Fraction Representations

Representation #1

Strengths: I think that this visual representation, demonstrated in chapter one of Lamon's textbook, can be very beneficial to students who are first learning the basics of proportions and ratios. This type of representation allows students to visualize what a 4:2 --> 2:1 ratio looks like. When students first begin to learn about proportions and ratios, a common confusion may be how a unit of a much larger size can be proportional (have the same ratio), as a smaller unit. This visual, where our unit can be determined as the size of the bag/number of jelly beans the bag holds, allows students to more clearly see how we can have ratios that are proportional to one another even when the unit sizes (in this case the size of the bags) are different.

Weaknesses: I think a drawback to this visual is students being able to correctly color in the amount of pink jelly bellies to purple jelly bellies but still not really understanding the concept of ratio and why all the bags end up being proportional to one another. A method students may use to come up with the correct answer with still very little understanding on ratio and proportion could be to count the number of pink jelly bellies, color them in, and then count the number of purple jelly bellies and fill those in. Then, with the remaining empty spots, they can continue to color in using this mechanism- color in the number of pink jelly bellies in the first drawing, and then the number of purple until they have no jelly bellies left to color. The way this visual representation is set up, a student could simply use this method and still not understand that the reason they are doing this is because in doing that they are keeping the 2:1 ratio and therefore keeping the pink to purple ratio of each bag proportional. Therefore, when students arrive to their solution, it is vital for teachers to ask how they came up with that solution and why that solution makes the bags proportional, to make sure the students are really understanding the 2:1 ratio that is being kept throughout all three of the bags, making the three proportional to one another.

Another more minor drawback of this representation would be the time consumption it could take once larger ratios and proportions began to appear. Therefore, this representation is best to use when students are first learning the fundamentals of ratios and proportions when the ratios they are comparing are still quite small and simple.

Representation #2

Strengths: This representation, demonstrated in Chapter One of Lamon's textbook, is extremely handy for finding the constant proportion among a mix of different ratios. This representation allows students to visually see that even numbers that are much larger than the 1:4 ratio actually are proportional to 1:4. If we wanted to continue on, we could have gone to say, 234 cubes and 936 inches. If a student was initially asked if 234:936 was proportional to 1:4, they may initially feel that there is way too large of a difference between these numbers and that 234:936 is not proportional to 1:4. This table however visually shows students that a 1:4 ratio can can be represented through a a variety of numbers. It also may shine light to the fact that students, when given a larger ratio like 234:936, can write this ratio as a fractiong 234/936 and simplify it to see if it is equivalent to the 1:4 ratio--> 1/4. Another strength of this kind of table is that it allows for quick calculations/comparisons with large numbers. This is not a visual that takes a ton of time to draw out, instead, it is extremely time efficient and a good representation to use when students are working with larger numbers.

Weaknesses: I think a drawback to this table would simply be the complexity and understanding that a student must have to understand what the problem is asking you to compare and how to navigate/understand the table. I feel that this table is very helpful but I do think that if you were trying to explain ratios and proportions to younger aged students, that you would want a more visual representation to begin with. This table is extrememly helpful to students who are not very visual learners and who understand how to read a table but for younger students who do not understand how a table really works or how the numbers in the table correlate to ratio's, the introduction of the table visual initially may prove to confuse many.

Representation #3

Strengths: The organization of the tape diagram is one major strength of this visual. The tape diagram allows individuals to break down the confusion of the long winded problem into simpler diagrams, comparing the two ratios. A problem like the one stated can lead students to feel like they have no idea where to begin, by using a tape diagram, students are able to more easily navigate where to start the problem and once they have the first initial portion of the diagram filled out, the remainder of problem is solved quite fluently and quickly.

The tape diagram is organized but is also visual, aiding those who learn better visually. By breaking each segment of the ratio into rectangles and stacking the ratio of girls to boys, students are able to visually see the difference between the 2:3 ratio and the 1:4 ratio.

The tape diagram in a way combines the concept of a visual and a visual table, resulting in it being a representation that helps a variety of students with different learning styles.

The tape diagram, while taking a little more time than the Table, does not take as much time as Representation #1, therefore, it is a fairly time efficient method that students can use with small values as well as large values.

When students use a tape diagram, they show that they really understand what is being asked of them. The tape diagram requires students to really understand the problem and what is being asked of them. Therefore, by using this representation, students really must work hard to solve the problem and have a thorough understanding of how ratios work and what a certain ratio will look like visually in comparison to another. This visual allows teachers to really gauge whether or not their students are understanding the concept of ratios.

Weaknesses: I think that one of the only weaknesses of the tape diagram is simply that it is typically used for problems comparing ratios, not really problems where we're trying to identify proportionality. While we could possibly use tape diagrams to show proportional values with different numbers in the rectangles, I do not think that this would be as visually helpful for students as representation 1 and 2, especially when comparing proportionality of different sized units. When it comes to comparing ratios tape diagrams are extremely beneficial but when it comes to solving for proportionality, a representation like #1 or #2 may be the better visual to use.