• Use the Multiplication Match resource to see how well students understand the different strategies and algorithms for solving multiplication problems.
• Observe students during the partner activity when students explore the related problem set to uncover any student misconceptions regarding multiplication.
• Use the Exit Ticket to assess student mastery of single-digit by double-digit multiplication.

“Different strategies and number sentences can be used to solve a multiplication problem. To start our lesson today, I am going to give each of you a strip of paper.”

Distribute the Multiplication Match resource (M-4-2-3_Multiplication Match.docx).

“Your job will be to find other students whose strips represent the same multiplication problem as you have. Sometimes the equations will be written as repeated addition, sometimes as a break-apart equation, and sometimes they will be represented as an array. When you find someone who has a strip representing the same multiplication problem as yours, stay with that person.”

Once students are in groups, confirm that everyone is correctly matched. Have students share the strategy represented on their strips of paper. While students are explaining the strategy represented on their strips, show the related equations on the overhead so all students can see the equations used. Give each group a piece of construction paper and have them attach equivalent equation strips to the construction paper. Between each equation strip write the words “is the same as” to reinforce that all the equations are equal. These visuals may be referred to throughout the unit.

Present the question: “What are the various ways we know to solve 23 × 45? Think about the activity we just completed. Are some strategies more reasonable to use than others? Why?” Have students share their ideas with each other and then have a class discussion. Guide student thinking toward the idea of using the break-apart method: 23 (20 + 3) and 45 (40 + 5) to find the product.

“Now we are going to learn another strategy for multiplying called the partial-products method. The partial-products method is an alternative to the customary multidigit multiplication method. Watch as I model the partial-products method with the same product we just discussed, 23 × 45.”

Model the following process for students to see how the partial-products method works.

Students should have a good sense of place value from practicing multiplication with tens; this method is an extension. After students recognize that 23 can break apart into 20 + 3, and 45 can break apart into 40 + 5, the multiplication process can begin. Use the think-aloud strategy as each step is being completed so students hear the reasoning behind each step.

“First the tens value on top is multiplied by the tens value on the bottom (20 × 40 = 800). Then the tens value on top is multiplied by the ones value on the bottom (20 × 5 = 100). Then the ones value on top is multiplied by the tens value on the bottom (3 × 40 = 120). Then the ones value on top is multiplied by the ones value on the bottom (3 × 5 = 15). The numbers are then added up (800 + 100 + 120 + 15 = 1,035) to find the product (23 × 45 = 1,035).”

Allow students to practice the partial-products method with some additional multiplication examples. Here is one that uses a two-digit factor and a one-digit factor:

Present the following problem:

• A team of four girls ran in a relay race. Each girl ran her segment of the race in exactly 12 seconds. The winning time in the race was 45 seconds. Do you think the girls won the race? Why or why not?

Encourage students to solve the problem in different ways (base-ten blocks, pictures, arrays, number sentences). Ask students to share their solution strategies. Post the different strategies for other students to see. Connect the number sentences to their corresponding solutions. For example, if students draw a rectangular array or use base-ten blocks to demonstrate the decomposition of the factors to compute the product, the number sentence they should write is
12 × 4 = (10 × 4) + (2 × 4).

Present a related problem set like the one below:

• 2 × 7 = ___
• 50 ×7 = ___
• 52 × 7 = ___

“Can the two equations we have on the board help us solve this multiplication problem
(52 × 7)? If so, how?” Guide students to see that the sum of these products (2 × 7 and 50 × 7) is 364 (14 + 350), which is also the product of 52 × 7. Discuss the benefits of thinking about the problem this way (easier to calculate products in our heads, reminds us of place values, etc.). Relate the discussion to this idea, “Let’s think about 52 × 7 as 52 groups of objects with 7 objects in each group. We can break the 52 groups into 50 groups of 7 objects and 2 groups of 7 objects.”

A rectangular array can be used to show the above problem.

Write the number sentence: 52 × 7 = (2 × 7) + (50 × 7). Encourage students to discuss that it does not matter if we multiply 50 × 7 first or 2 × 7 first. Continue to expand on this idea, emphasizing the different properties.

Have pairs of students continue to explore other related problem sets*. Observe and listen to student interactions. When necessary, intervene and model using a think-aloud strategy to show how the problem sets are related. Examples of related problem sets could be:

*Additional printable problem sets are available at this Web site:

http://worksheetplace.com/index.php?function=DisplayCategory&showCategory=Y&links=3&id=21&link1=40&link2=45&link3=21

After students share their thinking processes, ask, “Can you summarize the ideas you need to keep in mind as you consider different ways to break apart a multiplication problem?”

Ask students to examine this related problem set:

• 23 × 10 = ____
• 23 × 1 = ____
• 23 × 9 = ____

“How can you use 23 × 10 and 23 × 1 to help determine the product of 23 × 9? Would using 23 × 9 and 23 × 1 be helpful in determining 23 × 10?” (23×10 - 23×1 = 23×9) Have students share their thinking processes. Contrast these processes with the previous examples. Use a visual (rectangle) or the concept of equal groups to help explain their thinking.

Then, working with a partner, students should come up with a related problem set similar to those presented. Have students consider different ways to break apart a multiplication problem. Then ask partners to switch their related problem sets and identify the relationship between each other’s problem sets. Students can provide feedback to one another. Observe student interactions and provide verbal prompting where necessary.

Before the end of class, give students the Exit Ticket (M-4-2-3_Exit Ticket.docx and M-4-2-3_Exit Ticket KEY.docx) to demonstrate their mastery. Collect the tickets from students on the way out the door. This quick activity will help identify which students need additional work to master the skill.

Extension:

• Routine: Entrance tickets may be used to refresh students’ memory of the partial-products method. Or a problem could be put on the board with a think-pair-share strategy to focus students on the partial-products method.
• Expansion: Students who have mastered the partial-products method of multiplication using two-digit numbers can work with three-digit numbers. They should document their work and explain why they did each step.
• Alternative Method: Students who have mastered the partial-products method can also explore the lattice method of multiplication. This alternative multiplication algorithm organizes the partial products in a visual display that emphasizes place value.

The lattice is set up using diagonals to represent place value. The lattice structure is determined by the number of digits in each factor. For example, 61 × 3 would have a lattice with two boxes (2 by 1); 234 × 9 would have a lattice with three boxes (3 by 1); 897 × 398 would have a lattice with 9 boxes (3 by 3). Look at the example above. Once the lattice is created, calculate all the partial products and place them in their corresponding boxes: 3 × 4 = 12, 3 × 9 = 27, 8 × 4 = 32, and 8 × 9 = 72. Then starting from the right, add along the diagonals. The ones place is 7; the tens place is 2 + 2 + 2 = 6; the hundreds place is 1 + 2 + 7 = 10 (0 stays in the hundreds place, and the 1 goes in the thousands place); and the thousands place is 1 + 3 = 4. 83 × 49 = 4,067. This algorithm is similar to the partial-products method. It also does all the multiplication first and then adds the partial products at the end to determine the product of the original multiplication problem. Once students show understanding of the lattice method for multiplication using two-digit numbers, have them work with three-digit numbers.

• Technology Connection: Students may practice multiplication with one- and two-digit numbers using this game. They receive immediate feedback. http://www.kidport.com/grade4/Math/NumberSense/G4-M-NS-Mult100.htm