CCGPS Mathematics Unit-by-Unit Grade Level Webinar Third Grade Unit 3: Operations and Algebraic Thinking: Properties of Multiplication and Division September 11, 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.

CCGPS Mathematics Unit-by-Unit Grade Level Webinar Grade Three Unit 3: Operations and Algebraic Thinking: Properties of Multiplication and Division September 11, 2012 Turtle Toms– Elementary 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 to share your feedback. After reviewing the remaining units, please contact us with content area focus/format suggestions for future webinars. Turtle Gunn Toms– Elementary 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 :

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. Accommodation 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?

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? The student may know the commutative property of multiplication but fails to apply it to simplify the “work” of multiplication. Example: Student states that 9 × 4 = 36 with relative ease, but struggles to find the product 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 apply it 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 ngscience.com/cri tical-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 to simplify the “work” of multiplication. Example: Student labors to find the product of three or more numbers, such as 8 × 13 × 5,, because he fails to recognize that it is much easier to multiply the numbers in a different order.

Misconceptions multiplication-problems-using-associative-property multiplication-problems-using-associative-property

Misconceptions The 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. or The 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. or The student has reasonable facility with multiplication facts but cannot multiply 6 × 23.

Explanations and Examples = = = =

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.

Misconceptions The 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

Misconceptions The 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.

Misconceptions The student has overspecialized during the learning process so that she recognizes some multiplication and/or division situations as multiplication or division and fails to classify others appropriately. Example: Student recognizes that a problem in which 4 children share 24 grapes is a division situation but states that a problem in which 24 cherries are distributed to children by giving 3 cherries 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- Inside math- Illustrative Mathematics- Edutopia- Teaching channel- Math Solutions-

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 :

Activate your Brain Algorithm A: Take a number Add 1 Double your answer ngscience.com/cri tical-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 3 What 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 3 Where does this understanding lead students? Look in your unit and find the Enduring Understandings.

Coherence and Focus – Unit 1 View across grade bands K-6 th  Operations with whole numbers and fractions.  Numbers and their opposites. 8 th -12 th  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.

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 Teachers Misconceptions 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 Teachers Teacher Misconceptions : As long as students are getting the correct answers, the students are understanding the material.

Basic Understandings for Teachers Students 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 3 MCC.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) = = 56. (Distributive property.)

Examples & Explanations Standards addressed in Unit 3 MCC.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 3 MCC.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

Examples & Explanations Standards addressed in Unit 3 MCC.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 3 MCC.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

Resources which work with Unit 3: Color tiles Beads Blocks Anchor charts Number talks

Examples & Explanations Resources which work with Unit 3:

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 h/number//index.html h/number//index.html

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

Examples & Explanations Standards: Illustrative Mathematics- MD cluster- one illustration SEDL- Tools: Tools for the Common Core: standards/ On the wiki: Discussion threads Unpacked standards from other states. Proceed with caution.

Assessment

Race to the Top Assessment 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 76

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 77

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 78

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. 79

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 80

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,

82 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 83

Mathematics Sample Item – Grade HS an extended response item 84

Example Rubric 85

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 86

Mathematics Sample Item – Grade 3 a 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 88

Where do you Find the Items? 89 rt student

Georgia Department of Education Assessment and Accountability Melissa Fincher Associate Superintendent Assessment and Accountability Dr. Melodee Davis Director Assessment Research and Development Robert Anthony Assessment Specialist Formative Item Bank Race to the Top Jan Reyes Assessment Specialist Interim Benchmark Assessments Race to the Top Dr. Dawn Souter Project Manager Race to the Top

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.

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?

Very Third Grade Wiki- 5.wikispaces.com/ 5.wikispaces.com/ Inside math- Edutopia- Teaching channel- Blogs/websites   problems-using-associative-property problems-using-associative-property

Resources Books  Van De Walle and Lovin, Teaching Student- Centered Mathematics, K-3 and 3-5  Parrish, Number Talks  Shumway, Number Sense Routines  Wedekind, Math Exchanges

Resources Common Core Resources  SEDL videos -  Illustrative Mathematics -  Dana Center's CCSS Toolbox -  Arizona DOE - practices/mathematics-standards/ practices/mathematics-standards/  Inside Mathematics-  Common Core Standards -  Tools for the Common Core Standards -  Phil Daro talks about the Common Core Mathematics Standards -

Resources Professional Learning Resources  Inside Mathematics-  Edutopia –  Teaching Channel -  Annenberg Learner - Assessment Resources  MARS -  MAP -  PARCC -

As you start your day tomorrow…

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