5.NBT.B.6
*This standard is part of a major cluster
Standard
Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors,
using strategies based on place value, the properties of operations, and/or the relationship between
multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays,
and/or area models.
Note: The study of division in 5th grade is to help students develop an understanding of division. The
standard algorithm is not expected until 6th grade (6.NS.B.2).
Unpacked
This standard references various strategies for division. Division problems can include remainders. Even
though this standard leads more towards computation, the connection to story contexts is critical. Make
sure students are exposed to problems where the divisor is the number of groups and where the divisor is
the size of the groups. In fourth grade, students’ experiences with division were limited to dividing by onedigit divisors. This standard extends students’ prior experiences with strategies, illustrations, and
explanations. When the two-digit divisor is a “familiar” number, a student might decompose the dividend
using place value.
Language plays an enormous role in thinking about division conceptually. We might be accustomed to the
“goes into” language that can be quite mysterious to students. In the example 583÷4, we might say “4 goes
into 5 one time.” Preferably, we would want students to think of 583 as 5 hundreds 8 tens, and 3 ones.
Example:
There are 1,716 students participating in Field Day. They are put into teams of 16 for the competition. How
many teams get created? If you have left over students, what do you do with them?
Using expanded notation 2682 ÷ 25 = (2000 + 600 + 80 + 2) ÷ 25 Using understanding of the relationship
between 100 and 25, a student might think ~
• I know that 100 divided by 25 is 4 so 200 divided by 25 is 8 and 2000 divided by 25 is 80.
• 600 divided by 25 has to be 24.
• Since 3 x 25 is 75, I know that 80 divided by 25 is 3 with a reminder of 5. (Note that a student
might divide into 82 and not 80)
• I can’t divide 2 by 25 so 2 plus the 5 leaves a remainder of 7.
• 80 + 24 + 3 = 107. So, the answer is 107 with a remainder of 7.
Using an equation that relates division to multiplication, 25 x n = 2682, a student might estimate the
answer to be slightly larger than 100 because s/he recognizes that 25 x 100 = 2500.
Base Ten Blocks:
Example: 187 ÷ 11 = n
Using base ten models, a student can represent 187 and use the models to make an array with one
dimension of 11. The student continues to make the array until no more groups of 11 can be made.
Remainders are not part of the array.
Area Model:
Example: 9984 ÷ 64
An area model for division is shown below. As the student uses the area model, s/he keeps track of how
much of the 9984 is left to divide.
Explicit Trade Method (Van de Walle):
Instead of the somewhat mysterious “bring down” step of the standard algorithm, the traded pieces are
crossed out, as is the number of existing pieces in the next column. In the example below, the hundreds are
traded for one 10 tens, combined with the 8 that were there for a total of 18 tens, the 18 is therefore written
in the tens column. Wide place value columns are highly recommended for this method.
Questions to check for understanding and increase the rigor:
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If the quotient is 43, what could a possible dividend and divisor be? (use either a three- or fourdigit dividend). Model and explain your reasoning.
How do you know that 34 is not the quotient of 1,216 ÷ 4?
Find a number that when divided by either 2, 3, or 5 has a remainder of 1. Model and explain your
reasoning
List three numbers that when divided by 5 each have a remainder of 1. (Extend by using different
number combinations). Model and explain your strategies.
Using the digits 4, 9, 7, 1 and 5, create a division sentence with a two-digit divisor and the greatest
possible quotient.
Write a division problem that has a four-digit dividend and a one-digit divisor with a quotient that
is even.
What is the relationship between multiplication and division? Provide examples to show your
thinking.
Use two different division strategies to solve 9,754 ÷ 5.
How does understanding place value help when solving 439 ÷14 with the partial quotient strategy?
Explain by using examples.