4.NBT.B.6
*This standard is part of a major cluster
Standard:
Find whole-number quotients and remainders with up to four-digit dividends and onedigit 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.
Unpacking:
In fourth grade, students build on their third grade work with division within 100.
Students need opportunities to develop their understandings by using problems in and out
of context.
Note: The study of division in 4th grade is to help students develop an understanding of
division. The standard algorithm is not expected until 6th grade (6.NS.B.2).
General methods for computing quotients of multi-digit numbers and one-digit numbers
rely on the same understandings as for multiplication, but cast in terms of division.
One component is quotients of multiples of 10, 100, or 1000 and one-digit numbers.
For example, 42÷6 is related to 420÷6 and 4200÷6. Students can draw on their work with
multiplication and they can also reason that 4200 6 means partitioning 42 hundreds into 6
equal groups, so there are 7 hundreds in each group. 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.
Another component of understanding general methods for multi-digit division
computation is the idea of decomposing the dividend into like base-ten units and finding
the quotient unit by unit, starting with the largest unit and continuing on to smaller units.
As with multiplication, this relies on the distributive property. This can be viewed as
finding the side length of a rectangle (the divisor is the length of the other side) or as
allocating objects (the divisor is the number of groups).
Multi-digit division requires working with remainders. In preparation for working with
remainders, students can compute sums of a product and a number, such as 4 x 8+3. In
multi-digit division, students will need to find the greatest multiple less than a given
number. For example, when dividing by 6, the greatest multiple of
6 less than 50 is 6x8=48. Students can think of these “greatest multiples” in terms of
putting objects into groups. For example, when 50 objects are shared among 6 groups, the
largest whole number of objects that can be put in each group is 8, and 2 objects are left
over. (Or when 50 objects are allocated into groups of 6, the largest whole number of
groups that can be made is 8, and 2 objects are left over.) The equation 6x8+2= 50 (or 8 x
6 + 2 = 50) corresponds with this situation.
Cases involving 0 in division may require special attention.
(Progressions for the CCSSM; Number and Operation in Base Ten, CCSS Writing Team,
April 2011, page 14)
Example Explicit Trade (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.
This method can be modeled along with base ten blocks. 7-5 tells how many hundreds are
left. Next trade 2 hundreds for 20 tens plus 6 tens already there, making 26 tens. 26-25=1
tells how many tens are left. Trade 1 ten for 10 ones plus 3 ones already there are now 13
ones. There are 2 groups of five in 13 ones with 3 remaining.
Example Partial Quotients:
In the below example, the numbers on the side indicate the quantity of the divisor being
subtracted from the dividend. The divisor can be subtracted repeatedly from the dividend
in groups of any amount, there may be many approaches to this particular algorithm for
each student.
Using Base Ten Blocks:
Students can then divide the remaining ones into the three groups, and find the remainder.
Questions to check for understanding/increase rigor:
• What is the relationship between multiplication and division? Provide examples to
show your thinking.
• How does knowing 5 x 5 help you to solve 75 ÷ 5? Explain.
• How many different ways can you solve 84 ÷ 6?
• If the quotient is 15, what could your possible dividend and divisor be?
• How does changing the value of your divisor affect the quotient? (e.g. 350 ÷ 5 vs.
350 ÷ 50?)
• Using the digits 4, 9, 7, and 5, create a division sentence with the greatest possible
quotient.
• Which division strategy (partial quotients, rectangular array, area model) do you
think is best? Justify your answer.