Teaching Division to Elementary Students Math Methods Spring 2006

Constructing Meaning for Division Division and Multiplication are so closely related, division doesn’t have to start from scratch. When we do division problems, we use multiplication to “think division.”

Thinking Division a x 9 = 63 a= number of bags of oranges (or number of sets) 9 is the number of oranges in each bag (number in each set) 63 is the total number of oranges (total number of objects)

Interpretations for Division SUBTRACTIVE DISTRIBUTIVE

Subtractive Division The easiest for children to grasp. We know the total number of objects and the number of objects in each set. We need to find the number of sets.

Subtractive Division Examples If 6 crackers come in a package, how many packages will it take to get 30 crackers? If hot dogs come in packages of 8, how many packages will 56 hot dogs make? There are 7 days in a week. How many weeks are there in 49 days?

Subtractive Division (cont’d) To solve these types of problems, give the students counters and containers to work through the problem. For example, with the cracker problem, the containers would represent the packages and the students would place 6 counters in containers until the counters are gone. The number of containers used would be the number of packages– the answer!

Subtractive Division (cont’d) The Number Sentence would be: ____ x 6= 30 Total number of crackers Number of packages Number of crackers in each package So, ______ = 5

Subtractive Division (cont’d) If there had been 32 crackers, the Number Sentence would be: (_____ x 6) + 2 = 32

Subtractive Division (cont’d) Repeated Subtraction is a part of Sub. Div. Students will be able to use this once they have worked with the manipulatives and understand these problems.

Subtractive Division (cont’d) REPEATED SUBTRACTION: 22 -5 17 12 7 2 4 groups of 5 Will not make a set of 5. Number Sentence: (4 x 5) + 2 = 22 Number of boxes; # of items in boxes; # of remaining; total number of items

Activity using Subtractive Division Materials: worksheet showing a table like the one here; plastic links; large, colored paper clips Directions: Make small chains as indicated by this table. Use what you find to fill in the blanks. Be sure to measure and count carefully. Write number sentences after you have completed the table. Use the big chain that is this long: Make as many smaller chains as possible that are this many links: How many new small chains do you have? How many links are left? 35 cm 9 cm 52 cm 6 cm (etc)

Distributive Division In these types of problems, we know the total number of objects and the number of sets. The goal is to find the greatest number of objects that can be placed in each set if the objects are distributed equally among the sets.

Distributive Division Examples There are 42 marbles. There are 6 children. How many for each child? There are 27 chocolate kisses. There are 9 children. How many kisses for each child? There are 35 cat treats. There are 7 cats. How many treats for each cat?

Distributive Division (cont’d) The strategy used for Dist. Div. is different than the Sub. Div. Instead of removing sets of equal numbers from the total number of objects, the total number of objects is distributed equally among a given number of sets until there are not enough objects to “go around again.” If there are leftovers, they are the remainder.

Distributive Division MARBLE PROBLEM Let’s solve the marble problem. Give the students 42 counters (for marbles) and 6 containers (for children). Have the students distribute the marbles among the containers, keeping the number of counters in the containers equal, until there aren’t enough to “go around again.”

Distributive Division MARBLE PROBLEM (cont’d) Then have the students draw pictures to show “what happened” and write a number sentence to describe their findings. Number Sentence: 6 x _____ = 42 # of children # of marbles in all Marbles for each child

When you divide, you don’t always get a WHOLE number Once students understand that division is derived from multiplication, it is time to introduce the division sign. Sometimes the idea of fractions needs to be introduced before or during division to explain a problem such as 11 divided by 5. There is no whole number answer. But using multiplication and addition helps. (2 x 5) + 1 = 11

When you divide, you don’t always get a WHOLE number Then explain it as a division problem and that there is 1 leftover. That is the remainder.

Division by Zero What if one of the factors is 0 and the product given is not a 0? 0 x ____ = 10 or ____ x 0 = 10 Is this possible? Why or why not?

Division by Zero If we have: 0x___=0 or ___x0=0, then what replacements can we find for ______? Would 0 work? How about 1? How about 4? How about 10? Why?

It is not possible to divide by zero! Division by Zero It is not possible to divide by zero!

Review of all 4 Basic Operations For every addition fact, there are 2 related subtraction facts. For every multiplication fact, there are 2 related division facts (except where 0 is involved). Addition is an associative and commutative operation. Multiplication is an associative and commutative operation. Multiplication can be thought of as repeated addition. Division can be thought of as repeated subtraction.