Strategies for Division
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4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends
and one-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.
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 drawon their work with multiplication and they can also reason that 4200 6 means partitioning 42 hundreds into 6equal groups, so there are 7 hundreds in each group. Another component of understanding general methods formulti-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 6 x 8 = 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 6 x 8 + 2 = 50 (or
8 x 6 + 2 = 50) corresponds with this situation.
Cases involving 0 in division may require special attention.
- 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.
- 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 drawon their work with multiplication and they can also reason that 4200 6 means partitioning 42 hundreds into 6equal groups, so there are 7 hundreds in each group. Another component of understanding general methods formulti-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 6 x 8 = 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 6 x 8 + 2 = 50 (or
8 x 6 + 2 = 50) corresponds with this situation.
Cases involving 0 in division may require special attention.