Proportion Sense

Recently in class we began discussing fractions. We began by looking and discussing proportions without the use of numbers. It seemed like it would be simple, but without the use of fractions at our disposal it became difficult.

We were given a picture of two boxes of chocolates(like below) and asked which one was more nutty?

It was difficult to stop yourself from using and comparing fractions - since we already have that knowledge. Our future students, however, will be just developing how to use fractions and relate them. Instead, we had to rely on the picture and ways in which we could manipulate it. What we ended up finding most helpful was the idea of buying enough of each box so you have the same number of chocolates and then comparing how many nuts each one has. An idea  that relates back to finding a common denominator. Once you had the same number of each, making a comparison between the two was logical.

However, after accomplishing this, when faced with another proportional problem we fell into mistakes. We were given a problem comparing mixtures of blue dye with water. Given certain a number of beakers of dye and water, we had to decide which would produce a stronger blue color. It was interesting to see how different we reacted to this problem. Many of us ignored the idea of getting to the same number of beakers and began to cancel out blue dye and water combos. The result was a false belief that the two mixtures shown below would be the same color.

We failed to take into account the ratio of water to blue dye. It was startling to find ourselves making this mistake.

These two activities really helped me to come to understand the importance of proportional sense. My future students will likely be making similar mistakes, but having a discussion on these ideas and working through proportions without numbers will be helpful. Introducing students first to proportions and making sense of the relationships between them will get students asking important questions:

• What quantities are being compared?
• What other factors do I have to consider? 
• What roles do these variables play/what's their relationship?
• How is it affecting the goal(if it's nuttier or more blue)?

Through this, students can begin to develop a sense of how to compare proportions and what to take into account. Then, when dealing with the numbers and fractions they should to able to reason if the answer they arrive at is a logical one or not. They can rely on asking questions to led them to the correct calculations and hopefully dealing with fractions will not seem as daunting.