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: Third Grade Fourth Grade Fifth Grade Sixth Grade
Students focus on understanding the meaning and properties of division.
Fourth Grade
Students use methods based on place value and properties of division supported by suitable representations to divide multi-digit numbers.
Fifth Grade
Students extend their understanding and reason about dividing whole numbers with two-digit divisors, decimals, and fractions.
Sixth Grade
Students extend their fluency with the division algorithm with decimals and divide fractions by fractions.

6. # of groups × amount in each group = total amount
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:
“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 one group gets.
“# 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.

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:
_____________________
______________________
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?
Mrs. McCullough has 36 students. She needed to put them into 3 groups. How many students will be in each group?

10. Two Types of Division Problems:
Partitive
Measurement
Write a partitive word problem for 45 ÷ 9:
Write a measurement 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 drawings and 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
136 16

12. Traditional vs. Partitive Algorithms
3,658 ÷ 5
Partitive Algorithm
Traditional Algorithm
Eliminates “goes into” misconception
Digits maintain their place value.
Allows for conservative estimation.
Many pathways to the quotient.
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.

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 7
2100
140 28

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

16. Division as a Fraction 128 ÷ 16 = = =
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 99 3
 12

21. Partitive Division of Fractions
Write a partitive word problem and draw a partitive model 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
Russell Geisner