Big Idea:

Operational sense: Students use a variety of strategies to solve a problem involving the multipli-cation of a two-digit number by a two-digit number. After solving the problem, students discuss how the distributive property can be used in multiplication.

Relationships: The activity allows students to recognize relationships between operations (e.g., the relationship between repeated addition and multiplication).

Expectations:

B2.6 represent and solve problems involving the multiplication of two-digit whole numbers by two-digit whole numbers using the area model and using algorithms, and make connections between the two methods

teacher background

In this learning activity, students solve a multiplication problem using strategies that make sense to them. When students apply their own methods, they develop a deeper understanding of the operation and of the efficiency of different strategies. This learning activity provides an opportunity for teachers to introduce the use of the distributive property in multiplication: for example, to multiply 29 × 20, students might decompose 29 into 20 + 9, then multiply 20 × 20 and 9 × 20, and then add the partial products (400 +180 = 580). Using compensation is also an application of the distributive property. With this strategy, students multiply more than is needed, and then remove the “extra” amount. To multiply 29× 20, students might recognize that 29 is close to 30 and multiply 30× 20 to get 600. They then subtract 20 (the difference between 30 × 20 and 29 × 20) to get 580. The experience of solving problems using their own methods allows students to apply the distributive property in informal, yet meaningful, ways.

Note: It is not necessary for students to define the “distributive property”, but they should learn how it can be applied to facilitate multiplication.

Describe the following scenario to the class:

“29 students are going on a field trip to a museum. The field trip costs $20.00 per student. For this fee, each student will receive bus transportation to and from the museum, an entrance ticket to the museum, and a picnic lunch. How much will it cost for 29 students to go on the field trip?”

Divide the class into pairs. Ask students to discuss important information about the problem with their partners. Have students summarize this information. Record the following on the board:

• 29 students

• $20 per student

• cost for 29 students

Ask students to solve the problem with their partners using a strategy that makes sense to both partners. Provide each pair of students with a sheet of paper on which they can record their work.

As students work on the problem, observe the various strategies they use to solve it. Pose questions to help students think about their strategies and solutions:

• “What strategy are you using to solve the problem?”

• “Why are you using this strategy?”

• “Did you change or modify your strategy? Why?”

• “What materials are you using? How are these materials helpful?”

• “How could you solve the problem in a different way?”

• “How could you represent your strategy so that others will know what you are thinking?”

STRATEGIES STUDENTS MIGHT USE

USING REPEATED ADDITION: Students might record $20 twenty-nine times, and repeatedly add 20 until they reach a solution.

USING SKIP COUNTING: Students might count by 20’s twenty-nine times.

USING GROUPINGS OF $100: Students might recognize that 5× $20 = $100 and determine the cost for 25 students.

Students would then add the cost for 4 students (4 × $20= $80) to determine the cost for 29 students ($500 + $80 = $580).

USING A NUMBER LINE:

USING DOUBLING: Students might continue to double the number of students and the related costs.

2 students - 2 × $20 = $40

4 students - 2 × $40 = $80

8 students - 2 × $80 = $160

16 students - 2 × $160 = $320

32 students - 2 × $320 = $640

Students will realize that 32 students is 3 more than 29 students, and might:

• subtract the cost for 3 students from $640 ($640 – $60 = $580); or

• combine the costs for 16, 8, 4, and 1 student(s), to calculate the total cost for 29 students ($320 + $160 + $80 + $20 = $580).

USING A T-CHART:

APPLYING THE DISTRIBUTIVE PROPERTY:

Some students might apply the distributive property in any of the following ways:

• Decompose 29 into smaller parts, then multiply each part by $20, and then add the partial products.

Example 1:

29 × $20 is the same as (10 +10 + 9) × $20.

(10 × $20) + (10 × $20) + (9 × $20) = $200 + $200 + $180 = $580

Example 2:

29 × $20 is the same as (20 + 9) × $20.

(20 × $20) + (9 × $20) = $400 + $180 = $580

• Use an open array to model partial products.

• Use a partial product algorithm.

USING COMPENSATION: Some students might recognize that 29 is close to 30. They might determine the total cost of the field trip for 30 students, and then subtract 20 (the extra $20 fee for one student), to calculate the total cost of the field trip for 29 students.

30 × $20 = $600

$600 – $20 = $580

This compensation strategy can be modelled using an open array.

After 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 demonstrate how they solved the problem. Make a note of groups that might share their strategies and solutions during Reflecting and Connecting. Include groups who used various methods that range in their degree of sophistication (e.g., using repeated addition, using doubling, applying the distributive property).

Reconvene the class. Ask a few groups to share their problem-solving strategies and solution, and post their work. Try to order the presentations so that students observe inefficient strategies (e.g., using repeated addition, using skip counting) first, followed by more efficient methods.

As students explain their work, ask questions that probe their thinking:

• “How did you determine the total cost of the field trip for 29 students?”

• “Why did you use this strategy? How did the numbers in the problem help you choose a strategy?”

• “Was your strategy easy or difficult to use? Why?”

• “Would you use this strategy if you solved another problem like this again? Why or why not?”

• “How would you change your strategy the next time?”

• “How did you record your strategy?”

• “Is your strategy similar to another strategy? Why or why not?”

• “How do you know that your solution is correct?”

Following the presentations, ask students to observe the work that has been posted, and to consider the efficiency of the various strategies. Ask:

• “Which strategy, in your opinion, is an efficient strategy?”

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

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

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.

Refer to students’ work to emphasize important ideas about the distributive property:

• Two-digit factors can be decomposed into parts to facilitate multiplication (e.g., 29 can be decomposed into 20 and 9, which are friendly numbers that are easy to multiply). Each part is multiplied by the other factor in the multiplication expression (20 × 20 = 400, 9 × 20=180), and then the partial products are added to calculate the total product (400 +180 = 580).

• An open array shows how factors in a multiplication expression can be decomposed into two or more parts. Partial products are recorded on the open array, and then added to calculate the total product.

• The parts in an open array can be represented in a partial product algorithm.

Provide an opportunity for students to solve a related problem. Explain that along with the 29 students, 7 adults will be going on the field trip to help supervise. The fee for each adult is also $20 per person. Ask students to give the multiplication expression related to the problem.

Record “36× 20” on the board.

Have students work in pairs. Encourage them to consider the various strategies that have been discussed and to apply a method that will allow them to solve the problem efficiently. After pairs have solved the problem, ask students to independently record a solution on a sheet of paper or in their math journals.

Observe students as they solve the problem, and assess how well they:

• represent and explain the problem;

• apply an appropriate strategy for solving the problem;

• explain their strategy and solution;

• judge the efficiency of various strategies;

• modify or change strategies to find more efficient ways to solve the problem;

• explain ideas about the distributive property (e.g., that 29× 20 can be decomposed into (20× 20) + (9× 20), and that the partial products, 400 and 180, can be added to determine the final product).

Collect the math journals or sheets of paper on which students recorded their strategies for determining the cost of a trip for 29 students and 7 adults. Observe students’ work to determine how well they apply an efficient strategy for solving the multiplication problem.

SEL Self-Assessments (English) and Teacher Rubric

Encourage students to use strategies that make sense to them. Recognize that some students may need to rely on simple strategies, such as repeated addition and skip counting, and may not be ready to apply more sophisticated strategies. Some students may benefit from using play money (e.g., $20 bills) to represent the problem and to find a strategy.

Scaffold the problem by asking:

• “How many $20 bills will be needed for 1 student?”

• “How many $20 bills will be needed for 5 students? What will the cost be for 5 students?"

• "For 10 students? For 20 students? For 29 students?”

Guide students in using more efficient strategies when you observe that they are ready to do so. Provide opportunities for these students to work with classmates who can demonstrate the use of more efficient strategies. Challenge students to solve the problem in different ways.

For example, if students use an algorithm, ask them to explain how the algorithm works and the meaning of the numbers in the algorithm within the context of the problem.