The challenge of equivalent ratios (repost)

It’s been one of those weeks. I’m trying to drag my way out of this flu that has been viciously attacking the Puget Sound area, and it’s not working. So if you’re wondering why the blog has been a bit quiet, it’s because Dead Bunny has been keeping Rebecca wrapped up in her blankets with hot tea and a great book.

This morning, though, I was at work. I had a normal math student working on skills for class and an SAT Prep student sitting across from him just starting out. Oddly enough, both students were working on fraction skills, and the one thing that tripped them up the most was creating the equivalent fractions they needed to get their work done.

I gave them my crash course in creating equivalent fractions, and then thought it might be nice to share an article I wrote on this skill. (It’s still a work-in-progress, so let me know if it’s clear in the comments.)

This week has been quite the interesting experience at my tutoring job. One technique keeps showing up in every single one of my math students. It’s one of those skills absolutely necessary to succeed in math, and 100% of my math and algebra students do not have this skill.

The skill in question is the creation of equivalent ratios.

I think the main problem is the fact that a fraction is involved. Once you involve fractions in anything, the student pales, sweats, and says, “I can’t do this.” I’ve often thought that if math education presented math concepts in the order suggested by my old math education methodologies professor, this fear of fractions wouldn’t be quite so great,  but for now I have to work with what I have.

To create an equivalent fraction, we must first start with a fraction. Let’s use 3/5 for the purpose of this example. At this point. I have the student model this either by using fraction tiles or by drawing it.

Next, we have to actually change this fraction so that it has different number but is still the same size. Hence, the “equivalent” in equivalent ratios.  At this point, I allow the student to pick the factor we’re going to change the numbers in the fraction by. Many choose 2 or 3, so we will use 3 for this example.

Here is where it usually gets tricky for students. We are going to change 3/5 without changing what it represents. That’s a little metaphysical for your average third grader, even more so for your average tenth grader. We want to multiply 3/5 by 1 so we don’t change what it represents. We know that 1 can also be written as 3/3 (see? we’re including those 3s we wanted to use to change our fraction), so we substitute 3/3 for 1. Our new number sentence looks like this: 3/5 x 3/3.

From here, most kids can figure out how to handle it and will come up with 9/15 for their answer. I have them model this number and then tell me if the model is similar to the first model they created. They typically agree it does.

One of the major pitfalls I’m seeing in the creation of equivalent ratios is that students are not effectively taught why you must multiply both the numerator and the denominator of the original fraction by the same number. I think teaching by showing the “x 1″ step helps eliminate that confusion by showing them why they both have be the same. When I used this technique last night, the students in question were creating equivalent ratios correctly on their own by the end of the hour.

Another pitfall in our program is that we teach students how to create equivalent ratios in one lesson. Within one or two lessons, we teach them how to check for equivalent ratios with larger numbers using the cross-product method. This seems to be where most of my students are getting lost. They learn that they can use cross-product to determine of two ratios are equivalent, and assume they can use this method to create equivalent ratios.

They’re right. They could, if they applied the technique properly. Most of them don’t, so I tend to limit them to the “x 1″ method until they’re more comfortable.