Big Idea:

The focus in this learning activity is on having students use strategies that make sense to them, rather than on applying learned procedures.

Expectations:

B2.7 represent and solve problems involving the division of three-digit whole numbers by two-digit whole numbers using the area model and using algorithms, and make connections between the two methods, while expressing any remainder appropriately.

Explain to students that their help is needed in organizing a Family Math Fair at the school. Tell them that the principal has conducted a survey and that 165 people from the community have indicated that they will attend the fair. Explain to students that the first part of the math night involves a presentation in the gym and that people will sit at tables in groups of 6.

Ask:

“How many tables need to be set up?”

STRATEGIES STUDENTS MIGHT USE

COUNTING: Students might draw diagrams (e.g., make tally marks) to represent 165 people, and then group the people (e.g., by circling tally marks) into sets of 6. Students then count the number of sets to determine the number of tables that are needed. Students will find that there are 27 groups of 6 people with 3 people left over and conclude that another table will be needed.

USING REPEATED ADDITION: Students might draw tables (e.g., rectangles) and indicate 6 people at each table (e.g., by sketching 6 chairs at each table, by recording “6” at each table). As they draw the tables, students keep a running count of the number of people by repeatedly adding 6 until they reach 162. Students might realize that 168, the next multiple of 6, is greater than 165, but that an extra table will be needed to accommodate the last 3 people. Students then count the number of groups of 6. Students might also use repeated addition without the use of a diagram. For example, they might repeatedly add 6, and then count the number of 6’s that were added together.

USING PROPORTIONAL REASONING: Students might use proportional reasoning, for example, a doubling strategy – 1 table for 6 people, 2 tables for 12 people, 4 tables for 24 people, and so on. Students might organize this information in a table.

If students use a doubling strategy, they will observe that 16 tables are too few and that 32 tables are too many. They might combine different table-people ratios to determine the total number of tables needed; for example, 16+ 8+4 tables (28 tables) will seat 96+48+24 people (168 people).

USING REPEATED SUBTRACTION: Students might begin with 165 and repeatedly subtract groups of 6 until they reach 3. To determine the number of tables, students count the number of times that 6 was subtracted and include an extra table for the remaining 3 people.

USING “CHUNKING”: Students might subtract “chunks” (multiples of 6) from 165.

USING PARTIAL QUOTIENTS: Students might use a strategy in which they calculate partial quotients by using their knowledge of multiplication. For example, they might know that 20 tables would seat 120 people (20 × 6 = 120), and then determine that another 8 tables (8 × 6 = 48) would be needed for the remaining 45 people. The partial quotients, 20 and 8, are added to determine the number of tables.

USING AN ALGORITHM: Students might have been taught an algorithm and apply these procedures to solve the problem. If students are unable to explain the meaning of the procedures and numbers in the algorithm, suggest that they select a method that they can explain.

When students have solved the problem, provide each pair with markers and a sheet of chart paper or a large sheet of newsprint. Ask students to record their strategies and solutions on the paper and to clearly show how they solved the problem. Make a note of pairs who might share their strategies and solutions during Reflecting and Connecting. Aim to include pairs who used various methods that range in their degree of efficiency (e.g., using counting, using repeated addition, using proportional reasoning, using partial quotients).

Gather the class. Ask a few pairs to share their problem-solving strategies and solutions. Try to order the presentations so that students observe inefficient strategies (e.g., counting, using repeated addition) first, followed by increasingly efficient methods. Post students’ work following. each presentation. Avoid making comments that suggest that some strategies are better than others – students need to determine for themselves which strategies are meaningful and efficient, and which ones they can make sense of and use.

As students explain their work, ask questions that encourage them to explain the reasoning behind their strategies:

• “How did you determine the number of tables that are needed?”

• “Why did you use this strategy?”

• “What worked well with this strategy? What did not work well?”

• “How do you know that your solution makes sense?”

Following the presentations, encourage students to consider the effectiveness and efficiency of the various strategies that have been presented. Ask the following questions:

• “In your opinion, which strategy worked well?”

• “Why is the strategy effective in solving this kind of problem?”

• “How would you explain this strategy to someone who has never used it?”

Provide an opportunity for students to extend their understanding of division strategies by posing the following problem:

“After the presentation in the gym, the 165 math fair participants will be divided into. teams of 4 people to play math games. How many teams will there be?”

Have students work independently to solve the problem. Encourage them to think back to the differ ent strategies presented by their classmates, and to use an efficient strategy that makes sense to them. Have students show their strategies and solutions on a sheet of paper or in their math journals.

Gather the class. Ask a few pairs to share their problem-solving strategies and solutions. Try to order the presentations so that students observe inefficient strategies (e.g., counting, using repeated addition) first, followed by increasingly efficient methods. Post students’ work following each presentation. Avoid making comments that suggest that some strategies are better than others – students need to determine for themselves which strategies are meaningful and efficient, and which ones they can make sense of and use. As students explain their work, ask questions that encourage them to explain the reasoning behind their strategies:

• “How did you determine the number of tables that are needed?”

• “Why did you use this strategy?”

• “What worked well with this strategy? What did not work well?”

• “How do you know that your solution makes sense?”

Following the presentations, encourage students to consider the effectiveness and efficiency of the various strategies that have been presented. Ask the following questions:

• “In your opinion, which strategy worked well?”

• “Why is the strategy effective in solving this kind of problem?”

• “How would you explain this strategy to someone who has never used it?”

Provide an opportunity for students to extend their understanding of division strategies by posing the following problem:

“After the presentation in the gym, the 165 math fair participants will be divided into teams of 4 people to play math games. How many teams will there be?”

Have students work independently to solve the problem. Encourage them to think back to the different strategies presented by their classmates, and to use an efficient strategy that makes sense to them. Have students show their strategies and solutions on a sheet of paper or in their math journals.

Observe students as they solve the problem to assess how well they:

• understand the problem;

• apply an appropriate problem-solving strategy;

• judge the efficiency and accuracy of their strategy;

• find and explain a solution;

• determine whether the solution is reasonable;

• explain their strategies and solutions clearly and concisely, using mathematical language.

Collect students’ solutions to the problem in which they determined the number of math teams of 4 people. Observe the work to determine how well they apply an efficient strategy to solve the division problem.

SEL Self-Assessments (English) and Teacher Rubric

Encourage students to solve the problem by using a strategy that makes sense to them. Recognize that some students may need to use simple strategies (e.g., counting, using repeated addition, using repeated subtraction). It may be necessary to model the use of manipulatives and simple. counting strategies for students who experience difficulty in solving the problem. These students might also benefit from working with a partner who is able to explain different strategies. For students requiring a greater challenge, have them solve the problem in different ways, and ask them to explain how the various strategies are alike and different.

The following problem could also be used as an extension to the learning activity:

“78 children and 87 adults are planning to attend the math fair. Each child will receive 2 glasses of juice, and each adult will receive 1 glass of juice. A jug of juice holds 7 glasses. How many jugs of juice will be needed?”