ccgps mathematics unit-by-unit grade level webinar fifth grade unit 3: multiplying and dividing with...
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CCGPS MathematicsUnit-by-Unit Grade Level Webinar
Fifth Grade Unit 3: Multiplying and Dividing with Decimals
September 13, 2012
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CCGPS MathematicsUnit-by-Unit Grade Level Webinar
Grade Five Unit 3: Multiplying and Dividing with Decimals
September 13, 2012
Turtle Toms– [email protected] Mathematics Specialist
These materials are for nonprofit educational purposes only. Any other use may constitute
copyright infringement.
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/
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.
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
• 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.
• After reviewing the remaining units, please contact us with content area focus/format suggestions for future webinars.
Turtle Gunn Toms– [email protected] Mathematics Specialist
Activate your Brain Four 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
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.
• 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.
New Concept: Multiplication using decimals.
Existing idea: Multiplication makes numbers larger.
Accomm
odationMisconception: Multiplying a whole number by a decimal will always result in a larger number.
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.
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.”
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”.
Diagnosis of misconceptions.
Misconception: Multiplication always makesnumbers larger.
Challenge: Require explanations that include diagrams or manipulatives to illustrate thinking.
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.
Two ways to teach...
M. Swann, Improving Learning in Mathematics, DFES
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.
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.
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?
Misconceptions
It is important to realize that inevitably students will develop misconceptions…
Askew and Wiliam 1995; Leinwand, 2010; NCTM, 1995; Shulman, 1996
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;
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.
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.”
Misconception – Invented Rule? Student misapplies procedure for rounding whole numbers when rounding decimals androunds to the nearest ten instead of the nearest tenth, etc.
Example: Round 3045.26 to the nearest tenth. Student responds, “3050” or “3050.26”
Misconceptions
bit.ly/OsvoV7
learnzillion.com
MisconceptionsStudent 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.
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
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 cents100 cents is $0.100The decimal point is used to separate units of measure1.5 feet is 1 foot 5 inches
Misconception- Invented Rule?When adding a sequence, adding the
decimal part separately from the whole number part
Example: Adding .25 beginning with .50.50, 0.75, 0.100 rather than 0.50, 0.75, 1.00
Misconception- Invented Rule?
Adding or subtracting without considering placevalue, or starting at the right as with whole numbers
Example: 4.15 + 0.1 = 34.16 or 12 – 0.1 = 11
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”
Misconception- Invented Rule?Misunderstanding the use of zero as a placeholderExample1.5 is the same as 1.05Thinking that decimals with more digits are smaller because tenths are bigger than hundredths and thousandthsExample.845 is smaller than .5Thinking that decimals with more digits are larger because they have more numbers1,234 is larger than 34 so 0.1234 is larger than 0.34
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 wholenumbersExample:600 > 6, so 0.600 > 0.6
Misconception- Invented Rule?Believing that placing zeros to the right of the decimal number change the value of the numberExample: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 thousandthsExample:0.5 > 0.936 because 0.936 has thousandths and 0.5 has only tenths
Misconception- Invented Rule?
Not recognizing the denseness of decimals.
Example:There are no numbers between 3.41 and 3.42There are a finite number of expressions that will add or subtract to get a given decimal number
Misconception- Invented Rule?When multiplying by a power of ten, multiplying both sides of the decimal point by the power of ten6.9 × 10 = 60.90
When dividing by a power of ten, dividing both sides of the decimal point by the power of ten70.5 ÷ 10 = 7 1/2
Vocabulary Development
Vocabulary Development• What vocabulary have we used in our discussion of
misconceptions today?
Just remember:
What’s the big idea?•Multiplying and dividing with decimals•Deepening understanding of decimals and place value•Powers of 10•Whole number exponents
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/ • Inside math- http://bit.ly/Mg07ml• Games- http://bit.ly/vJEbdG• Edutopia- http://bit.ly/o1qaKf• Teaching channel- http://bit.ly/wm0OcJ• Math Solutions- http://bit.ly/MqPf6w
Activate your Brain Four 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
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
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?
What’s the big idea?
• Enduring Understandings• Essential Questions• Common Misconceptions•Strategies for Teaching and Learning• Overview•Standards
Coherence and Focus – Unit 3What 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
Coherence and Focus- Unit 3Where does this understanding lead students?
• Look in your unit and find the Enduring Understandings.
Coherence and FocusView across grade bands
• K-6th Whole numbers, fractions, fractions represented as decimals.Place value understanding.Operations with whole numbers, decimals, and fractions.Numbers and their opposites.
• 8th-12th Everything!
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
Revision-ish Unit 3
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.
Basic Understandings for Teachers
• Build on informal understandings of sharing and proportionality.
• Students need to understand that decimals are numbers with magnitudes.
Explanations and Examples
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.
Basic Understandings for Teachers
Teachers should present decimalproblems in real-world contexts withplausible numbers.
Basic Understandings for Teachers
Context matters.
Think carefully about money and measurement contexts.
Basic Understandings for Teachers
Context matters.
Think carefully about money and measurement contexts.
Basic Understandings for Teachers• Visual representations of decimals help developconceptual understanding of computationalprocedures.• Students should be taught to estimate answersto problems before computing the answers, sothat they can judge the reasonableness of theircomputed answers.
Basic Understandings for Teachers• http://nzmaths.co.nz/equipment-animations
• http://extranet.edfac.unimelb.edu.au/DSME/decimals/slimversion/teaching/models/lab.html#labmab
• http://www.atm.org.uk/resources/gaps-misconceptions/fractions/fractions-objectives.html
Basic Understandings for Teachers
Basic Understandings for Teachers
Teachers should discuss and correctcommon misconceptions aboutdecimal arithmetic.
Basic Understandings for TeachersTeacher Misconception:
As long as students are getting the correct answers, the students are
understanding the material.
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.
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.
Examples & Explanations Resources which work with Unit 3:
Number cubesBase ten blocksArraysGrid paper
Metric measurement toolsAnchor chartsNumber talks
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
Examples & Explanations
•http://www.learner.org/courses/learningmath/number//index.html
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
Not sure about the math yourself?• VandeWalle
Shameless Plug
Fractions: A Vertical ViewGaDOE presentation
GCTM October 18 and 19
http://gctm-resources.org/drupal/
Examples & ExplanationsStandards:Illustrative Mathematics-http://illustrativemathematics.org/standards/k8#SEDL-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 threadsUnpacked standards from other states. Proceed with caution.
Assessment
Race to the TopAssessment Toolbox
Update Fall 2012
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
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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
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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
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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.
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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
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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.
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Item Types
–Multiple Choice (MC)
–Extended Response (ER)
–Scaffolded Item (SC)
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
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Mathematics Sample Item – Grade HSan extended response item
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Example Rubric
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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
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Mathematics Sample Item – Grade 3a scaffolded item
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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
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Where do you Find the Items?
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student
Georgia Department of Education Assessment and Accountability
Melissa FincherAssociate SuperintendentAssessment and [email protected]
Dr. Melodee Davis Director Assessment Research and [email protected]
Robert Anthony Assessment SpecialistFormative Item BankRace to the [email protected]
Jan ReyesAssessment SpecialistInterim Benchmark AssessmentsRace to the [email protected]
Dr. Dawn SouterProject ManagerRace to the [email protected]
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
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.
Have you visited the wiki yet?
http://ccgpsmathematicsk-5.wikispaces.com
Very Fifth Grade• Wiki-
http://ccgpsmathematicsk-5.wikispaces.com/ • Inside math- http://bit.ly/Q5Wb8f• Edutopia- http://bit.ly/o1qaKf• Teaching channel- http://bit.ly/LZ5DJR• Blogs/websites
http://www.projectapproach.org/grades_1_to_4.php http://teacherslifeforme.blogspot.com/
Resources• Books
Van De Walle and Lovin, Teaching Student-Centered Mathematics, K-3 and 3-5
Parrish, Number TalksFosnot and Dolk, Young Mathematicians at WorkShumway, Number Sense RoutinesWedekind, Math Exchanges
Resources Common Core Resources
SEDL videos -http://secc.sedl.org/common_core_videos/ Illustrative Mathematics - http://www.illustrativemathematics.org/ Dana Center's CCSS Toolbox - http://www.ccsstoolbox.com/ Arizona DOE - http://www.azed.gov/standards-practices/mathematics-standards/Inside Mathematics- http://www.insidemathematics.org/ Common Core Standards - http://www.corestandards.org/ Tools for the Common Core Standards - http://commoncoretools.me/ Phil Daro talks about the Common Core Mathematics Standards - http://serpmedia.org/daro-talks/index.html
Resources• Professional Learning Resources
Inside Mathematics- http://www.insidemathematics.org/ Edutopia – http://www.edutopia.org Teaching Channel - http://www.teachingchannel.orgAnnenberg Learner - http://www.learner.org/
• Assessment Resources MARS - http://www.nottingham.ac.uk/~ttzedweb/MARS/ MAP - http://www.map.mathshell.org.uk/materials/index.php PARCC - http://www.parcconline.org/parcc-states
As you start your day tomorrow…
Thank You!Please visit http://ccgpsmathematicsK-5.wikispaces.com/ to provide us with
your feedback!
Turtle Gunn TomsProgram Specialist (K-5)[email protected]
These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement.
Join the listserve! [email protected]
Follow on Twitter!@GaDOEMath & @turtletoms
