1. CCGPS Mathematics Unit-by-Unit Grade Level Webinar Fifth Grade Unit 3: Multiplying and Dividing with Decimals September 13, 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 Five Unit 3: Multiplying and Dividing with Decimals September 13, 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 5. 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 F our girls washed the neighbor's dog for 50 cents. They didn't know how to divide the money, so the dog owner said: "I will give the four of you.8 of the total amount. To the first one to tell me how much that is, I will give.5 of the other.2". If someone gave the dog owner the right answer, how did the money get divided up? 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: Multiplication using decimals. Existing idea: Multiplication makes numbers larger. Accommodation Misconception: Multiplying a whole number by a decimal will always result in a larger number.

13. Misconception: Multiplication makes numbers larger. Cognitive conflict: When confronted with multiplication of a whole number by a decimal less than one, the student thinks the product must be a larger number than either factor.

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: Multiplication always makes numbers larger. 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? Misconceptions from America’s Choice Thinking that decimals are bigger than fractions because fractions are really small things. Thinking that you cannot convert a fraction to a decimal—that they can not be compared because they are different things.

25. Misconception- Invented Rule? Student misapplies knowledge of whole numbers when reading decimals and ignores the decimal point. Example: Student reads the number 45.7 as, “four fifty- seven” or “four hundred fifty-seven.”

26. Misconception – Invented Rule? Student misapplies procedure for rounding whole numbers when rounding decimals and rounds to the nearest ten instead of the nearest tenth, etc. Example: Round 3045.26 to the nearest tenth. Student responds, “3050” or “3050.26”

27. Misconceptions bit.ly/OsvoV7 learnzillion.com

28. Misconceptions Student has restricted his definition of decimals to one type of situation or model, such as base ten blocks. Example: Student does not recognize decimals as points on a number line, representations of fractions, or as division calculations.

29. Misconception – Invented Rule? Student misapplies rules for comparing whole numbers in decimal situations. Examples: 0.058 > 0.21 because 58 > 21 2.04 > 2.5 because it has more digits

30. Misconception – Invented Rule? Thinking that a decimal is just two ordinary numbers separated by a dot The decimal point in money separates the dollars from the cents 100 cents is $0.100 The decimal point is used to separate units of measure 1.5 feet is 1 foot 5 inches

31. Misconception- Invented Rule? When adding a sequence, adding the decimal part separately from the whole number part Example: Adding.25 beginning with.5 0.50, 0.75, 0.100 rather than 0.50, 0.75, 1.00

32. Misconception- Invented Rule? Adding or subtracting without considering place value, or starting at the right as with whole numbers Example: 4.15 + 0.1 = 34.16 or 12 – 0.1 = 11

33. Misconception- Invented Rule? Believing that two decimals can always be compared by looking at their “lengths” Example: “longer numbers are always bigger,” or “shorter numbers are always bigger”

34. Misconception- Invented Rule? Misunderstanding the use of zero as a placeholder Example 1.5 is the same as 1.05 Thinking that decimals with more digits are smaller because tenths are bigger than hundredths and thousandths Example.845 is smaller than.5 Thinking that decimals with more digits are larger because they have more numbers 1,234 is larger than 34 so 0.1234 is larger than 0.34

35. Misconception- Invented Rule? Mistakenly applying what they know about fractions; Example: 1/204 > 1/240, so 0.204 > 0.240 Mistakenly applying what they know about whole numbers Example: 600 > 6, so 0.600 > 0.6

36. Misconception- Invented Rule? Believing that placing zeros to the right of the decimal number change the value of the number Example: 0.4 is smaller than 0.400 because 4 is smaller than 400, or 0.81 is closer to 0.85 than 0.81 is to 0.8 Believing that a number that has only tenths is larger than a number that has thousandths Example: 0.5 > 0.936 because 0.936 has thousandths and 0.5 has only tenths

37. Misconception- Invented Rule? Not recognizing the denseness of decimals. Example: There are no numbers between 3.41 and 3.42 There are a finite number of expressions that will add or subtract to get a given decimal number

38. Misconception- Invented Rule? When multiplying by a power of ten, multiplying both sides of the decimal point by the power of ten 6.9 × 10 = 60.90 When dividing by a power of ten, dividing both sides of the decimal point by the power of ten 70.5 ÷ 10 = 7 1/2

39. Vocabulary Development

40. What vocabulary have we used in our discussion of misconceptions today?

41. Just remember:

42. What’s the big idea? Multiplying and dividing with decimals Deepening understanding of decimals and place value Powers of 10 Whole number exponents

43. What’s the big idea? Standards for Mathematical Practice What might this look like in the classroom? Wiki- http://ccgpsmathematicsk- 5.wikispaces.com/5th+Grade/ http://ccgpsmathematicsk- 5.wikispaces.com/5th+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

45. Activate your Brain F our girls washed the neighbor's dog for 50 cents. They didn't know how to divide the money, so the dog owner said: "I will give the four of you.8 of the total amount. To the first one to tell me how much that is, I will give.5 of the other.2". If someone gave the dog owner the right answer, how did the money get divided up? Bonus for the curious : http://www.parentingscience.com/critical-thinking-in-children.html

46. Activate your Brain Jean needs to cut a board into.15 meter pieces for her FIRST Lego League team. She starts with a 5 meter board. How much of the board will she have left after she makes the cuts? How many.15 meter pieces will she have? How many cuts does she have to make? Bonus for the curious : http://www.parentingscience.com/critical-thinking-in-children.html

47. Work through 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?

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

49. Coherence and Focus – Unit 3 What are students coming with from Unit 2? Whole number and whole number operations understandings Base ten understandings Experience modeling mathematical thinking Experience with area model Experience with fractions having denominators of 10 and 100. Experience using money as a context for problem solving

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

51. Coherence and Focus View across grade bands K-6 th  Whole numbers, fractions, fractions represented as decimals.  Place value understanding.  Operations with whole numbers, decimals, and fractions.  Numbers and their opposites. 8 th -12 th  Everything!

52. 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

53. Revision-ish Unit 3

54. Questions from the Wiki Why so many essential questions? How on earth can we get all this done in the time we have? Powers of ten notation.

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

56. Explanations and Examples

57. Basic Understandings for Teachers If children do not have a secure understanding of place value, ordering and rounding whole numbers they will not have the prerequisite skills and understanding to move onto decimals.

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

59. Basic Understandings for Teachers Context matters. Think carefully about money and measurement contexts.

60. Basic Understandings for Teachers Context matters. Think carefully about money and measurement contexts.

61. Basic Understandings for Teachers Visual representations of decimals 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.

62. Basic Understandings for Teachers http://nzmaths.co.nz/equipment-animations http://extranet.edfac.unimelb.edu.au/DSME/decimals/slim version/teaching/models/lab.html#labmab http://extranet.edfac.unimelb.edu.au/DSME/decimals/slim version/teaching/models/lab.html#labmab http://www.atm.org.uk/resources/gaps- misconceptions/fractions/fractions-objectives.html http://www.atm.org.uk/resources/gaps- misconceptions/fractions/fractions-objectives.html

63. Basic Understandings for Teachers

64. Teachers should discuss and correct common misconceptions about decimal arithmetic.

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

66. Examples & Explanations Standards addressed in Unit 3 CCGPS.5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.

67. Examples & Explanations Standards addressed in Unit 3 CCGPS.5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

68. Examples & Explanations Resources which work with Unit 3: Number cubes Base ten blocks Arrays Grid paper Metric measurement tools Anchor charts Number talks

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

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

71. 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

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

74. Examples & Explanations Standards: Illustrative Mathematics- 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.

75. Assessment

76. Race to the Top Assessment Toolbox Update Fall 2012

77. 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 77

78. 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 78

79. 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 79

80. 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. 80

81. 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 81

82. 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. 82

83. 83 Item Types – Multiple Choice (MC) – Extended Response (ER) – Scaffolded Item (SC)

84. 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 84

85. Mathematics Sample Item – Grade HS an extended response item 85

86. Example Rubric 86

87. 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 87

88. Mathematics Sample Item – Grade 3 a scaffolded item 88

89. 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 89

90. Where do you Find the Items? 90 rt1234567890 student

91. 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

92. 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

93. 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.

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

95. Very Fifth 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://teacherslifeforme.blogspot.com/ http://teacherslifeforme.blogspot.com/

96. 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

97. 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

98. 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

99. As you start your day tomorrow…

100. 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! @GaDOEMath & @turtletoms