Spacing out on Proportions and Fractions

As we progress through our education and enhance our knowledge, we then must take a step back and place ourselves within the shoes of students. What are some strategies and tools that they would use to understand a problem? How would they go about solving it? Little did I know that this would be one of the hardest things to do within my college classes. During a recent class, we focused on problems that were related to proportions and fractions. The strategies that we used were on a level that middle grade students would have a hard time wrapping their mind around. The real issue is that we need to realize that the ways that we solve a particular problem may not be on the same level as our students. We, as teachers, need to have enough tools to formulate multiple strategies for our students in order for them to master the concepts presented to them. This post has been created to put in writing certain strategies and tools that I plan to pass along to my students referring to proportions and fractions so that they are not simply getting frustrated and spacing out.

Within the recent class period, the problem referred to the diluting of blue chemicals for blue jeans. The problem was asking which solution was "bluer" based on how many blue bottles and clear bottles were in each solution. The strategy that I found most useful was cancelling to find out what was left in the larger solution to see which one was "bluer." However, I soon realized that this is not a well thought out strategy to use. By using this manipulation, the proportions are changed which would be hard for a student to understand and realize. Although it worked for me and I was able to determine which solution was bluer, if I were teaching this to students, it would not be the strategy that I would focus on to teach. So what are some other ways that students would be able to have "click" in their head? After this class, some thoughts raced through my mind that I would try to implement into my lesson plans.

The use of examples that do not have an emphasis on numerical values. One of the main problems that have rose to the surface through my volunteering and observations of classrooms and other mathematical education activities, is the emphasis on correct numerical answers. What if we used questions that required more reasoning? Consider an example that talks about the concentration of orange juice. When mixing with 3 cans of water, would a container of orange juice with 3 cans of orange concentrate have more or less compared to a container with 4 cans of orange concentrate? These types of questions require much more in-depth understanding and comprehension compared to a question asking strictly about a comparison of numbers.

When I was in mathematics classes, nothing excited me more than money. Since this was the case I sincerely believe that by including more examples that the students would be able to relate to and be excited about, they're more apt to stay within the learning part. An example including this would be an exchange rate. The students would be able to use hands on money that have different exchange rates. They could then write out proportions and fractions that model what they are trading with one another based on a system that tells which exchanges they are allowed to make. For example, we could say that 3 US dollars are exchanged for 2 British pounds. As described in the previous strategy and example, the students would have a more in-depth understanding and comprehension by using these hands-on activities regarding proportions and fractions. Below are a few examples that could be used throughout the activity. It is important to note that planning questions are not concrete. Students may struggle or fly through specific parts of an activity, which is why the teacher must be adaptive.

1.  Represent the exchange rates as fractions for US currency and British Pounds. 
2. How many US dollars would you have if you had 14 British Pounds? 
3. How many British Pounds would you have if you had 26 US dollars? 
4. If you had 1 British Pound, how much US dollars would you have? 
5. Create your own proportion. 

Overall it is known that proportions and fractions are a hard concept for a lot of students to wrap their head around. It is our responsibility as teachers to provide different strategies and tools to our students so that they are able to comprehend at a deeper level. Using the above strategies and many more that are not elaborated on within this post, the students would be able to manipulate more on their own with the problems and be able to apply the correct operations when working with non-numerical and numerical problems.