Grade 1: Unit G. A. 1-3, Reason with shapes and their attributes

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  Review the Progressions for Geometry at: http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g_k6_2012_06_27.pdf to see the development of the understanding of Geometry in Kindergarten stated by the Common Core Standards Writing Team, which is also the guiding information for the PARCC Assessment development.
  
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  When implementing this unit, be sure to incorporate the Enduring Understandings and Essential Questions as the foundation for your instruction, as appropriate.
  
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  Students should engage in well-chosen, purposeful, problem-based tasks. A good mathematics problem can be defined as any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific correct solution method (Hiebert et al., 1997). A good mathematics problem will have multiple entry points and require students to make sense of the mathematics. It should also foster the development of efficient computations strategies as well as require justifications or explanations for answers and methods.
  
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  Learning about Geometry does not progress in the same way as learning about number, where the size of the number gradually increases and new kinds of numbers are considered later. Instead, students’ reasoning about Geometry develops through five sequential levels in relation to understanding spatial ideas. In order to progress through the levels, instruction must be sequential and intentional. These levels were hypothesized by Pierre van Hiele and Dina van Hiele-Geldof. For more information about the van Hiele Levels of Geometric Thought listed below, please go to: http://images.rbs.org/cognitive/van_hiele.shtml or http://gogeometry.com/mindmap/van_hiele_geometry_level.html.
  

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  Level 0: Visualization
  
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  Level 1: Analysis
  
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  Level 2: Informal Deduction
  
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  Level 3: Deduction
  
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  Level 4: Rigor
  

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  Attributes refer to any characteristic of a shape.
  
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  Use many real-world examples and non-examples of shapes in order to provide greater depth of understanding as well as to begin noticing them based on similar characteristics and defining attributes. Encourage students to distinguish between defining attributes and non-defining attributes when discussing shapes and the characteristics of shapes.
  
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  When experiencing the properties of shapes kinesthetically, students should be encouraged to participate in free play as well as more directed exploration when composing and decomposing two- and three-dimensional shapes. Allow time for students to test and share their ideas in collaborative groups as they work.
  

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  Through your discussions and interactions with students, emphasize reasoning with shapes and their attributes as emphasized in the Maryland Common Core Standards, as opposed to simply identifying figures, which is typically only a vocabulary exercise.
  
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  Students can learn the correct names of shapes as they are exploring with them. If a student is holding and discussing a diamond, you can tell them that this is a called a rhombus. Similarly, when they are pointing to or manipulating an oval, you can share with them that this is an ellipse.
  
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  Students may describe groups of shapes but may not use conventional classifications in grade 1. By incorporating the Standards for Mathematical Practice into instruction, students will learn to justify and reason why a shape fits into one category but not another.
  
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  When partitioning rectangles and circles into two and four equal shares using the words halves, fourths, and quarters, students should learn these phrases through hands-on exploration. They are being introduced to the idea of fractional parts of the whole (the parts that result when the whole or unit has been partitioned into equal sized portions or fair shares).
  
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  Students develop geometric concepts and spatial reasoning from experience with two perspectives on space: the shapes of objects and the relative positions of objects. Combining the teaching of Geometry with number concepts reinforces the fact that mathematical content is related.
  
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  When allowing students to explore with fair shares, it is important that children are aware that the number of equal parts or fair shares that make up the whole determines the name of the shares, in this case halves, fourths, or quarters.
  
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  Introducing the phrase ‘quarter of’ might be confusing to students who have had experiences with money. While money is not formally introduced until grade 2 in the Maryland Common Core State Curriculum, it is important to note that students who have experience understanding a quarter as representing 25 cents may find this term confusing when learning as ‘a quarter of’ as meaning a fourth of. Therefore, there should be careful and clear explanation to students. It is important to show that circles and rectangles can be divided into four equal parts, shares, or quarters just as a dollar represents the whole and equals four quarters. Students need to understand that thinking that the phrase ‘quarter of’ is 25 of the whole is incorrect.
  

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  Geometry helps us understand the structure of space and the spatial relations around us.
  
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  Through geometry we can analyze the characteristics and properties of two- and three-dimensional shapes, as well as develop mathematical arguments concerning geometric relationships.
  
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  Geometry helps us develop and use rules for two- and three-dimensional shapes.
  

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  Where in the real world can I find shapes?
  
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  How can objects be represented and compared using geometric attributes?
  
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  How can I put shapes together and take them apart to form other shapes?
  
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  How can I identify and describe solid figures?
  
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  How can I compare and contrast two- and three-dimensional shapes?
  

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  Major Clusters
  

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  Supporting Clusters
  

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  Additional Clusters
  

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  Represent and solve problems involving addition and subtraction.
  
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  Understand and apply properties of operations and the relationship between addition and subtraction.
  
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  Add and subtract within 20.
  

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  Work with addition and subtraction equations.
  

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  Extend the counting sequence.
  

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  Understand place value.
  
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  Use place value understanding and properties of operations to add and subtract.
  

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  Measure lengths indirectly and by iterating length units.
  

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  Tell time and write time
  

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  Represent and interpret data.
  

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  Reason with shapes and their attributes.
  

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  1.G.A.3 Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
  

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  Have the opportunity to become engaged in problem solving that is about thinking and reasoning.
  
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  Collaborate with peers in an environment that encourages student interaction and conversation that will lead to mathematical discourse.
  
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  Describe in their own words why a shape belongs to a given category.
  
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  Use reasoning to differentiate between geometrically defining attributes (e.g., triangles have three sides) and non-defining attributes e.g., (color, overall size, or orientation).
  
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  Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular prisms) to create a composite shape. Students do not need to learn formal names such as ‘right rectangular prism.’
  
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  Compose new shapes from composite shapes they have composed.
  
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  Partition circles and rectangles into two and four equal shares.
  
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  Describe the shares they partition using the words ‘halves’, ‘fourths’, and ‘quarters’, and use the phrases ‘half of’, ‘fourth of’, and ‘quarter of’.
  
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  Describe the whole as two of, or four of the shares and understand that for these examples decomposing into more equal shares creates smaller shares.
  
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  Become engaged in problem solving that is about thinking and reasoning.
  
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  Collaborate with peers in an environment that encourages student interaction and conversation that will lead to mathematical discourse about geometry.
  

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  The Progressions for Geometry at: http://commoncoretools.files.wordpress.com/2012/06/ccss_progression_g_k6_2012_06_27.pdf stated by the Common Core Standards Writing Team, which is also the guiding information for the PARCC Assessment development.
  

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  Key Advances from Previous Grades:
  

Students in Prekindergarten:

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  Match like (congruent and similar) shapes.
  
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  Group shapes by attributes.
  
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  Correctly name shapes (regardless of their orientations or overall size).
  

Students in Kindergarten:

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  Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
  
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  Correctly name shapes regardless of their orientations or overall size.
  
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  Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
  
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  Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts and other attributes.
  
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  Model shapes in the world by building shapes from components and drawing shapes.
  
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  Compose simple shapes to form larger shapes.
  

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  Additional Mathematics:
  

In grade 2, students:

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  Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.
  
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  Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
  
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  Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.
  
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  Partition circles and rectangles into two, three, or four equal shares.
  
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  Describe shares using the words halves, thirds, half of, a third of, etc. and describe the whole as two halves, three thirds, four fourths.
  
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  Recognize that equal shares of identical wholes need not have the same shape.
  

In grade 3, students:

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  Understand concepts of area and relate area to multiplication and to addition.
  
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  Recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
  
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  Understand that shapes in different categories may share attributes and that the shared attributes can define a larger category.
  
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  Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.
  
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  Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.
  

In Grades 4 and beyond, students:

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  Draw and identify lines and angles, and classify shapes by properties of their lines and angles.
  
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  Graph points on the coordinate plane to solve real-world and mathematical problems.
  
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  Classify two-dimensional figures into categories based on their properties.
  
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  Solve real-world and mathematical problems involving area, surface area, and volume.
  
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  Draw, construct, and describe geometrical figures and describe the relationships between them.
  
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  Solve real-life and mathematical problems involving angle measure, are, surface area, and volume.
  
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  Understand congruence and similarity using physical models, transparencies, or geometry software.