CCGPS MathematicsUnit-by-Unit Grade Level Webinar

Third Grade Unit 3: Operations and Algebraic Thinking: Properties of Multiplication

and DivisionSeptember 11, 2012

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CCGPS MathematicsUnit-by-Unit Grade Level Webinar

Grade Three Unit 3: Operations and Algebraic Thinking: Properties

of Multiplication and Division September 11, 2012

Turtle Toms– tgunn@doe.k12.ga.usElementary Mathematics Specialist

These materials are for nonprofit educational purposes only. Any other use may constitute

copyright infringement.

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– tgunn@doe.k12.ga.usElementary Mathematics Specialist

Activate your Brain Sue bought 3 lunch tickets on Monday and 5 lunch tickets on Friday. If each ticket costs $2, how much did she spend in all?

Write to help explain your best thinking using words, numbers, or pictures.

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: I can flip the numbers in a multiplication problem and get the same answer.

Existing idea: I can also do this when I’m adding numbers. Accom

modation

Misconception: This will work for subtraction and division.

Misconception: Commutative property applies when I divide numbers.

Cognitive conflict: The answers don’t make sense anymore.

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: I can use commutative property when dividing.

Challenge: Flip a contextual situation, and see what happens.

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?

MisconceptionsIt is important to realize that inevitably

students will develop misconceptions…

Askew and Wiliam 1995; Leinwand, 2010; NCTM, 1995; Shulman, 1996

MisconceptionsTherefore 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? The student may know the commutative

property of multiplication but fails to apply it tosimplify the “work” of multiplication.

Example:

Student states that 9 × 4 = 36 with relative ease, but struggles to find theproduct of 4 × 9.

Misconceptions from America’s Choice

Activate your Brain Does order count in multiplication? Can you tell by looking at the “shape” of the multiplication table?

Misconceptions

The student does not understand the distributive property and does not know how to applyit to simplify the “work” of multiplication.

Example:Student has reasonable facility with multiplication facts but cannot multiply 12 × 8 or 23 × 6.

Activate your Brain Algorithm A: •Take a number•Add 1•Double your answer

http://www.parentingscience.com/critical-thinking-in-children.html

Algorithm B: • Take a

number• Double it• Add 2 to

your answer

Explain why algorithms A and B will always give the same output if you give them the same input.

From learner.org

Activate your Brain In what sense are algorithms A and B “the same”? In what sense are they different?

Learner.org/courses/learning math/algebra

Misconception – Invented Rule? The student may know the associative property

of multiplication but fails to apply it tosimplify the “work” of multiplication.

Example:Student labors to find the product of three or more numbers, such as8 × 13 × 5, , because he fails to recognize that it is much easier to multiplythe numbers in a different order.

Misconceptions

• http://learnzillion.com/lessons/46-solve-multiplication-problems-using-associative-property

MisconceptionsThe student can state and give examples of properties of multiplication but does not apply them to simplify computations.

The student multiplies 6 × 12 with relative ease but struggles to find the product 12 × 6. orThe student labors to find the product 12 × 15 because he does not realize that he could instead perform the equivalent but much easier computation, 6 × 30.orThe student has reasonable facility with multiplication facts but cannot multiply 6 × 23.

Explanations and Examples

• 3 + 8 = 8 + 3• 31 + 42 = 42 + 13• 25 + 46 = 46 + 25• 165 + 785 = 785 + 868

Misconception – Invented Rule?The student has overspecialized his knowledge of multiplication or division facts and restricted it to “fact tests” or one particular problem format.Student completes multiplication or division facts assessments satisfactorily but does not apply the knowledge to other arithmetic and problem-solving situations.

MisconceptionsThe student knows how to multiply but does not know when to multiply (other than because he was told to do so, or because the computation was written as a multiplication problem).

Example:Student cannot explain why he should multiply or connect multiplication to actions with materials.

Misconception – Invented Rule? The student knows how to divide but does not know when to divide (other than because she was told to do so, or because the computation was written as a division problem).

Example: Student cannot explain why she should divide or connect division to actions with materials.

Misconceptions• Thinking that division is commutative, for example • 5 ÷ 3 = 3 ÷ 5• Thinking that dividing always gives a smaller number• Thinking that multiplying always gives a larger number• Always dividing the larger number into the smaller: • 4 ÷ 8 = 2

MisconceptionsThe student sees multiplication and division as discrete and separate operations.

His conception of the operations does not include the fact that they are linked as inverse operations. Student has reasonable facility with multiplication facts but cannot master division facts. He may know that 6 × 7 = 42 but fails to realize that this fact also tells him that 42 ÷ 7 = 6.

Student knows procedures for dividing but has no idea how to check the reasonableness of his answers.

MisconceptionsThe student has overspecialized during the learning processso that she recognizes some multiplication and/or divisionsituations as multiplication or division and fails to classifyothers appropriately.Example: Student recognizes that a problem in which 4 children share24 grapes is a division situation but states that a problem inwhich 24 cherries are distributed to children by giving 3cherries to each child is not.

Misconceptions• What vocabulary have we used in our discussion of

misconceptions today?

Just remember:

What’s the big idea?•Usefulness of properties of operations•Strategies for fact fluency•Data representation and interpretation

What’s the other big idea?Standards for Mathematical Practice

• What might this look like in the classroom?• Wiki- http://ccgpsmathematicsk-5.wikispaces.com/1st+Grade/ • Inside math- http://bit.ly/RgKGrz• Illustrative Mathematics-

http://illustrativemathematics.org/standards/k8• Edutopia- http://bit.ly/o1qaKf• Teaching channel- http://bit.ly/LZ5DJR• Math Solutions- http://bit.ly/MqPf6w

Activate your Brain Sue bought 3 lunch tickets on Monday and 5 lunch tickets on Friday. If each ticket costs $2, how much did she spend in all?

Write to help explain your best thinking using words, numbers, or pictures.

Bonus for the curious: http://www.parentingscience.com/critical-thinking-in-children.html

Activate your Brain Algorithm A: •Take a number•Add 1•Double your answer

http://www.parentingscience.com/critical-thinking-in-children.html

Algorithm B: • Take a

number• Double it• Add 2 to

your answer

Explain why algorithms A and B will always give the same output if you give them the same input.

Activate your Brain In what sense are algorithms A and B “the same”? In what sense are they different?

Learner.org/courses/learning math/algebra

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?A developing understanding of number and quantity, deeper ideas about addition and

subtraction, developing ideas about multiplication and division.

Coherence and Focus- Unit 3Where does this understanding lead students?

• Look in your unit and find the Enduring Understandings.

Coherence and Focus – Unit 1View across grade bands

• K-6th Operations with whole numbers 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• Page 47- MATHEMATICAL not MATHEMATCIAL

• A revision look ahead- Unit 5, page 26. Correct definition: A trapezoid has at least one set of parallel sides.

Questions from the Wiki•Suggestions abound for variations on Arrays on the Farm• What’s with all those Essential Questions? •The tasks require students to do difficult things before they are fluent. •Where is the practice? •If you slow down to teach the skill, you get behind on pacing.

Basic Understandings for TeachersMisconceptions Some common misconceptions that teachers hold about multiplication and division: Multiplication facts should be taught in numerical order starting with 0 and 1 and then facts less than 5, and then, in order, facts 6 through 12. Multiplication should be taught in isolation with no connection to addition. Multiplication should be taught separately from division. (Wallace & Gurganus, 2005)

Basic Understandings for TeachersTeacher Misconceptions:

As long as students are getting the correct answers, the students are

understanding the material.

Basic Understandings for TeachersStudents need to realize that

multiplication has many meanings and is not just a list of isolated facts. This

can be hard for students to realize and comprehend.

(Wallace & Gurganus, 2005).

Examples & Explanations Standards addressed in Unit 3MCC.3.OA.5 Apply properties of operations as strategies to multiply and divide.Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.)Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

Examples & Explanations Standards addressed in Unit 3MCC.3.OA.6 Understand division as an unknown-factor problem.For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Basic Understandings for Teachers

Examples & Explanations Standards addressed in Unit 3MCC.3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.

Basic Understandings for Teachers

Basic Understandings for Teachers

Examples & Explanations Standards addressed in Unit 3MCC.3.MD.3 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two-step “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets.

Examples & Explanations Standards addressed in Unit 3MCC.3.MD.4 Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units – whole numbers, halves, or quarters.

Examples & Explanations

Examples & Explanations

Examples & Explanations Resources which work with Unit 3:Color tilesBeadsBlocksAnchor chartsNumber talks

Examples & Explanations Resources which work with Unit 3:

Examples & Explanations

Examples & Explanations Mathematically Flexible Thinking

•Look for likenesses and differences.•Expansiveness of thought•Understanding of properties 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

Examples & ExplanationsStandards:Illustrative Mathematics- MD cluster- one illustrationhttp://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

84

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

87

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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rt1234567890

student

Georgia Department of Education Assessment and Accountability

Melissa FincherAssociate SuperintendentAssessment and Accountabilitymfincher@doe.k12.ga.us

Dr. Melodee Davis Director Assessment Research and Developmentmedavis@doe.k12.ga.us

Robert Anthony Assessment SpecialistFormative Item BankRace to the Topranthony@doe.k12.ga.us

Jan ReyesAssessment SpecialistInterim Benchmark AssessmentsRace to the Topjreyes@doe.k12.ga.us

Dr. Dawn SouterProject ManagerRace to the Topdsouter@doe.k12.ga.us

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 Third 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://learnzillion.com/lessons/46-solve-multiplication-problems-

using-associative-property

Resources• Books

Van De Walle and Lovin, Teaching Student-Centered Mathematics, K-3 and 3-5

Parrish, Number TalksShumway, 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)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)