Division

There are two different types of division. One is sharing or partitioning, which involves dividing a collection of objects into a given number of equal parts. Sharing twelve candies among three friends is a division situation of this type. The second type is grouping, dividing a collection of objects into groups of a known size. Figuring how many $0.15 pencils can be purchased with $1.00 is an example of division by grouping.

Children should become familiar with both types of division. Their formal instruction with division should begin with problem situations for them to solve. The situations can be related to the standard symbolic representations.


 * Day One:**

Introducing Division Through Problem Situations

Presenting children with problem situations helps develop their familiarity with the language and process of division. Be sure to present problems for each type of division. Have children work with partners or small groups so that they have opportunities to talk about their thinking. For all problems, when children present answers, have them explain their reasoning orally and in writing. Link the situations and solutions to mathematical symbols so that children learn the standard notation.

//Division **SHARING** Problems:

While walking to school one day, four children found a $5.00 bill. When they arrived at school, they told their teacher, and she asked them to tell the principal. The principal thanked the children and told them she would try to find who lost the money. A week later, the principal called the children back into her office, and told them no one had claimed the money and that it was theirs to keep. However, they first had to figure how to share it equally among the four of them.//

Have the children work together to solve the problem. Ask them to record their solution and explain thie reasoning in their Math Journals. It may help to write the following prompt on the board for the children to follow:

Each person gets. We think this because ___.

Have children share their solutions. Show them how to record the stories and solutions mathematically.

Continue with stories about the four children who find things on the way to school. They can find a bag of marbles with 64 marbles, a train of interlocking cubes, a bag with 20 apples, etc. Also, give problems with other divisors. For example:

//Suppose someone gives the class a gift of 100 pencils. If they are divided among all of the students, how many pencils will each student get?//

//If eight children were at a birthday party, and there were a box of cookies with four packages of ten cookies each, how many cookies could each child get?//

It is ok to have problems with remainders as long as you exaplin the meaning of remainders to children.

If there is time:


 * The Doorbell Rang:**

//The Doorbell Rang// provides children the opportunity to think about several division problems. The book begins with two children who are about to have a snack. There is a plate of a dozen cookies. Have the class figure how many cookies each child should eat. Just as the children are about to divide the cookies between them, however, the doorbell rings. Two more children enter. Have the class figure how many cookies they could then have. Then just before the four children begin to eat the cookies, the doorbell rings again, and two more children enter. Have the class figure how many cookies they could then have. The six children are about to eat their cookies but the doorbell rings once more. This time, six more children arrive. There are now twelve children and twelve cookies. The doorbell rings again and the children sit, frozen. When the door opens, it is Grandma with a plate of freshly baked cookies!

As you read the story, record the mathematical sentences for each problem. Have children practice retelling the story, referring to the mathematical sentences for cues. In this way, they see mathematical symbols related to the action of a story. If you would like manipulatives for this, then print the following on cardstock and laminate (cute alert):




 * Day Two:**

//Division **GROUPING** Problems:

There were two cartons in the refrigerator with a dozen eggs in each, plus three extra eggs in the holders in the refrigerator door. Mom liked to eat an omelette each day and used two eggs in each omelette. How many days could she make omelettes before she had to buy more eggs?//

As with the other problems, have the children work in pairs or small groups, figure the answer, and explain their reasoning. Represent the story and solution with the correct mathematical symbols.

Present other grouping problems. Keep the emphasis on the children's thinking. Have them share their reasoning processes and explain why their methods and answers make sense. As children present their solutions, represent the problems and their solutions mathematically on the board so that children see the mathematical symbols related to a variety of problem situations.

Here are some real world "problems" you can pose for your students:
 * Bring a loaf of sliced bread to class. Choose a loaf that is packaged in a clear wrap so children can count the slices. Ask them to figure out how many sandwiches can be made from the loaf.
 * Tell the class that for a math activity, each student needs eight tiles. A box has two hundred tiles. Have them figure out if everyone in the class can participate in the activity at one time.
 * Talk with the children about the way paper is packaged. Tell them that a ream contains 500 sheets. Present this problem: //If each student needs 20 sheets of paper to make a recording book, how many books can be made from a ream of paper?//
 * Tell the children about a person with an apple tree who had a giveaway celebration to get rid of fifty extra apples. She offered three apples to each person who asked. How manypeople could get free apples?


 * Day Three:**

Relating Division to Mulitplication

Seeing the realtionship between division and multiplication helps deepen children's understanding.

Manipulatives and visual aids are important when teaching multiplication and division. Students have used arrays to illustrate the multiplication process. Arrays can also illustrate division.



Since division is the inverse, or opposite, of multiplication, you can use arrays to help students understand how multiplication and division are related. If in multiplication we find the product of two factors, in division we find the missing factor if the other factor and the product are known.

In the multiplication model below, you multiply to find the number of counters in all. In the division model you divide to find the number of counters in each group. The same three numbers are used. The model shows that division “undoes” multiplication and multiplication “undoes” division. So when multiplying or dividing, students can use a fact from the inverse operation. For example, since you know that **4 x 5 = 20**, you also know the related division fact **20 ÷ 4 = 5** or **20 ÷ 5 = 4**. Students can also check their work by using the inverse operation.



There are other models your students can use to explore the relationship between multiplication and division. Expose your students to the different models and let each student choose which model is most helpful to him or her. Here is an example using counters to multiply and divide.



Here is an example using a number line.



Another strategy your students may find helpful is using a related multiplication fact to divide. The lesson Relating Multiplication and Division focuses on this strategy. Here is an example. Think: **6 x ? = 18** Six times what number is 18? so **18 ÷ 6 = 3**.
 * 18 ÷ 6 = ?**
 * 6 x 3 = 18**,


 * Day Four:**

Fact Families:

Students will already have met fact families for addition and subtraction. They will not necessarily recognize the links between multiplication and division. Students can feel overwhelmed by how many multiplication and division facts there are to learn, unless they see the links between them.

Understanding about fact families builds connections in mathematics and reduces the amount of material that students need to learn. The key teaching strategy is to emphasise that there are related facts that belong together. Once a student knows one fact, they can use this to solve related number sentences with missing numbers. The activities below are illustrated with the fact family of 3 × 4 = 12, but teachers can use fact families from the multiplication tables currently being learned.

//Arrays in common objects:// Many objects found at home are arranged in arrays, for example, egg cartons, muffin trays, trays for organizing nails and screws, and boxes of chocolates. As well as using real objects, a digital camera can be used to bring pictures into the classroom. Ask the students to choose an object or picture and write down as many multiplication and division number sentences as they can about their array. For example, consider a muffin tray which is a four by three array. Ask students to write as many number sentences as they can, for example, 3 × 4 = 12 and 12 ÷ 3 = 4. They should also write a sentence or story about these facts (e.g. there are 3 rows of muffins with 4 in each row, so there are 12 muffins on the tray). //Arrays from materials and squared paper:// Demonstrate the fact 4 × 3 = 12 using an array of counters. Use both array language (3 rows of 4 items / 4 columns of 3 items) and equal groups language (3 groups of 4 items / 4 groups of 3 items) so that students build links between previous "equal groups" understandings and the powerful array model of multiplication. Note that the array can also be rotated to appear as 4 rows of three, so that 4 × 3 = 12 and 3 × 4 = 12. Rearrangement does not change the relationship between the three numbers. Cover portions of the pictures or materials and ask questions such as “how many rows of four makes twelve?” (? × 4 = 12) as a missing multiplication question and as a division question (12 ÷ 4 = ?). Give students some counters and get them to write other number sentences, also explained in words.
 * Activity 1 - Fact families from arrays.**

Give students a set of numbers (e.g. 3, 4, 5, 12, 15, 20) and ask them to write as many different multiplication or division number sentences as they can using only numbers from the set (e.g. 3 × 4 = 12 and 12 ÷ 4 = 3). Ask students to group all the number sentences from the same family together. For example, here are eight number sentences from one fact family (there are two more fact families possible with the numbers provided):
 * Activity 2 - Recognizing different fact families.**
 * 3 × 4 = 12 || 4 × 3 = 12 || 12 ÷ 4 = 3 || 12 ÷ 3 = 4 ||
 * 12 = 3 × 4 || 12 = 4 × 3 || 3 = 12 ÷ 4 || 3 = 12 ÷ 4 ||

Note that the last row contains number sentences that some students think are "backwards". They are just as valid as the number sentences in the top row, and it is important that these are included to assist students with the notion that an equals sign can mean "balance" as well as "give an answer". Include multiplication facts that the students are currently learning. Stress the usefulness of these links in calculating: “I can’t remember 9 sixes, but I do know 6 nines.”




 * Day Five:

Day Six, Seven, Eight and Nine:**

Independent Activities (Menu)

The independent activities give children additional experiences with division.


 * Leftovers With 20:**

You need: a partner, die, 20 color tiles, 6 (3 x 3 inch) square of construction paper

How to play:
 * 1) The first player rolls the die, takes that number of 3 x 3 inch squares, and divides the tiles among them. If there are leftover tiles, the first player keeps them.
 * 2) The second player writes the math sentence on a piece of paper.
 * 3) Switch roles. Continue until there aren't any tiles left.


 * Division Stories:**

Write a division story that follows two rules:
 * 1) It must end in a question.
 * 2) The question must be one that's possible to answer by dividing.

Solve your story problem in as many ways as you can. Exchange papers and solve each other's problems.
 * The Kings-and-Elephants Problem:**


 * You need: //17 Kings and 42 Elephants//, by Margaret Mahy

This book tells the story of seventeen kings and forty-two elephants going on a journey through the jungle. Figure out how seventeen kings could divide up the work of taking care of forty-two elephants. Explain your reasoning. Write other math problems that use the information in the story. For example: How many elephant feet were there altogether?

You need: //The Doorbell Rang//, by Pat Hutchins Write another version of this story. You may change the number of cookies on the plate and also the number of children who arrive each time. Illustrate your story. Then read it to someone else.
 * The Doorbell Rings Again:**

You need: picture of 5 (chocolate bars) Some chocolate bars are scored to make them easy to break apart and share. Figure out how to share the five chocolate bars shown equally among four people. Explain how you did this.
 * Chocolate Bars:**

Review and Test - The test is not here because I had to cut and paste to get it to be less than 6 pages long (YIKES). It has been sent to print shop.
 * Day Ten:**


 * Links:**

@http://oswego.org/ocsd-web/games/Mathmagician/mathsdiv.html

@http://www.math-drills.com/division.shtml


 * Activities that go with picture books:**


 * Powerpoints:**


 * Envision:**


 * More Division Word Problems for Practice:**




 * Multiplication and Division Review Sheets:**