Conceptualizing the Distributive Property

1. Conceptualizing the Distributive Property Math Alliance June 1, 2010 Melissa Hedges Beth Schefelker And DeAnn Huinker

2. Sharing Strategies Warm-up Walking Review: • Walk around and find 1 person that partitioned the 7×9 array (problem 6) differently than you. • 2. Using the mathematical language of multiplication justify why your partner’s partitioning strategy makes sense. • 3. Move to another partner who has a different strategy than you and your first partner and repeat.

3. The Language of Multiplication • Array – rectangular arrangement of rows and columns • Natural connection between multiplication and the areas of rectangles. • Area: The number of one unit by one unit squares it takes to cover a region without gaps or overlaps • Visually demonstrates the meaning of the commutative property of multiplication. • ___ rows of ____ • ___ columns of ___ • ___ groups of ___ • ___ sets of ____

4. Modes of representation of a mathematical idea Pictures As children move between and among these representations for concepts, there is a better chance of a concept being formed correctly and understood more deeply. Written symbols Manipulative models Real-world situations Oral/Written language Lesh, Post & Behr (1987)

5. Multiplication “Start-with” Thinking As a table team… • Turn over one card. • Think about the number sentence on the card and the suggested start-with expression. • Draw an open array for the number sentence. • Use the start-with to partition your array into two partial arrays. • Use math language to make sense of the partition. • Complete your array by labeling each area. • Write a number sentence that reflects the mathematics in your partial arrays.

6. Bridging for Success In what way might “start-with” thinking support students who are struggling with multiplication facts? “Students who struggle often have difficulty applying knowledge from one situation to another…using prior knowledge in new situations is automatic for successful students but needs to be taught to students who struggle in mathematics” ---My Kids Can: Making Math Accessible to All Learners K-5.

7. If students haven’t Mastered multiplication facts by 5th or 6th grade need something other than more drill. What Can Be done? Recognize More Drill Work Won’t help. Inventory the known and unknown Facts Suggest Strategies Diagnose strengths and weakness Basic Facts for Upper-Grade Students

8. Final Thoughts Students who have command of basic facts do not Necessarily reason better than those who haven’t Mastered their facts yet. Mathematics is about reasoning, patterns and making sense of things. There is no reason why a child who has not mastered basic facts should be excluded from mathematical experiences.

9. Math Homework • Beckman p. 245 problems #1 & #2 • Inventory “known” and “unknown” basic multiplication facts with your case study student. • Pick 1 unknown fact to work on with your case study student. • In one page (typed) state the process you followed to help your student learn this fact. Strategies learned in this class must be the basis of your instruction.