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Building Foundations:

Building Foundations:. Conceptual Division. Ashley McCullough Russell Geisner Fifth Grade Teachers Promenade Elementary. Warm Up: Dividing Fractions.

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Building Foundations:

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  1. Building Foundations: Conceptual Division Ashley McCullough Russell Geisner Fifth Grade Teachers Promenade Elementary

  2. Warm Up: Dividing Fractions • Rules: Find the value of the following expression without using the standard algorithm. That’s right, no flipping and multiplying. Write a word problem to match the situation. • 3 ÷ ¼

  3. The Shifts in Common Core • Why do we all need to see the whole progression? • “Apply and Extend Previous Understandings…” • From the Progressions for the Common Core State Standards in Mathematics (2013). • “Mathematics is the practice of defining concepts in terms of a small collection of fundamental concepts rather than treating concepts as unrelated.” • “As number systems expand from whole numbers to fractions in Grades 3-5, to rational numbers in Grades 6-8, to real numbers in high school, the same key ideas are used to define operations within each system.”

  4. The Shifts in Common Core • Standards for Mathematical Practices: • The “varieties of expertise that mathematics educators at all levels should seek to develop in their students.” • 1-Make sense of problems and persevere in solving them. • 5-Use appropriate tools strategically. • 7-Look for and make use of structure. • 8-Look for and express regularity in repeated reasoning.

  5. Division Overview:

  6. Where does it begin? • The foundation for division is found in multiplication. • Grade 2: CCSS2.OA Work with equal groups to gain foundations for multiplication. • Grade 3: CCSS3.OA.A.1-Interpret products of whole-numbers, e.g., 5 × 7 as the total number of objects in 5 groups of 7 objects each. # of groups × amount in each group = total amount lllllll lllllll lllllll lllllll lllllll

  7. Two Types of Division Problems: Partitive: Measurement: “# of groups unknown” Shelley has 24 inches of ribbon. She needs 8 inches to make a bow. How many bows can she make? 24 ÷ 8 Quotient is how many groups were made. • “group size unknown” • Shelley has 24 books to put onto 8 shelves. How many books will go on each self? • 24 ÷ 8 • Quotient is what onegroup gets.

  8. Two Types of Division Problems: Partitive Measurement The model 24 ÷ 8: • The model 24 ÷ 8: lll lll lll lll 24 lll lll lll lll

  9. Two Types of Division Problems: _____________________ ______________________ Mrs. McCullough has 36 students. She needed to put them into 3 groups. How many students will be in each group? • Mr. Geisner had 36 hours of community service to complete. He could volunteer at the library for 3 hours a day. How many days will he have to volunteer to complete all his community service?

  10. Two Types of Division Problems: Partitive Measurement Write a measurement word problem for 45 ÷ 9: • Write a partitive word problem for 45 ÷ 9:

  11. Partial Quotient Division • A partitive procedure • Algorithm develops out of manipulative use • Russell has 536 gold doubloons. He and his 3 pirate friends are sharing them equally. How many gold doubloons will each pirate get? • “By reasoning repeatedly about the connection between math drawingsand written numerical work, students can come to see division algorithms as summaries for their reasoning about quantities.” Progressions for the Common Core State Standards in Mathematics (2013) 4 ) 536 -400 100 136 -12030 16 -16 + 4 0 134

  12. Traditional vs. Partitive Algorithms 3,658 ÷ 5 Partitive Algorithm Traditional Algorithm Encourages “goes into” misconception (eg. The divisor is not going into the dividend.) Place value is lost. One pathway to the quotient. Not present in CCSS until grade 6. • Eliminates “goes into” misconception • Digits maintain their place value. • Allows for conservative estimation. • Many pathways to the quotient.

  13. Partitive Division Practice • 6,732 ÷ 4

  14. Area Model Division • Builds on previous understanding of Multiplication Area Model which is explicitly referenced by CCSS. 7 × 324 300 + 20 + 4 7 2100 140 + 28 2268 2100 140 28

  15. Area Model Division • Students find side length for a rectangle with a known area. 869 ÷ 7 100 + 20 + 4 869= 7 × 124 + 1 7 700 140 28 + 1 Known area: 869

  16. Division as a Fraction • Every division problem is a fraction at . 128 ÷ 16 = = = • The quotient is the numerator when the denominator is one. • CCSS.Math.Practice.MP8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

  17. Fifth Grade Student Work: Simplifying Division Problems

  18. Fifth Grade Student Work: Simplifying Division Problems

  19. Fifth Grade Student Work: Simplifying Division Problems

  20. Division of Fractions: 9 ÷ • Story Context - Partitive • “9 being shared into 3/4 of a group • It took Sarah 9 hours to finish 3/4 of her homework. How long will it take Sarah to do her entire homework at this rate? • Visual model: 3 3 3 3 3 3 3  12 • 99

  21. Partitive Division of Fractions • Write a partitiveword problem and draw a partitivemodel for the following expression: 12 ÷ 2/3

  22. Division of Fractions: 9 ÷ • Story Context- Measurement • “Groups of 3/4 being taken from 9” • Steve had 9 candy bars. A recipe for s’mores calls for 3/4 of a candy bar. How many s’mores can he make? • Visual model: He can make twelve s’mores.

  23. Measurement Division of Fractions • Write a measurement word problem and draw a measurement model for the following expression: 12 ÷ 2/3

  24. Contact Information • Ashley McCullough • ashley.mccullough@alvord.k12.ca.us • Russell Geisner • russell.geisner@alvord.k12.ca.us

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