Understanding remainders in division problems is crucial for students, but it can be a challenging concept. Often, students can perform the calculation but struggle to interpret the remainder in the context of a real-world problem. This difficulty stems from a lack of focus on making sense of the problem itself. Students may rush to compute an answer without considering whether it logically fits the situation.
A real-world example illustrates this challenge. Students were presented with a scenario: Mrs. Gannon is planning a picnic and needs to buy bottled water and hot dog rolls. She needs enough for 36 people. Bottled water comes in packs of 8, and hot dog rolls come in packs of 8.
The initial question posed to the students was what they noticed and wondered about the scenario. Their responses were recorded, and then they were given the information that Mrs. Gannon needs 40 bottles of water and 36 hot dog rolls.
The focus shifted to how many packs of hot dog rolls Mrs. Gannon needed to buy. This problem was designed to highlight the importance of remainders. Several different approaches emerged from the students.
Some students used division, arriving at the answer 4 with a remainder of 4. However, they often didn’t connect this answer back to the context of the problem. They didn’t explain what the remainder meant in terms of buying packs of hot dog rolls.
Other students utilized skip counting to determine the number of packs needed. These students tended to arrive at the correct answer of 5 packs because they understood that 4 packs (32 rolls) wouldn’t be enough for all 36 people. Their reasoning was inherently tied to the context of the problem.
A third group of students attempted to use multiplication, with varying degrees of success. Some correctly multiplied to find the necessary number of rolls, while others made errors in their calculations.
The key observation was that students who used skip counting were more likely to get the correct answer on the first try because they inherently considered the context of the problem. They understood that having leftover people without hot dog rolls wasn’t a viable solution. In contrast, students who focused solely on the division calculation often missed the real-world implication of the remainder.
To further explore the concept, students were then asked how many people Mrs. Gannon could invite if she wanted to have no leftover hot dog rolls. This prompted them to think about divisibility and factors.
This exercise highlighted the difference between procedural fluency and conceptual understanding in mathematics. While efficient division is a valuable skill, it’s equally important for students to understand how to interpret mathematical results in real-world scenarios. This ability to apply mathematical reasoning to practical problems is essential for success in both academic and everyday life. Understanding “how many are left” requires more than just calculation; it requires critical thinking and contextual awareness.
