Visual Models in Math Connecting Concepts with Procedures for Whole Number & Decimal Multiplication and Division Tuesday, April 7, 2015 Presented by Sara Delano Moore, Ph.D. Director of Mathematics and Science at ETA hand2mind Join our community on edWeb.net Building Understanding in Mathematics www.edweb.net/math

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Visual Models in Math: Connecting Concepts with Procedures April 7, 2015: Base Ten Multiplication & Division Sara Delano Moore, Ph.D. Director of Mathematics & Science ETA hand2mind

Visual Models in Math: Series Overview January 6: Connecting Concepts with Procedures Overview February 3: Connecting Concepts with Procedures for Whole Number & Decimal Addition & Subtraction March 3: Connecting Concepts with Procedures for Fraction Addition & Subtraction April 7: Connecting Concepts with Procedures for Whole Number & Decimal Multiplication & Division May 5: Connecting Concepts with Procedures for Fraction Multiplication & Division

Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems. PtA, page 42

Being fluent means that students are able to choose flexibly among methods and strategies to solve contextual and mathematical problems, they understand and are able to explain their approaches, and they are able to produce accurate answers efficiently. PtA, page 42

Hands-On Learning Instructional Cycle Concrete Representational This instructional cycle is how we make the connections between work with manipulatives (the Concrete gear in the graphic) and abstract mathematics (the A gear in the graphic). If you remove the representations gear, the concrete and abstract ones don’t connect. When we use this instructional method with fidelity, what do we see in terms of student learning? [Research supporting this CRA cycle appears in Appendix D of the RTI report.] Abstract

Key Ideas for Base Ten Multiplication & Division Reminder: procedural focus in this series Models for Multiplication Repeated addition Area model Models for Division How many groups? How many in each group?

Strategies and Methods: Multiplication Repeated Addition Opportunity to show young learners about efficiency – “there has to be a better way” Good transition from addition to multiplication See Spring 2014 series for more information This method becomes inefficient with larger numbers and numbers which are not whole numbers. Arrays serve as a transition to area model.

Repeated Addition Example Susan gets 10₵ per day allowance. How much allowance does she receive in one week? 10₵ + 10₵ + 10₵ + 10₵ + 10₵ + 10₵ + 10₵ = 70₵ For 7 days, she receives 10₵ each day. 7 × 10₵ = 70₵ What happens over two weeks?

Strategies and Methods: Multiplication Area Model Connects the operation to one of its applications Translates more easily to a wider range of factor types e.g., fractions or algebraic expressions Most effective for one or two terms/digits in each factor

Area Model Examples 7 × 10 = 70

Area Model Examples 7 × 10 = 70 12 × 14 = 168 12 × (10 + 4) = (12 × 10) + (12 × 4) = 120 + 48 = 168

Moving from Concrete to Abstract: Partial Products Multiplication 12 × 14 20 8 2 + 10 8 40 20 + 100 100 40 168 10 + 4

Strategies and Methods: Division Repeated subtraction Inverse of repeated addition Find the missing factor Inverse of area model Both models are appropriate for both forms of division How many groups? How many in each group? As with multiplication, each model has situations where it is a better fit.

Repeated Subtraction Example 30 ÷ 6 = 5 How many times can I subtract 6 from 30? 1 2 3 4 5

Missing Factor Example 30 ÷ 6 = 5 6 × = 30

Moving from Concrete to Abstract: Partial Quotients Division 21 12)252 10 -120 132 10 -120 1 12 - 12

Working with Decimals Multiplication Extend the models we have to more decimal places.

Multiplying Decimals 1.2 × 32.4 = 30 + 2 + 1 + 30 2 6 38 + + = 38.88

Working with Decimals Multiplication Division Extend the models we have to more decimal places. Division Understand the fraction bar as a representation of division. Use knowledge of equivalent fractions & multiplicative identity.

Dividing Decimals 25.2 ÷ 1.2 = × = . . 12

Visual Models in Math: Series Overview January 6: Connecting Concepts with Procedures Overview February 3: Connecting Concepts with Procedures for Whole Number & Decimal Addition & Subtraction March 3: Connecting Concepts with Procedures for Fraction Addition & Subtraction April 7: Connecting Concepts with Procedures for Whole Number & Decimal Multiplication & Division May 5: Connecting Concepts with Procedures for Fraction Multiplication & Division

Building Understanding in Mathematics Join our community on edWeb.net Building Understanding in Mathematics Invitations to upcoming webinars Webinar recordings and resources CE quizzes Online discussions Join the community www.edweb.net/math

Recognition for your participation today! Attending Live? Your CE Certificate will be emailed to you within 24 hours. Viewing the Recording? Join the community at www.edweb.net/math Go to the Webinar Archives folder Take the CE Quiz to get a personalized CE Certificate CE Certificate provided by

Visual Models in Math: Join us for the next webinar Tuesday, May 5th – 4 PM Eastern Time Visual Models in Math: Connecting Concepts with Procedures for Fraction Multiplication and Division For an invitation to the next webinar Join Building Understanding in Mathematics www.edweb.net/math

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