Strategies are listed from the earliest strategies up through the standard algorithm. Many are used side by side, but it is important to understand that the variety of strategies are used to build a deeper conceptual understanding and move to a more procedural model backed by conceptual understanding of division. Keep in mind that mastery of the standard algorithm of division is not expected until grade 6 per the Maine Learning Results and Common Core State Standards, however students will begin practicing the standard algorithm along side other strategies much earlier than grade 6.
Using models for division are going to be important to help students understand what division represents. The context of a division problem is going to be helpful in selecting an appropriate strategy and model and the context will also help students understand what to do with any remainders - can the remainders be broken up equally, left as remainders, or do you have to do something else. Most of the strategies below are using a model to solve division problems, so specific information about each model will be given there. As a reminder, to build flexibility in understanding division and strategies, students need practice with missing dividends, missing divisors, and missing quotients. These different types of problems should be introduced at different times within their learning progression. For more information about varying the types of division word problems students should use, check out the Glossary Table 2 from the Common Core State Standards for Mathematics.
When beginning to learn about division, we often refer to equal groups or fair shares. Students can physically act out sharing a variety of manipulatives as they make small groups or make equal groups of a certain size. This is where the context of the problem is important. This strategy is also referred to as repeated subtraction because you are taking away each group or one for each group as you equally share.
Much like equal groups, using arrays is a way to visualize an organized arrangement to see equal rows. Using this strategy, students would create rows the length of the divisor and see how many rows they could create. Once created, the array model shows the rows and columns representing the quotient and divisor with a total amount of objects representing the dividend. Watch the quick video to see how manipulatives can be used to create the array for division.
Area Models (Connects to Multiplication)
The area model can be a great strategy for modeling division with small or large numbers. It can easily transition to decimals as well. This strategy also connects well to the area model strategy for multiplication and shows how the two operations are related. In the area model, students start with a rectangular are model with the divisor listed on one side as one of the dimension of the rectangle. The student can then use break out pieces of the model to show smaller chunks of the dividend, using facts they know or are easier to use. The process continues until the entire dividend value is reached. The quotient is then the total of all the lengths on the side of the rectangle adjacent to the divisor. This strategy is great for large numbers that are harder to model with manipulatives. Watch the video for several demonstrations of this strategy in action.
The partial quotient strategy utilizes facts that students know and often connects to place value. It is similar to the standard algorithm, however students can use the divisor and break out small partial quotients and work toward the final quotient. This strategy can rely on any facts, however most students use multiples of 2, 5, 10, or 100 to help them successfully divide out smaller partial quotients. As students become stronger with this strategy, they become more efficient with their partial quotients. For a detailed example of how this strategy can work in several ways, please watch the video below.
US Standard Algorithm
The US Standard Algorithm is the strategy most people think of when they hear long division. It probably also creates a bit of anxiety for some people. The standard algorithm for division is a set of procedural steps that when followed correctly can give you a quotient, but there is rarely understanding or meaning behind the steps. There are even several mnemonic device strategies that you may have learned as a student to help you remember the steps. We want students to have a more conceptual understanding of division, so these procedural steps need other strategies (from above) to support what is happening beyond just following a list of step. The partial quotient strategy is a great stepping stone to the standard algorithm.
Graham Fletcher Video - Progression of Division