Developing Geometric Reasoning K-2 Common Core State Standards (CCSS) Class March 14, 2011 Paige Richards and Dana Thome
Learning Intentions We Are Learning To: • Recognize how Mathematical Practices 1 and 2—sense making and reasoning, —are connected to a selected standards’ content progression for geometric reasoning. • Identify how students will develop and demonstrate Practices 1 and 2 in their work and discussions.
Success Criteria • We will know we are successful when we can articulate how Mathematical Practice Standards 1 and 2 —sense making and reasoning—are infused in mathematical tasks or lessons for a standards’ content progression.
Content strand across grades: Operations & Algebraic Thinking Domain “Big Idea” that groups together a set of related standards. Cluster Statements that define what students should understand and be able to do at a grade level. Standards
A Content Standards Progression Domain: Geometry Clusters: K: Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). Analyze, compare, create, and compose shapes. 1 & 2: Reason with shapes and their attributes. Standards: K.G.1; K.G.2; K.G.3; K.G.4; K.G.5; K.G.6; 1.G.1; 1.G.2; 2.G.1
Laying the Foundation in PK-K Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres). 1. 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. 2. Correctly name shapes regardless of their orientations or overall size. 3. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”). Analyze, compare, create, and compose shapes. 4. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/“corners”) and other attributes (e.g., having sides of equal length). 5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes. 6. Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides touching to make a rectangle?”
What is a Triangle? On the 3-by-5 card write a definition of a triangle.
Considering Triangle Characteristics • Turn the card over. • Draw a triangle. • Draw another triangle that is different from the triangle you just drew. • Draw a third triangle that, again, is different than the previous triangles. • Finally, draw a fourth triangle that is different than any of the previous triangles. What is different about the triangles? What is similar?
van Hiele Levels of Geometric Reasoning • Level 0: Visualization Recognize figures as total entities, but do not recognize properties. • Level 1: Analysis (Description) Identify properties of figures and see figures as a class of shapes. • Level 2: Informal Deduction Formulate generalizations about relationships among properties of shapes; Develop informal explanations.
van Hiele Levels of Geometric Reasoning • Level 3: Deduction Understand the significance of deduction as a way of establishing geometric theory within an axiom system. See interrelationship and role of undefined terms, axioms, definitions, theorems and formal proof. See possibility of developing a proof in more than one way. • Level 4: Rigor Compare different axiom systems (e.g., non-Euclidean geometry). Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.
“I believe that development is more dependent on instruction than on age or biological maturation and that types of instructional experiences can foster, or impede, development.” Pierre M. van Hiele
How do students progress in developing geometric reasoning? • How would you recognize each of these levels of thinking in your students’ work? • Considering the first three levels, where would you place the majority of the lessons that you teach?
Tricky Triangles Envelope. . . has a selection of shapes. Goal... Sort the shapes into 2 groups. “Triangles” and “Not Triangles” Process... • Pull out a card (without looking). • Show it to the table group. • Explain why it is or isn’t a triangle. Pass the Envelope... to the next person and repeat the process until all the shapes are sorted. Group Discussion...What defines a triangle?
Reviewing Student Work Assign Roles Facilitator: Give all a voice. Recorder: Take notes on record sheet. Directions Distribute one or two work samples to each group member. Review what “student understands” and for“student misconceptions” (silently). Taking turns, present your observations. Table Group Discussion: What are some instructional implications you will take back to your school?
Tricky TrianglesPercent of Correct Responses for MPS Students in Grades 3-11
Revisiting Your Triangle Definition • Review and revise your definition of a triangle. • Highlight the main ideas you want to emphasize with your students. • Share definitions of triangles.
Reflect • How do these tasks engage you in the content learning infused with practices? (Mathematical Practices Standards 1, 2, 5) • How do these tasks help you to better understand the mathematics? (Content Standards K.G.1; K.G.2; K.G.3; K.G.4; K.G.5; K.G.6; 1.G.1; 1.G.2; 2.G.1)
Big Ideas of Geometry • Two- and three-dimensional objects can be described, classified and analyzed by their attributes. • Objects can be oriented in an infinite number of ways. The orientation of an object does not change the other attributes of the object. • Some attributes of objects (e.g. area, volume, perimeter, surface area) are measurable and can be quantified using unit amounts. • Objects can be constructed from or decomposed into other objects. In particular, any polygon can be decomposed into triangles.
Development Through the van Hiele Levels • Level is not affected by biological age. • Level is affected by degree of experience. • In order to progress through the levels, instruction must be sequential and intentional. • When instruction (or materials or vocabulary, etc.) is at an inappropriate level, students will not be able to understand the instruction. They may be able to memorize it, but with no understanding of material.
What other practices were infused in the content learning?Provide specific examples.
Summary We were learning to recognize three of the Standards for Mathematical Practices—sense making, reasoning, and tools— within a chosen Content Standards progression. We will know we are successful when we can articulate how both a Content Standard and a Standard for Mathematical Practice are infused in a math lesson in the classroom.
Grade 1 Reason with shapes and their attributes. 1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size) ; build and draw shapes to possess defining attributes. 2. 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 cylinders) to create a composite shape, and compose new shapes from the composite shape.1 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. _________________ 1 Students do not need to learn formal names such as “right rectangular prism.”
Grade 2 Reason with shapes and their attributes. 1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. 2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. 3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. _________________ 1 Sizes are compared directly or visually, not compared by measuring.