Explore Multiplication by Multidigit Numbers

Est. Class Sessions: 2–3

Developing the Lesson

Part 2. Two-Digit Multiplication Using Rectangles

Explore Break-Apart Rectangles. Show students the rectangle in Question 1 on a display of the How Many Squares pages in the Student Activity Book. Direct them to find the number of small squares in the rectangle.

Then ask:

What multiplication problem does this rectangle represent? (18 × 26)
How do you know? (There are 18 rows and 26 columns.)

Allow students several minutes to solve the problem. Remind them to show their method on the page. Pay attention to the strategies students are using.

If the majority of students are using a similar strategy or algorithm, ask:

What is a second method for solving the problem? Show how your strategy uses the rectangle.
How can you divide the larger rectangle into smaller rectangles to make the problem easier?

Ask students to share their strategies with the class. Students should record their strategies so that they can be compared and discussed by the class. Students can use rulers to help them partition the rectangles. Students who use the grid to solve the problem will need to use a display of the Half-Centimeter Grid Paper Master. Others can use the board or chart paper.

Sample student responses are shown in Figure 2.

Discuss student strategies. Sample Dialog 2 has a possible discussion for the solutions shown in Figure 2.

Teacher: Nick, How did you decide to divide the rectangle? Explain your thinking.

Nick: I decided to break apart the 26 into 10 + 1 + 2 + 2 + 2 because I wanted to multiply by easier numbers like 10s and 2s.

Teacher: Ming, How did you divide your rectangle?

Ming: I broke 26 into 10 + 10 + 6 and then I broke 18 into 10 + 8 because I like to multiply by 10, so I found as many 10s as I could.

Teacher: What numbers did you multiply to get the numbers inside the rectangles?

Nick: I multiplied 10 × 18 to get the 180 in the two big rectangles and then I multiplied 2 × 18 to get the 36 in the bottom three small rectangles.

Ming: I decided to break the 26 into 10 + 10 + 6, but I also broke the 18 into 10 + 8 because I wanted to make as many 10s as possible. I multiplied 10 × 10 = 100 for two of the rectangles but then I multiplied 10 × 8 = 80 for two other rectangles. I multiplied 6 × 10 = 60 and then 6 × 8 = 48 for the two smaller rectangles.

Teacher: How are Nick's and Ming's strategies similar?

Shannon: They both got the same answer.

Teacher: What else is similar?

John: They both divided the big rectangle into smaller rectangles using 10s. They also both added the answers they found for each of the smaller rectangles.

Teacher: How are their strategies different?

Grace: Ming broke apart both the 26 and the 18 so he had more rectangles. He had to add more numbers to get his answer.

The thinking and strategy involved in the rectangle model, all-partials, and expanded form help students estimate products. The rectangle model, even a sketch, can help students experience the magnitude of the product produced by a pair of factors. The rectangle model also helps students develop conceptual and visual models for the distributive property. Students will continue to use and build on this model to apply the distributive property in other situations.

Ask students to do Question 2 on the How Many Squares pages using a different method for dividing up the rectangle from the one they used for the previous problem. If students are having difficulty thinking of new methods on their own, suggest that they try one of the methods that was shared for the previous problem. After giving students several minutes to solve the problem, ask them to explain how they solved the problem to a partner. Then ask two or three students to explain their strategies to the class. You may want to refer to a method by the name of the student who explained it initially, such as “Tanya's Method.”

Continue recording students' methods on a display of Question 2 in the How Many Squares pages for any methods that have not yet come up. A sample solution is shown in Figure 3.

Use Rectangles to Multiply. Read the vignette in the Setting Up for the Play section on the Explore Multiplication by Multidigit Numbers pages in the Student Guide. Ask students to study the rectangles Michael and Roberto drew to represent 38 × 24.

Ask:

Why did Michael and Roberto break up their rectangles into tens and ones? (Possible response: Using tens and ones makes it easier to multiply the numbers. You can use what you know about place value to help you.)
What is different about the two rectangles? (Michael's model shows all the squares and shows the number sentences for each partition. Roberto's can be drawn without grid paper and is quicker, and it does not need to be drawn exactly to scale.

Discuss Questions 3–6 with the whole class. In Question 3, students identify each part of the rectangle as a partial product of the multiplication problem.

Question 4 develops the concept that for a two-digit by two-digit problem, the numbers in the tens × tens rectangle always result in the largest partial product. The numbers in the ones × ones rectangle place result in the smallest partial product. This concept will be revisited again in Lesson 5.

Question 5 leads students to consider a method of estimation that flows from the rectangular array model. Students may respond that an estimate of 600 is not very close when the actual answer is 912. This model presents a visual sense of why the estimate is low (much of the product comes from the other three partial products).

In Question 6, students need not solve the problem, only decide how many smaller parts (or partial products) there would be if they multiplied a three-digit number by a two-digit number using the rectangle method. Suggest to students that they draw the rectangle model.

Ask students to complete the Using Rectangles to Multiply pages in the Student Activity Book with a partner.

Student Guide: 173 174

Answers