This activity will teach students how to deal with remainders in real life by either rounding, splitting them evenly, or simply ignoring them.
Invitation to Learn
Additional Resources
Books
Teaching with the Brain in Mind, by Eric Jensen; ISBN 1-4166-0030-2
It often takes a leap of understanding for students to apply the procedural algorithm of division with remainders to real-world situations where remainders are encountered. A child who can easily calculate 40 divided by 6 = 6R4 will too often state 6R4 as the answer to the number of cars necessary to transport 40 children to a baseball game if 6 children can fit in each car. The activities in this section will first review the concept of division as proportional reasoning involving equal shares and then they will lead children to discover the three usual ways of dealing with remainders in real life: they are either used to round up to the next whole number, they are dropped and discarded, or they are split evenly among the participants.
Before beginning this lesson, students must be able to express remainders as fractions and decimals.
Invitation to Learn
Distribute a set of 12 counting objects to each child. (They may be cubes, blocks, chips, etc.) Tell the students that they each have a set of 12 objects. Then ask the students to divide their sets into four fair shares. Guide them to create four sets with three objects in each set. Discuss the term "fair shares" if it is not part of your usual vocabulary. It means every set has the same number of objects, the dividend is divided equally by the divisor. Then write the following equation on the board and ask the children to copy it into their math journals.
Ask what is different from the usual way of writing a division problem. They should notice that the number 1 is written above the divisor. What is significant about the number 1? Take several ideas from students. Lead them to discover that the 1 is implied in every division problem, because the quotient is how many items are in 1 fair share. Then have the children write the following two statements in their journals:
Explore with the children the relationships between the numbers as they discover the proportions: 1/4 = 3/12; 1/3=4/12 and 1X12=3x4. Write all the true statements on the board and have the children list them in their journals.
Next, copy these three equations on the board:
Ask what is the question in the first equation. (The students are asked to form 4 fair shares from 12 objects.) What is the question in the second equation? (The students are asked to find how many fair shares of 3 each can be made with 12 objects.) What is the question in the third equation? (Students are asked to find how many objects must be used to make 4 fair shares containing 3 objects each. This case involves multiplication rather than division.) Have the children build each situation with their manipulatives, knowing that even though the problem looks the same each time, in the first instance the question is the number of fair shares in each set. In the second instance, the question is the number of sets, and in the third instance, the question is the total number of objects.
Introducing division as proportional reasoning prepares children for equivalency in fractions; proportionality in ratios, proportions and percents; and provides a more concrete understanding of division as the process of creating fair shares.
Instructional Procedures
Family Connections
Wiebe, A., (1989). Proportionality: A major concept in mathematics--part II: Remainders--what are we to do with them? Aims newsletter, volume iii, No. 7, 6-7.
Dr. Wiebe explores the gap between abstract answers to division problems with remainders and real-life situations where students encounter remainders. Expressing remainders as fractions and decimals are explored and applied, and the choices of rounding up, dropping, and sharing remainders are introduced.
Martinez, J.G.R., (2000). Look smart. Early years, January 2000. Retrieved January 12, 2007 from http://www.findarticles.com.
Engaging children in math story problems is easier when the stories have real plots and good endings. By engaging students in the plot, they become interested in solving the math situations, rather than routinely solving a page of "word problems." Additionally, the enthusiasm generated motivates students to write their own stories, developing new problems within the story context, and acting out the story line.