Big Idea(s):

Operational sense: Students solve a problem involving the multiplication of a two-digit number by a one-digit number using a variety of strategies (e.g., using repeated addition, using doubling, using the distributive property). The learning activity focuses on informal strategies that make

sense to students, rather than on the teaching of multiplication algorithms.

Relationships: The learning activity allows students to recognize relationships between operations (e.g., the relationship between repeated addition and multiplication). Working with arrays also helps students to develop an understanding of how factors in a multiplication

expression can be decomposed to facilitate computation. For example, by applying the distributive property, 7× 24 can be decomposed into (7× 20) + (7× 4).

Expectations:

B2.5 represent and solve problems involving the multiplication of two-or three-digit whole numbers by one-digit whole numbers and by 10, 100, and 1000, using appropriate tools, including arrays

teacher background

Students are capable of solving multiplication problems before they develop an understanding of algorithms. When students apply strategies that make sense to them, they develop a deeper understanding of the operation and of different multiplication strategies.

This learning activity allows students to explore multiplication by using an array. The organization of items in rows and columns allows students to observe arrays as models of multiplication.

Dividing arrays into parts is an effective way to show how the distributive property can be applied to facilitate multiplication.

Discussions about various multiplication strategies and the use of arrays to represent multiplication are important components of this learning activity. These conversations allow students to learn strategies from one another and to recognize the power of the array as a tool for representing multiplication.

Describe the following scenario to the class: “The custodian at our school needs to set up chairs in the gym for a parents’ meeting. He plans to arrange the chairs in 7 rows with 24 chairs in each row. He is wondering, though, whether there will be enough chairs for 150 parents. How many chairs will there be altogether? Will there be enough chairs?”

On the board, record important information about the problem:

• 7 rows

• 24 chairs in each row

• How many chairs altogether?

• Are there enough chairs for 150 parents?

Divide the class into groups of three. Ask students to work together to solve the problem in a way that makes sense to everyone in their group. Suggest that students use materials such as square tiles and grid paper. Provide each group with a sheet of paper on which students 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?”

• “How are you representing the rows of chairs? Is this an effective way to represent the chairs?”

Students might use manipulatives (e.g., square tiles), draw on grid paper, or make a diagram to represent the arrangement of chairs. Concrete arrays and pictorial arrays help students to think of and apply strategies for determining the total number of chairs.

Strategies Students Might Use:

• COUNTING

Although inefficient, counting the chairs is a strategy some students might use if they are not ready to consider the array as a representation of multiplication.

• USING REPEATED ADDITION

The creation of a 7× 24 array might prompt some students to use repeated addition – adding 7 twenty-four times, or adding 24 seven times.

• DOUBLING

Students might use a doubling strategy similar to the following:

• DECOMPOSING THE ARRAY (USING THE DISTRIBUTIVE PROPERTY)

Some students might decompose 7× 24 into smaller parts, then use known multiplication facts to determine the products of the smaller parts, and then add the partial products to determine the total number of chairs. Students might divide the array into two or more parts without considering whether the resulting numbers can be easily calculated. Other students might think about ways to divide the array to work with “friendly” numbers.

• USING MENTAL COMPUTATION (APPLYING THE DISTRIBUTIVE PROPERTY)

Students might think of these steps:

• 7 rows of 20 chairs is 140 chairs.

• To account for the extra 4 chairs in each row, multiply 7× 4.

• 140 chairs + 28 chairs = 168 chairs

When students have solved the problem, provide each group with markers and a sheet of chart paper or large sheets of newsprint. Ask students to record their strategies and solutions on the paper, and to clearly demonstrate how they solved the problem. If students used grid paper, they could cut out their arrays and glue them to the sheet of paper. Make a note of groups who might share their strategies and solutions during Reflecting and Connecting. Include groups who used various methods that range in their degree of efficiency (e.g., counting; using repeated addition; using doubling; using the distributive property without considering whether the resulting numbers can be easily calculated; using the distributive property to find friendly numbers).

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., counting, using repeated addition) first, followed by increasingly efficient methods.

As students explain their work, ask questions that help them to describe their strategies:

• “What strategy did you use to determine the total number of chairs?”

• “Why did you use this strategy?”

• “How does your strategy work?”

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

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

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

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

If students describe a mental computation strategy that is based on the distributive property, you might model their thinking by drawing an open array (i.e., an array in which the interior squares are not indicated).

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 ideas about the distributive property:

• Arrays can be decomposed into two or more parts.

• The product of each part can be calculated and the partial products added together to determine the product of the entire array.

• An array can be decomposed into parts that provide friendly numbers, which are easy to calculate with.

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

Provide an opportunity for students to solve a related problem. Explain that bottles of water will be set on a table for the parent meeting, and that the bottles will be arranged in 6 rows with 31 bottles in each row. Ask students to give the multiplication expression related to the problem.

Record “6 × 31” on the board.

Have students work in groups of three. (You can use the same groups as before, or form different groups.) Encourage students, in their groups, to consider the various strategies that have been discussed, and to apply a method that will allow them to solve the problem efficiently. After groups have solved the problem, ask students, independently, to 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 situation (e.g., using an array made with square tiles or grid paper, using a drawing);

• 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 7× 24 can be decomposed into (7× 20) + (7× 4), and that the partial products, 140 and 28, can be added to calculate the final product).

Collect the math journals or sheets of paper on which students recorded their strategies and solutions for the water bottle problem. 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 counting or using repeated addition, and may not be ready to apply more sophisticated strategies. Ensure that students use concrete materials or a drawing to represent the multiplication situation and to connect it to an array. Guide students in using more efficient strategies when you observe that they are ready to do so. For example, students who use a counting strategy could be encouraged to use repeated addition. Provide opportunities for students who experience difficulties to work with students who can support them in understanding the arrangement in an array and how it can be divided into smaller parts. 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.