4.1: Common Core and Division

The Common Core State Standards (CCSS) define division as an operation that is the inverse of multiplication. It is described as the process of partitioning a number into equal parts or groups (partitive division) or finding how many times one number is contained within another (measurement division).

Key Concepts of Division in the Common Core:

1. Division as Repeated Subtraction:
   • Division is sometimes described as repeatedly subtracting the divisor from the dividend until there is nothing left or a remainder. For example, dividing 12 by 4 can be seen as subtracting 4 from 12 repeatedly:
     $$12 – 4 = 8, \quad 8 – 4 = 4, \quad 4 – 4 = 0.$$ 
   • The number of times 4 was subtracted from 12 (3 times) is the quotient.
2. Division as Equal Grouping (Partitioning):
   • One of the primary ways to understand division is as partitioning a number into equal groups. If you have 12 objects and need to divide them into 4 equal groups, how many objects will be in each group?
     $$\frac{12}{4} = 3.$$ 
   • This can be visualized as dividing a set of 12 objects into 4 groups, with 3 objects in each group.
3. Division as Measurement:
   • Division is also understood as measurement division, where you are determining how many times a number (the divisor) fits into another number (the dividend). For example, how many 4s fit into 12?
     $$12 \div 4 = 3.$$ 
   • In this context, division is seen as the process of measuring how many units of the divisor are needed to reach the dividend.
4. The Relationship Between Division and Multiplication:
   • Division is understood as the inverse (opposite) operation to multiplication. If
     $4 \times 3 = 12$ 

     , then $12 \div 4 = 3$ 

     , and vice versa. This relationship is key to understanding the fact families and reversal of operations.

5. Long Division and Remainders:
   • In more advanced stages, division involves breaking up large numbers and using long division. Students learn to divide, multiply, subtract, and bring down numbers to find the quotient and, when necessary, the remainder.
   • For example, dividing 29 by 5 involves seeing how many times 5 fits into 29, with a remainder of 4.
6. Understanding Division with Fractions:
   • Division also extends to fractions, where dividing by a fraction is equivalent to multiplying by its reciprocal. For example:
     $$\frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = 1.5.$$ 
   • This concept is introduced in later grades.

Common Core Standards for Division (K-5):

• Kindergarten and 1st Grade: Students are not explicitly taught division in formal terms, but they explore the concept of fair sharing and equal grouping through practical experiences.
• 2nd Grade: Students start to formally understand division as repeated subtraction and partitioning into equal groups, as well as relate division to multiplication (e.g., using division facts like
  $8 \div 4 = 2$ 

  ).

• 3rd Grade: Students begin to develop fluency with division facts and understand division as both partitioning and measurement. They learn to divide numbers up to 100 using one-digit divisors, and they begin to work with remainders.
• 4th Grade: Students divide larger numbers (up to four digits) using more advanced strategies like long division, with an emphasis on understanding remainders and interpreting them correctly.
• 5th Grade: In 5th grade, division extends to dividing fractions and decimals, and students refine their understanding of division with larger numbers and more complex divisors.

Summary of Common Core Division:

1. Division is a Process of Fair Sharing: Students learn to divide numbers into equal parts, groups, or units.
2. Connection to Multiplication: Division is taught as the inverse of multiplication, reinforcing the relationship between the two operations.
3. Multiple Models: Division is understood through different models, such as repeated subtraction, equal grouping (partitioning), and measurement.
4. Long Division and Remainders: As students advance, they learn more formal methods of division, including long division and working with remainders.

Attributions and more Reading

Ginsburg, L., Decker, Ginsburg, Steen, Schmidt, R., Houang, R., National Research Council, & Confrey. (n.d.). Common Core State Standards for Mathematics. In Common Core State Standards for Mathematics. https://corestandards.org/wp-content/uploads/2023/09/Math_Standards1.pdf