Scroll to find several challenges and misconceptions often seen when teaching fractions at the intermediate level, and some strategies to regulate these issues:
“The numerator of a fraction is difficult to visualize because it can’t be equally divided.”
As teachers we know that fractions can be divided into even smaller parts of the “whole” fraction, but how do we explain this? We can work interactively with students on a whiteboard or SMART Board to draw, shade, and erase parts of a whole circle to demonstrate equal division of the fraction’s numerator.
The fraction 2/7 is easily divisible into 1/7 and 1/7 by looking at this visual representation.
“The numerator and denominator of a fraction each represent a separate value.”
We must ask “Why is that?”, because perhaps there is some sound thinking that we could use to transform that logic into proper fractions logic. Use a number line or boxes to demonstrate the denominator as the whole, and the numerator as parts of that whole. We can then move forward to express that the denominator acts as somewhat of a base to say how much of, let’s say, a pizza or a pie we have left after eating some of it.
“Greater denominator means a greater value, while a lesser denominator means a lesser value.”
Let’s divide one pizza into three slices, and the other into five slices. Take one slice away from each pizza. Which pizza has more left over—not necessarily slices—but pizza itself? The pizza cut into five? Why do you think the pizza with the greater denominator ended up having more pizza left over?
“To divide a fraction I just have to divide the numerators and the denominators.”
NCTM Resource re: Inverting and Multiplying
An interesting way to go about justifying the process of inverting and multiplying is to compare the problem to a problem where we are asked to subtract a negative integer from another integer. Just as we would invert the negative integer to positive and invert the subtraction to addition, we invert the fraction and invert the division to multiplication.
“I can simply add the numerators and denominators to find the sum of multiple fractions.”
If we go about adding fractions in this manner, we would all of a sudden have denominators that would fundamentally make no sense to the problem. How can we use visuals to support the learning of common denominators for addition? Drawing a shaded circle to represent each fraction in the problem is a sound strategy to visualize how we should logically go about solving an addition problem.
Fractions Help 