Big Idea(s):

Operational sense: Students solve division problems by using strategies that make sense to them. They discover that solutions to division problems sometimes involve a remainder and that the remainder must be dealt with within the context of the problem.

Relationships: By solving division problems, students explore the relationship involving a quantity, the number of groups the quantity can be divided into, and the size of each group. They also explore the relationships between the operations, particularly between division and repeated subtraction.

Expectations:

B2.6 represent and solve problems involving the division of two-or three-digit whole numbers by one-digit whole numbers, expressing any remainder as a fraction when appropriate, using appropriate tools, including arrays

teacher background

Explain to students that their help is needed in solving a problem about organizing intramural teams. Present the context for the problem:

“Seventy-eight students signed up for intramural sports. All the students will play both soccer and four-square. Ms. Boswell [Note: You might want to use the name of a teacher in your school who is involved in organizing intramural teams] would like to create the different teams and then make a chart with the names for each team. The chart will allow students to see which teams they belong to. But first of all, Ms. Boswell needs to solve some problems.”

Present the following problems on the board or on chart paper:

• There will be 4 soccer teams. How many players will there be on each team?

• There are 4 players on each four-square team. How many teams will there be?

Ask students to work with a partner. (You might decide to have students work with the same partner as in Getting Started, or you might form different pairs.) Explain that students may solve the problems in any order. Encourage students to consider whether any of the various strategies that were demonstrated earlier could help them solve the problems. Encourage them, as well, to modify any of the strategies or to develop new ones. Ensure that manipulatives (e.g., counters, base ten blocks, square tiles) are available, and invite students to use them.

Provide each student with a copy of.BLM1: : Intramural Dilemmas.

Explain that although they are working in pairs, each student is responsible for recording solutions to the problems. Remind students to think about ways to use words, symbols, and/or diagrams to explain their ideas.

STRATEGIES STUDENTS MIGHT USE:

COUNTING

Students might use manipulatives (e.g., counters, base ten blocks, square tiles) to represent the students in the class. To solve the first problem, they might count out the number of manipulatives that corresponds to the number of students in the class, then arrange the manipulatives into groups of two, and then count the number of groups. For the second problem, students could divvy the

manipulatives into two equal groups and count the number of manipulatives in each group.

USING REPEATED SUBTRACTION

Students might use repeated subtraction to solve the first problem. If there are 26 students in the class, they might subtract 2 from 26, and then continue to subtract 2 from the remaining difference. Finally, students would determine that there were 13 groups of two by counting the number of times they subtracted 2. The second problem does not lend itself to a process of repeated subtraction.

Instead, students might use a halving process (e.g., recognizing that one half of 26 is 13).

DECOMPOSING A NUMBER INTO PARTS

Students can determine the number of groups in the first problem by decomposing the number of students into tens and ones. For example, if there are 26 students, they might break 26 into 2 tens and 6 ones. There are 5 pairs in each group of 10 students and 3 pairs in the group of 6 students.

10 students → 5 pairs

10 students → 5 pairs

6 students → 3 pairs

13 pairs altogether

Gather students together after they have had sufficient time to solve the problems. Select a few pairs to present and discuss their solutions, choosing students who used different strategies. During students’ presentations, avoid making comments that suggest that some strategies are better than others. Instead, encourage students to consider the effectiveness and efficiency of each strategy by asking the following questions after each presentation:

• “Was it easy to find a solution using your strategy?”

• “What worked well?”

• “What did not work well?”

• “How would you change your strategy if you solved the problem again?”

After students have solved the problems, use an inside-outside circle strategy (described below) so that they can share their solutions with each other. Have students use their completed copy of BLM1: Intramural Dilemmas, and organize them for the activity:

• Divide the class into two equal groups.

• Ask one group to form a circle with students facing outwards, away from the centre of the circle.

• Ask the other group to form another circle around the first circle. Each student in the inside circle faces a partner in the outside circle.

To begin the activity, ask students to discuss with their partner how they determined the number of players on each soccer team. Encourage students to refer to their work on Div4.BLM1: Intramural Dilemmas as they explain their strategies. Remind students to be courteous by allowing time for their partners to present their ideas, by listening attentively, and by making positive comments (e.g., “I think you had a clever idea!”). Allow three to four minutes for students to share their strategies for this problem. Next, ask the outside circle to move counterclockwise by three people, and have students share their strategies and solutions for the soccer teams problem with their new partners. Ask the inside circle to move counterclockwise by four people. Have students, with their new partners, discuss the strategies they used to determine the number of four-square teams. Again, allow three to four minutes for students to share their strategies. Conduct another rotation to provide an opportunity for students to share their strategies and solutions for the four-square teams problem with another partner. Following the inside-outside circle activity, discuss the problems, one at a time, with the entire group by asking the following questions.

QUESTIONS FOR THE SOCCER TEAMS PROBLEM

• “How many players will there be on each soccer team?”

• “Was there a remainder (a leftover quantity) when you divided 78 by 4? What does this leftover quantity represent? How did your solution include these 2 extra students?”

• “Is it reasonable to have 19 (or 20, if the leftover students are placed on teams) players on each team?” (Students might suggest that a team of 19 players is too large and that someplayers would have little opportunity to play.)

• “If a team of 19 players is too large, what would you recommend to the teacher?”

QUESTIONS FOR THE FOUR-SQUARE TEAMS PROBLEM

• “How many four-square teams will there be?”

• “Was there a remainder when you solved the problem? What does this leftover quantity represent? How did your solution include these 2 extra students?”

• “Do you think that it is appropriate to have 19 four-square teams? Why or why not?”

Ask a few students to explain their strategies to the entire group. Select different strategies, and emphasize the idea that a variety of strategies are possible. Encourage students to think about the efficiency of different strategies by asking the following questions:

• “Which strategies work well? Why?”

• “Which strategy makes the most sense to you? Why?”

• “Which strategies are similar? How are they alike?”

• “How could you change a strategy to make it more efficient?”

Next, focus on the relationships between the problems and between their solutions. Ask:

• “Which problems are similar?”

Students might observe that the problems are alike because 4 is the divisor in both situations. Discuss the differences between the problems (e.g., in the soccer teams problem, the number of teams was known, but the size of each team was unknown; in the four-square teams problem, the size of each team was known, but the number of teams was unknown).

Observe students while they are solving the problems. Assess how well they are able to determine and apply a strategy that allows them to solve the problems effectively. Ask questions such as the following:

• “How are you solving these problems?”

• “What is working well with your strategy?”

• “What is not working well?”

• “Did you change your strategy? If yes, how?”

• “How are the problems the same?”

• “How are the problems different?”

• “How are you recording your solution?”

• “How do you know that your solution will be understood by others?”

• “Did solving one problem help you solve another? If yes, how?”

Provide the following problems for students to solve individually:

• Suppose we need to divide our class into 3 groups for science activities. How many students would there be in each group?

• Suppose we need to divide our class into groups of 3 for a rock-paper-scissors challenge. How many groups would there be?

Observe completed solutions to assess how well students:

• apply an appropriate strategy to solve the problems;

• use an efficient strategy;

• explain their strategies;

• recognize and apply the relationship between the problems.

SEL Self-Assessments (English) and Teacher Rubric

Some students may need to work with smaller numbers. Rather than dealing with problems that involve 78 students, they could determine how 24 students could be arranged on teams. Encourage these students to use manipulatives (e.g., counters, cubes, square tiles) to represent and model the problems. Extend the activity for students requiring a greater challenge by asking them to determine different ways in which 78 students could be divided exactly into equal teams (i.e., with no remainders).