CCGPS Mathematics Unit-by-Unit Grade Level Webinar Fourth Grade Unit 6: Geometry February 13, 2013 Session will be begin at 3:15 pm While you are waiting, please do the following: Configure your microphone and speakers by going to: Tools – Audio – Audio setup wizard Document downloads: When you are prompted to download a document, please choose or create the folder to which the document should be saved, so that you may retrieve it later.

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What’s Unit 6 all about? Organizing thinking Geometric properties of a variety of polygons with a focus on lines, angles, and symmetry Examples and non-examples of a variety of plane and solid figures Hierarchy of quadrilaterals Properties of triangles Modeling problem solving Multiple solution paths

What should students bring from before? Knowledge of triangles, quadrilaterals, pentagons, hexagons, cubes, cones, cylinders, spheres, rectangular prisms, and more. Familiarity with angle, face, parallel, perpendicular Two-dimensions, three dimensions Defining attributes of various shapes Foundational ability to compose and decompose quantities and shapes Foundational organizational ability Foundational ability to problem solve Use of the 8 Standards for Mathematical Practice

Organizing Thinking: Three approaches to teaching geometry Introduce and explain concepts Emphasis on student discovery, with no conceptual analysis of discoveries Discuss and extend concepts and procedures that come up in children’s problem solving (from Saxe et al., 1999, slightly modified by T. Toms)

The goldilocks dilemma… problem-solving?fd=1

Levels of geometric thinking: Visual/syncretic- Students recognize shapes, e.g., a rectangle “looks like a door.” Descriptive- Students perceive properties of shapes, e.g., a rectangle has four sides, all its sides are straight, opposite sides have equal length. Analytic- Students characterize shapes by their properties, e.g., a rectangle has opposite sides of equal length and four right angles. Abstract- Students understand that a rectangle is a parallelogram because it has all the properties of parallelograms.

Students find that some combinations of properties signal certain classes of figures and some do not; thus the seeds of geometric implication are planted. However, only at the next level, abstraction, do students see relationships between classes of figures (e.g., understand that a square is a rectangle because it has all the properties of rectangles) Competence at this level a ff ords the learning of higher-level geometry, including deductive arguments and proof.

Thus, learning geometry cannot progress in the same way as learning number, where the size of the numbers is gradually increased and new kinds of numbers are considered later. In learning about shapes, it is important to vary the examples in many ways so that students do not learn limited concepts that they must later unlearn. ( 06/ccss_progression_g_k6_2012_06_27.pdf)

Geometric properties

Geometric Properties Draw and identify: right angle acute angle obtuse angle straight angle segment line ray parallel lines perpendicular lines

Shape analysis How many acute, obtuse and right angles are in this shape?

Parallel and Perpendicular

Identify which of these shapes have perpendicular or parallel sides and justify your selection. A possible justification that students might give is: The square has perpendicular lines because the sides meet at a corner, forming right angles.

Angle measurement Benchmark angles: 90º, 180º, 360º Acute- less than 90º Obtuse- greater than 90º Isosceles right triangle - Scalene right triangle -

Symmetry For each figure, draw all of the lines of symmetry. What pattern do you notice? How many lines of symmetry do you think there would be for regular polygons with 9 and 11 sides? Sketch each figure and check your predictions. Polygons with an odd number of sides have lines of symmetry that go from a midpoint of a side through a vertex.

Sorting based on attributes

Drawing shapes using attributes Draw and name a figure that has two parallel sides and exactly 2 right angles. For example: For each of the following, sketch an example if it is possible. If it is impossible, say so, and explain why or show a counter example. A parallelogram with exactly one right angle. An isosceles right triangle. A rectangle that is not a parallelogram. Every square is a quadrilateral. Every trapezoid is a parallelogram.

Using and sorting examples and non-examples of a variety of plane and solid figures

Regular vs irregular polygons Regular- equal sides, equal angles Irregular- unequal sides and/or angles. Is a rectangle regular or irregular?

Hierarchy of quadrilaterals

Subset thinking:

Trapezoid? In the U.S., that the term “trapezoid” may have two different meanings. In their study, The Classification of Quadrilaterals (Information Age Publishing, 2008), Usiskin et al. call these the exclusive and inclusive definitions: T(E): a trapezoid is a quadrilateral with exactly one pair of parallel sides T(I): a trapezoid is a quadrilateral with at least one pair of parallel sides.

These different meanings result in different classifications at the analytic level. According to T(E), a parallelogram is not a trapezoid; according to T(I), a parallelogram is a trapezoid. Both definitions are legitimate. However, Usiskin et al. conclude, “The preponderance of advantages to the inclusive definition of trapezoid has caused all the articles we could find on the subject, and most college- bound geometry books, to favor the inclusive definition.” Based on the above, Georgia will use the inclusive definition. A trapezoid is a quadrilateral with at least one pair of parallel sides.

Properties of triangles

Geometry progression Where does this begin? Where is it going? Please read pages 14 and 15 of the Geometry Progression. I pushed it out to you, and it is posted on the wiki.

Conceptual Understanding Matters 31

4 th grade?

Where is this going?

Common Core progression for Geometry: _progression_g_k6_2012_06_27.pdf

Modeling problem solving

What about teaching? A teacher asks: How do you envision how math class would be taught with the Common Core Standards? I think many teachers teach math in a fairly “traditional” way – instructing students on how to do (whatever) and then assign problems to be completed. How is our “mode of business,” if you will, going to change? Bill McCallum says: Dear Teacher, I don’t see the standards as dictating any particular teaching method, but rather setting goals for student understanding. Different people have different ideas about what is the best method for achieving that understanding. That said, I think it’s pretty clear that classrooms implementing the standards should have some way of fostering understanding and reasoning, and classrooms where students are just sitting and listening are unlikely to achieve that.

One of the strongest results in recent research is that the most important feature in effective teaching is giving students "opportunity to learn". Teachers can set expectations, time, kinds of tasks, questions, acceptable answers, and type of discussions that will influence students' opportunity to learn. This must involve both skill efficiency and conceptual understanding. Hiebert, James; Grouws, Douglas The Effects of Classroom Mathematics Teaching on Students' Learning 2007

Horizontal and vertical connections Strategies apply everywhere, in multiple contexts Integration of content areas ensures connections and relational thinking Multiple steps builds understanding and deeper thinking Work the culminating task collaboratively with colleagues so you know where your kids need to go, and what they might have difficulty with

Focus on Making Student Thinking Explicit need to be able to use student strategies as the center of the workgroup conversation, as one of the tools teachers interact with expertise shared by students change in power structures, from teacher as expert Journaling, anchor charts provide an explicit trace of the group’s thinking extends to other communities of practice in other content areas Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Focus on Making Student Thinking Explicit Ask children how they solved problems. Probe their understanding. Introduce symbols, number sentences, and mathematical language to go with strategies Extending Children's Mathematics Fractions and Decimals: Innovations in Cognitively Guided Instruction By Susan B. Empson and Linda Levi

Resources standards-elementary-school?fd=1 standards-elementary-school?fd=1

As part of the continuing implementation of CCGPS in the year , the current GADOE mathematics units are being augmented. We would like your assistance with this critical process. Student work samples are a vital component of the frameworks which only you can provide. If you have been using the GADOE frameworks and have student work which you would be willing to share, please send it our way. We will remove any identifiers, and include selected student work samples in the revised frameworks which are slated to be released July 1, If your student work sample is selected for inclusion we will notify you of its placement in the units via . Call for Student Work Samples

Submission guidelines : Attach the work sample(s) to an in any format. Whatever works for you, works for us. Indicate the grade level, unit, and task in the body of your , and on the work sample in the upper left corner. You may cover any student/school identifiers if you wish, and we will do the same if any remain. Send the with student work attached to the appropriate team member. To be considered for inclusion, work samples must be submitted by May 17, We look forward to seeing your students’ work. Call for Student Work Samples

As part of the continuing implementation of CCGPS in the year , the current GADOE mathematics frameworks and units are being reviewed, revised, and augmented. We are offering an opportunity for educators to assist in this critical process. The challenge: Create a career-based mathematics task using guidelines provided to supplement and/or address gaps in the existing CCGPS frameworks units. If your task is selected for addition to a unit, you will receive a $200 honorarium per task. All work is to be original using support structures provided by the Georgia Department of Education Mathematics Team. If you are interested in participating in this challenge, please view the task creation guidelines at and get started! Task submission period begins now and closes May 1, We look forward to seeing your tasks. Career-Based Mathematics Task Challenge

As part of the continuing implementation of CCGPS, the current CCGPS mathematics frameworks and units are being reviewed, revised, and augmented. The Georgia Department of Education is seeking qualified math educators to become part of the 2013 CCGPS Mathematics Resource Revision Team which will assist in this critical process. The scope of the CCGPS Mathematics Resource Revision Team work will include, but is not limited to: evaluating newly submitted tasks assessing the need for additional tasks assessing the order of current units and tasks editing of current units and tasks creating additional tasks to address gaps, if necessary 2013 Resource Revision Team

All work will be completed collaboratively with support structures provided by the Georgia Department of Education Mathematics Team. All work is to be completed at the Georgia Department of Education, June 3rd-June 6, and June 10-13, Team members will be compensated for contracted work in the amount of $2000 and travel expenses will be reimbursed. If you are interested in becoming a part of the CCGPS Mathematics Resource Revision Team, please respond to the appropriate Georgia Department of Education contact below by March 1, In your response, please indicate grade level interest, why you would like to be part of this team, related experience, and the contact information for two references Resource Revision Team Grades K-5 Turtle Toms Mathematics Program Elementary Specialist

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Thank You! Please visit to provide us with your feedback! Turtle Gunn Toms Program Specialist (K-5) These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Join the listserve! Follow on Twitter! (yep, I’m tweeting math resources in a very informal manner)