1. CCGPS Mathematics Unit-by-Unit Grade Level Webinar Fourth Grade Unit 3: Adding and Subtracting Fractions September 12, 2012 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.

2. CCGPS Mathematics Unit-by-Unit Grade Level Webinar Grade Four Unit 3: Adding and Subtracting Fractions September 12, 2012 Turtle Toms– tgunn@doe.k12.ga.ustgunn@doe.k12.ga.us Elementary Mathematics Specialist These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.

3. Expectations and clearing up confusion This webinar focuses on CCGPS content specific to Unit 3, Grade 4. For information about CCGPS across a single grade span, please access the list of recorded GPB sessions on Georgiastandards.org. For information on the Standards for Mathematical Practice, please access the list of recorded Blackboard sessions from Fall 2011 on GeorgiaStandards.org. CCGPS is taught and assessed from 2012-2013 and beyond. A list of resources will be provided at the end of this webinar and these documents are posted on the K-5 wiki. http://ccgpsmathematicsk-5.wikispaces.com/

4. Expectations and clearing up confusion The intent of this webinar is to bring awareness to:  the types of tasks contained in the unit.  your conceptual understanding of the mathematics in this unit.  approaches to tasks which provide deeper learning situations for your students. We will not be working through each task during this webinar.

5. Welcome! Thank you for taking the time to join us in this discussion of Unit 3. At the end of today’s session you should have at least 3 takeaways:  The big idea of Unit 3  Something to think about… food for thought  How can I support student understanding?  What is my conceptual understanding of the material in this unit?  a list of resources and support available for CCGPS mathematics

6. Please provide feedback at the end of today’s session.  Feedback helps us all to become better teachers and learners.  Feedback helps as we develop the remaining unit-by-unit webinars.  Please visit http://ccgpsmathematicsK-5.wikispaces.com/ to share your feedback.http://ccgpsmathematicsK-5.wikispaces.com/ After reviewing the remaining units, please contact us with content area focus/format suggestions for future webinars. Turtle Gunn Toms– tgunn@doe.k12.ga.ustgunn@doe.k12.ga.us Elementary Mathematics Specialist

9. Activate your Brain Which of the following sums are equal to 22/17? 5/17 +4/17 +3/17 +10/17 3/17 +8/17 +3/17 +10/17 6/17 +4/17 +3/17 +5/17 +2/17 +2/17 12/17 +10/17 1/17 +1/17 +9/17 +3/17 Find another way to write 22/17 as a sum of fractions. Bonus for the curious : http://www.parentingscience.com/critical-thinking-in-children.html

10. Why do learners make mistakes? Lapses in concentration. Hasty reasoning. Memory overload. Not noticing important features of a problem. or…through misconceptions based on: alternative ways of reasoning; local generalisations from early experience.

11. A pupil does not passively receive knowledge from the environment - it is not possible for knowledge to be transferred holistically and faithfully from one person to another. A pupil is an active participant in the construction of his/her own mathematical knowledge. The construction activity involves the reception of new ideas and the interaction of these with the pupils existing ideas.

12. New Concept: Fractions in which the numerator is greater than the denominator (improper fractions) Existing idea: Whole numbers are larger than fractions. Accommodation Misconception: Mixed numbers are larger than improper fractions.

13. Misconception: Mixed numbers are larger than improper fractions. Cognitive conflict: When confronted with an improper fraction, the student says it is not a fraction because in a fraction, the numerator is always less than the denominator.

14. What do we do with mistakes and misconceptions? Avoid them whenever possible? "If I warn learners about the misconceptions as I teach, they are less likely to happen. Prevention is better than cure.” Use them as learning opportunities? "I actively encourage learners to make mistakes and to learn from them.”

15. Diagnostic teaching. Source: Swann, M : Gaining diagnostic teaching skills: helping students learn from mistakes and misconceptions, Shell Centre publications “ Traditionally, the teacher with the textbook explains and demonstrates, while the students imitate; if the student makes mistakes the teacher explains again. This procedure is not effective in preventing... misconceptions or in removing [them]. Diagnostic teaching..... depends on the student taking much more responsibility for their own understanding, being willing and able to articulate their own lines of thought and to discuss them in the classroom”.

16. Diagnosis of misconceptions. Misconception: Mixed numbers are larger than improper fractions. Challenge: Require explanations that include diagrams or manipulatives to illustrate thinking.

17. Example of dealing with a misconception. One way to contrast or challenge this misconception might be to get agreement among students via discussion of the various answers and explanations of answers.

18. Two ways to teach... M. Swann, Improving Learning in Mathematics, DFES

19. Importance of dealing with misconceptions: 1) Teaching is more effective when misconceptions are identified, challenged, and ameliorated. 2) Pupils face internal cognitive distress when some external idea, process, or rule conflicts with their existing mental schema. 3) Research evidence suggests that the resolutions of these cognitive conflicts through discussion leads to effective learning.

20. Some principles to consider Encourage learners to explore misconceptions through discussion. Focus discussion on known difficulties and challenging questions. Encourage a variety of viewpoints and interpretations to emerge. Ask questions that create a tension or ‘cognitive conflict' that needs to be resolved. Provide meaningful feedback. Provide opportunities for developing new ideas and concepts, and for consolidation.

21. Look at a task from the unit What major mathematical concepts are involved in the task? What common mistakes and misconceptions will be revealed by the task? How does the task: – encourage a variety of viewpoints and interpretations to emerge? – create tensions or 'conflicts' that need to be resolved? – provide meaningful feedback? – provide opportunities for developing new ideas?

22. Misconceptions It is important to realize that inevitably students will develop misconceptions… Askew and Wiliam 1995; Leinwand, 2010; NCTM, 1995; Shulman, 1996

23. Misconceptions Therefore it is important to have strategies for identifying, remedying, as well as for avoiding misconceptions. Leinwand, 2010; Swan 2001; NBPTS, 1998; NCTM, 1995; Shulman, 1986;

24. Misconception – Invented Rule? Student writes fraction as part/part instead of part/whole. Student says that 3/5 are shaded. Misconceptions from America’s Choice

25. Misconception- Invented Rule? Student does not understand that when finding fractions of amounts, lengths, or areas, the parts need to be equal in size. Student says that ¼ of the square is shaded with stars.

26. Misconception – Invented Rule? Student overgeneralizes and thinks that “all 1/4s (for example) are equal”; she does not understand that the size of the whole determines the size of the fractional part. Example: Amir and Tamika both went for hikes. Amir hiked 2 miles and Tamika hiked 8 miles. Student thinks that when both students had completed1/4 of their hikes, they have each walked the same distance because 1/4=1/4.

27. Misconceptions bit.ly/OGA5q4 learnzillion.com

28. Misconceptions Student has restricted his definition of fractions to one type of situation or model, such as part/whole with pieces. Example: Student does not recognize fractions as points on a number line or as division calculations.

29. Explanations and Examples

31. Misconception – Invented Rule? Misapplies rules for comparing whole numbers in fraction situations. Example: 1/8 is bigger than 1/6 because 8 is bigger than 6

32. Misconception – Invented Rule? Overgeneralizes the idea that “the bigger the denominator, the smaller the part” by ignoring numerators when comparing fractions. Example: ¼ > 3/5 because fourths are greater than fifths.

33. Misconception – Invented Rule? Restricts interpretation of fractions inappropriately and does not understand that different fractions that name the same amount are equivalent. Example: 2/3 and 4/6 cannot name the same amount because they are different fractions.

34. Misconception- Invented Rule? Misapplies additive ideas when finding equivalent fractions. Example: 3/8=4/9 because 3 + 1 = 4 and 8 + 1 = 9

35. Misconception- Invented Rule? When adding fractions, generalizes the procedure for multiplication of fractions by adding the numerators and adding the denominators Example: 1/4+1/4 =2/8 Note that this error can also be caused by the alternative “conception” that fractions are just two whole numbers that can be treated separately.

36. Misconception- Invented Rule? When writing a fraction, comparing two parts to each other rather than comparing one part to the whole. When adding two fractions, adding the numerators and multiplying the denominators Example: 3/5+1/2=4/10 When subtracting mixed numbers, always subtracting the smaller whole number from the larger whole number and subtracting the smaller fraction from the larger fraction,

37. Misconception- Invented Rule? When writing a fraction, comparing two parts to each other rather than comparing one part to the whole. When adding two fractions, adding the numerators and multiplying the denominators Example: 3/5+1/2=4/10 When subtracting mixed numbers, always subtracting the smaller whole number from the larger whole number and subtracting the smaller fraction from the larger fraction,

38. Vocabulary Development What vocabulary have we used in our discussion of misconceptions today?

39. Vocabulary Development What vocabulary have we used in our discussion of misconceptions today?

40. Just remember:

41. What’s the big idea? Addition and subtraction of fractions. Deepening understanding of fractions.

42. What’s the big idea? Standards for Mathematical Practice What might this look like in the classroom? Wiki- http://ccgpsmathematicsk- 5.wikispaces.com/4th+Grade/ http://ccgpsmathematicsk- 5.wikispaces.com/4th+Grade/ Inside math- http://bit.ly/Mg07mlhttp://bit.ly/Mg07ml Games- http://bit.ly/vJEbdGhttp://bit.ly/vJEbdG Edutopia- http://bit.ly/o1qaKfhttp://bit.ly/o1qaKf Teaching channel- http://bit.ly/wm0OcJhttp://bit.ly/wm0OcJ Math Solutions- http://bit.ly/MqPf6whttp://bit.ly/MqPf6w

44. Activate your Brain Which of the following sums are equal to 22/17? 5/17 +4/17 +3/17 +10/17 3/17 +8/17 +3/17 +10/17 6/17 +4/17 +3/17 +5/17 +2/17 +2/17 12/17 +10/17 1/17 +1/17 +9/17 +3/17 Find another way to write 22/17 as a sum of fractions. Bonus for the curious : http://www.parentingscience.com/critical-thinking-in-children.html

45. Activate your Brain Abigail picked 2 3/4 pounds of peaches from the tree in her backyard. She gave 1 1/4 pounds to her neighbor Madeline. How many pounds of peaches does Abigail have left? Bonus for the curious : http://www.parentingscience.com/critical-thinking-in-children.html

46. What’s the big idea? Enduring Understandings Essential Questions Common Misconceptions Strategies for Teaching and Learning Overview Standards

47. Coherence and Focus – Unit 3 What are students coming with from Unit 2? A developing understanding of fraction equivalency. Whole number and whole number operations understandings. Partitioning experience.

48. Coherence and Focus- Unit 3 Where does this understanding lead students? Look in your unit and find the Enduring Understandings.

49. Coherence and Focus View across grade bands K-6 th  Whole numbers and fractions  Operations with whole numbers and fractions.  Numbers and their opposites. 8 th -12 th  Everything!

50. Navigating Unit Three The only way to gain deep understanding is to work through each task. No one else can understand for you. Make note of where, when, and what the big ideas are. Make note of where, when, and what the pitfalls might be. Look for additional tools/ideas you want to use Determine any changes which might need to be made to make this work for your students. Share, ask, and collaborate on the wiki. http://ccgpsmathematicsk-5.wikispaces.com/Home

51. Revision-ish Unit 3 Missing task- Snacks in a Set. I pushed out a present (Mega-Fun Fractions) to make up for this- see pages 28 and 29 for a similar task, perhaps.

52. Questions from the Wiki Why so many essential questions? How on earth can we get all this done in the time we have?

53. Basic Understandings for Teachers Build on informal understandings of sharing and proportionality. Students need to understand that fractions are numbers with magnitudes.

54. Basic Understandings for Teachers Visual representations of fractions help develop conceptual understanding of computational procedures. Students should be taught to estimate answers to problems before computing the answers, so that they can judge the reasonableness of their computed answers.

55. Basic Understandings for Teachers Teachers should present fraction problems in real-world contexts with plausible numbers.

56. Basic Understandings for Teachers Teachers should discuss and correct common misconceptions about fraction arithmetic.

57. Basic Understandings for Teachers Teacher Misconception : As long as students are getting the correct answers, the students are understanding the material.

58. Examples & Explanations Standards addressed in Unit 3 MCC.4.NF.3 Understand a fraction a / b with a > 1 as a sum of fractions 1/ b. a.Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b.Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

59. Examples & Explanations Standards addressed in Unit 3 c.Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. d.Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

60. Examples & Explanations Resources which work with Unit 3: Color tiles Pattern Blocks Fraction strips Fraction circles Anchor charts Number talks

61. Examples & Explanations Mathematically Flexible Thinking Look for likenesses and differences. Expansiveness of thought Understanding of fractions at an appropriate developmental level Reasoning and articulating thought both verbally and in journals

62. Examples & Explanations http://www.learner.org/courses/learningmat h/number//index.html http://www.learner.org/courses/learningmat h/number//index.html

63. How to develop all of these? Hold number talks regularly, making sure to include ideas that support development of relevant understanding. http://bit.ly/OYVpKN http://bit.ly/OYVpKN Not sure about the math yourself? VandeWalle

64. Shameless Plug Fractions: A Vertical View GaDOE presentation GCTM October 18 and 19 http://gctm-resources.org/drupal/

66. Examples & Explanations Standards: Illustrative Mathematics- MD cluster- one illustration http://illustrativemathematics.org/standards/k8# SEDL-http://secc.sedl.org/common_core_videos/http://secc.sedl.org/common_core_videos/ Tools: Tools for the Common Core: http://commoncoretools.me/2012/04/02/general-questions-about-the- standards/ On the wiki: Discussion threads Unpacked standards from other states. Proceed with caution.

67. Assessment

68. Race to the Top Assessment Toolbox Update Fall 2012

69. RT3 Assessment Initiatives Purpose – To support teachers in preparing the students for the Common Core Assessment that is to occur in spring 2015 – To provide assessment resources that reflect the rigor of the CCGPS – To balance the use of formative and summative assessments in the classroom 69

70. RT3 Assessment Initiatives Development of a three-prong toolkit to support teachers and districts and to promote student learning – A professional development opportunity that focuses on assessment literacy – A set of benchmarks in ELA, Math, and selected grades for Science and Social Studies – An expansive bank of formative assessment items/tasks based on CCGPS in ELA and Math as a teacher resource 70

71. Formative Assessment Conducted during instruction (lesson, unit, etc.) Identifies student strengths and weaknesses Helps teacher determine next steps – Review – Differentiation – Continuation Supplies information to provide students with detailed feedback Assessment for the purpose of improving achievement LOW stakes 71

72. Purpose of the Formative Item Bank The purpose of the Formative Item Bank is to provide items and tasks used to assess students’ knowledge while they are learning the curriculum. The items will provide an opportunity for students to show what they know and show teachers what students do not understand. 72

73. Formative Item Bank Assessments Aligned to CCGPS Format aligned with Common Core Assessments Open-ended and constructed response items as well as multiple choice items Holistic Rubrics Anchor Papers Student Exemplars 750+ Items Available in OAS by late September 73

74. Formative Item Bank Availability All items that pass data review will be uploaded to the Georgia OAS at Level 2. Formative Item Bank will be ready for use by all Georgia educators mid-September, 2012. 74

75. 75 Item Types – Multiple Choice (MC) – Extended Response (ER) – Scaffolded Item (SC)

76. Extended Response Items Performance-based tasks May address multiple standards, multiple domains, and/or multiple areas of the curriculum May allow for multiple correct responses and/or varying methods of arriving at a correct answer Scored through use of a rubric and associated student exemplars 76

77. Mathematics Sample Item – Grade HS an extended response item 77

78. Example Rubric 78

79. Scaffolded Items Include a sequence of items or tasks Designed to demonstrate deeper understanding May be multi-standard and multi-domain May guide a student to mapping out a response to a more extended task Scored through use of a rubric and associated student exemplars 79

80. Mathematics Sample Item – Grade 3 a scaffolded item 80

81. Mathematics Items Assess students’ conceptual and computational understanding Tasks require students to – Apply the mathematics they know to real world problems – Express mathematical reasoning by showing their work or explaining their answer 81

82. Where do you Find the Items? 82 rt1234567890 student

83. Georgia Department of Education Assessment and Accountability Melissa Fincher Associate Superintendent Assessment and Accountability mfincher@doe.k12.ga.us Dr. Melodee Davis Director Assessment Research and Development medavis@doe.k12.ga.us Robert Anthony Assessment Specialist Formative Item Bank Race to the Top ranthony@doe.k12.ga.us Jan Reyes Assessment Specialist Interim Benchmark Assessments Race to the Top jreyes@doe.k12.ga.us Dr. Dawn Souter Project Manager Race to the Top dsouter@doe.k12.ga.us

84. Navigating Unit Three The only way to gain deep understanding is to work through each task. No one else can understand for you. Make note of where, when, and what the big ideas are. Make note of where, when, and what the pitfalls might be. Look for additional tools/ideas you want to use Determine any changes which might need to be made to make this work for your students. Share, ask, and collaborate on the wiki. http://ccgpsmathematicsk-5.wikispaces.com/Home

85. Resource List The following list is provided as a sample of available resources and is for informational purposes only. It is your responsibility to investigate them to determine their value and appropriateness for your district. GaDOE does not endorse or recommend the purchase of or use of any particular resource.

86. Have you visited the wiki yet? http://ccgpsmathematicsk-5.wikispaces.com

87. Very Fourth Grade Wiki- http://ccgpsmathematicsk- 5.wikispaces.com/http://ccgpsmathematicsk- 5.wikispaces.com/ Inside math- http://bit.ly/Q5Wb8fhttp://bit.ly/Q5Wb8f Edutopia- http://bit.ly/o1qaKfhttp://bit.ly/o1qaKf Teaching channel- http://bit.ly/LZ5DJRhttp://bit.ly/LZ5DJR Blogs/websites  http://www.projectapproach.org/grades_1_to_4.php http://www.projectapproach.org/grades_1_to_4.php  http://learnzillion.com/lessons http://learnzillion.com/lessons

88. Resources Books  Van De Walle and Lovin, Teaching Student- Centered Mathematics, K-3 and 3-5  Parrish, Number Talks  Fosnot and Dolk, Young Mathematicians at Work  Shumway, Number Sense Routines  Wedekind, Math Exchanges

89. Resources Common Core Resources  SEDL videos -http://secc.sedl.org/common_core_videos/http://secc.sedl.org/common_core_videos  Illustrative Mathematics - http://www.illustrativemathematics.org/http://www.illustrativemathematics.org/  Dana Center's CCSS Toolbox - http://www.ccsstoolbox.com/http://www.ccsstoolbox.com/  Arizona DOE - http://www.azed.gov/standards- practices/mathematics-standards/http://www.azed.gov/standards- practices/mathematics-standards/  Inside Mathematics- http://www.insidemathematics.org/http://www.insidemathematics.org/  Common Core Standards - http://www.corestandards.org/http://www.corestandards.org/  Tools for the Common Core Standards - http://commoncoretools.me/http://commoncoretools.me/  Phil Daro talks about the Common Core Mathematics Standards - http://serpmedia.org/daro-talks/index.html http://serpmedia.org/daro-talks/index.html

90. Resources Professional Learning Resources  Inside Mathematics- http://www.insidemathematics.org/http://www.insidemathematics.org/  Edutopia – http://www.edutopia.orghttp://www.edutopia.org  Teaching Channel - http://www.teachingchannel.orghttp://www.teachingchannel.org  Annenberg Learner - http://www.learner.org/http://www.learner.org/ Assessment Resources  MARS - http://www.nottingham.ac.uk/~ttzedweb/MARS/http://www.nottingham.ac.uk/~ttzedweb/MARS/  MAP - http://www.map.mathshell.org.uk/materials/index.phphttp://www.map.mathshell.org.uk/materials/index.php  PARCC - http://www.parcconline.org/parcc-stateshttp://www.parcconline.org/parcc-states

91. As you start your day tomorrow…

92. Thank You! Please visit http://ccgpsmathematicsK-5.wikispaces.com/ to provide us with your feedback!http://ccgpsmathematicsK-5.wikispaces.com/ Turtle Gunn Toms Program Specialist (K-5) tgunn@doe.k12.ga.us These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. Join the listserve! join-mathematics-k-5@list.doe.k12.ga.us Follow on Twitter! Follow @turtletoms (yep, I’m tweeting math resources in a very informal manner)