrams sets (two different colors per pair of players)
  • Cheesy Triangle sheet or game board (could be run on heavier stock paper) (Pg. 19)
/
  • Go through most of the book. As you approach the section with “other polygons” – you may want to stop after sharingthe 8-sided shape (octagon).
Cheesy Triangles:
  • This activity gives players a chance to use strategy.
  • Players will become familiar with all of the various ways that the Tangram shapes can be broken down into smaller triangles.

VIII. Press and Play
(“face impressions”)
Time: 15 minutes /
  • Provide a geometric solid for each group of
2 -4 participants. (May wish to limit selection to the cube, rectangular prism, pyramid, cylinder, cone, sphere, hemisphere)
  • Ask participants to consider the following questions about their group’s geometric solid. Discuss.
Does it roll?
Could it be stacked?
Why can’t some of the solids be stacked?
  • Provide play dough for each table and ask participants to explore stamping or making an impression of any part of their solid they can press into the clay and create an image.
  • Continue with a discussion of:
Which solids were you able to make an impression and of what part(s) of that solid? Why were some portions of the solids difficult or impossible to imprint? This should lead to a mini lesson on flat and curved surfaces. It would also be an appropriate time to introduce the terms face, edge, and corner (vertex) as discussion progresses.
  • Discuss how many of each type of face they can find on their solid. (The beginnings of a study of “net”)
/
  • Sets of geometric solids in wood or plastic
  • Play dough
/
  • This activity is a good starting point for children to discover the 2 dimensional faces of solids.
  • Emphasize properties and characteristics of these solids as they arise in discussion.
  • Pay close attention to language. Some children may interchange words for 2 and 3 dimensional shapes as they are learning. Teachers should always use correct shape names as they respond to the children.
  • An extension of this activity could involve using plastic or sponge solids that can be dipped in paint and printed on paper. With solids such as the cube, triangular prism, or rectangular prism encourage students to make a copy of eachface onto paper. Thus, they are creating the beginnings of a “net” or covering for the solid.

IX. Fat and Flat Shapes
Time: 15 minutes /
  • Give participants a collection of geometric solids and thin attribute block pieces. Paper attribute shapes cut out from Ellison die could be used.
  • Ask them to categorize (sort) the collection in several different ways and share their findings.
  • Record these various sorts on a dry erase board or on the Elmo (overhead).
/
  • Geometric solids
  • Attribute Blocks (thin shapes) Note: Teachers could also use paper shapes cut from Ellison die.
/
  • After acknowledging the many sorting variations, suggest that participants sort by dimension and find “partners” across groups. Possible pairs may be the cylinder and circle or the cube and square.
  • Ask why some geometric solids have more than one partner. (i.e. pyramid, cone) whereas others (cube and sphere) have only one partner.
  • Throughout this tactile exploration rich conversation and new vocabulary will surface.
  • With teachers, a discussion may arise over the fact that the thin Attribute Blocks are actually geometric solids. Children may not even consider this and accept them as “flat shapes”.

X. The 7 Piece Mosaic Puzzle
(Van Hiele model)
Time: 25 minutes /
  • Provide the Mosaic model (large rectangle that has been broken into 7 pieces; each piece being numbered)
  • Begin by asking “What can we do with these pieces?” Give exploration time.
  • Direct participants to make a house with 2 pieces. Then make a house with 2 other pieces.
  • Can you make a house with 3 pieces? Can you find two ways to do it?
  • Can you make a tall house?
  • Is it possible to make a house with 4 pieces?
  • Direct participants to build enlargements of the equilateral triangle (piece 2).
  • How many different rectangles can you make? Can you make any in more than one way?
/
  • Mosaic model (Pg. 20) Best to run on heavy stock paper
  • Scissors
/
  • The Mosaic model includes the following:
1 isosceles triangle
1 equilateral triangle
2 right triangles
3 quadrilaterals made up of a rectangle, a trapezoid, and an isosceles trapezoid
  • Children will naturally want to have free play or discovery time with building various creations before expecting them to work through a directed activity with these pieces.
  • Children may have joined two pieces to make another. You may want to encourage or challenge them to find all the pieces that can be made from 2 other pieces.
  • Encourage children to flip pieces. This ability will help broaden their thinking later on about rotation of shapes.

M-GLAnCe – 2nd Grade – Session 5 – Geometry – Participant Packet
Page 1 Revised 10.15.07

Geometry Matching Exercise

1. A broken angleA. Polygon

2. Place where people are sent for committing crimesB. Pi

3. King of the jungleC. Prism

4. An angle that is never wrongD. Postulate

5. Used to tie up packagesE. Sphere

6. What girls want to find at the beachF. Geometry

7. Person who voted “yes” on tractorsG. Chord

8. Mathematician’s favorite dessertH. Protractor

9. A sharp weaponI. Tangent

10. What little acorns say when they grow upJ. Center

11. The one in chargeK. Coincide

12. What the forgetful professor did with the letter L. Inverse

he carried for a week before mailing

13. What the husband did when his mother-in-lawM. Rectangle
wanted to go home

14. What a person should do when it rainsN. Right angle

15. The way the poet wrote his love lettersO. Line

16. A dead parrotP. Ruler

M-GLAnCe – 2nd Grade – Session 5 – Geometry – Participant Packet
Page 1 Revised 10.15.07

The van Hiele Theory of Geometric Thought

0. Visualization 1. Analysis 2. Informal Deduction 3. Deduction 4. Rigor

Adapted from Elementary and Middle School Mathematics, Teaching Developmentally, John Vande Walle

M-GLAnCe – 2nd Grade – Session 5 – Geometry – Participant Packet
Page 1 Revised 10.15.07

Instruction for the VanHiele Levels

  • A major goal is to advance students to the next level of thinking.
  • It is crucial that instruction is at the student’s level of thought. Students must move through all prior levels before advancing.
  • Almost any activity can be modified to span two levels of thinking.
  • Using physical materials, drawings, and computer models is a must at any level. Activities that encourage exploration and discussion are the greatest single factor in moving students to higher levels.
  • Levels are not age dependent, as in the developmental Piaget stages. It is reasonable for K-2 students to be at a level 0, as well as many 3rd and 4th graders. Even a high school student may still be at a level 0.
  • If instruction or language is at a level higher than that of a student, then learning is very superficial and temporary. A student may, for example, memorize that all squares are rectangles without having constructed that relationship.

Adapted from Elementary and Middle School Mathematics, Teaching Developmentally, John VandeWalle

Level 0: Visualization

Students at this level:

  • Recognize and name shapes based on visual characteristics Students make “visual measurements”. It is appearance alone that defines a shape. A square is a square because it “looks like a square”.
  • Use shape appearance rather than properties to classify a shape

A square, for example, when rotated 45 degrees may now be a new shape – a diamond.

  • Sort and classify based solely on appearance

Their descriptions do not necessarily contain mathematical language. For example, shapes may be classified together because they are all “fat” or “pointy” or “look like a house” or are “dented in, sort of”.

  • Need to explore shapes in 2 and 3 dimensions

Students should be given many opportunities to explore how shapes are alike and different and use this knowledge to create classes of shapes (rectangles, triangles, prisms, cylinders, etc.)

  • Begin to notice shape properties

Properties of shapes such as right angles, parallel sides, symmetry, etc. might be discovered at this level but mainly in an informal, observable way.

  • Operate on what they see in front of them

What clearly defines this level is not the presence or absence of traditional geometric properties or terms, rather operating merely on what is directly observable in front of them.

  • Need many opportunities to build, make, draw, put together, and take apart shapes.

M-GLAnCe – 2nd Grade – Session 5 – Geometry – Participant Packet
Page 1 Revised 10.15.07

Level 1: Analysis

Students at this level:

  • Are able to consider all shapes within a class rather than a single shape

They can focus on, for example, what makes a rectangle a rectangle (4 sides, opposite sides parallel, opposite sides the same length, 4 right angles, congruent diagonals, etc.)

  • Not yet able to see subclasses of one another

They are able to list all properties of squares, rectangles,

parallelograms, but not yet able to see, for example, that all

squares are rectangles and all rectangles are parallelograms.

  • Should focus on analyzing classes of figures to discover new properties

Instruction should, for example, focus on finding all ways to sort a set of triangles. This activity will help students discover various types of triangles (equilateral, isosceles, scalene, right, acute, obtuse).

  • Can comfortably work with classes of shapes

Students at Level 1 are able to work with finding attribute properties that apply to ALL shapes in a particular class of shapes and not just to those shapes in front of them, as with Level 0 students.

M-GLAnCe – 2nd Grade – Session 5 – Geometry – Participant Packet
Page 1 Revised 10.15.07

Tips for Avoiding Misconceptions in Geometry

  • Allow plenty of exploration time for students to experience the properties of shape kinesthetically. (i.e. Physically manipulate solids to explore which can roll and which can be stacked.)
  • Encourage participation in more directed exploration, such as sorting by attributes and characteristics.
  • Provide examples as well as non-examples even though a child may not be ready to specifically name the non-example. Recognizing that a specific shape is not a circle requires the same knowledge as naming a shape as a circle.
  • Varying a shape’s position during presentations will help children understand that a shape remains constant regardless of its placement in space.
  • Pay close attention to language use. Facility with language should grow naturally from exploration and experience. Many young children will interchange words for two and three dimensional shapes as they are learning. Accept their wording, but always respond with correct shape names.
  • Challenge understanding and broaden generalizations about shapes and properties. Do this with questions such as: How do you know that this shape is not a square? What could you change to make it a square? What do these shapes have in common? How are these shapes different?

M-GLAnCe – 2nd Grade – Session 5 – Geometry – Participant Packet
Page 1 Revised 10.15.07

M-GLAnCe – 2nd Grade – Session 5 – Geometry – Participant Packet
Page 1 Revised 10.15.07

Grade 2 – Geometry Session 5

Instructional Sequence

GLCE Connections