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Applications of the Distributive Property

Applications of the Distributive Property. Math Alliance June 22, 2010 Beth Schefelker , DeAnn Huinker , Melissa Hedges, Chris Guthrie. Learning Intention (WALT) & Success Criteria. We are learning to…

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Applications of the Distributive Property

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  1. Applications of the Distributive Property Math Alliance June 22, 2010 Beth Schefelker, DeAnnHuinker, 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. The Distributive PropertyA × (B + C) = A × B + A × C Think about 9x7 and the use of the distributive property. • What is being partitioned? • What is being distributed? 5 + 2 Use the Notetaking Guide…. 9 x 2 9 9 x 5 Make a quick sketch of a 9×7 open array. Start with 9×5 to partition it. Label all dimensions and each partial product.

  7. A × (B+C) = A × B + A ×C B + C Visualize 9x7, starting with 9x5. Think, then turn to your neighbor: In the above equation, what is the value of A? B? C? A x B A x C On your Guide, make the second open array using letters to label dimensions and partial products. A Write equations to show the partitioning and distributing. 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 7 • 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 usingthe 5x7 “start with” fact.Shade and label the array to represent the dimensions and the partitions. 5 + 4 5 x 7 4 x 7

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

  10. Variation 2A × (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 2 is 18 9 groups of 3 is 27

  11. Variation 2: The Distributive PropertyA × (B+C+D) = A × B + A × C + A × D 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 B + C + D A x B A x C A x D A

  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! Consider 9 x 7 Complete Variation 3.• sketch and label both arrays• write the equations

  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

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