Building Foundations:

Building Foundations:Conceptual Division

Ashley McCulloughRussell GeisnerFifth Grade TeachersPromenade Elementary

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

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.”

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.

Division Overview:

Third Grade

Students focus on understanding the meaning and properties of division.

Fourth GradeStudents use methods based on place value and properties of division supported by suitable representations to divide multi-digit numbers.

Fifth GradeStudents 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.

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

Two Types of Division Problems:

Partitive:

• “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.

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.


Partitive• The model 24 ÷ 8:

Measurement• The model 24 ÷ 8:

lll

lll

lll

lll

lll

lll

lll

lll

24


_____________________

• 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?


Partitive• Write a partitive word

problem for 45 ÷ 9:

Measurement• Write a measurement

word problem for 45 ÷ 9:

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 -120 30 16 -16 + 4 0 134

Traditional vs. Partitive Algorithms

Partitive Algorithm• Eliminates “goes into”

misconception• Digits maintain their

place value.• Allows for conservative

estimation. • Many pathways to the

quotient.

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.

3,658 ÷ 5

Partitive Division Practice

• 6,732 ÷ 4

300 + 20 + 4 7

Area Model Division

• Builds on previous understanding of Multiplication Area Model which is explicitly referenced by CCSS.

7 × 324

2100 140 + 28

2268

2100 140 28

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

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.

12816 64

8 81

Fifth Grade Student Work: Simplifying Division Problems



• 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:

Division of Fractions: 9 ÷

3 3 3

993 3 3 3 12

34

Partitive Division of Fractions

• Write a partitive word problem and draw a partitive model for the following expression:

12 ÷ 2/3

• 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.

Division of Fractions: 9 ÷ 34

Measurement Division of Fractions• Write a measurement word problem and draw a

measurement model for the following expression:

12 ÷ 2/3

Contact Information• Ashley McCullough• [email protected]

• Russell Geisner• [email protected]

Documents

Building Foundations: