Problem Solving and Operations: Transitioning from Addition to Multiplication Tuesday, February 11, 2014 Presented By Sara Delano Moore, Ph. D. Director of Mathematics and Science at ETA hand2mind Join our community Implementing Common Core Standards in Math www.edweb.net/math

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Director of Mathematics & Science Problem Solving and Operations: Transitioning from Addition to Multiplication Sara Delano Moore, Ph.D. Director of Mathematics & Science ETA hand2mind

Problem Solving Series January - Problem Solving and Operations: Addition & Subtraction February – Problem Solving and Operations: Transitioning from Addition to Multiplication March – Problem Solving and Operations: Multiplication and Division

Understanding Multiplication

How many eggs? How can we figure it out?

How many eggs? What did you see first? What do we need to count? How many cartons of eggs? (38) 12 eggs, 38 times = 456 eggs altogether

Multiplication in Context Think about real world situations where we have repeated groups of objects Help students think about what’s happening in those situations How is it different from addition? Describe the situations with numbers in context 12 eggs in each carton and I have 38 cartons 38 cartons of a dozen eggs each There are 12 eggs, 38 times 38 groups of 12 eggs each

Multiplication Conventions Dictionary definitions The multiplicand multiplied by the multiplier equals the product How many items multiplied by how many times equals the result So…… 12 x 38 = 456 or take 12, 38 times. In the US, our convention is 38 x 12 = 456 or 38 groups of 12 is 456.

Language of CCSS Standards tend to talk about factors rather than multiplicand and multiplier However, this convention is still described 3.OA.A. 1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.

Commutative Property Focus on computation and value more than meaning. There are twelve cats in the room, each with four legs. How many legs? 4 legs per cat x 12 cats = 48 legs Does this make sense? 12 legs per cat x 4 cats = 48 legs My doctor gave me 20 antibiotic pills and told me to take 2 per day for 10 days. Can I take ten per day for two days instead?

What does CCSSM say? 2.OA.C - Work with equal groups of objects to gain foundations for multiplication. 3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

And in grade three…. 3.OA.A - Represent and solve problems involving multiplication and division. 3.OA.B - Understand properties of multiplication and the relationship between multiplication and division. 3.OA.C - Multiply and divide within 100. 3.OA.D - Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Important to Notice Addition/subtraction have a gradual progression over three years Bulk of multiplication and division happen in grade three. Equal groups perspective All facts

Grade Four Multiplicative Comparison 4.OA.A.1 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. Computation up to four digits by one digit or two digits by two digits

Grade Five Multiplication as Scaling 5.NF.B.5 - Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1. Multidigit algorithm

Grade 7 7.SP.C.8 – Find probabilities of compound events using organized lists, tables, tree diagrams, and simulations This standard depends on an understanding of Cartesian products to describe possible outcomes.

Progression of Meanings Equal Groups (grade 3) Comparison (grade 4) Scaling (grade 5) Cartesian Cross Products (grade 7) Always focus on context Use commutative property to help with computations.

Implementing Common Core Standards in Math A CE certificate for today’s webinar will be emailed to you 24 hours after the live session Join our community Implementing Common Core Standards in Math www.edweb.net/math Invitations to upcoming webinars Webinar archives and resources Online discussions CE quizzes for archived webinars

Join us for the next webinar Problem Solving and Operations: Tuesday, March 11, 2014 at 4PM Eastern Time Problem Solving and Operations: Multiplication and Division   Join the Implementing Common Core Standards in Math community for an invitation www.edweb.net/math

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