1. Applications of the Distributive Property Math Alliance June 22, 2010 Beth Schefelker, DeAnn Huinker, Melissa Hedges, Chris Guthrie

2. Learning Intention (WALT) & Success Criteria We are learning to… Make connections between the distributive property, the use of arrays, and the area model for multiplication. We will be successful when… We can explain how splitting arrays, “start with facts,” and the partial products algorithm are grounded in the distributive property.

3. Our Journey with Multiplication Grounded ourselves in the foundations of a conceptual understanding of multiplication Viewed multiplication as more than basic facts Learned flexible strategies for multiplication Expanded our understanding of representations Learned new vocabulary Factor Product Partial product Array ___ groups of ____ ___ rows of _____ ____ sets of ______

4. The Distributive Property of Multiplication Over Addition The distributive property is the most important and computationally powerful tool in all of arithmetic.  Beckmann (2005) For all real numbers, A, B, and C, A × (B + C) = (A × B) + (A × C) “A times the quantity (B + C), is the same as A times B plus A times C.” … or conceptually “partitioning & distributing.” NOTE: Real numbers are all the numbers that have a spot on the number line.

5. Revisiting Splitting Arrays and “Start-with facts” 9 × 7 = ___ Use concept-based language to describe the meaning of this equation. Write down two different “start with” facts that could help you solve 9x7. Visualize how your start-with fact works on this array.

6. Think about 9x7 and the use of the distributive property. What is being partitioned? What is being distributed? 9 5 + 2 9 x 5 9 x 2 Make a quick sketch of a 9×7 open array. Start with 9×5 to partition it. Label all dimensions and each partial product. Use the Notetaking Guide…. The Distributive Property A × (B + C) = A × B + A × C

7. A × (B+C) = A × B + A ×C Write equations to show the partitioning and distributing. A B + C A x BA x C Visualize 9x7, starting with 9x5. Think, then turn to your neighbor: In the above equation, what is the value of A? B? C? On your Guide, make the second open array using letters to label dimensions and partial products. 9 x 7 = 9 x (5+2) = 9 x 5 + 9 x 2

8. Variation 1: Splitting the Array ( A + B) × C = A × C + B × C Use concept-based language to describe the relationship between the expressions. 9 x 7 5 x 7 On your guide, sketch a 9x7 array, partition using the 5x7 “start with” fact. Shade and label the array to represent the dimensions and the partitions. 5 x 7 4 x 7 5+45+4 7

9. Variation 1: The Distributive Property (A+B) × C = A × C + B × C 9 ×7 = (5+4) × 7 = 5 × 7 + 4 × 7 (A+B) × C = A × C + B × C C ABAB A x C B x C Visualize 9x7, start with 9x5. On your recording sheet, draw the second array with letters to label dimensions and partial products. Write equations to show the partitioning and distributing.

10. Variation 2 A × (B+C+D) = A × B + A × C + A × D Complete Variation 2 on your guide sketch and label both arrays write the equations. Consider 9 x 7 What is being partitioned? What is being distributed? 9x7 9 groups of 7 9 groups of 2 is 18 9 groups of 3 is 27

11. 9 x 7 = 9 x (2+2+3) = 9x2 + 9x2 + 9x3 = 18 + 18 + 27 = 63 A × (B + C + D) = A × B + A × C + A × D A B + C + D A x B A x C A x D Variation 2: The Distributive Property A × (B+C+D) = A × B + A × C + A × D

12. Variation 3: A x (B–C) = A x B – A x C or(B–C) x A = B x A – C x A “9 groups of 7”… Too hard!Think…10x7…10 groups of 7. Much better! Complete Variation 3. sketch and label both arrays write the equations Consider 9 x 7

13. Variation 3: A x (B–C) = A x B – A x C or(B–C) x A = B x A – C x A 9 groups of 7 “10 groups of 7 less 1 group of 7.” 9 x 7 = (10 – 1) x 7 = 10 x 7 – 1 x 7 = 70 – 7 = 63

14. Quick Quiz 7× 8 Match the algebraic notation of the distributive property and each of its variations to corresponding number sentences and arrays. A × (B + C) = A × B + A × C 7 × 8 = 7 × (5 + 3) = 7×5 + 7×3 (A+B)× C= A×C +BxC 7 × 8 = (5+2) × 8 = 5×8 + 2×8 A × (B+C+D) = A×B + A×C + A×D 7× 8 =7 × (2+2+4) = 7×2 + 7×2 +7×4 (B-C)×A = B×A – C×A 7 × 8 = (8-1) × 8 =(8×8) - (8-1)

15. Learning Intention (WALT) & Success Criteria We are learning to… Make connections between the distributive property, the use of arrays, and the area model for multiplication. We will be successful when… We can explain how splitting arrays, “start with facts,” and the partial products algorithm are grounded in the distributive property.

16. Exam Next Week! Review study guide