Geometry: K6 (CCSSM)
Common Core Content Domain Geometry: K6 (CCSSM)
Contributors: Katie Quinlan, Lindsey Gatfield, Suheyla Sarac, and Hannah Suh
Contents 
Background: Levels of Geometric Thinking
 Lead Contributor: Suheyla Sarac
 Distinguish between Examples vs. NonExamples
The first step in the development of geometric shapes is students' ability to be able to distinguish between the examples and nonexamples of shapes (K.G.4).
 Visual/Syncretic: Recognize Shapes
Students recognize shapes as wholes (i.e. a figure is a rectangle because it looks like a door) rather than identifying the properties of the shapes.
 Descriptive: Perceive Attributes of Shapes
Students classify shapes by their attributes, e.g. a square has four sides. They differentiate between geometrically defining attributes (hexagons have six sides) and nondefining attributes (color, size, or orientation)
 Analytic: Characterize Shapes by their Properties
Students characterize shapes by their properties, e.g. the sum of the angles in a triangle is 180 degrees.
 Abstract: See Relationship between Classes of Figures
They can classify shapes hierarchically by their properties, i.e. accept classes within classes. e.g. a rectangle is a parallelogram because it has all the properties of parallelograms. e.g. a square is a rectangle because it has all the properties of rectangles
Common Core Strands
Geometric Shapes & Components
2D Shapes
 Lead Contributor: Kathryn Quinlan
Common Core Progression for 2D figures
Summary of the 2D Shape Progression:
Kindergarten and first grade are devoted to the general concept of shapes, and students learn about shape vocabulary, recognition, comparison, and attributes. Students learn to distinguish between 2D and 3D, then focus on 2D attributes and a wider range of shapes (ex: hexagons) until 3rd grade. The category of quadrilaterals are introduced in 3rd grade as are a greater range of shapes (ex: rhombuses). Students move to into the concept of angles and delve even deeper into attributes of 2D shapes in 4th grade. RIght angles and right triangles are formally introduced in the 4th grade. Graphing begins in 5th grade with an expectation that students learn to graph points on the coordinate plane to solve realworld and mathematical problems. Students continue to classify 2D figures into categories based on their properties. In 6th grade, 3D shapes are reintroduced for the first time since 1st grade, and continue to focus mainly on graphing and representing and measuring 2D shapes on the coordinate plane. Kindergarten:
1st Grade:
2nd Grade:
3rd Grade:
4th Grade:
5th Grade:
6th Grade:
Understanding the Concept of Shape
Threesided 2D Shapes
Foursided 2D Shapes
CirclesAngles and Other Attributes
Graphing and the Coordinate Plane
Technology Links for 2D Shapes
 

Assessment Resources for 2D Shapes
 A geometry assessment that includes 2D shapes assessment items:
 NCTM's Teaching Mathematics Problem generator is a resource for geometry assessment items. Teachers can select the "shapes" topic and generate anywhere from 530 assessment items for shapes.
 A shapes item example:
 The supermarket is full of objects having different shapes. The next time you are in the grocery store, take time to notice the various food items, packaging, and storage displays. What are some of the different shapes you can see? How many different shapes of crackers can you fi nd? Choose some crackers that have interesting shapes and bring these to school to compare them with crackers your classmates found. Group the crackers by their attributes, such as shape and size.
 A shapes item example:
 NCTM's Teaching Mathematics Problem generator is a resource for geometry assessment items. Teachers can select the "shapes" topic and generate anywhere from 530 assessment items for shapes.
3D Shapes
 Lead Contributor: Suheyla Sarac
Common Core Progression for 3D figures
3D shapes is introduced in the primary grades as a way to talk about the world we live in being threedimensional. Then the concentration shifts to 2D figure in grades 2 through 5. After they get a grasp of 2D shapes, the focus is back to 3D shapes starting in 6th grade.
Kindergarten:
 Students begin their exploration of 3D figures (solid) by firstly distinguishing them apart from 2D figures (K.G.3), using informal language, e.g. balls, boxes (K.G.4).
 Students then begin to name and describe 3D shapes with mathematical vocabulary, e.g. sphere, cylinder (K.G.1).
1st Grade:
 Compose 3D shape to create a composite shape and compose new shapes from the composite shape (1.G.2).
6th Grade:
 Students make 3D shapes from nets. They describe the shapes of the faces and number of faces, edges, and vertices solid figures have.
 Students develop visualization skills for components of 3D shapes that are not visible from a given viewpoint.
 Distinguish between units used to measure volume and area/length.
Primary Grades K1
Sorting Activity
Geometric thinking begins with play (Van Hiele, 1999). It is important to make the lessons, especially at primary grades, play like. A useful activity for distinguishing between examples and nonexamples of shapes Sorting activities are helpful for students to distinguish between 2D and 3D shapes. Tangible, reallife objects are recommended before a computer software program is used. 

Photo Sort Activity
Sorting can also be used to help students distinguish between the different 3D shapes. They can sort photos of real life objects into different categories of 3D shapes. 

Block Center
Building in the Block Center is helpful at this stage to help students have experience with the shapes. Draw their attention to the solid figures and their faces. They can begin to model objects in their environment, e.g. building a simple representation of the classroom using unit blocks and/or other solids.
Identify and describe the plane shapes found in objects. Explore and describe faces, edges, and corners of 2D and 3D objects make, name and describe polygons and other plane shapes 

Making Shapes
Making their own shapes help students understand the properties of shapes better. Different materials can be used. 

Feely Bags
To help students recognize 3D figures, describe attributes of solids and to make comparisons between different solids using mathematical language feely bags activity is a great tool. Place solid figures into a bag, have kids feel it, describe it, and name it. 

Technology Links:
 Video that explains solid figures, the difference between 2D and 3D. There are also additional resources like lesson plans and activities.
 2D to 3D morphing  helps students see the connection between the two.
Upper Grades – 6th
Guess What Shape I can Make
A great activity to relate plane shapes to solid figures is to have students first guess all the plane shapes a solid figure has. Then have the students press 3D figures into clay to make impressions of each face of the 3D figure.
Shapes with sticks Classify plane shapes and prisms by their spatial features. Construct models of polyhedra using everyday materials and use the terms faces, edges and vertices to describe models of polyhedra 

Technology Links:
 A great site that covers the difference between 2D and 3D shapes as well as different attributes of 3D shapes. Includes questions to assess student understanding.
Match Nets and Solids
Students are required to make 3D figures from nets and be able to distinguish what figure a net will result in and viceversa. After giving students concrete opportunities in making 3D figures from different nets, you can assess their knowledge more abstractly. 

Technology Links:
 A great game that covers faces, edges, and vertices of 3D shapes as well as nets of 3D shapes.
 Endof Unit Assessment Tool  Jeopardy Games
3D shapes have faces (sides), edges and vertices (corners). A net is what a 3D shape would look like if it were opened out flat.
Draw Pictures of Solids
Students need to develop visualization skills for components of 3D shapes that are not visible from a given viewpoint. Again, it is best to start with more concrete examples. Blocks or connecting cubes can be used to give students the opportunity to grasp the idea and move on to abstract representations on paper. 

Technology Links:
 Website that explains 3D shapes in detail with real world examples.
 A great game that asks students to figure out the viewpoints of different characters in the game.
Composing & Decomposing
Main Contributor: Lindsey Gatfield
Background
Composing and Decomposing geometric shapes is one of the major components of elementary geometry. In geometry, composing is the idea of putting two or more shapes together to create a new, larger shape. Decomposing then is taking away a part of a shape by covering the total shape with a part or taking shapes apart to see what smaller shapes they can be decomposed into. Composing and decomposing geometric shapes builds a knowledge of properties in students, such as being able to recognize equal length and equal angles in various shapes.
Primary Grades (K2) Progression Overview
 In the first three years of composing geometric shapes, students progress from being unable to compose shapes at all to developing the skills required to build and recognize more difficult composite shapes.
 In Kindergarten, students build shapes from components, compose shapes to build pictures and designs, combine shapes, solve problem such as deciding which piece will fit into a space in a puzzle, and use geometric motions such as sliding, flipping, and turning.

 In 1st grade, students learn to perceive a combination of shapes as a single new shape and to solve shape puzzles and construct designs with shapes.
 By the end of 2nd grade,students can build the same shape from different parts, can solve puzzles with numerous pieces, and intentionally substitute arrangements of smaller shapes for larger shapes.

Technology Resources
1. Students can match shapes with their outlines in online puzzles using pbskids.org. This allows them to begin to recognize the ways to match angles and edges of shapes.

2. Students can use multiple smaller shapes to compose larger shapes using online Tangrams. This is another application on pbskids.org.

3. Students can use online pattern blocks to compose and decompose shapes. This allows them to see how many smaller shapes can make up a larger shape. This online tool is provided by Learning Today.

Curriculum Resources
1. Students can use pattern block to compose larger shapes. They can also decompose the larger shapes to see which smaller shapes fit in the picture. This is a resource from McRuffy Press.

2. Teachers can purchase a Geometry Activity Box from LakeShore Learning Store. With this activity box, students will be able to use pattern blocks to complete puzzles, as well as compose smaller shapes to build a house. This box will also allow students to master other important concepts in Geometry

3. LakeShore Learning Store also has Geometry folder games for the elementary level. There are games that include different shapes, and help students to master the concepts of composing and decomposing. This helps students by actually being able to manipulate the shapes.

Assessment Resources
1. An assessment like this one allows teachers to see how students can decompose a larger shape to identify smaller shapes that make up that larger shape.

2. This assessment is similar to the previous one, but gives students options to choose from. It also provides an example to use for 3d objects. This assessment could also be used in higher grades.

Upper Grades (36) Progression Overview
 In terms of composing and decomposing geometric shapes, the emphasis in the geometry curriculum for 3rd6th grade is highly developing these concepts in order to understand multiplication, area, volume, and coordinate plane.
 In 3rd grade, students begin with finding all different compositions of a set of shapes involved and illustrate properties of multiplication by viewing rows and columns. Skills developed from 3rd5th grades provide the foundation for students to be able to understand formulas, decompose to find area, and compose and decompose polyhedral solids by the end of 6th grade.
Curriculum Resources
1. This puzzle demonstrates the 3rd grade skill in which students find all possible compositions of a set of shapes. This puzzle has at least three ways that students can compose the pieces to make a rocket ship image. This puzzle is from Big Cartel, but puzzles like these can be found at any toy or learning store.

2. This geometry kit from LakeShore Learning is an example of many geoboards that students can use to compose shapes and figure out area.

3. Illustrations like this one demonstrate the 6th grade skill of students being able to decompose a parallelogram into two triangles. This helps students to understand the concept of decomposing shapes to find the area of the composed shape.

Techology Resources
1. This online puzzle provided by Kids Math Games Online allows students to compose a larger shape using many smaller shapes. Students have to take into consideration angles and sides to see which pieces go where. This is a similar puzzle to ones a teacher may use in the primary grades, but its complexity is at a level for older students.

2. This online tool from Shodor allows student to decompose shapes into square units to find the area. Different shapes are given to the students and then they can check their answer to see if they are correct. This could also be used as an assessment tool.

3. Khan Academy is an extraordinary math courseware that teaches students new math skills, reviews with them, and assesses their learning. It also keeps track of their progress and understanding, so that the student or teacher/parent can follow along. Khan Academy has a lot of amazing resources, videos, and assessments for geometry subjects, especially for area and volume.

Assessment Resources
1. An assessment tool where teachers can give students questions such as this one. This is for higher grade levels, where students are decomposing shapes to find area. This resource is called Aleks Assessments.

2. This online investigation program is great for fifth or sixth graders. It goes through six geometry concepts and then assesses the student's knowledge afterwards. This tool is provided by the University of Connecticut.

Spatial Relations & Structuring
 Lead Contributor: Hannah Suh
Background
 Early composition and decomposition of shape is a foundation for spatial structuring
 Spatial structuring is the mental operation of constructing an organization or form for an object or set of objects in space, a form of abstraction, the process of selecting, coordinating, unifying, and registering in memory a set of mental objects and actions.
Primary Grades (K2) Progression Overview
 Students discuss shapes, orientations, and the relative positions of objects using terms such as "above, below, next to, behind, in front of, and beside".
 Students use spatial reasoning competencies, their growing knowledge of 3D shapes, and their ability to compose them to model objects in their environments
 For example: Building a simple representation of the classroom using unit blocks and/or other solids
 Students learn to see an object, such as a row, in two ways:
 As a composite of multiple squares
 As a single entity (a row of units)
 Such spatial structuring precedes meaningful mathematical use of the structures, including multiplication, area, volume, and the coordinate plane.
 Spatial structuring can be further developed with several activities with grids, including:
 Battleship Game
 Copying designs drawn on grid paper by placing manipulative squares and right triangles onto copies of the grid
 Creating and drawing designs on grid paper; Exchange with a partner and try to copy each others' designs
 Tessellations
Upper Grades (36) Progression Overview
 Students develop more competence in spatially structuring rectangular arrays
 Students explore line segments, lengths, perpendicularity, and parallelism on different types of grids (rectangular and triangular grids)
 Students extend their spatial structuring in two ways:
 Learn to spatially structure in three dimensions and use this understanding to find volume
 Extend their knowledge of the coordinate plane, understanding the continuous nature of 2D space and the role of fractions in specifying locations in that space
 Students solve mathematical and realworld problems using coordinates (whole number ordered pairs)
 Students develop visualization skills connected to their mathematical concepts as they recognize the existence of, and visualize, components of 3D shapes that are not visible from a given viewpoint
 Use drawings and physical models to learn to identify:
 lines perpendicular to a plane
 lines parallel to a plane
 plane passing through three given points
 plane perpendicular to a given line at a given point
Curriculum Examples
Students practice drawing tessellations 


Students practice copying and creating designs on grid paper 

Students practice drawing square units inside a rectangular plane 

Students practice counting square units on a grid 

Curriculum Resources
PreK through 2nd Grade
 Going Places Geometry Unit Measuring and Mapping to Explore Spatial Relations, Principles of Locations, and Navigation
Students practice moving a ladybug on a plane 

 Ladybug Adventures Geometry Unit Learning Geometry and Measurement Concepts
 Tangram Puzzles Geometry Unit Developing Geometric Understanding and Spatial Visualization Skills
3rd through 6th Grade
 Archimedes' Puzzle Geometry & Measurement Unit Learning about Symmetry and Transformation with an Ancient Greek Puzzle, the Stomachion
 Junior Architects Geometry & Measurement Unit Designing your Dream Clubhouse Using Measurement and Geometry
Students use pattern blocks to create a design for their clubhouse 

 Paper Quilts Geometry Unit Using Translation, Reflection, Rotation, and Line Symmetry to Make FourPart Quilt Squares
Technology Resources
 NCTM General Purpose Tool Offers a free downloadable interactive geometry platform that constructs, measures, manipulates, transforms (translations, reflections, rotations, dilations) and animates geometric figures including lines, circles, angles, perpendiculars, bisectors, tangents and vectors
Geometry Games
 Shape Inlay Create a geometric work of art by adding shapes to the board as quickly as possible
 Building Blocks Arrange the blocks to fit in the given shapes
 Shape Mods Reflect, rotate, and translate your way through 18 levels of shape shifting geometry fun
 Design a Party Use area and perimeter skills on a gird to design a part
 Cube Perspective Count the number of cubes on the board. Don't forget the hidden ones
 Locate the Aliens Identify the coordinates of points on a graph
 Space Boy to the Rescue Locate the coordinates on a plane
Assessment Items
 Smarter Balanced Assessment Consortium Provides sample problems and solutions
Example assessment problem requiring students to apply skills and knowledge of arranging shapes on a plane 



Useful Links
http://nzmaths.co.nz/numeracyprojects
http://illuminations.nctm.org/
References
http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g_k6_2012_06_27.pdf
http://www.dpi.state.nc.us/acre/standards/commoncoretools/#unmath
http://ime.math.arizona.edu/progressions/
Barkley, C. A. & Cruz, S. (2001). Geometry through beadwork designs. Teaching Children Mathematics, 7(6), 362367.
 This article is about ethnomathematics  studying mathematics within the context of culture. It goes through a 5stage lesson based on Native American beadwork design. It address an inquiryinvestigative approach to teaching.
Battista, M. T. (2002). Learning geometry in a dynamic computer environment. Teaching Children Mathematics, 8(6), 333339.
 This article discussed the benefits of students developing personally meaningful geometric concepts and ways of reasoning that enable them to carefully analyze spatial problems and situations. The author elaborated on the van Hiele levels of geometric thinking and also discussed the benefits of using the Shape Makers Interactive Geometry Microworld program, which is a special addon to Geometer's Sketchpad.
Hannibal, M. (1999). Young children's developing understanding of geometric shapes. Teaching Children Mathematics, 5(6), 353357.
 This article outlines the ways that early childhood educators should introduce basic shapes to young children. The author provides researchbased activities and suggestions for fostering an initial understanding of shape categories and attributes.
Munier, V., Devichi, C., & Merle, H. (2008). A physical situation as a way to teach angle. Teaching Children Mathematics, 14(7), 402407.