Before I get into some examples, let me just say that the main message of this post is this: When modeling fractions, always do so with accuracy and precision, and make sure your students do the same. That might mean asking students to erase their work and start again, or it might mean supplying students with multiple blank pages so they can practice a few times before they get it right.
OK, here are a couple of examples. The first involves using area models of rectangles to demonstrate multiplication of fractions and mixed numbers. Have a look at the image below:
Notice that the model has been designed to show that the portion of the rectangle representing 1 is larger than the portion of the rectangle representing 3/4. Likewise, the portion of the rectangle representing 2 is larger than the portion representing 1/3. They are not exactly proportionally accurate, but they show the fraction as smaller than the whole. It is a geometric model, and should serve as a visual representation of multiplying fractions and not as merely another algorithm for multiplying.
Now observe the image below.
Here is the same model, but drawn as a square and partitioned as though all parts are the same size. With this version, I have reduced the model to an abstract representation of the multiplication task. And worse, it looks like lattice, which is an algorithm for calculating only that is entirely devoid of conceptual representation of quantity. We do not want students confusing rectangular area models with lattice.
It is important as a teacher when using rectangular area models for representation, to do your best to partition the rectangle so that the smaller quantities are represented with smaller rectangular partitions, and the larger quantities are represented with larger rectangular partitions. It may be difficult for students to follow this protocol when they depict their own rectangular area models, but if their teacher does, it will help them make better use and sense of the models.
For the next example, let's look at fraction strips. The Common Core emphasizes fractions on a number line a great deal, and Everyday Math gives students a lot of opportunities to work with fractions on a number line. But many will benefit from making their own fraction strips to visualize fractional partitions of a whole before they start placing fractional representations on a number line. Accuracy and precision are very important in both scenarios, otherwise the actual quantities that fractions represent can be lost on some students. Unfortunately, it is super easy to add to students' confusion when we have them work with fraction strips. So for our students to gain the most benefit from this crucial tool for understanding, we need them to make accurate fraction strips. Absolute perfection to the millimeter is not the goal, but accurate visual representation is. Observe the image below:
The top and bottom fraction strips represent visually accurate representations of 1/3 and 1/4, respectively, but the middle strip is a little bit off. As a sample of student work, we might let the middle one slide, as it appears to be almost accurate, and after all, it is divided into three parts, which is the important part, right? Not so much.
Look at the first partition in the middle strip, labeled 1/3. Notice how close it is in length to the representation of 1/4 just below it. For students who are not fully grasping the concept of fractional partitioning, the middle strip reinforces the misconception that any whole partitioned into three parts is divided into "thirds," and two parts are always halves, and four parts are always fourths, no matter how much of the whole they actually represent. Partitioning a whole into thirds means three EQUAL parts, and their models must represent that.
The more confident they are with fractions, the more competent they will be, and vice-versa. The more we can prevent those misconceptions from settling into their brains for the long-haul the better, and fractions so often represent, along with long division, one of the first truly intimidating concepts students encounter in their mathematical journeys. But it doesn't have to be that way (!) if we do our best to keep accuracy and precision at the forefront of our mathematical modeling.