• Observe students using the Observation Checklist to determine how well students are grasping the concepts of the lesson.
• Checking student answers during the index card activity and Problem-Solving Card presentation will indicate how well students are mastering the content.

“During this lesson, we will be exploring what it means to divide two numbers. Division is important because it allows us to determine how many groups we have to make or how many objects we need to place in each group.”

Bring in a bag of small objects or wrapped candy. Explain to the class that you would like to make prize bags for an upcoming event. “What kind of information might you want to know in order to help me in making these prize bags? I will record your responses on chart paper.” (Possible answers: How many bags do you need to make? How many items are in the bag? Does each bag need to have the same amount? What happens to the leftovers? Are all the items that will be in the prize bag the same? Do you have to use up all the items in the bag?)

“Now let’s look at our list and decide which information is essential to know in order to complete this task.” (Essential information: the number of bags that need to be made, the total number of items to be used, and the fact that each bag must have an equal number of items.) Place all items on a table for students to see. Explain that four prize bags need to be made.

“Do you have any suggestions on how I can divide these items to make four equal bags?” Allow students time to dialogue and discuss what can be done. (Possible responses: Take the items one at a time from the large pile and drop one item into each of the four bags; repeat the process until all the items have been placed. Estimate how many items can go in each bag by looking at the pile and placing that amount in each bag. Sort the items into piles on the table and then place each pile into a bag. Count the total number of items; decide how many you think can go into each bag; place that number of items in each bag; and adjust if necessary.)

“These are all very good responses. Today we will look at what it means to divide. I am trying to find how many items will be in each bag. If I tell you that I know I have four bags and that there are 20 items to start with, you know that you have to break 20 into four equal groups. I will take one item at a time and place it in a different bag and repeat the process until all the items are used up. I will then count how many I have in each bag. Each of my four prize bags will have five items. We can write the division equation that represents this scenario by writing 20 ÷ 4 = 5. That means that 20 items (total) are placed into 4 prize bags with 5 items in each bag. The dividend is 20; 4 is the divisor; and 5 is the quotient for this division problem. The dividend represents the total number of items we have, the divisor for this problem represents the number of groups we have, and the quotient for this problem represents the number of items in each group.

“Let’s represent this process using a graphic organizer (M-4-4-1_Division Organizer.doc). We put the total in the middle. Then we shade in four rectangles to represent the four prize bags, or groups, that we have. We equally distribute the items from the center to each of the four bags. We know how many are in the center, 20. I can ‘wipe out’ one at a time as I place an item in each of the rectangles, distributing the items equally.”

“This type of division problem is called a sharing problem.” (Another word used to describe this type of problem is partitioning.) “A sharing problem is one in which you know the number of groups you are starting with, and your task is to determine how many will be in each group.”

“Let’s try a different problem. This time I have a different bag. I know I have 20 marshmallows in the bag. We are going to do a science experiment and each group will need five marshmallows. How many groups can I have for the science experiment using the marshmallows I have in my bag? Does anyone have a suggestion for how we can solve this problem?” (Possible answers: Keep counting out five marshmallows and put them into separate piles. Make one line of five marshmallows and line up as many rows as you can with the same number.)

“These are good suggestions. Let’s see if we could use the organizer to help us. We know how many marshmallows we have in all (20), so we will put that in the center. We know there will be five in a group, so instead of starting with the rectangles, which represent the groups, we have to start with the actual items. We take five from the center because that is the size of the group we are working with, according to the problem. Then we place these items in the first rectangle, and cross off five in the center so we know we have used them. This tells us we have one group of five. Since we have more left in the center, we put five in the next rectangle and cross out five more from the center. We now have two groups of five with items left in the center. We continue this process until we have no items left in the center. We then count up how many rectangles we were able to fill with five items.” (See the diagram below.)

“The answer is four groups with five items in each bag. We can write this as 20 ÷ 5 = 4. Twenty is the dividend; 5 is the divisor; and 4 is the quotient for this division problem. The dividend represents the total number of items we have. The divisor for this problem represents the number of items in each group. The quotient for this problem represents the number of groups we can make.

“Another way we could think about division is to use repeated subtraction. Start with the total, which is 20. Then subtract five items from the total, repeating the process until you have none left.” Model and record this process on the board while thinking aloud. “I start with 20 items, and I start by subtracting five items each time. 20 − 5 = 15, 15 − 5 = 10, 10 − 5 = 5, 5 − 5 = 0. If I count how many fives I subtracted, I see that it is four. So I can make four groups for the science experiment.”

“Notice that we are using the same numbers as we did in the previous example: 20, 5, and 4. The question, however, was asking something different. When we are solving to find out how many groups we can make with a given number of items in each, it would be classified as a grouping problem. We can solve this type of problem using an organizer, repeated subtraction, and/or a multiplication sentence with an unknown factor.

“Let’s review the different types of division problems. Remember there are two types of division problems: sharing and grouping.” Give each student an index card.

“On one side of your index card write ‘sharing’; on the other side of the index card write ‘grouping.’ I am going to show a division problem on the overhead.” Use the transparency of Overhead Problems (M-4-4-1_Overhead Problems.doc). Continue, “Decide which type of division problem it is. Then hold up the index card showing me which type of division problem you think it is. If you are not sure, you may quietly ask your neighbor, but it is important that you understand why. Be ready to share your reasoning once I give everyone a chance to read and think about the problem.” Do several examples until students show proficiency in understanding the two types of division problems.

You can prepare problem-solving cards ahead of time. The problem-solving cards should be created as pairs (M-4-4-1_Problem Solving Cards and KEY.doc). Each pair will use the same numbers. One card in the pair will be labeled sharing; the other card in the pair will be labeled grouping. Students will find their match and, on two separate pieces of chart paper, they will visually represent the two different types of division problems. You can monitor student interaction and performance. Using the Observation Checklist (M-4-4-1_Observation Checklist-Lesson 1.doc), ask each student to explain how s/he knows the type of division problem s/he is working on. Clarification and additional support can be given during this process. Once the students’ work is finished, check it. Ask each student whose work is accurate to complete a Division Organizer (M-4-4-1_Division Organizer.doc) to show the process s/he used to solve the problem. Students can hang their chart papers under the correct category by division type. Similarities should be seen in the processes used to solve division problems within the same category.

Using the observations made during the index-card activity and on the checklist, you can identify those students who need additional support in small groups. Those students who show proficiency can work on an expansion.

Extension:

• Routine: Anytime groups need to be made or multiple items need to be distributed to students, relate it to a division problem. Pose timely word problems to students that show the relevance of division in their everyday lives. Try to incorporate real-world situations into these problems as well. Have students create a vocabulary section in their math notebook if they do not have one already. Definitions, examples, and nonexamples should follow each vocabulary word for the lesson. A visual representation would also be helpful with many of the words in this unit. Suggested words include: dividend, division, divisor, estimation, grouping division models, sharing division models, and quotients.
• Expansion: Students can take any of the problem-solving cards used earlier in the lesson or student-created problems and write related multiplication sentences with an unknown factor. Doing this will strengthen student understanding of the relationship between division and multiplication. For example, students can use the following word problem found in the problem-solving cards: The soccer team had 27 raffle tickets to hand out to its 9 players. Each player got the same number of tickets. How many raffle tickets did each player get? The division equation used to solve this problem is: 27 ÷ 9 = □. A related multiplication equation with an unknown factor is 9 × □ = 27 or □ × 9 = 27 (reinforcing the commutative property of multiplication). Once students demonstrate an understanding of this relationship, they can be given a multiplication equation with an unknown factor and can write a related division equation and a word problem to go along with the division equation. For example, if students were given this multiplication equation, □ × 5 = 35, a related division equation is 35 ÷ 5 = □. A word problem might be: Janet has 35 stamps. She puts 5 stamps on each page in her stamp book. How many pages will Janet fill if she uses all 35 stamps?
• Small Group: Provide multiple bags of items with varying numbers of items inside. Provide enlarged laminated organizers so students who need additional practice can actually move the items to the proper locations to work through a division problem. Students could also draw a pictorial representation on an organizer. Students should write the equation to go along with the problem. Problem-solving cards and student-created problems could be used as a resource bank of problems.