This lesson will teach students about two sorts of division problems: repeated subtraction and sharing. The students will: 
- distinction between the two types of difficulties and detect and categorize division problems as grouping or sharing. 
- use small numbers to clearly indicate the division of two different categories. 
- create word problems with division that demonstrate both grouping and sharing. 

- How are relationships represented mathematically?
- How can mathematics help us communicate more effectively?
- How may patterns be used to describe mathematical relationships?
- How can mathematics help to quantify, compare, depict, and model numbers?
- What does it mean to analyze and estimate numerical quantities?
- What makes a tool and/or strategy suitable for a certain task?
- When is it appropriate to estimate versus calculate?

- Division: The operation of making equal groups (e.g., there are 3 groups of 4 in 12). 
- Factor: The number or variable multiplied in a multiplication expression.

- bag of small objects or wrapped candy
- plastic or paper bags
- marshmallows
- index cards
- transparency of Overhead Problems (M-4-4-1_Overhead Problems)
- copies of Division Organizer (M-4-4-1_Division Organizer)
- copies of Sharing and Grouping Problem-Solving Cards (M-4-4-1_Problem Solving Cards and KEY)
- copies of Observation Checklist (M-4-4-1_Observation Checklist-Lesson 1)

- Observe students using the Observation Checklist to determine how well they understand the instructional contents. 
- Checking student answers during the index card activity and problem-solving card presentation will reveal how well students understand the content of the lesson. 

Active Engagement, Modeling, and Explicit Instruction 
W: The lesson will teach students the distinction between grouping and sharing division problems. 
H: Introduce the idea of creating prize bags for an event to engage students in the lesson. 
E: Remind students that there are two sorts of division problems: grouping and sharing. Show and explain how to utilize graphic organizers for different types of division problems. 
R: Have students utilize problem-solving cards to practice division and go over the two types of division issues. 
E: Use the Observation Checklist to assess and document student grasp of division principles. 
T: Suggestions in the Extension section can be utilized to further challenge students or to reinforce concepts as needed. 
O: Rather than just solving problems with algorithms, the lesson is aimed to employ basic division facts to assist students gain a conceptual understanding of division. 

"In this lesson, we will learn what it means to divide two numbers. Division is significant because it tells us how many groups we need to form and how many objects we need to place in each group." 

Bring in a bag containing little objects or wrapped candy. Explain to the class that you want to create prize bags for an upcoming event. "What kind of information would you like to know in order to help me in creating these prize bags? I'll write your comments on chart paper." (Potential responses: How many bags do you need to make? How many items are inside the bag? Does each bag have to contain the same amount? What happens with 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 to be made, the total number of items to be used, and the requirement that each bag include an equal number of items.) Place everything on a table for students to see. Explain that four prize bags must be constructed.

"Do you have any suggestions on how I can divide these items to make four equal bags?" Allow students time to talk about what can be done. (Possible answers: Take one item at a time from the main pile and place it in each of the four bags; repeat the process until all of the items are placed. Look at the pile and estimate the number of items that can fit in each bag. Sort the items into piles on the table and place each pile in a bag. Count the total number of items; determine how many you believe can fit into each bag; arrange that number of items in each bag; and adjust as needed.)

"These are all excellent responses. Today we will discuss what it means to divide. I am trying to figure out how many items will be in each bag. If I tell you that I have four bags and that there are 20 items to begin with, you understand that you must divide 20 into four equal groups. I'll take one item at a time and place it in a different bag, repeating the process until all of the items are used. I'll then count how many are in each bag. Each of my four reward bags will contain five items. We can write the division equation that reflects this circumstance as 20 ÷ 4 = 5. This means that a total of 20 items are divided into four reward bags, each containing five items. This division issue has a dividend of 20, a divisor of 4, and a quotient of 5. The dividend represents the total number of items; the divisor indicates the number of groups; and the quotient reflects the number of items in each group. 

"Let's visualize this procedure with a graphic organizer (M-4-4-1_Division Organizer). We placed the total in the middle. Then we shade in four rectangles to symbolize our four prize bags, or groups. We distribute the things from the center evenly to each of the four bags. We know how many are in the center: 20. I can ‘wipe out’ one at a time by placing an object in each of the rectangles, distributing the items equally."



"This type of division problem is called a sharing problem." (Another term for 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've got a different bag. I know I have 20 marshmallows in the bag. We're going to undertake a science experiment, so each group needs five marshmallows. How many groups can I create for the science experiment using the marshmallows I have in my bag? Does anyone have any suggestions for how we can solve this problem?" (Possible responses: Continue counting out five marshmallows and dividing them into separate piles. Make one line of five marshmallows, then line up as many rows as possible with the same number.)

"These are great suggestions. Let's see if the organizer can help us. We know how many marshmallows we have in total (20), so we'll place them in the center. We know there will be five in each group, so instead of starting with the rectangles that represent the groups, we must begin with the real items. We choose five from the center because the problem specifies the size of the group we're working with. Then we arrange the items in the first rectangle, crossing off five in the center to indicate that we have utilized them. This means we have one group of five. Because we have more space in the middle, we put five in the next rectangle and cross off five more from the center. We now have two groups of five, with items remaining in the center. We continue this process until there are no more things in the center. We then counted how many rectangles we could fill with five things.” (See the diagram below.)



"The solution is four groups, each with five articles in their bag. This can be written as 20 ÷ 5 = 4. This division issue has a dividend of 20, a divisor of 5, and a quotient of 4. The dividend indicates the total number of items we have. The divisor for this problem indicates the number of elements in each category. The quotient for this problem represents the number of groups we can make.”

"Another approach to think about division is through repeated subtraction. Begin with the sum, which is 20. Then subtract five items from the total, continuing until there are none left." Model and record this process on a whiteboard while thinking aloud. "I begin with 20 items and proceed 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 find that there are four. So I can divide into four groups for the science experiment."

"Notice how we're utilizing the same numbers as in the last example: 20, 5, and 4. The question, however, asked for something different. A grouping problem is one in which we need to determine how many groups we can make with a certain number of items in each. We can answer this type of issue with an organizer, repeated subtraction, and/or a multiplication statement involving an unknown factor."

"Let's look at the various types of division problems. Remember, there are two kinds of division problems: sharing and grouping." Give each student an index card.

"Write 'sharing' on one side of your index card, and 'grouping' on the other. I'm going to show a division problem on the overhead." Use the transparency of Overhead Problems (M-4-4-1_Overhead Problems). Continue, "Determine which type of division problem this is. Then show me the index card indicating which type of division problem you believe it is. If you are unsure, gently question your neighbor, but make sure you understand why. Be prepared to explain your reasons once I've given everyone a chance to read and reflect on the challenge." Repeat multiple examples until students demonstrate skill in grasping the two types of division problems.

You can create problem-solving cards ahead of time. Problem-solving cards should be generated in pairs (M-4-4-1_Problem Solving Cards and KEY). Each pair will utilize the same numbers. One card in the pair will be labeled "sharing," while the other will be labeled "grouping." Students will discover their match and physically portray the two types of division problems on two separate sheets of chart paper. You may track student involvement and performance. Using the Observation Checklist (M-4-4-1_Observation Checklist-Lesson 1), have each student explain how he or she understands the sort of division problem they are working on. Clarification and further support can be provided during this process. Check the students' work once they have completed it. Ask each student whose work is correct to fill out a Division Organizer (M-4-4-1_Division Organizer) to demonstrate how he or she solved the problem. Students can hang their chart papers in the correct category based on division type. Similarities should be observed in the methods used to solve division issues within the same category. 

Using the observations made during the index-card activity and the checklist, you can identify those students who require additional help in small groups. Students who demonstrate proficiency can work on an expansion.

Extension:

Routine: When creating groups or distributing many items to students, relate it to a division problem. Pose timely word problems to students that demonstrate the importance of division in their daily lives. Try to include real-life situations in these problems as well. If students do not already have a vocabulary section in their math notebook, they should establish one. Definitions, examples, and nonexamples should be provided for each vocabulary word in the lecture. A visual representation would also be useful for many of the words in this unit. Words to consider include: dividend, division, divisor, estimation, grouping division models, sharing division models, and quotients.

Expansion: Students can utilize any of the problem-solving cards from earlier in the class or their own problems to develop related multiplication sentences with an unknown factor. This will improve students' grasp of the relationship between division and multiplication. For example, students can use the following word problem from the problem-solving cards: The soccer team had 27 raffle tickets to give away to its nine players. Every player received the same number of tickets. How many raffle tickets did every player receive? The division equation for this problem is 27 ÷ 9 = □. A related multiplication equation with an unknown component is 9 × □ = 27 or □ × 9 = 27. This reinforces the commutative feature of multiplication. Once students have demonstrated a knowledge of this relationship, they will be given a multiplication equation with an unknown factor and asked to produce a similar division equation as well as a word problem. For instance, if students were taught the multiplication equation □ × 5 = 35, a corresponding division equation is 35 ÷ 5 = □. A word problem could be: Janet has 35 stamps. She stamps five times on each page of her stamp book. What number of pages will Janet fill if she uses all 35 stamps?

Small Group: Provide several bags of items with different numbers within. Provide larger laminated organizers so that children who need more practice can actually move items to the correct locations while working through a division problem. Students could also create a pictorial depiction using an organizer. Students should write the equation that goes along with the problem. Problem-solving cards and student-created problems could be used to create a problem resource bank.