Choose Appropriate Strategies. Begin by reading together the opening vignette on the Multiplication Strategies pages in the Student Guide. Ask students to complete Question 1. They begin by drawing a chart like the one in the Student Guide and sorting problems into those that can be solved using mental math and those that need paper and pencil. Students should be prepared to discuss why they placed each problem where they did. Before your students share their ideas, read what Jerome, Jessie, and Roberto said about the problems.
- Do you agree with the choices that Jerome, Jessie, and Roberto made? Why or why not?
Discuss each problem.
- Did you put the problem in the mental math or paper-and-pencil column? Why?
- What numbers in the problem helped you make your choice?
- For problems in the mental math column, show or tell us how you solved the problem.
- Use [student name]'s method to solve 295 × 8.
Possible mental math and invented strategies are shown in the Content Note.
As you discuss Question 2, students should recognize that often a problem can be solved with more than one mental-math strategy, depending on how an individual student views it. They should also realize that some students may use mental math to solve a particular problem, but other students may need to use paper and pencil. Ultimately, students will come to understand there are times when mental-math strategies are most efficient, and other times when pencil-and-paper methods are best used. In either case, students need to use strategies that will ensure accurate answers.
Sample Multiplication Strategies for Mental Math. Research has identified different types of strategies students frequently use to solve 1-digit by 2-digit multiplication problems, including partitioning strategies and compensation strategies.
In partitioning strategies, students break the numbers into the sum of parts and then multiply. This strategy depends on the distributive property. Figure 1 shows three natural ways to break apart 26 to solve 26 × 3. Besides breaking a number into tens and ones, it is also helpful to break a number so that one of the parts is 25 or 50 since multiples of 25 and 50 are easy to compute with mentally.
In compensation strategies, students adjust the numbers to make calculations easier. In Figure 2, the problem 37 × 9 is changed to the easier problem 37 × 10. Then 37 is subtracted to compensate for this change. Another compensation strategy is shown in Figure 3. There the problem 418 × 5 is solved by cutting one factor in half and doubling the other. This changes the problem so the task is to multiply by 10, which is easier than multiplying by 5. Figure 4 shows a strategy that uses both partitioning and compensation to solve 224 × 8.
Multiply with Paper-and-Pencil Methods. Ask students to solve 6 × 47 using a paper-and-pencil method. As students work, note those using different methods. Students will likely use the compact method, all-partials method, and other invented strategies such as the doubling strategy shown in Figure 5. As they work, ask students to record their strategies on the board and be prepared to explain their strategies to the class.
Lead a brief, whole-class discussion about the different strategies used to solve 6 × 47. Ask selected students to share the solutions they recorded.
- How is [student name]'s method similar to [student name]'s?
- How are they different?
- Which methods make sense to you? Choose one to explain to the class in your own words.
- Use [student name]'s method to solve 4 × 76.
The compact method (the traditional algorithm taught in the United States) will be reviewed in Lessons 4 and 5. Many students will be able to use it proficiently and will prefer it to the methods in this lesson. The goal of this lesson is to increase students' understanding of multiplication and expand their use of methods so they can solve problems flexibly, using more than one strategy.
Encourage students who are proficient with the compact method to demonstrate their understanding of multiplication by using and explaining the partitioning models and strategies in this lesson. It is through the use of these strategies that you will have insight as to whether students are using the traditional algorithm as a rote process or whether they really understand the process, including the place value of each digit and the use of the distributive property to break apart the products. Allowing students time to develop these ideas using the rectangle model and making connections among the methods will increase students' flexibility and enhance their computation skills.
Other students who have trouble with the compact method will benefit from using these methods where the steps are visible, so students can clearly understand the process.
Use Rectangles and Expanded Form. Read the Rectangle Model and Expanded Form section in the Student Guide. In this vignette, Mr. Moreno demonstrates how to use rectangles to model multiplication and how to use the expanded form to break apart products. Ask students to read Question 3 and to discuss their answer with a partner.
Use these or similar prompts to encourage students to share their thinking:
- What number did Mr. Moreno break apart to form the two rectangles? (Possible response: He broke 47 into 40 and 7.)
- What number did he break apart in the expanded form? (Possible response: He broke 47 into 40 and 7.)
- How is the rectangle model the same as expanded form? (Possible response: You break the numbers into tens and ones in both the rectangle model and in expanded form.)
- How are the two strategies different? (Possible response: In the rectangle model you draw a picture of the problem using rectangles to show the tens and ones. In expanded form, you break the number into tens and ones but you use a regular number sentence to multiply.)
Ask students to solve the problems in Questions 4–6 in two ways and share their results with a partner.
Direct students to the text in the Student Guide following Question 6.
Pose the same question to your class as Mr. Moreno did to his class:
- Can you multiply a three-digit number times a one-digit number using rectangles and expanded form? Try it with the problem 4 × 237.
Give students time to work with a partner to solve the problem using the two methods. Then lead a discussion using prompts similar to those in the Sample Dialog.
Use this Sample Dialog to explore Solving 4 × 237.
Teacher: How is this problem different from the problems you solved in Questions 4–6?
Jessie: We multiplied a two-digit number times a one-digit number for those problems. For this problem we are multiplying a one-digit by a three-digit number.
Teacher: How did you solve it?
Jessie: I drew a rectangle. I thought, if I can break apart a two-digit number into tens and ones and then multiply each part, I can break apart a three-digit number into hundreds, tens, and ones and multiply each one. That means multiply 4 × 200, 4 × 30, and 4 × 7. [Jessie draws a rectangle as shown in Figure 6.] I write 800, 120, and 28 inside.
Teacher: That's good thinking. What should Jessie do with the products after she multiplies?
Maya: She should add up the numbers inside the rectangles. [Maya goes to the board and writes: 800 + 120 + 28 = 948]
Teacher: Did anyone try expanded form?
Jerome: I used expanded form. It looks a lot like using rectangles without drawing the rectangle. I broke 237 into 200 + 30 + 7 like this. [See Figure 7.] Next I multiplied each part times 4, getting 800 + 120 + 28. When I added those numbers together, I got 948, too.
Teacher: Can someone show us how each number in Jerome's problem matches the numbers in Jessie's solution?
Students can use rulers to help them draw their rectangles.
Following your classroom discussion, ask students to compare their solutions to Maya's solution in the Student Guide above Question 7 and complete Questions 7–9 using rectangles and expanded form. Then challenge students to use a mental-math strategy to solve one of the problems.
Use the All-Partials Method. Ask students to refer to the All-Partials Method section of the Student Guide in which Mr. Moreno introduces his class to another paper-and-pencil strategy, the all-partials method.
Ask students to read this section. It is very important that students understand how each partial product is derived. Encourage students to say the multiplication problem for each partial product.
Students read that Ana wonders if the order of multiplying the partial products matters. Discuss Ana's discovery in Question 10. Then allow several minutes for students to determine the missing partial products in Questions 11–13. Students should be able to give the factors that determine each partial product.
Practice Multiplication. Assign Questions 14–16 in the Explore section of the Student Guide. One strategy is to have students complete the work independently and then meet with a partner to compare their work. Students should be encouraged to use a variety of strategies to solve the problems. Both partners should be able to explain and justify the steps they took to reach their answers.
Before the students begin their work, refer them to the Math Practices in the Student Guide Reference section. For these problem solving questions, focus on MPE2 and 4.
Ask students to discuss with their partner how they would apply the Math Practices to each of the problems.
- Look at Math Practice 2. What do you need to include in your explanation to show that you chose a good and efficient strategy for solving the problem?
- Look at Math Practice 4. What do you need to include in your explanation to show that you checked your calculations?
After students have had an opportunity to work on one or two problems, ask several students to share their solution strategies.
Assign Check-In: Questions 17–19. Students can complete these questions independently in order to check their understanding of multiplication strategies.
