Multi-digit multiplication and division differ in numerical complexity from single-digit multiplication and division because the use of any algorithm for the former requires students to have a knowledge of Base 10. For example, to multiply 11 by 2, a student has to understand that they are taking 2 groups of 11, or 2×1 (2 ones) and 2×10 (2 tens). The same is true with division.

In order to scaffold Base 10 understanding, one should always begin by teaching multiplication and division through direct modeling. This can be done by teaching the use of manipulatives and drawings to help students understand and visualize what is happening to the objects represented by the numerical notation in the operation. “Multiplication instruction traditionally has focused on two objectives: memorizing the multiplication facts and using one consistent, standard algorithm to multiply multi-digit numbers. Knowing the multiplication facts and computing efficiently are very important goals, but a deeper conceptual view of multiplication is essential” (Utah LessonPlans, 2003). This “deeper conceptual view of multiplication” can be achieved by connecting algorithms and operations to the real world situation that they represent.

“Traditionally in the United States and Canada, students have first learned how to compute with whole numbers and then have applied that kind of computation. This approach presents several problems. First, less-advanced students sometimes never reach the application phase, so their learning is greatly limited. Second, … This practice, plus the emphasis on teaching students to focus on key words in problems rather than to build a complete mental model of the problem situation, leads to poor problem solving because students never learn to read and model the problems themselves. …This isolation limits the meaningfulness of the operations and the ability of children to use the operations in a variety of situations.

“Research has indicated that beginning with problem situations yields greater problem-solving competence and equal or better computational competence. Children who start with problem situations directly model solutions to these problems. They later move to more advanced mathematical approaches as they progress through levels of solutions and problem difficulty. Thus, their development of computational fluency and their acquisition of problem-solving skills are intertwined as both develop with understanding.”

(Fuson, 2003)

Teachers should begin with teaching multi-digit multiplication, which will lead to an understanding of multi-digit division. Their instruction should begin with the concept of grouping. The grouping method of multiplication involves making a certain number of groups with a certain amount in each group, the total number of which is the answer to the problem. The number of groups is determined by one of the numbers in the equation (the multiplicand) and the amount in each group is determined by the other number in the equation (the multiplier). Either number can be used as the multiplicand or the multiplier, but the typical way to model a multiplication equation is to use the first number as the multiplicand (number of groups) and the second as the multiplier (amount in each group). The best way to help students understand how multi-digit multiplication algorithms work is to start your modeling of the process by making groups of certain amounts using Base 10 blocks. Since students need an understanding of Base 10 before beginning multi-digit multiplication, they should be familiar with Base 10 blocks. This can be done most effectively by having some type of visual aid for defining the groups, such as paper plates or a circles of string. The amount for each group (demonstrated by Base 10 blocks) is then placed on or in each defined group. Below you can see an example of how this was done for the problem 12 x 25.

Once understanding of the grouping method for multiplication has been solidified, students can successfully move toward an understanding of division in relation to grouping. Division as grouping can be seen in two different ways. These perspectives are partitive and measurement division. Partitive division involves placing a certain amount in each group and finding the total number of groups created, while measurement division involves making a certain number of groups and finding how many of each object are placed in a certain group. Making sure that students understand that there are two different ways to view division as grouping is important to their success. Allowing for students to practice seeing division problems in these two different ways allows them to 1) determine which method they are most comfortable with, and 2) see that there are more efficient ways for solving certain problems (for example, the problem 150 ÷ 50 is solved more easily putting 50 objects in each group rather than making 50 groups). Below you can see examples of modeling partitive and measurement division for the problem 132 ÷ 12.

Once students have displayed satisfactory performance and understanding using the concept of grouping in solving multiplication and division problems, they can be introduced to the array method of solving multiplication and division problems. The array method is very similar to the grouping method, except that instead of making separate groups of a certain amount of objects, students make a certain number of rows, with a certain amount of objects in each row. This method of solving multiplication and division problems is a scaffolding step towards area models (which are extremely important for the application of multiplication and division in many aspects of real-world application). The array method is a little more abstract than the grouping method because their is no literal separation between the groups/rows like there is in the grouping method, which is why it is important to teach the concept of grouping first. While using the array model, it is important to emphasize to students the importance of keeping tens and ones together within each row, so as to provide a way to connect the model with the traditional algorithm. Below you can see an example of the array model for the multiplication problem 12 x 11 and the division problem 132 ÷ 12. When starting out, have students separate the array model into portions with know facts. This can also be seen in the picture below.

After students have been exposed to the array model of multiplication and division, and have demonstrated satisfactory performance and understanding in relation to the array model to solve multiplication and division problems, they are ready to be introduced to the area model for solving multiplication and division problems. This method is essentially the same as the array model, but instead of using objects and manipulatives, the area model focuses on drawings and measurements (such as inches, feet, etc.). These drawings should usually be done on graph paper to scaffold the idea that the answer represents the total number of parts. Since there are no longer manipulatives, the method for connecting the area model with the traditional algorithm is to have students represent the lengths of each side using their Base 10 knowledge. They can do this by separating lengths into hundreds, tens, ones, etc. (See picture below for how this was done with 11 x 12). You can then have the students solve for the area for each part of the model which results from this separation (see below), and then add together the areas of each part to find the total area of the whole.

**Connections to Traditional Algorithms**

Through beginning your multi-digit multiplication/division teaching using the direct modeling strategies discessed above, your students will be able to create numerical algorithms that tie into each of the strategies. At each of the stages described above, you can also help students tie the traditional algorithms (the lattice method, the distributive multiplication method, long division methods) to these direct modeling strategies. It is important for students to be able to use traditional algorithms to solve problems because they are quicker and more efficient, but they must first understand the situation that the problems are based off of and how the algorithms are generated.

For example, the traditional method for solving long division problems can be tied to the grouping method by having students guesstimate how many times the given number (the divisor) fits into the original number (the dividend). This allows them to apply their knowledge of Base 10 to the long division method rather than having to memorize a specific procedure.

*Other Ties to Traditional Algorithms:*

Area model of multiplication –> partial products algorithm

Grouping model –> partial quotients (described above)

References

Bahr, D. L., & De Garcia, L. A. (2010). *Elementary Mathematics Is Anything but Elementary*. Belmont, CA:

Wadsworth Cengage Learning

Fuson, K. C. (2003, February 1). Toward computational fluency in multidigit multiplication and division.

In *Access my Library*. Retrieved April 17, 2009, from

http://www.accessmylibrary.com/coms2/summary_0286-3475344_ITM

Utah LessonPlans. (2003, August 29). Math 4 – Act. 03: Multi-digit multiplication. In *Utah Education*

* Network*. Retrieved April 15, 2009, from

http://www.uen.org/Lessonplan/preview.cgi LPid=6095

**Web Links**

Multiplication and Division Learning Progression

(pretty cool!)Video: Visual Representation of the Lattice Method

*Games*

*Multiplication Concepts and Tables Games*

*Lesson Ideas:*

*Visualizing Multiplication Lesson Plan*

*Teaching Multiplication Lesson Ideas*

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